Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators

This book reviews and examines how power system low-frequency power oscillations and sub-synchronous oscillations may be affected by grid connection of wind power generation. Grid connection of wind power generation affects the power system small-signal stability and has been one of the most actively pursued research subjects in power systems and power electronics engineering in the last ten years. This book is the first of its kind to cover the impact of wind power generation on power system low-frequency oscillations and sub-synchronous oscillations. It begins with a comprehensive overview of the subject and progresses to modeling of power systems and introduces the application of conventional methods, including damping torque analysis, modal analysis and frequency-domain analysis, presented with detailed examples, making it useful for researchers and engineers worldwide.

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Wenjuan Du · Haifeng Wang · Siqi Bu

Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators

Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators

Wenjuan Du • Haifeng Wang • Siqi Bu

Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators

Wenjuan Du School of Electrical and Electronic Engineering North China Electric Power University Beijing, China

Haifeng Wang School of Electrical and Electronic Engineering North China Electric Power University Beijing, China

Siqi Bu Department of Electrical Engineering The Hong Kong Polytechnic University Kowloon, Hong Kong

ISBN 978-3-319-94167-7 ISBN 978-3-319-94168-4 https://doi.org/10.1007/978-3-319-94168-4

(eBook)

Library of Congress Control Number: 2018947164 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Doubly fed induction generators and permanent magnet synchronous generators are the two main types of variable speed wind generators. Changes brought about by the grid connection of variable speed wind generators in an AC power system are twofold. Firstly, unlike conventional synchronous generators, which are linked directly to the power system, a variable speed wind generator is connected to the grid via controlled voltage source converters. The dynamic behavior of the variable speed wind generator is significantly different from that of a synchronous generator. The impact of dynamic interactions introduced by the grid-connected variable speed wind generator on power system stability needs to be examined carefully. In addition, wind generation is variable due to the stochastic fluctuation of wind speed. It is important to investigate how the random variation of wind speed may affect power system stability. Secondly, VSC-based HVDC lines and multi-terminal DC (MTDC) networks for large scale wind power generation, especially for the offshore wind power, have many technical advantages. Stability of a VSC-based DC/AC power system with wind power generation is an important aspect of the impact brought about by grid-connected variable speed wind generator and needs to be studied. We started the investigation on the small-signal stability of the power system affected by the grid-connected variable speed wind generators about a decade ago. Our investigation has covered the two important aspects mentioned above, i.e., the small-signal stability of the power system considering the impact of dynamic interactions introduced by the VSWGs, the stochastic variations of wind speed, and the integration of VSC-based HVDC lines and MTDC network. In this book, we introduce the small-signal stability analysis of the power system integrated with the variable speed wind generators by mainly presenting the results of our investigation made in the last decade. Two main methods for the investigation are the damping torque analysis and modal analysis. Their applications to examine the impact of gridconnected variable speed wind generators on power system small-signal stability are comprehensively introduced in the book. It is worth mentioning particularly one important phenomenon we have found recently. That is the possibility of strong v

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dynamic interactions introduced by the grid-connected variable speed wind generators to degrade the small-signal stability of the power system under the condition of open-loop modal resonance. Although we are still studying the phenomenon from the aspects of theoretical analysis and practical applications, we introduce the initial results of our investigation in the book. We hope the potential instability risk brought about by the open-loop modal resonance should be aware of and understood fully by more power system researchers and engineers. We would like to thank the contributions from our research students to the book. They are Mr. Xiao CHEN, Mr. Jintian BI, Mr. Qiang FU, Mr. Xubin WANG, and Mr. Wenkai DONG. Beijing, China Beijing, China Kowloon, Hong Kong April 2018

Wenjuan Du Haifeng Wang Siqi Bu

Contents

1

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability of Power System . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Impact of the VSWGs: A Brief Review . . . . . . . . . . . . . . 1.1.2 Impact of the VSWGs: Discussions . . . . . . . . . . . . . . . . . 1.2 Small-Signal Stability of the VSC-Based DC/AC Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Small-Signal Stability of a Single Grid-Connected VSC System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Small-Signal Stability of Integrated VSC-Based DC/AC Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearized Model of a Power System with a Grid-Connected Variable Speed Wind Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linearized Model of a PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Linearized Model of the Permanent Magnet SG . . . . . . . . 2.1.2 Linearized Model of Machine Side Converter and Associated Control System . . . . . . . . . . . . . . . . . . . . 2.1.3 Linearized Model of Grid Side Converter and Associated Control System . . . . . . . . . . . . . . . . . . . . 2.1.4 Linearized Model of the Entire PMSG System . . . . . . . . . 2.2 Linearized Model of a DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Linearized Model of the Induction Generator and Two-Mass Shaft Rotational System . . . . . . . . . . . . . 2.2.2 Linearized Model of the RSC and Associated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Linearized Model of the GSC and Associated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Linearized Model of the Entire DFIG . . . . . . . . . . . . . . .

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Linearized Model of the Power System . . . . . . . . . . . . . . . . . . . Closed-Loop Interconnected Model of the Power System with a Grid-Connected VSWG . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Closed-Loop Interconnected Model of the Power System with a PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Closed-Loop Interconnected Model of the Power System with a DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . .

Damping Torque Analysis of Small-Signal Angular Stability of a Power System Affected by Grid-Connected Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Impact of Grid-Connected Wind Power Induction Generators on Small-Signal Angular Stability . . . . . . . . . . . . . . . 3.2 System Modeling for Damping Torque Analysis of a Power System Affected by Grid-Connected Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Phillips-Heffron Model of a Power System with Wind Power Induction Generators as a Generic Open-Loop Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Aggregate Model and Transfer Function of Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . 3.3 Implementation Framework for Damping Torque Analysis of a Power System Affected by Grid-Connected Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Analytical Comparison on Damping Effectiveness And Robustness of Type 1 and 3 Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Step 1: Detailed DFIG Model . . . . . . . . . . . . . . . . . . . . . 3.4.2 Step 2: DFIG Model Without RSC Dynamics . . . . . . . . . 3.4.3 Step 3: DFIG Model with Offset Rotor Voltage Only . . . . 3.4.4 Step 4: DFIG Model with Constant Rotor Voltage . . . . . . 3.4.5 Step 5: FSIG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Main Findings in Analytical Comparison on Damping Mechanism of Type 1 and 3 Wind Power Induction Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Numerical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Example 3.1 (Base Case Comparison Study) . . . . . . . . . . 3.5.2 Example 3.2 (Comparison Study Under Different Wind Penetration Conditions) . . . . . . . . . . . . . . . . . . . . . 3.6 Summary of Analytical and Numerical Comparison Analysis . . .

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Appendix 3.1: A Typical Example of a SMIB System with Interface Equations of a WPIG . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3.2: Data of Examples 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . Example 16-Machine 68-Bus New York and New England Power System [36] (Tables 3.4, 3.5 and 3.6) . . . . . . . . . . . . . . . . Data of DFIG and FSIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Modal Analysis of Small-Signal Angular Stability of a Power System Affected by Grid-Connected DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on Power System Small-Signal Angular Stability . . . . . . . . 4.1.1 Method of Decomposed Modal Analysis . . . . . . . . . . . . . . 4.1.2 Example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on Power System Small-Signal Angular Stability . . . . . . . . 4.2.1 The Index of Dynamic Interactions (IDI) . . . . . . . . . . . . . . 4.2.2 The Impact of Dynamic Interactions Introduced by the DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4.1: Data of Example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16-Machine 68-Bus New York and New England Power System [3] (Tables 4.10, 4.11, 4.12) . . . . . . . . . . . . . . . . . Data of DFIG [4] (Tables 4.13, 4.14, 4.15) . . . . . . . . . . . . . . . . . . Appendix 4.2: Data of Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Data of New England Power System [13] (Tables 4.16, 4.17 and 4.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data of DFIG [16] (Tables 4.19, 4.20 and 4.21) . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small-Signal Angular Stability of a Power System Affected by Strong Dynamic Interactions Introduced from a Grid-Connected VSWG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Strong Dynamic Interactions Introduced by the Grid-Connected PMSG and the Impact . . . . . . . . . 5.1.2 Example 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Strong Dynamic Interactions Introduced by the Grid-Connected DFIG and the Impact . . . . . . . . . . 5.2.2 Example 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 97 104 110 111 117 124 138 138 143 143 143 146 146

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Appendix 5.1: Data of Examples 5.1 and 5.2 . . . . . . . . . . . . . . . . . . . Data of New England Power System . . . . . . . . . . . . . . . . . . . . . Data of DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data of PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

Small-Signal Stability of a Power System with a VSWG Affected by the PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Impact of Open-Loop Modal Resonance Caused by the PLL for a Grid-Connected PMSG . . . . . . . . . . . . . . . . . . 6.1.1 Closed-Loop Interconnected Model of a Power System with a PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Open-Loop Modal Resonance Caused by the PLL . . . . . . 6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Example 6.1: A PMSG Connected to an External Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Example 6.2: A PMSG Wind Farm Connected to an External Power System . . . . . . . . . . . . . . . . . . . . . 6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System Affected by the PLL . . . . . . . . . . . . . . . . . . . 6.3.1 Test 1: Changes of Power System Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Test 2: Variation of PLL Gains (SRF PLL for PMSG1) . . 6.3.3 Test 3: Variation of SCR and Wind Power Penetration (SRF PLL for PMSG1) . . . . . . . . . . . . . . . . . 6.3.4 Test 4: DSOGI PLL for PMSG . . . . . . . . . . . . . . . . . . . . 6.3.5 Test 5: The Case of DFIG . . . . . . . . . . . . . . . . . . . . . . . Appendix 6.1: Data of Example 6.1 (Tables 6.5, 6.6 and 6.7) . . . . . . . . Appendix 6.2: Data of Example 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . Data of PMSG1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 6.3: Data of Example 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . Data of New England Power System . . . . . . . . . . . . . . . . . . . . . Data of PMSG (Tables 6.8, 6.9 and 6.10) . . . . . . . . . . . . . . . . . . Data of SRF PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small-Signal Stability of a Power System Integrated with an MTDC Network for the Wind Power Transmission . . . . . . . . . . . . 7.1 Small-Signal Angular Stability of an AC Power System Affected by the Integration of an MTDC Network for the Wind Power Transmission . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Open-Loop Modal Resonance (OLMR) in the MTDC/AC Power System . . . . . . . . . . . . . . . . . . . 7.1.2 Example MTDC/AC Power Systems . . . . . . . . . . . . . . . .

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Contents

Small-Signal Stability of an MTDC Network for the Wind Power Transmission Affected by the Dynamic Interactions Between the VSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Small-Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . 7.2.2 Example MTDC Power Systems . . . . . . . . . . . . . . . . . . . 7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic Interactions Introduced by a Selected VSC Control on the Small-Signal Stability of an MTDC/AC Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Method of Open-Loop Modal Analysis . . . . . . . . . . . . . . 7.3.2 An Example MTDC/AC Power System for Applying the Open-Loop Modal Analysis . . . . . . . . . Appendix 7.1: Data of Example 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 7.2: Data of Examples 7.2 and 7.4 . . . . . . . . . . . . . . . . . . . Appendix 7.3: Data of Example 7.3 (Table 7.7) . . . . . . . . . . . . . . . . . Appendix 7.4: Data of Example 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Probabilistic Analysis of Small-Signal Stability of a Power System Affected by Grid-Connected Wind Power Generation . . . . . . . . . . . . 8.1 Probabilistic Analysis of Small-Signal Stability of a Power System: An Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Framework of Probabilistic Small-Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Methodologies of Probabilistic Small-Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Probabilistic Analysis of Small-Signal Stability of a Power System Affected by Directly Grid-Connected Onshore Wind Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Cumulant Theory Based Analytical Method of Probabilistic Small-Signal Stability Analysis of a Conventional AC Power System . . . . . . . . . . . . . . . . 8.2.2 Example 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Probabilistic Analysis of Small-Signal Stability of a Power System Affected by MTDC–Connected Offshore Wind Power Generation . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Cumulant Theory Based Analytical Method of Probabilistic Small-Signal Stability Analysis of a Hybrid AC/DC Power System . . . . . . . . . . . . . . . . . . 8.3.2 Example 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Variance Indices Analysis of Probabilistic Small-Signal Stability of a Hybrid AC/DC Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Example 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix 8.1: Data of Examples 8.1, 8.2 and 8.3 . . . . . . . . . . . . . . . . . 341 Example 16-Machine 68-Bus New York and New England Power System [34] (Tables 8.5, 8.6 and 8.7) . . . . . . . . . . . . . . . . . . . . . . 341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Chapter 1

Introduction

This chapter introduces the small-signal stability analysis of a power system integrated with variable speed wind generators (VSWGs) by reviewing the following two subjects which are covered in this book: (1) The impact of grid connection of the VSWGs on the small-signal angular stability of the power system, i.e., the low-frequency electromechanical power oscillations (LEPOs); (2) Small-signal stability of the integrated VSC-based DC/AC power systems.

1.1

Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability of Power System

The small-signal angular stability of the power system is mainly about the problem of low-frequency electromechanical power oscillations (LEPOs). In this section, the review starts with four early published papers which proposed two main strategies to examine the impact of the VSWGs on power system LEPOs: (1) The VSWGs displacing the SGs; (2) The VSWGs being connected to the power system. Following-on work to those four representative papers is reviewed separately according to the two different strategies of examination. Then, detailed analysis on each of strategies of examination is carried out. It revealed that in each of two strategies of examination used, inherently there are two factors affecting the results of examination. Mixture of two affecting factors has caused the diversification of results of examination achieved so far about the impact of grid-connected VSWGs on power system small-signal angular stability.

© Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_1

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1 Introduction

1.1.1

Impact of the VSWGs: A Brief Review

1.1.1.1

Representatives of Earlier Study

Representatives of earlier published study about the impact of grid-connected VSWGs on power system small-signal angular stability are [1–4]. Simple example power systems were used to examine the change of system small-signal angular stability when the VSWGs were connected. The examination was carried out by computing system electromechanical oscillation modes (EOMs). In [1], power output from the VSWGs displaced that of a SG gradually. The results indicated that the effect of the VSWGs on the damping of system inter-area EOM was “limited and varied” [1]. The approach adopted in [2] was different, where a SG was displaced by a DFIG. Damping of inter-area EOMs of the example system with the displaced SG and the displacing DFIG was compared. It was found that the maximum difference of the damping was about 15% which was significant. In [3, 4], the focus of study was the impact of change of system load flow caused by the gridconnected VSWGs. The results show that the impact on the damping of system oscillation modes was very small [3] and could be positive or negative [4]. Grid connection of a VSWG to a power system changes the system load flow. The change brought about by the VSWG is balanced by the power output from the SGs at the slack nodes in the power system. This is equivalent to having the power output from the VSWG to replace that of the SGs. Under certain circumstance, some SGs may withdraw from the power system to give way of generation to cleaner wind power. Earlier work presented in [1–4] represented some of the features of the consequence of grid connection of the VSWG. Slootweg and Kling [1] investigated the case with the power output from the VSWGs to replace that of the SGs at same locations in power systems. Thus, load flow of example power systems did not change. Sanchez-Gasca et al. [2] examined the case of displacement of the SGs by the VSWGs at the same locations. Load flow of example power systems remained the same after the displacement. While the case considered in [3] and [4] was the addition of the VSWGs in the example power systems. The change of load flow of the power systems was balanced by the power output from the SGs at the slack bus. Those three different operational scenarios of the grid connection of the VSWG can be illustrated by Figs. 1.1, 1.2, and 1.3. From the practical point of view of grid connection of wind power generation, the case of Fig. 1.3 is more general. Figure 1.1 represents the case of wind-thermal Fig. 1.1 Case considered in [1]: Pw0 replacing Pg0

Pw0 VSWG Pg0 SG

A power system

1.1 Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability. . .

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Fig. 1.2 Case considered in [2]: the VSWG replacing the SG Fig. 1.3 Case considered in [3, 4]: Addition of a VSWG in the power system

VSWG A power system Pw0 + jQw0

bundled power generation and supply. The case shown by Fig. 1.2 is rare in practice but useful for understanding the mechanism about how the VSWG affects power system small-signal angular stability differently with the SG. In principle, the cases considered in [1] (Fig. 1.1) and [3] (Fig. 1.3) are same that the wind power generation in Fig. 1.3 replaces the power output from the SGs in the slack node in the power system. Their difference is that the case considered in Fig. 1.1 may bring about much less change of system load flow than that in Fig. 1.3. Hence Fig. 1.1 may be considered as a special case of Fig. 1.3. For years, the investigation on the power system small-signal angular stability as affected by grid-connected VSWGs has been carried out in the similar ways to that used in [1–4]. Mainly two types of grid connection of wind power generation have been considered. The first one is the displacement of the SGs by the VSWGs at the same locations as shown by Fig. 1.2. The other is the simple addition of the VSWGs in the power system as shown by Fig. 1.3.

1.1.1.2

Representatives Study: The VSWG Displacing the SG

Strategy of investigation via a VSWG displacing a SG in a power system is shown by Fig. 1.2. The objective of investigation is to compare the dynamic function between the VSWG and the SG so as to understand the mechanism on how the VSWG affects the small-signal angular stability of the power system differently to the conventional SG. In [5, 6], relatively more complicated example power systems were used to examine the effect when a grid-connected DFIG displaced a SG. Computational results of modal analysis in [5, 6] indicated that the effect could be small or large as well as positive to enhance or negative to reduce the damping of power system EOMs. The results were dependent of the locations and dynamic importance of the SGs being displaced as well as the load conditions of the example power systems.

4

1 Introduction

Gautam et al. [7] is one of the most representative work adopting the approach of the displacement. Authors of [7] thoughtfully pointed out that a DFIG affects power system angular stability in four ways: 1. 2. 3. 4.

“displacing synchronous machines thereby affecting the modes; impacting major path flows thereby affecting the synchronizing forces; displacing synchronous machines that have power system stabilizers; DFIG controls interacting with the damping torque on nearby large synchronous generators” [7].

Approach proposed and used in [7] is “to convert the DFIG machines into equivalent conventional round rotor synchronous machines and then evaluate the sensitivity of the eigenvalues with respect to inertia. In this regard, modes that are both detrimentally and beneficially affected by the change in inertia are identified” [7]. Results presented in [7] indicated that the effect of displacement can be detrimental or beneficial to the system small-signal angular stability. The results were highly related with the locations of displacement and load conditions of power systems. Vittal et al. [8] and Vittal and Keane [9] further extended the strategy of displacement to investigate the effect of supplementary reactive power/voltage control of DFIGs on power system small-signal angular stability. Authors of [8, 9] considered that “the reactive power control strategy of the wind farm can have a significant impact on the rotor angle stability of conventional synchronous units in the system. Controlling reactive power to achieve a targeted voltage can improve the rotor angle stability of the existing synchronous units in the system”. It was hypothesized in [9] “that when the synchronous wind farms with high participation factors are replaced by a VSWT wind farm with terminal voltage control enabled, the rotor angle stability of the system will be improved”. However, the hypothesis was not generally proved to be true in [9].

1.1.1.3

Representatives Study: Adding the VSWG in the Power System

Strategy of investigation on the case that a VSWG is simply added in a power system is illustrated by Fig. 1.3. As it is pointed out previously, Fig. 1.1 could be a special case of Fig. 1.3. Tsourakis et al. [10] presents the study of a simple single-machine infinite-bus power system connected with two DFIGs for wind power generation. Power output from the DFIGs increased with the load and thus the power output from the SG remained unchanged. Under this circumstance of system operation, results of modal analysis indicated that the DFIGs with reactive power control or voltage control affected the system oscillation modes differently with the change of penetration level of the wind. Oliveira et al. [11] studied a Brazilian real power distribution network. Results in [11] showed that different operational mode of the DFIG, i.e., power factor control mode or voltage control mode, did not affect significantly the electromechanical oscillations.

1.1 Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability. . .

5

Quintero et al. [12] used a large-scale real power system as an example to study the effect of converter control-based generators (CCBGs) on system small-signal angular stability. The focus of study in [12] was the participation of CCBGs in power system electromechanical oscillation modes so as to examine the scale of dynamic interactions between the CCBGs and synchronous generators (SGs). Results in [12] indicated that at higher level of wind penetration (13.2%), participation of CCBGs in the electromechanical oscillation modes was observable. It was found that there were oscillation modes caused by the CCBGs which were highly sensitive to the control parameter variations of the CCBGs. Those modes were name as converter oscillation modes. It was reported that when the converter oscillation modes were of the frequency within the range of frequency of power system electromechanical oscillation modes, the SGs participated considerably in some of those converter oscillation modes. Ding et al. [13] studied the impact of adding a VSWG (DFIG and PMSG) into a power system and presented the comparative results of modal analysis of an example power system with the VSWG using models of different level of details. Main conclusion made in [13] was that when the VSWG was modelled as a constant power source, the results of modal analysis were close to that when dynamic models of VSWG were used. Knuppel1 et al. [14] and Kunjumuhammed et al. [15] examined both cases of displacement by and addition of the VSWGs using example power systems. Results in [14] showed that without the displacement, the participation of the SGs in the example power system did not change. While with the displacement, it changed significantly. Work in [15] used two-area test system and a representative Great Britain test system. Various possible operating scenarios with different active power combinations between synchronous generators and DFIGs, in some cases synchronous generator disconnected, were examined. It indicated that the reduction in output of synchronous generators with high participation in the power system electromechanical oscillation modes resulted in higher improvement in modes’ damping, and the change in inter-area power flow caused the change of frequency of interarea EOMs.

1.1.2

Impact of the VSWGs: Discussions

1.1.2.1

Strategy of Adding the VSWG in the Power System

This strategy reflects the real scenario that a VSWG is connected to a power system. Investigation is to compare the system EOMs before and after the VSWG is added as it did in [10–15]. The strategy can be illustrated by a simple power system shown by Fig. 1.4 when no VSWG is connected. After a VSWG is added to the power system of Fig. 1.4, the system configuration is shown by Fig. 1.5. Obviously, the addition brings about two changes that may affect system small-signal angular stability. Firstly, addition of the VSWG changes

6 Fig. 1.4 A single-machine infinite-bus power system

Fig. 1.5 Addition of a VSWG in power system of Fig. 1.4

1 Introduction

Infinite bus

SG1

Infinite bus

SG1

Pw + jQw VWSG

the load flow condition and configuration of the power system. Consequently, system small-signal angular stability may change accordingly. This affecting factor is well demonstrated by the results presented in [13]. When the VSWG was modelled as the constant power source, effect of the VSWG dynamics was not included [13]. Grid integration of the VSWG only changed the system load flow condition to affect the damping of the power system EOMs. Results of case study in [13] indicated the improvement of the damping of the EOMs when wind penetration increased. Secondly, addition of the VSWG introduces dynamic interactions of the VSWG with the SG in the power system which could affect system small-signal angular stability. In [13], results of modal computation with the VSWGs’ dynamics being considered were compared with that with the VSWGs being modelled as constant power sources (hence dynamic interactions were excluded). Comparison indicated that both results were very close. This meant that the impact of the VSWG’s dynamic interactions was small. Results of comparison presented in [13] fitted the usual recognition on the dynamic effect of the VSWGs, because it is normally believed that VSWGs are inertia-less due to the function of the VSC control and hence the VSWGs’ dynamic interactions with SGs are weak. Results in [14] indicated that the addition of the VSWGs did not change the participation of the SGs in the EOMs. This implied that the dynamic impact of the SGs was not affected by the VSWGs’ dynamics. Hence results in [14] indirectly confirmed the recognition that normally the VSWGs dynamically interact with the SGs weakly and affect little the smallsignal angular stability of the power system. However, results presented in [12] challenged the normal recognition that the VSWGs’ dynamic involvement in the EOMs is small. When the wind penetration was high, [12] found the existence of considerable dynamic interactions between the

1.1 Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability. . .

7

VSWGs and the SGs from the computational results of the participation factors (PFs) of the EOMs [12]. The findings reported in [12] implied that the dynamic interactions introduced by the VSWGs may become strong under certain special conditions which may affect power system small-signal angular stability significantly. In the fifth chapter of the book, the special conditions under which the dynamic interactions between the VSWGs and the SGs may become strong are investigated. The impact of the strong dynamic interactions introduced by the VSWGs is examined.

1.1.2.2

Separate Examination of Affecting Factors

From the discussions in the above subsection, it can be concluded that when a VSWG is added in a power system, the effect of addition on system small-signal angular stability is the mixed result of two affecting factors: load flow change and dynamic interactions brought about by the VSWG. Both load flow change and dynamic interactions vary with the change of power output from the VSWG (wind penetration level). Results of investigation on the effect of adding the VSWG in the power system obtained so far have been varied. For example, [3, 4] indicated that effect of change of power output from the VSWGs was small; but according to [14] it was not. The reasons of such diversification are in two folds. Firstly, those results were dependent of example power systems used and their operating conditions. Secondly, they were mixed effect of two affecting factors. Strict theoretical analysis can provide example-independent results and conclusions. At present if this is difficult, separate examination of two affecting factors is a correct way forward towards more detailed and elaborated investigation. Quintero et al. [12] adopted the conventional tool of the PFs to examine the dynamic engagement of the VSWGs in power system dynamic and vice versa. It was an attempt to individually investigate the affecting factor of dynamic interactions between the VSWGs and the SGs. However, the PFs can only estimate the scale of dynamic involvement, not the direction to be positive to improve or negative to reduce the power system small-signal angular stability. This inherent drawback of the PFs may limit their applications in the investigation [13] used model of constant power source to represent the VSWG. This model effectively enabled the separate examination of the effect of power flow change introduced by the VSWG. In [16], a method of separate examination of two affecting factors of grid connection of the DFIGs on power system small-signal angular stability was proposed. The method considered the dynamic variation of power exchange between a grid-connected DFIG and power system to be ΔPw þ jΔQw. A closed-loop state space model of the power system was established as shown by Fig. 1.6, where the DFIG was treated as the feedback controller. Based on the model, impact of the DFIG’s dynamics on an EOM of interests was estimated to be Δλ by use of the damping torque analysis (DTA). In the third and fourth chapter of the book, how the DTA is applied to examine the impact of the DFIG is introduced and demonstrated in details.

8

1 Introduction

Fig. 1.6 Closed-loop state space model of power system with DFIG [16]

d ΔXg = Ag ΔXg dt

Power system

DPw DFIG DQw

DVpcc

GP (s) GQ (s)

When there is no dynamic interaction between the DFIG and power system, ΔPw þ jΔQw ¼ 0. The DFIG is degraded into a constant power source Pw0 þ jQw0 and the system is represented by its open-loop model. Hence it was concluded in [16] that with the constant power model of the DFIG, effect of load flow change introduced by the DFIG can be determined by calculating the EOM from system open-loop state matrix Ag to be λ0. Thus, two affecting factors of the DFIG can be examined separately. The total impact is Δλ þ λ0 ¼ λ. It was derived in [16] that the dynamic variation of power exchange between the DFIG and power system was ΔPw ¼ GP(s)ΔVpcc, ΔQw ¼ GQ(s)ΔVpcc where ΔVpcc is the dynamic variation of magnitude of voltage at the point of common coupling (PCC) of the DFIG. The derivation indicated that the dynamic variation of power exchange should normally be small, because in power system ΔVpcc is usually small. Hence normally the impact of the DFIG’s dynamics, Δλ, is small. Estimation of the impact from the constant power model of the DFIG should approximately be equal to the total impact, i.e., λ0  λ. This conclusion made in [16] confirmed the results presented in [13] and is useful in practical applications. For example, by use of the constant power model of the DFIG, the most dangerous scenarios of the grid connection of the DFIG can be identified. This will be very helpful in planning the grid connection of the wind farms when the dynamics of the wind farms are often unknown but their impact needs to be predicted. However, feasibility of such application still needs to be investigated. The obstacle is the finding reported in [12] that the considerable dynamic interactions between the VSWG and the power system may occur. Method proposed in [16] may be helpful in overcoming the obstacle because the estimation of the impact of the dynamic interactions between the VSWG and the power system by use of Δλ can tell not only the scale of impact as the PFs do (which was used in [12]), but also the direction of impact. For example, if the impact of dynamic interactions is predicted to be positive, in this case, the constant power model of the DFIG can definitely identify the most dangerous scenarios of grid connection of the DFIG. In the fourth chapter of the book, the method proposed in [16] is introduced and demonstrated.

1.1.2.3

Strategy of the VSWG Displacing the SG

A VSWG displacing a SG at a same location rarely happens in practical power systems. However, it is considered that this strategy of investigation is helpful in

1.1 Impact of Grid-Connected VSWGs on the Small-Signal Angular Stability. . . Fig. 1.7 Power system with SG2 to be displaced by a VSWG as shown by Fig. 1.5

9

Infinite bus

SG1

DPg + jDQg SG2

understanding the difference between the dynamic function of the VSWG and the SG. The strategy can be explained by use of the simple power system of Fig. 1.5 with the VSWG being displaced by SG2 as shown in Fig. 1.7. Modal analysis can be carried out to compare the small-signal angular stability of system shown by Figs. 1.5 and 1.7 to examine the impact of displacement. Obviously, load flow condition of system shown by Figs. 1.5 and 1.7 can be kept exactly same. The VSWG displacing the SG can happen without causing any change of system load flow condition and configuration. Thus the affecting factor of the load flow change caused by the grid connection of VSWG is excluded. The difference of the impact of the displacement is from the different dynamics of the VSWG and the SG. Results of investigation by the displacement so far were also diversified as reviewed in the previous section [2, 4–7]. However, it has been found that the dynamic importance of the SGs being displaced seemed playing more role in determining the results of the displacement [5, 6, 15]. In fact, the displacement can be considered being implemented in two separate steps. First, the dynamics of the displaced SG are withdrawn from the power system. Second, the dynamics of the displacing VSWG are added. Hence, there are two factors affecting the impact of the VSWG displacing SG on power system smallsignal angular stability. The first affecting factor is the dynamics of the SG and the second is the dynamics of the VSWG. If the impact of the VSWG’s dynamics is small as indicated by the results in [13, 16], the dynamics of the displaced SG are more influential on the total impact of the displacement as reported in [5, 6, 15]. In addition, if the impact of those two affecting factors is in the opposite direction, their impact may cancel each other to result in little impact. Therefore, indeed the problem is complicated and it is difficult to make a definite conclusion from the displacement. In order to carry out in-depth investigation on the impact of the displacement of the SG by the VSWG so as to understand the difference between the SG’s and the VSWG’s dynamic function, it looks that methods to rank the dynamic importance of the SGs need to be developed. The PFs can give indication of dynamic involvement of the SGs in the EOMs of interests. However, their relationship with the dynamic importance of the SGs has been unclear. In addition, they cannot give the indication of direction of the impact.

10

1 Introduction

Method proposed in [7] to calculate the modal sensitivity to the constant of inertia of the displaced SG was an attempt to develop an index to predict the impact of the displacement without knowing the dynamics of the displacing VSWG. The index can provide some aspects of dynamic importance of the SGs. However, some important aspects of the SGs’ dynamics, such as the function of the automatic voltage regulators (AVRs) and the power system stabilizers (PSSs), are excluded from the index. In addition, application of the index is based on the assumption that the relation between the EOMs of interests and the constant of inertia of the SGs is linear which may not be true sometimes.

1.1.2.4

Impact of the VSWG Reactive Power/Terminal Voltage Control

A grid-connected VSWG can adopt the fixed reactive power output control or fixed terminal voltage control. Impact of those two types of control on power system small-signal angular stability may be different. This important issue was discussed in [8–11] with diversified results. Conclusion made in [8, 9] was that the terminal voltage control was more beneficial to system small-signal angular stability. Results from [10] indicated that the impact was variable and dependent of system operating conditions. It was concluded in [11] that the difference of impact between the reactive power and terminal voltage control was small. When the VSWG operates with the fixed reactive power output or the fixed terminal voltage control, reactive power output from the VSWG at steady-state operation is different. Thus, change of system load flow condition brought about the VSWG is different when the reactive power or the terminal voltage control is used. In addition, dynamic interactions introduced by the VSWG should also be different. Hence, it is a problem mixed with two affecting factors (load flow condition and dynamic interactions) and more careful comparison needs to be carried out by examining each of the affecting factors individually. The comparison is important in planning the grid connection of wind farms. Hence, it is more desirable that the comparison is carried out without knowing the dynamics of wind farms and their associated control systems. In the fourth chapter of the book, an index to predict the difference of the impact between the reactive power and terminal voltage control is introduced and evaluated.

1.1.2.5

Effect of the Phase Locked Loop (PLL)

Vector control is the most widely implemented scheme by the VSCs for the grid connection of the VSWGs. Implementation of vector control relies on the phase tracking of the PCC voltage at the VSWGs’ terminal. A phase locked loop (PLL) is a control system being specially designed to fulfil the task of phase tracking of the voltage. It plays a very important role for the grid connection of the VSWGs. Functionally, the PLL links the VSWG with the power system. Hence, the impact of the PLL should be in two folds. On one hand, dynamics of the PLL will affect the

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

11

performance of grid-connected VSWG. Recent study has shown that when the VSWG is weakly connected with the power system, the PLL’s dynamics may impose negative impact on the VSC-based system. Improper tuning of the PLL’s parameters could even cause the small-signal instability of the VSWG itself. Hence, careful tuning of the PLL’s parameters is needed to consider its effect on the operational stability of the VSWG. In the study, the power system is represented by a three-phase voltage source connected to the VSWG via transmission impedance. Dynamics of power system are not considered. In Sect. 1.2.1, study on the small-signal stability of grid-connected single VSC system under weak system condition is reviewed. On the other hand, dynamics of the PLL may also affect power system dynamic performance, for example, the small-signal angular stability. So far, impact of the VSWGs on power system small-signal angular stability has been examined by assuming that the PLL dynamic performance is perfect, which can provide accurate and instantaneous phase tracking of terminal voltage of the VSWGs. Thus, the examination has been carried out under the known relative position of d-q coordinate of the VSWGs to that of power system common coordinate without the effect of the PLL’s dynamics being considered. Error of the PLL phase tracking has not been included in the examination. Normally, ignorance of the phase tracking error of the PLL may be acceptable if the PLL is designed properly. However, unusual cases may occur if parameters of the PLL are not tuned properly. In [17], impact of the PLL’s dynamics on power system small-signal angular stability was examined. Results of modal analysis based on an example power system were presented to indicate that a high-gain and fast-acting PLL affected little the electromechanical dynamics of the power system. However, when the frequency of the PLL oscillation mode was close to that of an EOM of the power system and at the same time the damping of the PLL oscillation mode was light, the PLL participated the power system electromechanical dynamics significantly. This implied considerable dynamic interactions between the PLL and the power system. The phenomenon reported in [17] is examined in the sixth chapter of the book.

1.2

Small-Signal Stability of the VSC-Based DC/AC Power System

Voltage source converter (VSC) based HVDC lines and multi-terminal DC (MTDC) networks are of many technical advantages over the conventional LCC HVDC lines. They are the favorable solution to the large-scale transmission of renewable power generation, such as the offshore wind power. In addition, the VSCs are the key components for the grid connection of renewable power generation. Hence, smallsignal stability of an AC power system integrated with the VSCs has been a topic of study attracted interests of many power system researchers and engineers. The potential instability risk that might be brought about by the integration of the

12

1 Introduction

VSCs, the VSC-based HVDC line or MTDC network, has been investigated in recent years from various standpoints. There have been two main concerned problems about the small-signal stability of an integrated VSC-based DC/AC power system. The first one is the small-signal stability of the VSC-based system itself, such as a single grid-connected VSC system or a MTDC network. The second problem is the small-signal stability of the integrated VSC-based DC/AC power system, such as the AC power systems integrated with the renewable power generation, the VSC-HVDC/AC or the MTDC/AC power systems. The key issue is how the grid connection of the VSCs may affect the small-signal stability of the integrated VSC-based DC/AC power system. Those two problems are not covered by the conventional theory and analysis of small-signal stability of normal AC power systems, where the dominant dynamic components are the synchronous generators (SGs). In the VSC-based DC/AC power system, the most important dynamic components are the VSCs, in addition to the SGs. Dynamic interactions among the multiple VSCs and SGs determine the small-signal stability of the VSC-based DC/AC power system. However, the conventional stability theory and analysis of AC power system did not tell how the VSCs may interact among themselves and with the SGs to affect the small-signal stability of the VSC-based DC/AC power system. This section reviews the research progress and main conclusions obtained so far about the small-signal stability of three types of integrated VSC-based DC/AC power systems. They are the grid-connected single VSC system, the AC power system integrated with the VSC-based HVDC line and the AC power system integrated with the MTDC network.

1.2.1

Small-Signal Stability of a Single Grid-Connected VSC System

A single VSC system is the simplest VSC-based DC/AC power system. It could be a grid-connected wind farm which is represented by a single VSWG being connected by the VSC to the AC power system. It could also be a VSC of the HVDC line or the MTDC network. Study on the small-signal stability of the single VSC system is about the stability of the VSC system itself without considering the impact of its dynamic interactions with the SGs in the external power system. The small-signal stability of the single VSC system is found being mainly affected by the converter control and the PLL.

1.2.1.1

Main Affecting Factors: Weak Grid Connection and the PLL

Figure 1.8 shows the configuration of a single grid-connected VSC system, where the external power system is represented by an infinite busbar at node b. An

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

13

VSC Renewable power generation or the HVDC line or the MTDC

Exteral power system b PWM

Inner loops

PLL

Outer loops

Power calculation

Fig. 1.8 Configuration of a single grid-connected VSC system

important factor affecting the small-signal stability of the VSC system is the short circuit ratio (SCR). Denote Prated as the rated power of the VSC system. The SCR is defined to be SCR ¼

Vpcc 2 XL Prated

ð1:1Þ

When the SCR is smaller than 3, connection of the VSC with the AC power system is considered being weak. It has been found that under the condition of weak grid connection, the VSC system may suffer from the problem of small-signal instability. Early study in [18–20] based on the state-space model of the VSC system by using the modal analysis indicated that the small-signal stability of the VSC system was affected by the dynamics of the PLL. Maximum power transmission capability decreased when the SCR was reduced. In [21, 22], Lennart Harnefors examined comprehensively the small-signal stability of the VSC system from the perspective of control system parameter conditions. The examination was conducted by using the passivity theory on the basis of input impedance matrix model of the VSC system. Results of examination demonstrated that increase of PI gains of current control inner loops of the VSC may cause system instability. In addition, high bandwidth of the DC voltage control outer loop and the PLL was unfavorable to the small-signal stability of the VSC system. Long-distance transmission of large-scale wind power was found to be prone to instability problem [23–25]. Study cases by modal analysis in [26] showed that when the SCR was reduced below a certain value, dominant poles of the PMSG system moved into the right half of the complex plane. Computation of participation factors indicated that unstable dominant poles were associated with the AC voltage control outer loop of the grid side converter (GSC). Similar conclusions were obtained from the study cases in [27]. However, main source of system instability was pointed to

14

1 Introduction

the PLL in [27]. Results presented in [18–27] consistently confirmed that the weak grid connection was the key condition under which the VSC system may lose the small-signal stability. Investigation in [28–30] was focused on finding the source of instability and confirmed that the PLL was the main dynamic component to cause the instability of the VSC system when the grid connection was weak. Based on the quasi-steady state model of the PLL, impact of transmission impedance and load impedance on the low-frequency dynamics of the PLL was investigated comprehensively in [28, 29]. Results of investigation showed that when the transmission impedance was high, the VSC system may suffer from the problem of growing oscillations. Results of root loci computation in [30] clearly indicated that when the SCR was small, increase of PI gains of the PLL led to the instability of the VSC system. Work in [28–30] confirmed the role played by the PLL to cause the loss of small-signal stability of weakly grid-connected VSC system. In [31], investigation based on the impedance model of the VSC system indicated that the range of negative resistance of the VSC impedance increased when the bandwidth of the PLL increased. The investigation attributed the impact of the PLL to the contribution of negative resistance. Work presented in [28–31] not only confirmed that the main factors affecting the small-signal stability of the VSC system are the weak grid connection and PLL’s dynamics, but also represented a series of effort to reveal the mechanism about how the PLL affects the small-signal stability of the VSC system. In order to understand why the PLL may affect the small-signal stability considerably, the impact of the dynamic interactions introduced by the PLL has become a special topic of study. It has been considered that the impact of dynamic interactions introduced by the PLL may be the in-depth reason and can reveal the mechanism about why the VSC system may lose the small-signal stability.

1.2.1.2

Impact of Dynamic Interactions Introduced by the PLL

Dynamic interactions are the mutual exciting and responding of two dynamic components in a system. A good illustration is the closed-loop system consisted of two open-loop subsystems shown by Fig. 1.9. Subsystem A responds to the excitation from subsystem B, u. At the same time, its output, y, is the excitation to subsystem B. If u ¼ 0 or y ¼ 0, dynamic interactions between subsystems are Fig. 1.9 Closed-loop system

G(s) + u

Subsystem A Subsystem B H( H s) H(s)

y

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

15

zero such that the closed-loop system is stable as long as the open-loop subsystems are stable. If u  0 or/and y  0, dynamic interactions between subsystems are weak and would normally affect little the closed-loop system stability. Only the existence of strong dynamic interactions may cause the system instability. Hence, by examining the degree of dynamic interactions introduced by the PLL, it is possible to find why the PLL may bring about the small-signal instability risk to the VSC system. The bandwidths of power control outer loops and current control inner loops of the VSC system in Fig. 1.1 are normally around 10 Hz and 100 Hz respectively. Hence, usually the dynamic interactions between the outer and inner control loops are weak and affect little the small-signal stability of the VSC system. The bandwidth of the PLL can vary in a wide range to possibly interact with either the outer loops or inner loops of VSC control. Hence, the dynamic interactions between the PLL and either the power control outer loops or current control inner loops may cause the small-signal instability of the VSC system. Study in [32] demonstrated that the PLL and outer loops of VSC control participated in the dominant oscillation modes of the VSC system which became unstable when the grid connection was weak. This indicated that the dynamic interactions between the PLL and outer loops caused system instability. In [33], damping torque analysis was applied to examine the impact of dynamic interactions between the DC and AC voltage control outer loop. Results of examination revealed that when the lag caused by voltage measurement was considered, the AC voltage control outer loop may contribute negative damping torque to the DC voltage control outer loop. Contribution of negative damping torque increased when the SCR was reduced. Results of modal analysis in [34] showed that under the condition of weak grid connection, dynamic interactions between the active, reactive control outer loop and the PLL became strong. Small-signal stability of a grid-connected PMSG was examined in [35]. The examination indicated that the dynamic interactions between the PLL, DC and AC voltage control outer loop were harmful to the stability of the PMSG system as the interactions were similar to the function of positive feedback. In addition, when the bandwidth of the PLL and the DC voltage control outer loop was close, the degree of dynamic interactions was maximum. Dynamic interactions between the PLL and current control inner loops of the VSC may also affect the small-signal stability of VSC system considerably. Results of modal analysis and simulation in [36, 37] demonstrated that when the grid connection was weak, the VSC system may lose the small-signal stability, which was related to the dynamic interactions between the PLL and current control inner loops. The small-signal stability of the PMSG with weak grid connection during the low voltage ride through was examined in [38]. Results of examination showed that the system stability was dependent of the dynamic interactions between the PLL and current inner loops. When the bandwidth of the PLL and inner loops was close, the degree of dynamic interactions increased.

16

1.2.1.3

1 Introduction

Summary: The Single VSC System

Study so far has confirmed the following two main conclusions about the smallsignal stability of the VSC system. 1. Under the condition of weak grid connection, dominant oscillation modes of the VSC system may move into the right half of complex plane. The PLL is highly related with the dominant modes. Hence, the PLL may very likely be responsible for the small-signal instability of the VSC system. 2. Under the condition of weak grid connection, the dynamic interactions between the PLL and either outer loops or inner loops of VSC control may become strong. The strong dynamic interactions may be detrimental to the small-signal stability of the VSC system. The strong dynamic interactions occur when the bandwidth of the PLL and control loops is close. However, the study has been carried out mainly by numerical computation, simulation and experiment on the case-by-case basis. Damping torque analysis and frequency-domain analysis based on the impedance model of VSC system attributed the instability risk to the occurrence of negative damping torque or negative resistance under the condition of weak grid connection. The analysis was very helpful for understanding and exposing the insight about why the PLL may bring about the instability risk under the condition of weak grid connection. However, the analysis normally was also based on the numerical results on the case-by-case basis. Therefore, the general applicability of the conclusions still needs the confirmation by analytical examination, which may reveal the in-depth insight about why the VSC system may lose the small-signal stability as affected by the PLL under the condition of weak grid connection.

1.2.2

Small-Signal Stability of Integrated VSC-Based DC/AC Power Systems

Explanation is presented for Fig. 1.9 above about why the small-signal stability of the VSC system is determined by the dynamic interactions between the converter control and the PLL. The explanation is general and applicable to more complicated AC power system integrated with the VSC-HVDC line and the MTDC network. In either the VSC-HVDC/AC power system or the MTDC/AC power system, there are dynamic interactions between multiple VSCs or/and between the VSCs and the SGs, which affect the system small-signal stability. When the VSC-HVDC/AC power system is investigated, the main strategy applied to simplify the investigation has been to examine a portion of the VSC-HVDC/AC power system by ignoring some of dynamic interactions. When the MTDC/AC power system is examined, strategy of simplification has been to mainly consider either the MTDC or the AC network. With the simplification, the examination has become simpler and

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

A

B

VSC-1

C

D

VSC-2

17

E

F SG-2

SG-1

DC/AC interactions

DC/DC interactions between VSC-1 and VSC-2

DC/AC interactions

Fig. 1.10 Configuration of an AC power system integrated with a VSC-based HVDC line

manageable. However, the applicability of the conclusions obtained is constrained by the assumptions of simplification made in the examination.

1.2.2.1

Small-Signal Stability of AC Power System Integrated with a VSC-Based HVDC Line

Figure 1.10 shows the configuration of an AC power system integrated with a VSC-based HVDC line. Without loss of generality, it can be assumed that VSC-1 uses the DC voltage control and VSC-2 adopts the active power control. Although the point-to-point VSC-based HVDC line is the simplest DC power transmission system, the DC/AC power system shown by Fig. 1.10 is much more complicated than the single VSC system shown by Fig. 1.8. In addition to the various dynamic interactions within each of two VSCs, following dynamic interactions may also affect the small-signal stability of the DC/AC power system shown by Fig. 1.10: (1) the DC/AC interactions between VSC-1 and SG-1; (2) the DC/AC interactions between VSC-2 and SG-2; (3) the DC/DC interactions between VSC-1 and VSC-2. Hence, examination of small-signal stability of the AC power system with the VSC-based HVDC line is much more difficult. The difficulty is not in assessing the system stability, as the modal analysis can always be applied. For example, results of modal analysis in [39, 40] indicated that the integration of the VSC-HVDC line may impose positive or negative impact on the system small-signal stability to improve or reduce the damping of system sub-synchronous oscillation (SSO) modes. The difficulty is to investigate why and how the VSC-HVDC line may affect the small-signal stability of integrated power system, for example, the damping of the SSOs. For the investigation, the frequency-domain analysis can be used with some simplifications made to the DC/AC power system. In [41, 42], the small-signal stability of the VSC-HVDC line system between point B and E in Fig. 1.10 was studied. In the study, the VSC-based HVDC line system was divided into two parts at point C, the part with VSC-1 and that with VSC-2 separately. The output impedance of VSC-1 and input impedance of VSC-2 were derived. By using the Middlebrook stability criterion, the system small-signal stability was assessed. Stability region with the variation of parameters of VSC-based HVDC system was calculated and validated by simulation. Obviously, study in [41, 42] considered only the DC/DC dynamic interactions between VSC-1

18

1 Introduction

and VSC-2. The impact of DC/AC interactions between the SGs and VSCs were not examined. In [43], the DC/AC power system between point D and F in Fig. 1.10 was examined. The DC voltage across C2 was assumed being constant such that the dynamics of HVDC line and AC system on the left of point D were ignored. Results of investigation in [43] showed that the system was always stable even if the grid connection of VSC-2 was weak. This implied that the DC/AC dynamic interactions between SG-2 and VSC-2 with active power control affected little the small-signal stability of the DC/AC power system shown by Fig. 1.10. The implication is useful, because point E can be simply assumed to be an infinite busbar. The assumption should not change the small-signal stability of the DC/AC power system shown by Fig. 1.10. The DC/AC power system shown by Fig. 1.10 between point A and E was considered in [44]. VSC-2 was modelled as a constant power source and hence dynamic interactions between VSC-2 and SG-2 was ignored. This simplification was allowable as being confirmed by [43]. A closed-loop model was derived with two open-loop subsystems. One open-loop subsystem was comprised of control systems of VSC-1 and SG-1, which represented the DC/AC dynamic interactions. The other open-loop subsystem consisted of DC line. Interface variables between two openloop subsystems were variations of active power output from VSC-1 and DC voltage across C1. Complex torque analysis was applied to examine the small-signal stability of the AC power system with the VSC-based HVDC line. Results of examination indicated that when the load on the DC line, length of DC line or the bandwidth of the VSC control increased, stability of the AC power system with the VSC-based HVDC line may degrade to lead to the system instability. Frequency-domain analysis attributes the degradation of small-signal stability of the AC power system integrated with the VSC-based HVDC line to the contribution of negative damping torque or negative resistance from the VSC-HVDC line. Hence, the analysis provides the insight into why the VSC- HVDC line may affect power system stability negatively. However, the frequency-domain analysis is applied on the basis of the closed-loop model. In order to derive the closed-loop model, an appropriate point of separation in the system needs to be selected to divide the system into two interconnected open-loop subsystems. In [45, 46], point C in Fig. 1.10 was chosen and the derived closed-loop model was single-input and single-output (SISO) such that the frequency domain analysis can be applied. Normally, assumptions of simplification were made in order to derive a SISO closed-loop model if the point of division was not particularly selected. Typical such assumptions were to examine a portion of the DC/AC power system shown by Fig. 1.10 and ignored some of VSC control functions [41–46]. When all the dynamic interactions introduced from the entire VSC-HVDC line are considered, the closedloop model of the power system is multiple-input and multiple-output (MIMO) [47, 48], which would be challenging for the frequency-domain analysis to handle. In [49], the impact of SSO dynamic interactions introduced by a control loop of a VSC in the AC power system integrated with the VSC HVDC line was investigated. A SISO closed-loop model was derived, where the control loop of the VSC was the

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

Interactions through the MTDC network

Interactions through the AC power system

A VSC-1

V1

VSC-2

V2

SG-1

...

DC Network

19

...

...

AC Network

SG-2 ...

... VSC-M

VM SG-n

Fig. 1.11 Configuration of an AC power system integrated with a VSC-based MTDC network

subsystem in the feedback loop and rest of the power system was the subsystem in the forward path. The investigation indicated that if an open-loop SSO mode of the subsystem of the control loop of the VSC was close to an open-loop SSO mode of the subsystem of the rest of the power system on the complex plane, the small-signal stability of the closed-loop power system was degraded as caused by the control loop of the VSC. The closeness of open-loop SSO modes was referred to as the open-loop modal coupling. Study cases were presented in [49] to show that under the condition of open-loop modal coupling, strong dynamic interactions occurred between the VSC and the SG to cause the loss of system small-signal stability.

1.2.2.2

Small-Signal Stability of AC Power System Integrated with a VSC-Based MTDC Network

Figure 1.11 shows the configuration of an AC power system integrated with a VSC-based MTDC network. The DC/AC power networks are connected via multiple VSCs. Small-signal stability of the MTDC/AC power system is affected by the AC/DC dynamic interactions between the VSCs as well as between the VSCs and SGs. The dynamic interactions are very complicated because they are via the MTDC network and the AC network. In order to examine how those dynamic interactions affect the small-signal stability of the MTDC/AC power system, following simplifications have been made in the examination. 1. Dynamics of the AC power system were ignored by assuming the AC system to be an infinite busbar such that only the dynamics of the MTDC network was

20

1 Introduction

a

Interactions between VSCs VSC-1

V1 SG-1

...

...

Interactions between the VSCs and SGs

... ...

AC Network ... VSC-M

VM SG-n

b

Interactions between VSCs

Interactions between the VSCs and SGs VSC-1 V1

...

...

...

SG-1

DC Network ...

... VSC-M VM

SG-n Fig. 1.12 Configuration of simplified MTDC/AC power system. (a) Multi-infeed VSCs/AC power system. (b) MTDC power system with multi-independent AC systems

considered. The examination was about the small-signal stability of the MTDC network, the system on the left-hand side of line A in Fig. 1.11. Obviously the stability is affected only by the dynamic interactions between the VSCs through the MTDC network. 2. The MTDC network was not considered and the configuration of the MTDC/AC power system was simplified as shown Fig. 1.12a. This is the multi-infeed VSCs/ AC power system. The system small-signal stability was affected by the dynamic interactions between the VSCs as well as between the VSCs and SGs through the AC network. 3. The AC network was ignored and the simplified configuration of the MTDC/AC power system is shown by Fig. 1.12b. The system small-signal stability was affected by the dynamic interactions between the VSCs as well as between the VSCs and SGs through the MTDC network.

1.2 Small-Signal Stability of the VSC-Based DC/AC Power System

21

Two factors affecting the small-signal stability of a MTDC network are the load flow condition and the dynamic interactions between the multiple VSCs through the MTDC network. Study cases presented in [48] showed that when the direction of load flow in the MTDC network changed, the system may encounter the instability problem. In [50], load flow optimization was studied in order to increase the stability of the MTDC network. The dynamic interactions in the MTDC network are related with the control schemes adopted by the VSCs. Hence, modal analysis in [51] focused on the effect of droop coefficient on the small-signal stability of the MTDC network, which implemented the DC voltage droop control. In [52], more detailed modal analysis assisted by computational results of participation factors, etc. was conducted to identify the dynamic components in the MTDC network responsible for the instability risk. Work presented in [47–52] demonstrated how the modal analysis can be applied to examine the small-signal stability of the MTDC network. In [53], an impedance model of the MTDC network was derived. The results of small-signal stability of the MTDC network by using the frequency-domain analysis indicated that when the voltage droop control was implemented, impropriate setting of control parameters may lead to the loss of small-signal stability of the MTDC network. Dynamic interactions between the VSCs in the MTDC network were examined in [54] under the condition of open-loop modal coupling when an open-loop oscillation mode of a selected VSC was close to an open-loop oscillation mode of the rest of MTDC network. Analysis in [54] indicated that there existed no open-loop oscillation mode in the VSC which adopted the active power control. Hence, it was concluded in [54] that when the MTDC network implemented the master-slave control, there was no possibility of open-loop modal coupling in the MTDC network to cause the loss of system small-signal stability. However, when the voltage droop control was used, the open-loop modal coupling may possibly happen between the VSCs adopting the DC voltage droop control. In [54], study cases were presented to show that the MTDC network with the DC voltage droop control lost the stability under the condition of open-loop modal coupling. In the seventh chapter of this book, detailed analysis about the impact of the open-loop modal coupling is introduced. In [55], a simple two-infeed VSC/AC power system shown by Fig.1.12a was considered. Dynamic interactions between two VSCs through the AC network were studied and routes of dynamic interactions were identified. Study by frequencydomain analysis indicated that when the bandwidth of VSC control was close to the frequencies of high-frequency oscillation modes of AC power system, system instability may occur. However, the mechanism of such instability risk was not examined in [55]. In the simplified MTDC/AC power system shown by Fig. 1.12a, the impact of dynamic interactions between the VSCs through the MTDC network is expected to be similar to that in the MTDC network being connected to the infinite AC busbar. Hence, the effect of DC/AC dynamic interactions through the MTDC network should be the focus of examination. Results of simulation presented in [47, 56] indicated that when the VSCs used the DC voltage droop control, DC/AC dynamic

22

1 Introduction

interactions between the VSC and SG propagated through the DC network. This implied that the two independent AC systems (the SGs) may interact via the DC network. However, no further analysis was carried out in [47, 56] to indicate why the interactions may happen and what their impact on system stability was. In the seventh chapter of the book, how the MTDC network may function as the media of DC/AC dynamic interactions to affect system stability is studied. In order to handle the complicated MTDC/AC power system shown by Fig. 1.11, a systematic and simple way to establish the linearized model for the study of smallsignal stability is essentially important. Work presented in [57] is the representative of pioneering effort along this particular direction of research. In [57], a closed-loop interconnected model of the MTDC/AC power system was derived, where the MTDC network and the AC power system were modelled as two separate openloop subsystems. The interconnected model derived in [57] made it possible to examine the dynamic interactions between the MTDC and the AC power system. In [50, 58], the open-loop subsystems of the VSCs, the SGs, DC and AC networks, were separately derived and interconnected to establish the model of the MTDC/AC power system. This not only reduced the complexity in modelling the MTDC/AC power system considerably, but also linked the system stability with the dynamic interactions between the VSCs and SGs. Whilst in [59], a generic state-space model of the MTDC/AC power system was derived. Stability of an example MTDC/AC system was examined using the derived model. In [60], analysis was carried out to investigate the effectiveness of damping control implemented by a pair of VSCs in a MTDC/AC power system. Results of analysis provided insight into the small-signal stability of the MTDC/AC power system. Based on the closed-loop model of the MTDC/AC power system, damping torque analysis was applied in [61] to examine the small-signal stability of the MTDC/AC power system. Results of analysis indicated that under the condition of open-loop modal coupling, i.e., an oscillation mode of the open-loop MTDC network being close to an electromechanical oscillation mode of the open-loop AC power system on the complex plane, dynamic interactions between the MTDC and AC power system may become strong to degrade the small-signal stability of the MTDC/AC power system. Thus, the open-loop modal coupling between the open-loop oscillation mode of the MTDC network and the electromechanical oscillation mode of the open-loop AC power system was a modal condition under which the MTDC/AC power system may lose the small-signal stability. Details of this study are to be introduced in Chap. 7 of the book. In [62], dynamic interactions between a selected control loop of a VSC and rest of a MTDC/AC power system was examined. Due to the selection of the control loop of the VSC, a SISO closed-loop model of the MTDC/AC power system was established. Based on the SISO closed-loop model, theoretical proof was presented to indicate that under the condition of open-loop modal coupling, the MTDC/AC power system may possibly loss the small-signal stability as caused by the dynamic interactions between the VSCs and the SGs. This study is also to be introduced in Chap. 7 of the book.

References

1.2.2.3

23

Summary: The HVDC/AC and MTDC/AC Power System

Key issues in the study of small-signal stability of the VSC-based HVDC/AC or MTDC/AC power system are the dynamic interactions between the VSCs and between the VSCs and SGs through both the DC and AC network. Results of study obtained so far can be summarized as follows. 1. Integration of VSC-HVDC line or MTDC network into an AC power system may bring about the small-signal instability risk. The instability risk may be caused by the contribution of negative damping torque or negative resistance from the VSC-HVDC line or the MTDC network. This is the conclusion obtained by using the frequency-domain analysis. The instability risk may occur under the condition of open-loop modal coupling when two open-loop oscillation modes of subsystems are close to each other on the complex plane. When the open-loop modal coupling happens, the dynamic interactions between the VSCs and the SGs may become strong to lead to the loss of small-signal stability of the integrated HVDC/AC or MTDC/AC power system. This is the explanation about why the VSC-based HVDC/AC or MTDC/AC power system may become unstable from the perspective of system open-loop modal condition. 2. In a MTDC network, dynamic interactions between the VSCs adopting the DC voltage droop control may cause the small-signal instability if the control parameters of the VSCs are not set carefully. For the MTDC network connected to the independent SGs (Fig. 1.5b), AC dynamic transient can propagate through the MTDC network. Hence, it seems that the MTDC network implementing the DC voltage droop control is more liable to the loss of small-signal stability because the DC voltage droop control links the dynamics of the VSCs implementing the DC voltage control. 3. The AC/DC dynamic interactions introduced by the VSC using the DC voltage control may become strong to negatively affect the small-signal stability of the HVDC/AC or the MTDC/AC power system. For the VSC adopting the active power control, the AC/DC dynamic interactions are weak and affect little the small-signal stability of the HVDC/AC or the MTDC/AC power system. Hence, as far as the DC side is concerned, the VSC control systems seem to be the main instability risk contributors. Design of the VSC control should consider the impact on power system stability especially when the DC voltage control is used.

References 1. Slootweg JG, Kling WL (2003) The impact of large scale wind power generation on power system oscillations. Elect Power Syst Res 67(1):9–20 2. Sanchez-Gasca JJ, Miller NW, Price WW (2004) A modal analysis of a two-area system with significant wind power penetration. In: Proc power systems conf Expo 2, New York, USA, pp 1148–1152 3. Tang H, Wu J, Zhou S (2004) Modelling and simulation for small signal stability analysis of power system containing wind farm. Power Syst Technol 28(1):38–41

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1 Introduction

4. Mendonca A, Lopes JAP (2005) Impact of large scale wind power integration on small signal stability. In: 2005 International Conference on Future Power Systems, Amsterdam, pp 1–5 5. Wang C, Shi LB, Yao LZ, Wang LM, Ni YX (2010) Small signal stability analysis of the largescale wind farm with DFIGs. Proc CSEE 30(4):63–70 6. Chen S, Chang X, Sun H, Zeng L (2013) Impact of grid-connected wind farm on damping performance of power system. Power Syst Technol 37(6):1570–1577 7. Gautam D, Vittal V, Harbour T (2009) Impact of increased penetration of DFIG-based wind turbine generators on transient and small signal stability of power systems. IEEE Trans Power Syst 24(3):1426–1434 8. Vittal E, Malley MO, Keane A (2012) Rotor angle stability with high penetrations of wind generation. IEEE Trans Power Syst 27(1):353–362 9. Vittal E, Keane A (2013) Identification of critical wind farm locations for improved stability and system planning. IEEE Trans Power Syst 28(3):2950–2958 10. Tsourakis G, Nomikos BM, Vournas CD (2009) Contribution of doubly fed wind generators to oscillation damping. IEEE Trans Energy Conver 24(3):783–791 11. Oliveira RV de, Zamadei JA, Hossi CH (2011) Impact of distributed synchronous and doublyfed induction generators on small-signal stability of a distribution network. In: Power and energy society general meeting (PES), San Diego, pp 1–8 12. Quintero J, Vittal V, Heydt GT, Zhang H (2014) The impact of increased penetration of converter control-based generators on power system modes of oscillation. IEEE Trans Power Syst 29(5):2248–2256 13. Ding N, Lu Z, Qiao Y, Min Y (2013) Simplified models of large-scale wind and their applications for small-signal stability. J Modern Power Syst Clean Energy 1(1):58–64 14. Knuppel T, Nielsen JN, Jensen KH, Dixon A, Ostergaard J (2012) Small-signal stability of wind power system with full-load converter interfaced wind turbines. IET Renew Power Gener 6 (2):79–91 15. Kunjumuhammed LP, Pal BC, Anaparthi KK, Thornhill NF (2013) Effect of wind penetration on power system stability. In: Power and energy society general meeting (PES), Vancouver, pp 1–5 16. Du WJ, Bi JT, Cao J, Wang HF (2016) A method to examine the impact of grid connection of the DFIGs on power system electromechanical oscillation modes. IEEE Trans Power Syst 31 (5):3775–3785 17. Wang Z, Shen C, Liu F (2014) Impact of DFIG with phase lock loop dynamics on power systems small signal stability. In: Power and Energy Soc General Meeting, National Harbor, pp 1–5 18. Konishi H, Takahashi C, Kishibe H, Sato H (2001) A consideration of stable operating power limits in VSC-HVDC systems. In: Seventh international conference on AC-DC power transmission, London, pp 102–106 19. Jovcic D, Lamont LA, Xu L (2003) VSC transmission model for analytical studies. In: IEEE power engineering society general meeting, vol 3, Toronto, p 1742 20. Durrant M, Werner H, Abbott K (2003) Model of a VSC HVDC terminal attached to a weak AC system. In: Proceedings of 2003 I.E. conference on control applications, vol 1, Istanbul, Turkey. pp 178–182 21. Harnefors L, Bongiorno M, Lundberg S (2007) Input-Admittance Calculation and Shaping for Controlled Voltage-Source Converters. IEEE Trans Ind Electron 54(6):3323–3334 22. Harnefors L, Wang X, Yepes AG, Blaabjerg F (2016) Passivity-based stability assessment of grid-connected VSCs—an overview. IEEE J Emerg Sel Topics Power Electron 4(1):116–125 23. Rasmussen M, Jorgensen HK (2005) Current technology for integrating wind farms into weak power grids. In: 2005 IEEE/PES transmission & distribution conference & exposition, Dalian, China, pp 1–4 24. Yuan X, Wang F, Boroyevich D, Li Y, Burgos R (2009) DC-link voltage control of a full power converter for wind generator operating in Weak-Grid systems. IEEE Trans Power Electron 24 (9):2178–2192

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25. Liserre M, Teodorescu R, Blaabjerg F (2006) Stability of photovoltaic and wind turbine gridconnected inverters for a large set of grid impedance values. IEEE Trans Power Electron 21 (1):263–272 26. Strachan NPW, Jovcic D (2010) Stability of a Variable-Speed Permanent Magnet Wind Generator With Weak AC Grids. IEEE Trans Power Del 25(4):2779–2788 27. Zhou Y, Nguyen DD, Kjær PC, Saylors S (2013) Connecting wind power plant with weak grid—challenges and solutions. In: 2013 I.E. power & energy society general meeting, Vancouver, BC, pp 1–7 28. Dong D, Li J, Boroyevich D, Mattavelli P, Cvetkovic I, Xue Y (2012) Frequency behavior and its stability of grid-interface converter in distributed generation systems. In: 2012 Twentyseventh annual IEEE applied power electronics conference and exposition (APEC) 1887–1893, Orlando, FL 29. Dong D, Wen B, Boroyevich D, Mattavelli P, Xue Y (2015) Analysis of phase-locked loop low-frequency stability in three-phase grid-connected power converters considering impedance interactions. IEEE Trans Ind Electron 62(1):310–321 30. Zhou JZ, Ding H, Fan S, Zhang Y, Gole AM (2014) Impact of short-circuit ratio and phaselocked-loop parameters on the small-signal behavior of a VSC-HVDC converter. IEEE Trans Power Del 29(5):2287–2296 31. Wen B, Boroyevich D, Burgos R, Mattavelli P, Shen Z (2016) Analysis of D-Q small-signal impedance of grid-tied inverters. IEEE Trans Power Electron 31(1):675–687 32. Zhou P, Yuan X, Hu J, Huang Y (2014) Stability of DC-link voltage as affected by phase locked loop in VSC when attached to weak grid. In: 2014 I.E. PES general meeting|conference & exposition, National Harbor, MD, pp 1–5 33. Huang Y, Yuan X, Hu J, Zhou P, Wang D (2016) DC-Bus voltage control stability affected by AC-bus voltage control in VSCs connected to weak AC grids. IEEE J Emerg Sel Topics Power Electron 4(2):445–458 34. Arani MFM, Mohamed YARI (2017) Analysis and performance enhancement of vectorcontrolled VSC in HVDC links connected to very weak grids. IEEE Trans Power Syst 32 (1):684–693 35. Huang Y, Yuan X, Hu J (2015) Modeling of VSC connected to weak grid for stability analysis of DC-link voltage control. IEEE J Emerg Sel Topics Power Electron 3(4):1193–1204 36. Midtsund T, Suul JA, Undeland T (2010) Evaluation of current controller performance and stability for voltage source converters connected to a weak grid. In: The 2nd international symposium on power electronics for distributed generation systems, Hefei, pp 382–388 37. Givaki K, Xu L (2015) Stability analysis of large wind farms connected to weak AC networks incorporating PLL dynamics. In: International conference on renewable power generation (RPG 2015), Beijing, pp 1–6 38. Hu J, Hu Q, Wang B (2016) Small signal instability of PLL-synchronized type-4 wind turbines connected to high-impedance AC grid during LVRT. IEEE Trans Energy Convers 31 (4):1676–1687 39. Prabhu N, Padiyar KR (2009) Investigation of subsynchronous resonance with VSC-based HVDC transmission systems. IEEE Trans Power Del 24(1):433–440 40. Joseph T, Ugalde-Loo CE, Liang J (2015) Subsynchronous oscillatory stability analysis of an AC/DC transmission system. In: 2015 I.E. Eindhoven PowerTech, Eindhoven, pp 1–6 41. Wildrick CM, Lee FC, Cho BH, Choi B (1995) A method of defining the load impedance specification for a stable distributed power system. IEEE Trans Power Electron 10(3):280–285 42. Wu XG, Sun YF, Li GQ (2016) Analysis of impedance stability of VSC-HVDC systems. Southern Power Syst Technol 10(5):75–79 43. Song Y, Breitholtz C (2016) Nyquist stability analysis of an AC-grid connected VSC-HVDC system using a distributed parameter DC cable model. IEEE Trans Power Del 31(2):898–907 44. Stamatiou G, Bongiorno M (2016) Stability analysis of two-terminal VSC-HVDC systems using the net-damping criterion. IEEE Trans Power Del 31(4):1748–1756

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Chapter 2

Linearized Model of a Power System with a Grid-Connected Variable Speed Wind Generator

Linearized model of a power system with a grid-connected variable speed wind generator (VSWG) is derived in this chapter in three steps. First, the linearized model of the VSWG is established. That includes the establishment of linearized model of a PMSG and a DFIG. Second, the linearized model of the power system is derived. Finally, the linearized model of the power system with the VSWG is established by combining the model of the VSWG and the power system.

2.1

Linearized Model of a PMSG

Figure 2.1 shows the configuration of a grid-connected PMSG, which is comprised of three main parts: (1) The permanent magnet SG; (2) The machine side converter (MSG) and associated control system; (3) The grid side converter (GSC) and associated control system. Linearized models for each of three parts of the PMSG are derived respectively in this section. Then, they are combined to form the linearized model of the grid-connected PMSG.

2.1.1

Linearized Model of the Permanent Magnet SG

The voltage equations of stator windings of the permanent magnet SG in Fig. 2.1 are: 8 d > > < ψpsd ¼ ω0 Rps Ipsd  ω0 Vpsd þ ω0 ωpr ψpsq dt ð2:1Þ > d > : ψpsq ¼ ω0 Rps Ipsq  ω0 Vpsq  ω0 ωpr ψpsd dt

© Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_2

27

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

28

Vpsd +jVpsq

MSC

N

Pps Cp

S

Ppc

Ipcd

GSC

X pf

Vpdc

Ipsd +jIpsq

+ jIpcq

Pp +jQ p

Vpsdref

Vpsqref

Vpd Control system

Control system

Permanent SG

w prref

Vpq

Ipsdref

Vpdcref

MSC and associated control system

Vpcd

Vpd + jVpq

+ jVpcq

(Vw q w )

Q pref

GSC and associated control system

Fig. 2.1 Configuration of a grid-connected PMSG

where Vpsd and Vpsq are the d and q component of voltage, ψpsd and ψpsq are the d and q component of magnetic flux, Ipsd and Ipsq are the d and q component of current of stator windings of the permanent magnet SG respectively; ω0 is the synchronous speed; ωpr is the angular speed of the permanent magnet SG; Rpsis the resistance of stator windings. Motion equations of the rotor of the permanent magnet SG are: Jpr

dωpr ¼ Tpm  Tpe dt

ð2:2Þ

where Jpr is the constant of inertia of the rotor; Tpm andTpe are the mechanical torque input to and the electromagnetic power output from the permanent magnet SG respectively; and Tpe ¼ ψpsq Ipsd  ψpsd Ipsq

ð2:3Þ

Flux linkage equations of the permanent magnet SG are (

ψpsd ¼ Xpd Ipsd  ψpm ψpsq ¼ Xpq Ipsq

ð2:4Þ

where ψpm is the flux of permanent magnet, Xpd and Xpq are the d and q axis reactance of stator windings respectively. Linearization of Eq. (2.1) is 8 d > > < Δψpsd ¼ ω0 Rps ΔIpsd  ω0 ΔVpsd þ ω0 ωpr0 Δψpsq þ ω0 ψpsq0 Δωpr dt ð2:5Þ > d > : Δψpsq ¼ ω0 Rps ΔIpsq  ω0 ΔVpsq  ω0 ωpr0 Δψpsd  ω0 ψpsd0 Δωpr dt

2.1 Linearized Model of a PMSG

29

By neglecting the variation of the mechanical torque input to the permanent magnet SG such that ΔTpm ¼ 0, linearization of Eq. (2.2) is obtained to be Jpr

dΔωpr ¼ ΔTpm  ΔTpe ¼ ΔTpe dt

ð2:6Þ

Substituting the linearization of Eq. (2.3) in Eq. (2.6), it can have Jpr

dΔωpr ¼ ψpsq0 ΔIpsd þ ψpsd0 ΔIpsq  Ipsd0 Δψpsq þ Ipsq0 Δψpsd dt

ð2:7Þ

ψpm in Eq. (2.4) is a constant. Hence, linearization of Eq. (2.4) is (

Δψpsd ¼ Xpd ΔIpsd Δψpsq ¼ Xpq ΔIpsq

ð2:8Þ

From Eqs. (2.7) and (2.8), Jpr

ψpsq0 ψpsd0 dΔωpr ¼ Δψpsd þ Δψpsq  Ipsd0 Δψpsq þ Ipsq0 Δψpsd dt Xpd Xpq

ð2:9Þ

Writing Eqs. (2.5) and (2.9) together in the following form of state-space representation, d ΔXp1 ¼ Ap1 ΔXp1 þ bp1 ΔVpsd þ bp2 Vpsq dt

ð2:10Þ

where h i Xp1 ¼ Δψpsd Δψpsq Δωpr , 2

ω0 Rps Xpd

3

ω0 ωpr0 ω0 ψpsq0 7 6 2 3 2 3 7 6 ω0 0 7 6 7 6 ω0 Rps 6 7 6 7 ω0 ψpsd0 7 Ap1 ¼ 6 7, bp1 ¼ 4 0 5, bq1 ¼ 4 ω0 5: 6 ω0 ωpr0 X pq 7 6 7 6 0 0 5 4 ψpsq0 Ipsq0 ψpsd0 Ipsd0 þ  0 Jpr Xpd Jpr Jpr Xpq Jpr Active power output from the permanent magnet SG is Pps ¼ Vpsq Ipsq þ Vpsd Ipsd

ð2:11Þ

Linearization of (2.11) is ΔPps ¼ Vpsq0 ΔIpsq þ Vpsd0 ΔIpsd þ Ipsq0 ΔVpsq þ Ipsd0 ΔVpsd

ð2:12Þ

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

30

Substituting Eq. (2.8) in Eq. (2.12), ΔPps ¼

Vpsq0 Vpsd0 Δψpsq þ Δψpsd þ Ipsq0 ΔVpsq þ Ipsd0 ΔVpsd Xpq Xpd

ð2:13Þ

Thus, ΔPps ¼ cp1 T ΔXp1 þ dp1 ΔVpsd þ dp2 ΔVpsq where

 cp1 T ¼

Vpsd0 Xpd

ð2:14Þ

 0 , dp1 ¼ Ipsd0 , dp2 ¼ Ipsq0 :

Vpsq0 Xpq

By arranging Eqs. (2.10) and (2.14) together, the state-space model of the permanent magnet SG is obtained to be 8 < d ΔXp1 ¼ Ap1 ΔXp1 þ bp1 ΔVpsd þ bp2 Vpsq dt : ΔPps ¼ cp1 T ΔXp1 þ dp1 ΔVpsd þ dp2 ΔVpsq

2.1.2

ð2:15Þ

Linearized Model of Machine Side Converter and Associated Control System

The MSC implements the vector control. Configuration of its control system is shown by Fig. 2.2. It can be seen that there are two current control inner loops in

Rotor speed control outer loop q-axis current control inner loop

K pp1

w prref – + pr

+

K pp 2

I psqref + –

K pi1 + x p1 s

I psq

K pi 2

+ –

I psd

+



+

xp2

s

K pp 3

I psdref

w pr y pm

+

V

+ psqref –

w pr X pd I psd –

Vpsdref

K pi 3

+

+

s

x p3

w pr X pq I psq

d-axis current control inner loop Fig. 2.2 Configuration of control system of the MSC of the PMSG

MSC

MSC

2.1 Linearized Model of a PMSG

31

the control system to respectively control the d and q axis output current from the stator windings of the permanent magnet SG, Ipsd and Ipsq. The speed control outer loop controls the angular speed of the rotor of the permanent magnet SG, ωpr, to generate the reference for the control of q axis current control inner loop. Normally, the reference for the control of d axis current control inner loop is set to be zero, i.e., Ipsdref ¼ 0. In Fig. 2.2, the outputs of three integral controllers are xp1, xp2 and xp3. It can have 8   d > > xp1 ¼ Kpi1 ωpr  ωprref > > dt > > > <   d ð2:16Þ xp2 ¼ Kpi2 Ipsqref  Ipsq > dt > > > > >   > : d xp3 ¼ Kpi3 Ipsdref  Ipsd dt where Kpi1, Kpi2 and Kpi3 are the gains of integral controllers, Ipsdref and Ipsqref are the references of current control inner loops, ωprref is the reference of rotor speed control outer loop. From Eq. (2.2),   8 Ipsqref ¼ Kpp1 ωpr  ωprref þ xp1 > > <   Vpsqref ¼ Kpp2 Ipsqref  Ipsq  xp2  ωpr ψpm þ ωpr Xpd Ipsd ð2:17Þ > >   : Vpsdref ¼ Kpp3 Ipsdref  Ipsd  xp3  ωpr Xpq Ipsq where Kpp1, Kpp2 and Kpp3 are the gains of proportional controller, Vpsdref and Vpsqref are the output signals from the MSC control system, which are used as the d and q axis references of the terminal voltage of the permanent magnet SG. Ignoring the transient of the PWM of the MSC, it can have ( Vpsdref  Vpsd ð2:18Þ Vpsqref  Vpsq Linearization of Eq. (2.16) is (Δωprref ¼ 0,ΔIpsdref ¼ 0) 8 d > > Δxp1 ¼ Kpi1 Δωpr > > dt > > > <   d Δxp2 ¼ Kpi2 ΔIpsqref  ΔIpsq > dt > > > > > > : d Δxp3 ¼ Kpi3 ΔIpsd dt

ð2:19Þ

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

32

Linearization of Eq. (2.17) is 8 ΔIpsqref ¼ Kpp1 Δωpr þ Δxp1 > > >   > > < ΔVpsqref ¼ Kpp2 ΔIpsqref  ΔIpsq  Δxp2  ψpm Δωpr

ð2:20Þ

> þXpd Ipsd0 Δωpr þ ωpr0 Xpd ΔIpsd > > > > : ΔVpsdref ¼ Kpp3 ΔIpsd  Δxp3  Xpq Ipsq0 Δωpr þ ωpr0 Xpd ΔIpsd Substituting the first equation in Eqs. (2.20) and (2.8) in Eq. (2.19), 8 d > > Δxp1 ¼ Kpi1 Δωpr > > dt > > > > > > > > > Kpi3 Δψpsd d > > : Δxp3 ¼  dt Xpd

ð2:21Þ

The above equation can be written as d ΔXp2 ¼ Ap2 ΔXp2 þ Bp1 ΔXp1 dt

ð2:22Þ

where Δxp2 Δxp3 T , 2 0 0 6 Kpi2 6 0 6 Xpq ¼6 6 4 Kpi3 0 Xpd

ΔXp2 ¼ ½ Δxp1 2

Ap2

0 6 ¼ 4 Kpi2 0

3

0

0

0

7 0 5, Bp1

0

0

Linearization of Eq. (2.18) is (

ΔVpsdref ¼ ΔVpsd ΔVpsqref ¼ ΔVpsq

Kpi1

3

7 Kpi2 Kpp1 7 7 7: 7 5 0

ð2:23Þ

Substituting Eq. (2.22) and the first equation of Eq. (2.20) into the last two equations of Eq. (2.20), 8 ΔVpsd ¼ Kpp2 Kpp1 Δωpr  Kpp2 Δxp1 þ Kpp2 ΔIpsq  Δxp2  ψpm Δωpr > > < þXpd Ipsd0 Δωpr þ ωpr0 Xpd ΔIpsd ð2:24Þ > > : ΔVpsq ¼ Kpp3 ΔIpsd  Δxp3  Xpq Ipsq0 Δωpr  ωpr0 Xpq ΔIpsq

2.1 Linearized Model of a PMSG

33

Substituting Eq. (2.8) in Eq. (2.24), 8 Kpp2 > > Δψpsq ΔVpsd ¼ Kpp2 Δxp1  Δxp2 þ ωpr0 Δψpsd þ > > Xpq > > > <   þ Xpd Ipsd0  Kpp2 Kpp1  ψpm Δωpr > > > > > Kpp3 > > Δψpsd  ωpr0 Δψpsq  Xpq Ipsq0 Δωpr : ΔVpsq ¼ Δxp3 þ Xpd

ð2:25Þ

The above equation can be written as (

ΔVpsd ¼ cp2 T ΔXp2 þ cp3 T ΔXp1

ð2:26Þ

ΔVpsq ¼ cp4 T ΔXp2 þ cp5 T ΔXp1 where cp2 T ¼ ½ Kpp2  cp3 T ¼ ωpr0

Kpp2 Xpq





Kpp3 Xpd

0 ,

Xpd Ipsd0  Kpp2 Kpp1  ψpm

cp4 T ¼ ½ 0 0 cp5 T ¼

1

ωpr0

 ,

1 ,  Xpq Ipsq0 :

The state-space representation of the MSC and associated control system is obtained by writing Eqs. (2.22) and (2.26) together as 8 d > > > ΔXp2 ¼ Ap2 ΔXp2 þ Bp1 ΔXp1 > < dt ð2:27Þ ΔVpsd ¼ cp2 T ΔXp2 þ cp3 T ΔXp1 > > > > : ΔVpsq ¼ cp4 T ΔXp2 þ cp5 T ΔXp1

2.1.3

Linearized Model of Grid Side Converter and Associated Control System

From Fig. 2.1, the line voltage equations across the filter reactance, Xpf, can be written as

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

34

8 ω0 Vpcd ω0 Vpd d > > > < dtIpcd ¼ Xpf  Xpf þ ω0 Ipcq > ω0 Vpcq ω0 Vpq d > >   ω0 Ipcd : Ipcq ¼ dt Xpf Xpf

ð2:28Þ

where Ipcd and Ipcq are the d and q axis component of output current from the GSC respectively; Vpcd and Vpcq are the d and q axis component of terminal voltage of the GSC; Vpd and Vpq are the d and q axis component of the point of common coupling (PCC) of the PMSG. Equation of the voltage across the capacitor is Cp Vpdc

dVpdc ¼ Pps  Ppc dt

ð2:29Þ

where Vpdc is the DC voltage across the capacitor; Cp is the capacitance; Ppc is the active power output from the GSC; and Ppc ¼ Vpcd Ipcd þ Vpcq Ipcq

ð2:30Þ

The configuration of vector control system of the grid side converter (GSC) of the PMSG is shown by Fig. 2.3. The current control inner loops control the d and q axis output current from the GSC, Ipcd and Ipcq, respectively. The control outer loops control the DC voltage across the capacitor, Vpdc, and the reactive power output from the GSC, Qp, respectively to generate the current control references for the current control inner loops. In Fig. 2.3, the outputs of four integral controllers are xp4, xp5, xp6 and xp7. From Fig. 2.3, DC voltage control outer loop

K pp 4

V pdcref – +

V pdc

K pi 4

+

s

xp 4

K pp 6

Q pref – +

Qp

+

+

d-axis current control inner loop



I pcd

+

K pi 5

+

s

x p5

+



s

xp 6

I pcq

V + + pcdref

GSC

X pf I pcq V pq

K pp 7

I cqref +

K pi 6

Reactive power control outer loop

V pd

K pp 5

I cdref +

+

Vpcqref + +

K pi 7

+

+

s

xp7

X pf I pcd

q-axis current control inner loop

Fig. 2.3 Configuration of control system of the GSC of the PMSG

GSC

2.1 Linearized Model of a PMSG

35

8 d > > xp4 > > dt > > > > > > d > > < dtxp5 > d > > xp6 > > > dt > > > > >d > : xp7 dt

  ¼ Kpi4 Vpdc  Vpdcref   ¼ Kpi5 Ipcdref  Ipcd 

¼ Kpi6 Qp  Qpref



ð2:31Þ

  ¼ Kpi7 Ipcqref  Ipcq

where Qpref is the reference of reactive power control outer loop; Ipcdref and Ipcqref are the references of d and q axis current control inner loops respectively; Vpdcref is the reference of DC voltage control outer loop; Qp is the reactive power output from the GSC Qp ¼ Vpq Ipcd  Vpd Ipcq

ð2:32Þ

8   Ipcdref ¼ Kpp4 Vpdc  Vpdcref þ xp4 > > > >   > > < Vpcdref ¼ Kpp5 Ipcdref  Ipcd þ xp5  Xpf Ipcq þ Vpd   > Ipcqref ¼ Kpp6 Qp  Qpref þ xp6 > > > >   > : Vpcqref ¼ Kpp7 Ipcqref  Ipcq þ xp7 þ Xpf Ipcd þ Vpq

ð2:33Þ

From Fig. 2.3,

where Kpp4, Kpp5, Kpp6 and Kpp7 are the gains of proportional controllers. Ignoring the transient of the PWM of the GSC, (

Vpcdref ¼ Vpcd Vpcqref ¼ Vpcq

ð2:34Þ

Linearization of Eqs. (2.28), (2.29) and (2.30) respectively is 8 d ω0 ω0 > > > < dtΔIpcd ¼ Xpf ΔVpcd  Xpf ΔVpd þ ω0 ΔIpcq > d ω0 ω0 > > ΔVpcq  ΔVpq  ω0 ΔIpcd : ΔIpcq ¼ dt Xpf Xpf

ð2:35Þ

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

36

Cp Vpdc0

dΔVpdc ¼ ΔPps  ΔPpc dt

ΔPpc ¼ Vpcd0 ΔIpcd þ Vpcq0 ΔIpcq þ Ipcd0 ΔVpcd0 þ Ipcq0 ΔVpcq0

ð2:36Þ ð2:37Þ

Substituting Eq. (2.37) in Eq. (2.36), Cp Vpdc0

dΔVpdc ¼ ΔPps  Vpcd0 ΔIpcd  Vpcq0 ΔIpcq  Ipcd0 ΔVpcd  Ipcq0 ΔVpcq dt ð2:38Þ

Qpref and Vpdcref are constants. Hence, linearization of Eq. (2.31) is 8 d > > Δxp4 ¼ Kpi4 ΔVpdc > > dt > > > > >   > d > > < dtΔxp5 ¼ Kpi5 ΔIpcdref  ΔIpcd > d > > Δxp6 ¼ Kpi6 ΔQp > > > dt > > > >   > > : d Δxp7 ¼ Kpi7 ΔIpcqref  ΔIpcq dt

ð2:39Þ

Linearization of Eqs. (2.32), (2.33) and (2.34) respectively is ΔQp ¼ Vpq0 ΔIpcd  Vpd0 ΔIpcq þ Ipcd0 ΔVpq  Ipcq0 ΔVpd 8 ΔIpcdref ¼ Kpp4 ΔVpdc þ Δxp4 > > >   > > < ΔVpcdref ¼ Kpp5 ΔIpcdref  ΔIpcd þ Δxp5  Xpf ΔIpcq þ ΔVpd > ΔIpcqref ¼ Kpp6 ΔQp þ Δxp6 > > > >   : ΔVpcqref ¼ Kpp7 ΔIpcqref  ΔIpcq þ Δxp7 þ Xpf ΔIpcd þ ΔVpq (

Vpcdref ¼ Vpcd Vpcqref ¼ Vpcq

ð2:40Þ

ð2:41Þ

ð2:42Þ

Substituting the first and the third equation of Eq. (2.41) into the second and the fourth equation respectively and using Eq. (2.42),

2.1 Linearized Model of a PMSG

(

37

ΔVpcd ¼ Kpp5 Kpp4 ΔVpdc þ Kpp5 Δxp4  Kpp5 ΔIpcd þ Δxp5  Xpf ΔIpcq þ ΔVpd ΔVpcq ¼ Kpp7 Kpp6 ΔQp þ Kpp7 Δxp6  Kpp7 ΔIpcq þ Δxp7 þ Xpf ΔIpcd þ ΔVpq ð2:43Þ Substituting Eq. (2.40) in Eq. (2.43)

8 ΔVpcd ¼ Kpp5 ΔIpcd  Xpf ΔIpcq þ Kpp5 Kpp4 ΔVpdc þ Kpp5 Δxp4 þ Δxp5 þ ΔVpd > > <     ΔVpcq ¼ Kpp7 Kpp6 Vpq0 þ Xpf ΔIpcd  Kpp7 Kpp6 Vpd0 þ Kpp7 ΔIpcq > >   : þKpp7 Δxp6 þ Δxp7  Kpp7 Kpp6 Ipcq0 ΔVpd þ Kpp7 Kpp6 Ipcd0 þ 1 ΔVpq ð2:44Þ Substituting Eq. (2.44) in Eq. (2.35), 8 ω0 Kpp5 ω0 Kpp5 Kpp4 ω0 Kpp5 d ω0 > > ΔIpcd ¼  ΔIpcd þ ΔVpdc þ Δxp4 þ Δxp5 > > dt X X X X > pf pf pf pf > > >   > < ω0 Kpp7 Kpp6 Vpd0 þ Kpp7 ω0 Kpp7 Kpp6 Vpq0 d ΔIpcq ¼ ΔIpcd  ΔIpcq > dt Xpf Xpf > > > > > > ωo Kpp7 ω0 Kpp7 Kpp6 Ipcq0 ω0 Kpp7 Kpp6 Ipcd0 ω0 > > Δxp6 þ Δxp7  ΔVpd þ ΔVpq :þ Xpf Xpf Xpf Xpf ð2:45Þ Substituting Eq. (2.44) in Eq. (2.38), dΔVpdc ¼ ΔPps  Kpp5 Kpp4 Ipcd0 ΔVpdc  Kpp5 Ipcd0 Δxp4  Ipcd0 Δxp5 dt   Ipcq0 Δxp7 þ Kpp5 Ipcd0  Vpcd0  Kpp7 Kpp6 Vpq0 Ipcq0 þ Xpf Ipcq0 ΔIpcd   þ Xpf Ipcd0  Vpcq0 þ Kpp7 Kpp6 Ipcq0 Vpd0 þ Kpp7 Ipcq0 ΔIpcq  Kpp7 Ipcq0 Δxp6     þ Kpp7 Kpp6 I2pcq0  Ipcd0 ΔVpd  Kpp7 Kpp6 Ipcd0 þ 1 Ipcq0 ΔVpq Cp Vpdc

ð2:46Þ Substituting Eq. (2.40) into the third equation of Eq. (2.41), ΔIpcqref ¼ Kpp6 Vpq0 ΔIpcd  Kpp6 Vpd0 ΔIpcq þ Kpp6 Ipcd0 ΔVpq  Kpp6 Ipcq0 ΔVpd þ Δxp6

ð2:47Þ

38

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

Substituting Eq. (2.40), the first equation of Eqs. (2.41) and (2.47) in Eq. (2.39), 8d > Δx ¼ Kpi4 ΔVpdc > > > dt p4 > > > > d > > > Δxp5 ¼ Kpi5 Kpp4 ΔVpdc þ Kpi5 Δxp4  Kpi5 ΔIpcd > > dt < d Δxp6 ¼ Kpi6 Vpq0 ΔIpcd  Kpi6 Vpd0 ΔIpcq þ Kpi6 Ipcd0 ΔVpq  Kpi6 Ipcq0 ΔVpd > > > dt > > > >   d > > > Δxp7 ¼ Kpi7 Kpp6 Vpq0 ΔIpcd  Kpi7 Kpp6 Vpd0 þ Kpi7 ΔIpcq > > > : dt þKpi7 Kpp6 Ipcd0 ΔVpq  Kpi7 Kpp6 Ipcq0 ΔVpd þ Kpi7 Δxp6 ð2:48Þ Arrange Eqs. (2.45), (2.46) and (2.48) in the following matrix form d ΔXp3 ¼ Ap3 ΔXp3 þ bp3 ΔPps þ bp4 ΔVpd þ bp5 ΔVpq dt where ΔXp3 ¼ ½ ΔIpcd

ΔIpcq

ΔVpdc

Δxp4

Δxp5

Δxp6

Δxp7 ,

ð2:49Þ

2.1 Linearized Model of a PMSG

2.1.4

39

Linearized Model of the Entire PMSG System

Writing the first equation of Eq. (2.15), the first equation of Eqs. (2.27) and (2.49) together, 8 d > > ΔX ¼ Ap1 ΔXp1 þ bp1 ΔVpsd þ bp2 Vpsq > > dt p1 > > < d ð2:50Þ ΔXp2 ¼ Ap2 ΔXp2 þ Bp1 ΔXp1 > dt > > > > > : d ΔXp3 ¼ Ap3 ΔXp3 þ bp3 ΔPps þ bp4 ΔVpd þ bp5 ΔVpq dt Substituting the last two equations of Eq. (2.27) in Eq. (2.50), d ΔXp ¼ Ap ΔXp þ bpd ΔVpd þ bpq ΔVpq dt

ð2:51Þ

where 2 6 Ap ¼ 4

Ap1 þ bp1 cp3 T þ bp2 cp5 T

bp1 cp2 T þ bp2 cp4 T

Bp1

Ap2

dp2 bp3 cp5 T þ bp3 cp1 T þ dp1 bp3 cp3 T dp1 bp3 cp2 T þ dp2 bp3 cp4 T 2 3 2 3 0 0 6 7 6 7 bpd ¼ 4 0 5, bpq ¼ 4 0 5: bp4 bp5

3

0

7 0 5, Ap3

Active power output from the PMSG is Pp ¼ Vpd Ipcd þ Vpq Ipcq ¼ Ppc ¼ Vpcd Ipcd þ Vpcq Ipcq

ð2:52Þ

From the linearization of Eqs. (2.52) and (2.40), variations of power output from the PMSG can be obtained to be " # ΔPp ð2:53Þ ¼ Fpv ΔVpv þ Fpi ΔIpi ΔQp where ΔVpv

¼ ½ ΔVpd

 Ipcd0 ΔVpq T , ΔIpi ¼ ½ ΔIpcd ΔIpcq T , Fpv ¼ Ipcq0   Vpd0 Vpq0 Fpi ¼ : Vpq0 Vpd0

 Ipcq0 , Ipcd0

ΔIpcd and ΔIpcq are the state variables (see Eq. (2.49)). Hence, Eq. (2.26) can be written as

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

40

(

ΔPp ¼ cpp T ΔXp þ dpp1 ΔVpd þ dpp2 ΔVpq

ð2:54Þ

ΔQp ¼ cpq T ΔXp þ dpq1 ΔVpd þ dpq2 ΔVpq

Finally, the state-space model of the entire PMSG system is obtained from Eqs. (2.51) and (2.54) to be 8 d > > ΔX ¼ Ap ΔXp þ bpd ΔVpd þ bpq ΔVpq > < dt p ð2:55Þ ΔPp ¼ cpp T ΔXp þ dpp1 ΔVpd þ dpp2 ΔVpq > > > : ΔQp ¼ cpq T ΔXp þ dpq1 ΔVpd þ dpq2 ΔVpq

2.2

Linearized Model of a DFIG

Configuration of a DFIG for wind power generation is shown by Fig. 2.4. The DFIG is of three main modules of electrical function: the induction generator and two-mass shaft rotational system, the rotor side converter (RSC) and associated control system, the gird side converter (GSC) and associated control system. Mechanical power is converted to the electrical power by the induction generator via the shaft rotational system. Function of the RSC and associated control system is to control the output power from the DFIG. The GSC and associated control system functions to maintain the DC voltage across the capacitor for the operation of the RSG and GSC.

Two-mass shaft Induction generator rotational system Lowspeed shaft

Vvsd +jVvsq (Vw q w )

I dsd +jI dsq

Highspeed shaft

Pds +jQds

Induction generator and the shaft system

I drd +jI drq Pdc1

RSC

Vdrd + jVdrq

Cd

Vdrdref RSC and associated control system

Vdrqref

Vddc

Vdcd + jVdcq

Vdcdref Control system

Control system

Pdsref

GSC

Qdsref

Fig. 2.4 Configuration of a DFIG for wind power generation

Pdc +jQdc

Pdr

I dcd +jI dcq

X df

Vddcref

Vdcqref GSC and associated control system

2.2 Linearized Model of a DFIG

41

In this section, the linearized state-space representation of each functional module of the DFIG is derived. Then, the linearized model of the DFIG in the d  q coordinate of the DFIG is established by combining the derived models of three function modules. All the variables are expressed in per unit (p.u.) except that the unit of constant of inertia of the rotor is second.

2.2.1

Linearized Model of the Induction Generator and TwoMass Shaft Rotational System

The voltage equations of stator windings of induction generator in Fig. 2.4 are: 8 d > > < ψdsd ¼ ω0 Rds Idsd þ ω0 Vdsd þ ω0 ψdsq dt ð2:56Þ > d > : ψdsq ¼ ω0 Rds Idsq þ ω0 Vdsq  ω0 ψdsd dt where Vdsd and Vdsq are the d and q component of voltage, ψdsd and ψdsq the d and q component of flux, Idsd and Idsq the d and q component of current of stator windings of the induction generator respectively; ω0 is the synchronous speed; Rdsis the resistance of stator windings. The voltage equations of rotor windings of induction generator are: 8 d > > < ψdrd ¼ ω0 Rdr Idrd þ ω0 Vdrd þ ω0 ð1  ωdr1 Þψdrq dt ð2:57Þ > d > : ψdrq ¼ ω0 Rdr Idrq þ ω0 Vdrq  ω0 ð1  ωdr1 Þψdrd dt where Vdrd and Vdrq are the d and q component of voltage, ψdrd and ψdrd the d and q component of flux, Idrd and Idrq the d and q component of current of rotor windings of the induction generator respectively; ωdr1 is the angular speed of high-speed shaft in the two-mass shaft rotational system; Rdris the resistance of stator windings. Flux equations of stator and rotor windings of the induction generator are 8 ψdsd ¼ Xdss Idsd  Xdm Idrd > > > > < ψdsq ¼ Xdss Idsq  Xdm Idrq ð2:58Þ > > > ψdrd ¼ Xdrr Idrd  Xdm Idsd > : ψdrq ¼ Xdrr Idrq  Xdm Idsq where Xdss and Xdrr are the self-inductance of stator and rotor windings respectively; Xdm is the mutual-inductance between the stator and rotor windings. Two-mass rotational system of induction generator consists of high-speed and low-speed shaft. Two shafts are connected by a gear box. Motion equations of the two-shaft rotational system are:

42

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

8 d > > Jdr1 ωdr1 ¼ Kdm θdr  Tde  Ddm ðωdr1  ωdr2 Þ  Ddr1 ωdr1 > > dt > > < d Jdr2 ωdr2 ¼ Tdm  Kdm θdr  Ddm ðωdr2  ωdr1 Þ  Ddr2 ωdr2 > dt > > > > d > : θdr ¼ ω0 ðωdr2  ωdr1 Þ dt

ð2:59Þ

where Jdr2 and Jdr1 are constants of inertia of high-speed and low-speed shaft respectively; Tdm is the mechanical torque input to the low-speed shaft; ωdr2 is the angular speed of low-speed shaft; Kdm and Ddm are the elastic coefficient and mutual damping coefficient between the high-speed and low-speed shaft; Ddr2 and Ddr1 are the self-damping coefficients of high-speed and low-speed shaft respectively; θdr ¼ θdr2  θdr1 is the relative angular position of low-speed and high-speed shaft (θdr1 is the angular position of high-speed shaft, is the angular position of low-speed shaft); Tde is the electromagnetic output torque of the induction generator, which can be expressed as: Tde ¼ ψdsd Idsq  ψdsq Idsd Substitute Eq. (2.58) in Eq. (2.60),   Tde ¼ Xdm Idsd Idrq  Idsq Idrd Linearization of Eq. (2.56) is 8 d > > < Δψdsd ¼ ω0 Rds ΔIdsd þ ω0 ΔVdsd þ ω0 Δψdsq dt > d > : Δψdsq ¼ ω0 Rds ΔIdsq þ ω0 ΔVdsq  ω0 Δψdsd dt

ð2:60Þ

ð2:61Þ

ð2:62Þ

Linearization of Eq. (2.57) is 8 d > > < Δψdrd ¼ ω0 Rdr ΔIdrd þ ω0 ΔVdrd þ ω0 ð1  ωdr10 ÞΔψdrq  ω0 ψdrq0 Δωdr1 dt > > d Δψ ¼ ω R ΔI þ ω ΔV  ω ð1  ω ÞΔψ þ ω ψ Δω : 0 dr drq 0 drq 0 dr10 0 drd0 dr1 drd dt drq ð2:63Þ Linearization of Eq. (2.57) is 8 Δψdsd ¼ Xdss ΔIdsd  Xdm ΔIdrd > > > > < Δψdsq ¼ Xdss ΔIdsq  Xdm ΔIdrq > Δψdrd ¼ Xdrr ΔIdrd  Xdm ΔIdsd > > > : Δψdrq ¼ Xdrr ΔIdrq  Xdm ΔIdsq

ð2:64Þ

2.2 Linearized Model of a DFIG

43

Mechanical torque input to the induction generator is a constant, i.e., ΔTdm ¼ 0. Thus, linearization of Eq. (2.59) is 8 d > > > Jdr1 dtΔωdr1 ¼ Kdm Δθdr  ΔTde  Ddm ðΔωdr1  Δωdr2 Þ  Ddr1 Δωdr1 > > > < d ð2:65Þ Jdr2 Δωdr2 ¼ Kdm Δθdr  Ddm ðΔωdr2  Δωdr1 Þ  Ddr2 Δωdr2 > dt > > > > > : d Δθdr ¼ ω0 ðΔωdr2  Δωdr1 Þ dt Linearization of Eq. (2.61) is   ΔTde ¼ Xdm Idsd0 ΔIdrq  Idsq0 ΔIdrd þ Idrq0 ΔIdsd  Idrd0 ΔIdsq

ð2:66Þ

Substituting Eq. (2.66) in (2.65), 8   d > > > Jdr1 Δωdr1 ¼ Kdm Δθdr  Xdm Idsd0 ΔIdrq  Idsq0 ΔIdrd þ Idrq0 ΔIdsd  Idrd0 ΔIdsq > dt > > > > > D dm ðΔωdr1  Δωdr2 Þ  Ddr1 Δωdr1 < d > Jdr2 Δωdr2 ¼ Kdm Δθdr  Ddm ðΔωdr2  Δωdr1 Þ  Ddr2 Δωdr2 > > > dt > > > > d > : Δθdr ¼ ω0 ðΔωdr2  Δωdr1 Þ dt ð2:67Þ From Eq. (2.64) ΔId ¼ M1 Δψd

ð2:68Þ

where 2

Xdss

6 6 0 M¼6 6 X 4 dm 0 ΔId ¼ ½ Idsd

Idsq

Idrd

0

Xdm

Xdss

0

0

Xdrr

0

3

7 Xdm 7 7, 0 7 5

0 Xdrr  Idrq T , Δψd ¼ ψdsd ψdsq Xdm

ψdrd

ψdrq

T

:

From Eqs. (2.62), (2.63), (2.67) and (2.68), the state equation of the induction generator and two-mass shaft rotational system is obtained to be d ΔXd1 ¼ Ad1 ΔXd1 þ Bd1 Δzd1 þ bd1 ΔVdsq þ bd2 ΔVdsd dt

ð2:69Þ

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

44

where  ΔXd1 ¼ Δψdsd Δψdsq Δψdrd T Δzd1 ¼ ½ ΔVdrd ΔVdrq  ,

Δψdrq

Δωdr1

Δωdr2

Δθdr



Active power output from the rotor side of the induction generator is Pdr ¼ Vdrd Idrd þ Vdrq Idrq

ð2:70Þ

Linearizing Eq. (2.70), ΔPdr ¼ Vdrd0 ΔIdrd þ Vdrq0 ΔIdrq þ Idrd0 ΔVdrd þ Idrq0 ΔVdrq

ð2:71Þ

2.2 Linearized Model of a DFIG

45

From Eqs. (2.68) and (2.71), ΔPdr ¼ ½ 0

0

Vdrd0

Vdrq0 M1 Δψd þ Idrd0 ΔVdrd þ Idrq0 ΔVdrq

ð2:72Þ

According to Eq. (2.69), the above equation can be written as ΔPdr ¼ cd1 T ΔXd1 þ cd2 T Δzd1

ð2:73Þ

where cd1 T ¼ ½ K5 Vdrd0

K7 Vdrq0 K6 Vdrd0 K8 Vdrq0 cd2 T ¼ ½ Idrd0 Idrq0 :

0

0 ,

0

Writing Eqs. (2.69) and (2.73) together, the state-space model of the induction generator and two-mass shaft rotational system is obtained to be 8 < d ΔX ¼ A ΔX þ B Δz þ b ΔV þ b ΔV d1 d1 d1 d1 d1 d1 dsq d2 dsd dt ð2:74Þ : T T ΔPdr ¼ cd1 ΔXd1 þ cd2 Δzd1

2.2.2

Linearized Model of the RSC and Associated Control System

The vector control is implemented by the RSC. Configuration of the control system of the RSC is shown by Fig. 2.5, where current control inner loops control the d-axis and q-axis stator current, Idsq and Idsd, respectively; the active and reactive power control outer loops control the active and reactive power output from the stator side q-axis current control inner loop

Active power control outer loop

Pdsref

K dp1

+

+ – Pds

Qdsref

K di1 + s xd 1

K dp 3

+ – Qds

Idsqref

+ Idsdref

K di 3 + s xd 3

Reactive power control outer loop

I drqref

X – dss X dm

X – dss X dm

– sdr1 ( X drr – K dp 2

+

+ – I drq

+

Vdsq



+ Vdrqref

+ – I drd

X dm

RSC

K di 2 + s xd 2

K dp 4

I drdref +



2 X dm s X ) Idrd + dr1 dm Vdsq X dss X dss

K di 4 + s xd 4 sdr1 ( X drr –

d-axis current control inner loop

Fig. 2.5 Configuration of vector control system of the RSC of the DFIG

Vdrdref



RSC

+ 2 X dm ) Idrq X dss

46

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

of the induction generator, Pds and Qds respectively. sdr1 ¼ 1  ωdr1 is slip of highspeed shaft of the induction generator. There are total four PI controllers in the RSC control system shown by Fig. 2.2. Take the active power control outer loop as an example, the state equation related with the integral control is d xd1 ¼ Kdi1 ðPdsref  Pds Þ dt

ð2:75Þ

where xd1 is the state variable and output of integral control; Kdi1 is the integral gain; Pdsref is the control reference of active power output from the stator side of the induction generator. Similarly, the output of integral control of q-axis rotor current control inner loop, the reactive power control outer loop and d-axis rotor current control inner loop is denoted as xd2, xd3 and xd4 respectively. Following state equations can be obtained. 8   d > > xd2 ¼ Kdi2 Idrqref  Idrq > > dt > > < d ð2:76Þ xd3 ¼ Kdi3 ðQdsref  Qds Þ > dt > > > > > : d xd4 ¼ Kdi4 ðIdrdref  Idrd Þ dt where Kdi2, Kdi3 and Kdi4 are the gains of corresponding integral controllers; Qdsref, Idrqref and Idrdref are control reference of reactive power output from the stator side, q-axis and d-axis of rotor current respectively. From Fig. 2.5, following algebraic equations can be obtained. 8 Idsqref ¼ Kdp1 ðPdsref  Pds Þ þ xd1 > > > > > I ¼ Kdp3 ðQdsref  Qds Þ þ xd3 > > > dsdref > > > Xdss > > Idrqref ¼  Idsqref > > Xdm > > > > > Vdsq Xdss > > I ¼ Idsdref  > > < drdref Xdm Xdm   ð2:77Þ Vdrqref ¼ Kdp2 Idrqref  Idrq  xd2 > > > >

> > X2dm sdr1 Xdm > > > s X  Vdsq I þ > > dr1 drr Xdss drd Xdss > > > > > > Vdrdref ¼ Kdp4 ðIdrdref  Idrd Þ > > >

> > > X2 > : xd4 þ sdr1 Xdrr  dm Idrq Xdss where Kdp1, Kdp2, Kdp3 and Kdp4 are the proportional gains; Idsqref and Idsdref are the control reference of q and d component of current of stator windings; Vdrqref and Vdrdref are the control reference output of the RSC for controlling the q and d

2.2 Linearized Model of a DFIG

47

component of voltage of rotor windings respectively; Normally, transient of modulation control can be ignored such that Vdrdref ¼ Vdrd ð2:78Þ Vdrqref ¼ Vdrq Active and reactive power outputs from the stator side of the induction generator are

Pds ¼ Vdsq Idsq þ Vdsd Idsd Qds ¼ Vdsq Idsd  Vdsd Idsq

ð2:79Þ

Power control references are constant such that ΔPdsref ¼ 0 and ΔQdsref ¼ 0. Thus, linearization of Eqs. (2.75) and (2.76) is 8 d > > Δxd1 ¼ Kdi1 ΔPds > > dt > > > >   > d > > < Δxd2 ¼ Kdi2 ΔIdrqref  ΔIdrq dt ð2:80Þ > d > > Δxd3 ¼ Kdi3 ΔQds > > dt > > > > > > : d Δxd4 ¼ Kdi4 ðΔIdrdref  ΔIdrd Þ dt Linearization of Eq. (2.77) is 8

  > X2dm > > ΔVdrqref ¼ Kdp2 ΔIdrqref  ΔIdrq  Δxd2  sdr10 Xdrr  ΔIdrd > > Xdss > > > >

> 2 > > > þIdrd0 Xdrr  Xdm Δωdr1 þ sdr10 Xdm ΔVdsq  Vdsq0 Xdm Δωdr1 > > > Xdss Xdss Xdss > > >

> > > X2dm > > ΔVdrdref ¼ Kdp4 ðΔIdrdref  ΔIdrd Þ  Δxd4 þ sdr10 Xdrr  ΔIdrq > > Xdss > > > <

X2dm ð2:81Þ Idrq0 Xdrr  Δωdr1 > > Xdss > > > > > > ΔIdsdref ¼ Kdp3 ΔQds þ Δxd3 > > > > Xdss > > > ΔIdrqref ¼  ΔIdsqref > > Xdm > > > > > ΔVdsq Xdss > > ΔIdrdref ¼  ΔIdsdref  > > > X Xdm dm > > : ΔIdsqref ¼ Kdp1 ΔPds þ Δxd1 where Δωdr1 ¼  Δsdr1.

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

48

Linearization of Eq. (2.78) is

ΔVdrdref ¼ ΔVdrd ΔVdrqref ¼ ΔVdrq

Linearization of Eq. (2.79) is ΔPds ¼ Vdsq0 ΔIdsq þ Vdsd0 ΔIdsd þ Idsq0 ΔVdsq þ Idsd0 ΔVdsd ΔQds ¼ Vdsq0 ΔIdsd  Vdsd0 ΔIdsq þ Idsd0 ΔVdsq  Idsq0 ΔVdsd

ð2:82Þ

ð2:83Þ

From Eqs. (2.68), (2.80), (2.81), (2.82) and (2.83), following state equation of the RSC and associated control system is obtained 8 < d ΔX ¼ A ΔX þ B ΔX þ b ΔV þ b ΔV d2 d2 d2 d2 d1 d3 dsq d4 dsd dt ð2:84Þ : Δzd1 ¼ Cd1 ΔXd2 þ Cd2 ΔXd1 þ Cd3 ΔVdsq þ Cd4 ΔVdsd where ΔXd2 ¼ ½ Δxd1 Δxd2 Δxd3 Δxd4 , 3 2 0 0 0 0 7 6 Kdi2 Xdss 6 0 0 07 7 6 X 7 6 dm Ad2 ¼ 6 7, 7 6 0 0 0 0 7 6 5 4 Kdi4 Xdss 0 0 0 Xdm

2.2 Linearized Model of a DFIG

49

2 6 Cd1 ¼ 4 K

0

dp2 Xdss

Xdm

2

Cd2 T

0

Kdp4 Xdss Xdm

1

1

0

0

Kdp3 Kdp4 Xdss Xdrr Vdsq0 þ Kdp4 Xdm 2   6 Xdm Xdrr Xdss  Xdm 2 6 6 6 6 Kdp3 Kdp4 Xdss Xdrr Vdsd0 sdr10 Xdm  þ 6 6 Xdm Xdm 2  Xdss Xdrr Xdss 6 6 6 Kdp3 Kdp4 Xdss Vdsq0 þ Kdp4 Xdss 6 6 Xdm 2  Xdrr Xdss ¼6 6 Kdp3 Kdp4 Xdss Vdsd0 6  sdr10 6 6 Xdss Xdrr  Xdm 2 6

6 Xdm 2 6 Idrq0 Xdrr  6 6 Xdss 6 6 0 4

3 7 5,

3 Kdp2 Kdp1 Xdss Xdrr Vdsd0 sdr10 Xdm   Xdss 7 Xdm Xdrr Xdss  Xdm 2 7 7 27 Kdp2 Kdp1 Xdss Xdrr Vdsq0 þ Kdp2 Xdm 7   7 7 Xdm Xdss Xdrr  Xdm 2 7 7 Kdp2 Kdp1 Xdss Vdsd0 7 7 þ s dr10 7 Xdm 2  Xdrr Xdss 7 Kdp2 Kdp1 Xdss Vdsq0 þ Kdp2 Xdss 7 7 7 7 Xdm 2  Xdss Xdrr 7

2 Vdsq0 Xdm 7 Xdm 7 Idrd0 Xdrr   7 7 Xdss Xdss 7 7 0 5

0 0 3 2 3 Kdp4  Kdp3 Kdp4 Xdss Idsd0 Kdp3 Kdp4 Xdss Idsq0 6 7 6 7 Xdm Xdm 7, Cd4 ¼ 6 7: ¼6 4s X 5 4 Kdp2 Kdp1 Xdss Idsq0 Kdp2 Xdss Kdp1 Idsd0 5 dr10 dm  Xdss Xdm Xdm

2

Cd3

ΔXd1 and Δzd1 are defined by Eq. (2.69).

2.2.3

Linearized Model of the GSC and Associated Control System

Configuration of the vector control system of the GSC is shown by Fig. 2.6, which is comprised of q-axis and d-axis current control loop as well as the DC voltage control outer loop. The q-axis and d-axis current control loop controls the output current of the GSC, Idcq and Idcd, respectively. DC voltage control outer loop controls the DC voltage across the capacitor, Vddc. In the d-q coordinate, the voltage equations across the filter reactance, Xfd, on the side of the GSC are 8 d ω0 Vdcd ω0 Vdsd > >  þ ω0 Idcq < dtIdcd ¼ X Xdf df ð2:85Þ > ωV ω V d > : Idcq ¼ 0 dcq  0 dsq  ω0 Idcd dt Xdf Xdf

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

50

q-axis current control inner loop

DC voltage control outer loop

Kdp 5

Vddcref –

+ + Vddc

– I dcq

Kdi 5 + s xd 5

I dcdref

Vdsq

Kdp 6

I dcqref+

Kdi 6 s

+ +

xd 6

– I dcd

Kdi 7 s

GSC

Vdsd

Kdp 7

+

+ + Vdcqref + X df I dcd

+ +

xd 7

+ +Vdcdref – X df I dcq

GSC

d-axis current control inner loop Fig. 2.6 Configuration of vector control system of the GSC of the DFIG

where Xdf is the reactance of the output filter, Idcd and Idcq are the d and q component of output current of the GSC, Vdcd and Vdcq are the d and q component of terminal voltage of the GSC. Power balance equation associated with the charging and discharging of the capacitor is dVddc Cd Vddc ¼ Pdc1  Pdr ð2:86Þ dt where Cd is the capacitance, Vddc is the DC voltage across the capacitor, Pdr is the active power output from the rotor side of the induction generator and expressed by Eq. (2.70), Pdc1 is the active power injected from the DC side to the GSC and can be expressed as Pdc1 ¼ Vdcd Idcd þ Vdcq Idcq

ð2:87Þ

The output variables of integrators in Fig. 2.3 are xd5, xd6 and xd7. It can have 8d > x ¼ Kdi5 ðVddc  Vddcref Þ > > > dt d5 > <   d ð2:88Þ xd6 ¼ Kdi6 Idcqref  Idcq > dt > > > > : d x ¼ K ðI d7 di7 dcdref  Idcd Þ dt where Kdi5, Kdi6 and Kdi7 are the integral gains of PI controllers in Fig. 2.6, Vddcref, Idcqref and Idcdref are respectively the control reference of DC voltage control, q and d component of output current o the GSC.

2.2 Linearized Model of a DFIG

51

Following algebraic equations can be obtained from Fig. 2.6 8 ¼ Kdp5 ðVddcref  Vddc Þ þ xd5 I > < cqref   Vdcqref ¼ Kdp6 Idcqref  Idcq þ xd6 þ Vdsq þ Xdf Idcd > : Vdcdref ¼ Kdp7 ðIdcdref  Idcd Þ þ xd7 þ Vdsd  Xdf Idcq

ð2:89Þ

where Kdp5, Kdp6 and Kdp7 are the proportional gains of PI controllers in Fig. 2.6 Vdcdref and Vdcqref are the control reference output of the GSC for controlling the q and d component of terminal voltage of the GSC respectively; Normally, transient of modulation control can be ignored such that Vdcdref ¼ Vdcd ð2:90Þ Vdcqref ¼ Vdcq Linearization of Eq. (2.85) is 8 d ω0 ΔVdcd ω0 ΔVdsd > >  þ ω0 ΔIdcq < dtΔIdcd ¼ X X df

df

> ω ΔV ω ΔV d > : ΔIdcq ¼ 0 dcq  0 dsq  ω0 ΔIdcd dt Xdf Xdf

ð2:91Þ

Linearization of Eq. (2.86) is Cd Vddc0

dΔVddc ¼ ΔPdr  ΔPdc1 dt

ð2:92Þ

Linearization of Eq. (2.87) is ΔPdc1 ¼ Vdcd0 ΔIdcd þ Vdcq0 ΔIdcq þ Idcd0 ΔVdcd þ Idcq0 ΔVdcq

ð2:93Þ

Substituting Eq. (2.93) in Eq. (2.92), it can have Cd Vddc0

dΔVddc ¼ ΔPdr  Vdcd0 ΔIdcd  Vdcq0 ΔIdcq  Idcd0 ΔVdcd  Idcq0 ΔVdcq dt ð2:94Þ

Control references in Eqs. (2.88) and (2.89) are constant such that ΔVddcref ¼ 0 and ΔIdcdref ¼ 0. Thus, linearization of Eq. (2.88) is 8 d > > Δx ¼ Kdi5 ΔVddc > > dt d5 > > <   d ð2:95Þ Δxd6 ¼ Kdi6 ΔIdcqref  ΔIdcq > dt > > > > > : d Δxd7 ¼ Kdi7 ΔIdcd dt

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

52

Linearization of Eq. (2.89) is 8 ¼ Kdp5 ΔVddc þ Δxd5 ΔI > < dcqref   ΔVdcqref ¼ Kdp6 ΔIdcqref  ΔIdcq þ Δxd6 þ ΔVdsq þ Xdf ΔIdcd > : ΔVdcdref ¼ Kdp7 ΔIdcd þ Δxd7 þ ΔVdsd  Xdf ΔIdcq Linearization of Eq. (2.90) is

ΔVdcdref ¼ ΔVdcd

ð2:96Þ

ð2:97Þ

ΔVdcqref ¼ ΔVdcq Substituting the first equation in Eq. (2.96) in Eq. (2.95), 8 d > > Δxd5 ¼ Kdi5 ΔVddc > > dt > > < d Δxd6 ¼ Kdi6 Kdp5 ΔVddc þ Kdi6 Δxd5  Kdi6 ΔIdcq > dt > > > > > : d Δxd7 ¼ Kdi7 ΔIdcd dt

ð2:98Þ

Substituting Eq. (2.97) and the first equation of Eq. (2.96) in the second and third equation of Eq. (2.96), (   ΔVdcq ¼ Kdp6 Kdp5 ΔVddc þ Δxd5  ΔIdcq þ Δxd6 þ ΔVdsq þ Xdf ΔIdcd ΔVdcd ¼ Kdp7 ΔIdcd þ Δxd7 þ ΔVdsd  Xdf ΔIdcq ð2:99Þ Substituting Eq. (2.99) in Eq. (2.91), 8 ω0 Kdp7 ΔIdcd þ ω0 Δxd7 d > > > < dtΔIdcd ¼ Xdf   > ω0 Kdp6 Kdp5 ΔVddc þ Δxd5  ΔIdcq þ ω0 Δxd6 d > > : ΔIdcq ¼ dt Xdf

ð2:100Þ

The state equations of the GSC and associated control system is obtained from Eqs. (2.94), (2.98) and (2.100) to be d ΔXd3 ¼ Ad3 ΔXd3 þ bd5 ΔVdsd þ bd6 ΔVdsq þ bd7 ΔPdr dt where ΔXd3 ¼ ½ ΔIdcd

ΔIdcq

ΔVddc

Δxd5

Δxd6

Δxd7 ,

ð2:101Þ

2.2 Linearized Model of a DFIG

2.2.4

53

Linearized Model of the Entire DFIG

By combining the linearized models of three main modules of the DFIG derived in the above three subsections, the linearized model of the DFIG can be obtained as follows. Firstly, by writing the state equations of Eqs. (2.74), (2.84) and (2.101) together, it can have 8d > ΔX ¼ Ad1 ΔXd1 þ Bd1 Δzd1 þ bd1 ΔVdsq þ bd2 ΔVdsd > > > dt d1 > < d ð2:102Þ ΔXd2 ¼ Ad2 ΔXd2 þ Bd2 ΔXd1 þ bd3 ΔVdsq þ bd4 ΔVdsd > dt > > > > : d ΔX ¼ A ΔX þ b ΔV þ b ΔV þ b ΔP d3 d3 d3 d5 dsd d6 dsq d7 dr dt Substituting the second equation of Eq. (2.84) in the second equation of Eq. (2.74), ΔPdr ¼ cd1 T ΔXd1 þ cd2 T Cd1 ΔXd2 þ cd2 T Cd2 ΔXd1 þcd2 T Cd3 ΔVdsq þ cd2 T Cd4 ΔVdsd

ð2:103Þ

From the second equation of Eqs. (2.84) and (2.103), Δzd1 and ΔPdr in Eq. (2.102) can be replaced. With the replacement, Eq. (2.102) becomes

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

54

8d > ΔXd1 ¼ Ad1 ΔXd1 þ Bd1 Δzd1 þ bd1 ΔVdsq þ bd2 ΔVdsd > > > dt > < d ΔXd2 ¼ Ad2 ΔXd2 þ Bd2 ΔXd1 þ bd3 ΔVdsq þ bd4 ΔVdsd > dt > > > > : d ΔX ¼ A ΔX þ b ΔV þ b ΔV þ b ΔP d3 d3 d3 d5 dsd d6 dsq d7 dr dt where

ð2:104Þ

 T ΔXd ¼ ΔXd1 T ΔXd2 T ΔXd3 T , 2 3 Bd1 Cd1 0 Ad1 þ Bd1 Cd2 6 7 Ad ¼ 4 Bd2 Ad2 0 5, bd7 cd2 T Cd2 þ bd7 cd1 T bd7 cd2 T Cd1 Ad3 2 3 2 3 Bd1 cd4 þ bd2 Bd1 cd3 þ bd1 6 7 6 7 bdd ¼ 4 bd4 bd3 5, bdq ¼ 4 5: bd7 cd2 T cd4 þ bd5

bd6 þ bd7 cd2 T cd3

Active and reactive power injected from the DFIG to the power system are Pd ¼ Pds þ Pcd ¼ Vdsd Idsd þ Vdsq Idsq þ Vdsd Idcd þ Vdsq Idcq ð2:105Þ Qd ¼ Qds þ Qcd ¼ Vdsq Idsd  Vdsd Idsq þ Vdsq Idcd  Vdsd Idcq where Pcd and Qcd are the active and reactive power injected from the GSC to the power system. Linearization of Eq. (2.105) can be written as   ΔPd ð2:106Þ ¼ Fdsv ΔVdsv þ Fdsi ΔIdsi þ Fdci ΔIdci ΔQd where T T T ΔVdsv ¼ ½ ΔVdsq ΔVdsd  , ΔIdsi ¼ ½ ΔIdsq ΔIdsd  , ΔIdci ¼ ½ ΔIdcq ΔIdcd  ,     Idsq0 þ Idcq0 Idsd0 þ Idcd0 Vdsq0 Vdsd0 Fdsv ¼ , Fdsi ¼ , Idsd0 þ Idcd0 Idsq0  Idcq0 Vdsd0 Vdsq0   Vdsq0 Vdsd0 Fdci ¼ : Vdsd0 Vdsq0

Substituting Eq. (2.68) in Eq. (2.106) with ΔIdsi being replaced, Eq. (2.106) becomes ( ΔPd ¼ cdp T ΔXd þ ddp1 ΔVdsq þ ddp2 ΔVdsd ð2:107Þ ΔQd ¼ cdq T ΔXd þ ddq1 ΔVdsq þ ddq2 ΔVdsd

2.3 Linearized Model of the Power System

55

where

Writing Eqs. (2.104) and (2.107) together, the state-space model of the DFIG is 8 d > > > < dtΔXd ¼ Ad ΔXd þ bdq ΔVdsq þ bdd ΔVdsd ð2:108Þ ΔPd ¼ cdp T ΔXd þ ddp1 ΔVdsq þ ddp2 ΔVdsd > > > : ΔQd ¼ cdq T ΔXd þ ddq1 ΔVdsq þ ddq2 ΔVdsd

2.3

Linearized Model of the Power System

Figure 2.7 shows the configuration of a multi-machine power system with N synchronous generators, G1, G2. . .GN. The load is represented by impedance model. A variable speed wind generator, either the PMSG or the DFIG, is connected at node W. ΔPw and ΔQw denote the small variations of active and reactive power injected from the variable speed wind generator. According to the notations used in the previous sections, when the variable speed wind generator is the PMSG, ΔPw ¼ ΔPp and ΔQw ¼ ΔQp. For the DFIG, ΔPw ¼ ΔPd and ΔQw ¼ ΔQd. In Fig. 2.7 Configuration of a power system integrated with a PMSG or DFIG

G1

G2

Transmission network

GN W

ΔVw ΔPw ΔQw

Δqw

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

56

this section, linearized state-space model of the power system is derived, where ΔPw and ΔQw are treated as the input variables and small variations of the PCC voltage, ΔVw ∠ Δθw, is treated as the output variables. Derivation of the model is carried out in the common x-y coordinate of the power system. ( ΔPw ¼ Ix0 ΔVx þ Iy0 ΔVy þ Vx0 ΔIx þ Vy0 ΔIy ð2:109Þ ΔQw ¼ Iy0 ΔVx þ Ix0 ΔVy þ Vy0 ΔIx  Vx0 ΔIy where ΔIx and ΔIy are the small variations of d and q component of the output current from node W injected into the transmission network, ΔVx and ΔVy are the small variations of d and q component of the terminal voltage at node W. Express Eq. (2.109) in the following matrix form,   ΔPw ð2:110Þ ¼ Fw1 ΔI þ Fw2 ΔV ΔQw where  ΔI ¼

    Vx0 ΔVx ΔIx , ΔV ¼ , Fw1 ¼ ΔIy ΔVy Vy0

  Vy0 Ix0 , Fw2 ¼ Vx0 Iy0

Iy0 Ix0



Denote ΔIgxj and ΔIgyj as the small variations of d and q component of the output current from Gj. Denote ΔVgxj and ΔVgyj as the small variations of d and q component of the voltage at the terminal of Gj. Define the following output current vector ΔIgj and terminal voltage vector ΔVgj of Gj. ( T ΔVgj ¼ ½ ΔVgxj ΔVgyj  ð2:111Þ T ΔIgj ¼ ½ ΔIgxj ΔIgyj  Network equation is " # ΔIg ΔI

" ¼

Ygg

Ygw

Ywg

Yww

#"

ΔVg ΔV

#

" ¼Y

ΔVg ΔV

# ð2:112Þ

 T  T where ΔVg ¼ ΔVg1 T    ΔVgN T , ΔIg ¼ ΔIg1 T    ΔIgN T ,   Ygg Ygw Y¼ is the admittance matrix of the transmission network with only Ywg Yww the generators’ nodes being kept. The state-space model of Gj is 8 < d ΔX ¼ A ΔX þ B ΔV gj gj gj gj gj dt ð2:113Þ : ΔIgj ¼ Cgj ΔXgj þ Dgj ΔVgj

2.3 Linearized Model of the Power System

57

where ΔXgj is the vector of all the state variables of Gj. From Eq. (2.113), the statespace model for N generators can be obtained by writing N state-space models of Gj together as 8 < d ΔX ¼ A ΔX þ B ΔV g g g g g dt ð2:114Þ : ΔIg ¼ Cg ΔXg þ Dg ΔVg where  ΔXg ¼ ΔXg 1 T Ag ¼ diag½ Ag1

T  T ΔXg N T , ΔIg ¼ ΔIg 1 T    ΔIg N T ,  T ΔVg ¼ ΔVg 1 T    ΔVg N T , Ag2 . . . AgN , Bg ¼ diag½ Bg1 Bg2 . . . BgN ,

Cg ¼ diag½ Cg1

Cg2



. . . CgN , Dg ¼ diag½ Dg1

Dg2

. . . DgN ,

diag[] denotes either a diagonal matrix or a block diagonal matrix. Substituting Eq. (2.112) in Eq. (2.110),   1 ΔPw ΔV ¼ ðFw2 þ Fw1 Yww Þ  ðFw2 þ Fw1 Yww Þ1 Fw1 Ywg ΔVg ΔQw From Eq. (2.112) and the output equation in Eq. (2.114)   Ygg  Dg ΔVg þ Ygw ΔV ¼ Cg ΔXg

ð2:115Þ

ð2:116Þ

Substituting Eq. (2.115) in Eq. (2.116),  ΔVg ¼ F3 ΔXg þ F4

ΔPw ΔQw

 ð2:117Þ

where h i1 F3 ¼ Ygg  Dg  Ygw ðFw2 þ Fw1 Yww Þ1 Fw1 Ywg Cg , h i1 F4 ¼  Ygg  Dg  Ygw ðFw2 þ Fw1 Yww Þ1 Fw1 Ywg Ywg ðFw2 þ Fw1 Yww Þ1 : Substituting Eq. (2.117) in the first equation of Eq. (2.114),   d ΔPw ΔXg ¼ A1 ΔXg þ B1 ΔQw dt

ð2:118Þ

where A1 ¼ Ag + BgF3 and B1 ¼ BgF4. Substituting Eq. (2.117) in Eq. (2.115), " ΔV ¼ F5 ΔXg þ F6

ΔPw ΔQw

# ð2:119Þ

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

58

where F5 ¼ ðFw2 þ Fw1 Yww Þ1 Fw1 Ywg F3 , F6 ¼ ðFw2 þ Fw1 Yww Þ1  ðFw2 þ Fw1 Yww Þ1 Fw1 Ywg F4 : Small variations of the magnitude and phase of the voltage at node W where the variable speed wind generator is connected with the power system are ΔVw and Δθw. It can have 8   Vx0 Vy0 > > ΔV ¼ ΔV ¼ f w1 T ΔV > w < Vw0 Vw0 ð2:120Þ   > Vy0 Vx0 > T >  ΔV ¼ f w2 ΔV : Δθw ¼ V2w0 V2w0 Substituting Eq. (2.119) in Eq. (2.120), " # 8 ΔPw > > > ΔVw ¼ C1 ΔXg þ D1 > > < ΔQw " # > > ΔPw > > > : Δθw ¼ C2 ΔXg þ D2 ΔQw where C1 ¼ fw1TF5, D1 ¼ fw1TF6, C2 ¼ fw2TF5, D2 ¼ fw2TF6. Writing Eqs. (2.118) and (2.121) together, " # 8 ΔP > w d > > ΔXg ¼ A1 ΔXg þ B1 > > dt > ΔQw > > > > " # > < ΔPw ΔVw ¼ C1 ΔXg þ D1 > ΔQw > > > > " # > > > ΔPw > > > Δθw ¼ C2 ΔXg þ D2 : ΔQw

ð2:121Þ

ð2:122Þ

Hence, the state-space model of the power system can be expressed as 8 d > > > < dtΔXg ¼ Ag ΔXg þ bp ΔPw þ bq ΔQw ð2:123Þ ΔVw ¼ cgv T ΔXg þ dvp ΔPw þ dvq ΔQw > > > : Δθw ¼ cgθ T ΔXg þ dθp ΔPw þ dθq ΔQw

2.4 Closed-Loop Interconnected Model of the Power System. . .

2.4 2.4.1

59

Closed-Loop Interconnected Model of the Power System with a Grid-Connected VSWG Closed-Loop Interconnected Model of the Power System with a PMSG

Consider that the VSWG connected at node W in Fig. 2.7 is a PMSG. Figure 2.8 shows the relation between the d-q coordinate of the PMSG and the common x-y coordinate of the power system. It can be seen that the direction of the terminal voltage of the PMSG, ΔVw ∠Δθw, is taken as the direction of d axis in the common x-y coordinate. Hence, ΔVwd ¼ ΔVw and ΔVwq ¼ 0. According to the notations to derive the state-space model of the PMSG in Sect. 2.1, ΔPw ¼ ΔPp, ΔQw ¼ ΔQp, ΔVpd ¼ ΔVwd and ΔVpq ¼ ΔVwq. Thus, ΔVpd ¼ ΔVwd ¼ ΔVw and ΔVpq ¼ ΔVwq ¼ 0. The state-space model of the PMSG shown by Eq. (2.55) becomes 8 d > > > < dtΔXp ¼ Ap ΔXp þ bpd ΔVw ð2:124Þ ΔPw ¼ cpp T ΔXp þ dpp1 ΔVw > > > : ΔQw ¼ cpq T ΔXp þ dpq1 ΔVw From Eq. (2.124), the transfer function model of the PMSG can be written as ( ΔPw ¼ Hpp ðsÞΔVw ð2:125Þ ΔQw ¼ Hpq ðsÞΔVw where Hpp(s) ¼ cppT(sI  Ap)1bpd + dpp1, Hpq(s) ¼ cpqT(sI  Ap)1bpd + dpq1. The state-space model of the power system shown by Eq. (2.123) can be written as Fig. 2.8 Relation between the d-q coordinate of the PMSG and the common x-y coordinate

y q

d Vw qw qw

x

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

60

Fig. 2.9 Closed-loop interconnected model of the power system with the PMSG

Subsystem of rest of power system ΔPw

Gvp (s) +

ΔQw

ΔVw

Gvq (s)

Hpq (s) Hpp (s) PMSG subsystem

8 < d ΔX ¼ A ΔX þ b ΔP þ b ΔQ g g g p w q w dt : ΔVw ¼ cgv T ΔXg þ dvp ΔPw þ dvq ΔQw

ð2:126Þ

From Eq. (2.126), the transfer function model of the power system can be written as ΔVw ¼ Gvp ðsÞΔPw þ Gvq ðsÞΔQw

ð2:127Þ

where Gvp(s) ¼ cgvT(sI  Ag)1bp + dvp, Gvq(s) ¼ cgvT(sI  Ag)1bq + dvq. From Eqs. (2.125) and (2.127), the closed-loop interconnected model of the power system with the PMSG is obtained and shown by Fig. 2.9. The state-space model of the closed-loop interconnected system can be derived from Eqs. (2.124) and (2.126) to be

 where ΔXgp ¼ ΔXg T

d ΔXgp ¼ Agp ΔXgp dt T ΔXp T ,

ð2:128Þ

    3 dpq1 bq þ dpp1 bp cgv T dpq1 bq þ dpp1 bp dvp cpp T þ dvq cpq T T T 6 Ag þ 7 bp cpp þ bq cpq þ 6 7 1  dvp dpp1  dvq dpq1 1  dvp dpp1  dvq dpq1 7 Agp ¼ 6 6 7 4 5 bpd cgv T dvp bpd cpp T þ dvq bpd cpq T Ap þ 1  dvp dpp1  dvq dpq1 1  dvp dpp1  dvq dpq1 2

2.4 Closed-Loop Interconnected Model of the Power System. . .

2.4.2

61

Closed-Loop Interconnected Model of the Power System with a DFIG

When the VSWG connected at node W in Fig. 2.7 is a DFIG, a closed-loop interconnected model of the power system with the DFIG can be derived similarly. Normally, the direction of the terminal voltage of the DFIG, ΔVw ∠Δθw, is taken as the direction of q axis in the common x-y coordinate as shown by Fig. 2.10. Hence, ΔVwq ¼ ΔVw and ΔVwd ¼ 0. According to the notations to derive the state-space model of the DFIG in Sect. 2.2, ΔPw ¼ ΔPd, ΔQw ¼ ΔQd, ΔVdsq ¼ ΔVwq and ΔVdsd ¼ ΔVdsd. Thus, ΔVdsq ¼ ΔVwq ¼ ΔVw and ΔVdsd ¼ ΔVwd ¼ 0. The state-space model of the DFIG shown by Eq. (2.108) becomes 8 d > > > < dtΔXd ¼ Ad ΔXd þ bdq ΔVw ð2:129Þ ΔPw ¼ cdp T ΔXd þ ddp1 ΔVw > > > : ΔQw ¼ cdq T ΔXd þ ddq1 ΔVw From Eq. (2.129), the transfer function model of the DFIG is obtained to be ( ΔPw ¼ Hdp ðsÞΔVw ð2:130Þ ΔQw ¼ Hdq ðsÞΔVw where Hdp(s) ¼ cdpT(sI  Ad)1bdq + ddp1Hdq(s) ¼ cdqT(sI  Ad)1bdq + ddq1. The state-space and transfer function model of the power system are expressed by Eqs. (2.126) and (2.127) respectively. From Eqs. (2.127) and (2.130), the closedloop interconnected model of the power system with the DFIG is obtained and shown by Fig. 2.11. From Eqs. (2.126) and (2.129), the state-space model of the closed-loop interconnected system shown by Fig. 2.11 is derived to be d ΔXgd ¼ Agd ΔXgd dt Fig. 2.10 Relation between the d-q coordinate of the DFIG and the common x-y coordinate

ð2:131Þ y

q Vw qw d

qw x

2 Linearized Model of a Power System with a Grid-Connected Variable. . .

62

Fig. 2.11 Closed-loop interconnected model of the power system with the DFIG

Subsystem of rest of power system ΔPw

Gvp (s) +

ΔVw

Gvq (s)

ΔQw

Hdq (s) Hdp (s) DFIG subsystem

where  ΔXgd ¼ ΔXg T 2

  ddq1 bq þ ddp1 bp cgv T

6 Ag þ 6 1  dvp ddp1  dvq ddq1 Agd ¼ 6 6 4 bdq cgv T 1  dvp ddp1  dvq ddq1

ΔXd T 

bp cdp T þ bq cdq T þ Ad þ

T

,

ddq1 bq þ ddp1 bp



dvp cdp T þ dvq cdq T

1  dvp ddp1  dvq ddq1 dvp bdq cdp T þ dvq bdq cdq T 1  dvp ddp1  dvq ddq1

3 7 7 7 7 5

Chapter 3

Damping Torque Analysis of Small-Signal Angular Stability of a Power System Affected by Grid-Connected Wind Power Induction Generators

Induction generator based wind power generation has been dominating the wind market since the rise of wind power industry at the end of last century and will be continuously in a favorable position for large-scale grid connection given its lower cost and more mature technology compared with other wind generation for the foreseeable future [1]. Fixed-speed induction generator (FSIG-Type 1 Wind Gen Model) and doubly-fed induction generator (DFIG-Type 3 Wind Gen Model) are two main types of induction generator adopted for wind power generation especially considering the fact that DFIG is the most frequently-used technology to date. The increasing penetration of wind power generation has significantly affected power system dynamics. Due to the difference in rotor structures and excitation principles, FSIG and DFIG possess different dynamic behaviors during system disturbances and hence impact the power system dynamics differently, which has posed a critical challenge for the real-time system operation and therefore deserves a careful investigation.

3.1

Impact of Grid-Connected Wind Power Induction Generators on Small-Signal Angular Stability

The impact of the integration of FSIG and DFIG on power system small-signal angular stability have been extensively examined from early this century. A comprehensive study regarding the influence of FSIG on power system oscillation is presented in [2] by modal analysis, which considers multiple impact factors including length of transmission interface, load condition, wind penetration level and wind farm configuration etc. It is concluded that in most cases FSIG introduces a negative damping to the system and additional reactive power compensation could mitigate the negative impact of FSIG on small-signal angular stability. This conclusion is supported by modal analysis in [3] but contradicted by [4]. Compared with FSIG, © Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_3

63

64

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

DFIG is comparatively new and has a more flexible control in active and reactive power, and thus most of research efforts are devoted to the grid connection study of DFIG in recent decade. Various case studies have been implemented to address different aspects of DFIG in affecting the small-signal angular stability such as integration method [4–9], inertia or other sensitivity based approach [10–12], reactive power/voltage control [13–17], operating condition [18], virtual inertia control [19–21], additional damping control [22–30] and external energy storage system [31, 32]. It can be seen that: (1) Most of the existing research is actually case-by-case observation by using the two common ‘computation’ methods (modal analysis and time domain simulation), and thus the essential reason for inconsistent study results with different preconditions cannot be effectively and convincingly investigated by these two ‘black box’ methods. No proper theoretical method is seen so far to clearly reveal the essential damping mechanism of power system small-signal angular stability as affected by FSIG and DFIG; (2) Most of the published research tends to study the grid impact of FSIG and DFIG separately and there is no systematic analytical theory to compare the damping effectiveness and robustness of these two wind power induction generators (WPIGs) and dig deeper information about their essential difference and inner connection in affecting power system oscillations, which certainly provides a better understanding of their individual damping mechanisms. Taking account of the points above, a generic methodology based on damping torque analysis theory in order to analyze the damping mechanisms of different WPIGs is proposed in this chapter, with the aim of giving a physical insight that how the different rotor structures and excitation systems of FSIG and DFIG affect their damping mechanisms. The rest of the chapter is organized as follows: In Sect. 3.2, system modeling for damping torque analysis of a power system affected by gridconnected wind power induction generators is carried out. Then a general implementation framework of explicit damping torque analysis of Phillips-Heffron model based multi-machine power system with WPIGs is presented in Sect. 3.3. Hence, the closed-form solution of damping torque contribution from the main internal dynamic components of wind generators to each synchronous generator (SG) can be derived. In Sect. 3.4, two typical linearized models and explicit transfer functions of different WPIGs are proposed to facilitate the analytical comparison on the impact mechanisms of DFIG and FSIG, where FSIG is treated as a special case of DFIG with rotor side short-circuit (i.e., rotor voltage equal to zero). Unlike the existing numerical comparison (case-by-case study), the purely analytical comparison of damping mechanisms between different WPIGs can reveal their essential difference and inner connection in damping mechanisms. In Sect. 3.5, the proposed methodology is demonstrated in a 16-machine test system and then employed to perform a numerical comparison under different wind penetration conditions, so that the conclusions of analytical comparison analysis from Sect. 3.4 is testified. Time domain simulation is employed to prove the accuracy of the proposed methodology in frequency domain.

3.2 System Modeling for Damping Torque Analysis of a Power System. . .

3.2

65

System Modeling for Damping Torque Analysis of a Power System Affected by Grid-Connected Wind Power Induction Generators

Philips-Heffron model is widely used for damping torque analysis of a power system [33], which is able to facilitate a deeper understanding of damping mechanism of WPIGs. Therefore, different from the linearized model presented in Chap. 2 for modal analysis, Philips-Heffron model of a power system with grid-connected wind power induction generators is established in Sect. 3.2.

3.2.1

Phillips-Heffron Model of a Power System with Wind Power Induction Generators as a Generic Open-Loop Controller

The network equation of a multi-machine power system before the connection of a WPIG can be expressed as 2

2 Y11 6 7 6 4 0 5 ¼ 4 Y21 31 Ig Y 0

3

12 Y 22 Y

1 3 13 32 V Y 6 7 23 7 Y 54 V 25

32 Y

33 Y

ð3:1Þ

g V

g and Ig is the vector of terminal voltage and current associated with SGs, where V  is the system terminal 1 and 2 is the potential location for installing the WPIG, and Y admittance matrix. After the WPIG is connected, according to Fig. 3.1, the network can be written as

V1 X1L

Node 1

Multimachine Power System

XwL

I1L Node L I2L Node 2

WPIG Iw

VL

Vw

X2L

V2

Fig. 3.1 Diagram of a multi-machine power system connected with a WPIG

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

66

8" # " 0 0 Y > 11 > > ¼ > > < 0 0 > >  > >   > : Ig ¼ Y31

#" # " # " # 13 1 I1L 0 V Y g V þ þ 0 23 2 I2L Y V Y 22 " # 1  V 32 g 33 V Y þY 2 V

ð3:2Þ

0 and Y 0 exclude X12 ¼ X1L þ X2L. where Y 11 22 " # " #1 " # " # ! I1L 1 jXwL jðX1L þ XwL Þ 1 V w , it Since ¼  V I2L 2 jXwL jðX2L þ XwL Þ 1 V obtains " # " # ! " # 13 1 1 V Y 1 g L  w  Y V V ¼Y ð3:3Þ w 23 2 1 V Y where " w ¼ I þ Y

#"

jðX1L þ XwL Þ

jXwL

jXwL

jðX2L þ XwL Þ

 L ¼ jðX1L þ XwL Þ Y jXwL

0 Y 11

0 0 Y

0

# ,

22

 jXwL : jðX2L þ XwL Þ

By substituting (3.3) into the second equation of (3.2), it should have  31 Ig ¼ Y

    1 1   33  Y  31  Y V þ Y Y32 w w 1

    1 Y13 g    V Y Y Y32 w L 23 Y

ð3:4Þ

  Y13 ΔVg Y23

ð3:5Þ

By linearizing (3.3), it gives 

ΔV1 ΔV2

where ΔV1 ¼ ½ ΔV1x 1 0 1 0

 ¼

Y1 w

ΔV1y T



 YI ΔVw  YL

and same form applies to other variables.

T

1 and Y L YI ¼ 0 1 0 1 , Y1 and YL are the expanded form of Y w w respectively. Equation (3.4) can be also linearized to be ΔIg ¼ ½ Y31

 33  ½ Y31 Y32 Y1 Y ΔV þ Y I w w

Y32 Y1 w YL



Y13 Y23

 ΔVg ð3:6Þ

3.2 System Modeling for Damping Torque Analysis of a Power System. . .

67

 L ¼ V w  jXwL Iw , it can have As  Iw ¼  I1L þ I2L and V     2 V1 w þ Y Iw ¼ Y 1 V 2 V where

1 ¼  Y

jðX1L þ X2L Þ X1L X2L þ X2L XwL þ XwL X1L

j½ X2L X1L  . (3.7) can be linearized to be X1L X2L þ X2L XwL þ XwL X1L   ΔV1 ΔIw ¼ Y1 ΔVw þ Y2 ΔV2   ΔV1 By substituting (3.5) into (3.8) and eliminating , it gives ΔV2

ð3:7Þ and

2 ¼ Y

   Y13 1 Y  Y Y Y ΔIw ¼ Y1 þ Y2 Y1 ΔVg ΔV I w 2 L w w Y23

ð3:8Þ

ð3:9Þ

As the standard algebraic interface equations of a WPIG can be written as T ΔIw ¼ CwΔXw + DwΔVw, where ΔXw ¼ ½ Δs ΔEd ΔEq  , by eliminating ΔIw, (3.9) becomes      1 Y13 1 Y C ΔX þ Y Y Y ΔVw ¼ Y1  Dw þ Y2 Y1 ΔV I w w 2 w L g w Y23

ð3:10Þ

By substituting (3.10) into (3.6), ΔVw is eliminated  1 ΔIg ¼ ½ Y31 Y32 Y1 YI Y1  Dw þ Y2 Y1 YI Cw ΔXw þ w w 0 1 33  ½ Y31 Y32 Y1 YL Y13 þ ½ Y31 Y32 Y1 YI Y w w B C Y23 B CΔVg   @ A  Y13 1 1 Y1  Dw þ Y2 Y1 Y Y Y Y I 2 w L w Y23

ð3:11Þ

Then the reference frame transformation from x-y to d-q is applied to (3.11) by g ¼ E0 introducing Δδ. After substituting the linearized form of the SG equation V q  jX0d Ig  j Xq  X0d Iq into (3.11) under d-q frame, ΔVg in (3.11) is eliminated and (3.11) should have the form ΔIg ¼ Rδ Δδ þ RE0q ΔE0q þ RIq ΔIq þ RXw ΔXw

ð3:12Þ

Hence, ΔIg is converted to ΔId and ΔIq and (3.12) becomes ΔId ¼ Rδd Δδ þ RE0q d ΔE0q þ RIq d ΔIq þ RXw d ΔXw ΔIq ¼ Rδq Δδ þ RE0q q ΔE0q þ RIq q ΔIq þ RXw q ΔXw

ð3:13Þ

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

68

Phillips-Heffron model of a multi-machine power system [34] is Δδ_ ¼ ω0 Δω Δω_ ¼ M1 ðΔTE  DΔωÞ ΔE_0q ¼ T1 d0 ðΔEQ þ ΔE fd Þ  ΔE_0 ¼ ΔE  K ΔV T1 fd

fd

A

g

ð3:14Þ

A

where   ΔTE ¼ ΔIq E0q0 þ Iq0 ΔE0q þ ΔIq Xq  X0d Id0 þ Iq0 Xq  X0d ΔId ,  ΔEQ ¼ ΔE0q  Xd  X0d ΔId and ΔVgd ¼ XqΔIq, ΔVgq ¼ ΔE0q  X0d ΔId . By substituting (3.13) into (3.14), Phillips-Heffron model of a multi-machine power system with the interface equations of a WPIG can be derived. The above derivations can be extended to accommodate the case of multiple WPIGs. To facilitate the understanding, a typical example of a single machine infinite bus (SMIB) system with a WPIG is presented in Appendix 3.1. Therefore, on this basis, the general form of the explicit Phillips-Heffron linearized model of a multi-machine power system considering the algebraic interface equations of WPIGs can be established 32 3 2 _ 3 2 Δδ Δδ 0 ω0 I 0 0 7 6 76 7 6 6 Δω_ 7 6 M1 K1 M1 D M1 K2 0 76 Δω 7 7¼6 76 7 6 76 ΔE0 7þ 6 ΔE_0 7 6 T1 K -1 K 0 Td0 T1 q 5 3 54 4 4 q 5 d0 4 d0 T1 0 T1 TA 1 ΔE0fd ΔE_0fd A KA K5 A KA K6 2 3 2 3 2 3 0 0 0 6 M1 K 7 6 M1 K 7 6 M1 K 7 6 6 6 ω1 7 ω2 7 ω3 7 6 7 6 7 6 7 6 T-1 KE0 1 7Δs þ 6 T1 KE0 2 7ΔEd þ 6 T1 KE0 3 7ΔEq d0 d0 d0 4 5 4 5 4 5 q q q 1 1 1 T K K T K K T K K A

A

E fd 1

A

A

E fd 2

A

A

E fd 3

ð3:15Þ where state variables (Δδ, Δω, ΔE0q and ΔEfd) and matrix elements (ω0, M, D, K1~K6, Td0, KA and TA) of SGs are defined in Chapter 3.1 of [34], Δs, ΔEd and ΔEq is the vector of variation of slip and direct/quadrant-axis electromotive force of WPIGs, and the rest elements (Kω1, Kω2, Kω3, KE0q 1 , KE0q 2 , KE0q 3 , KE fd 1 , KE fd 2 and KE fd 3 ) can be determined by the above derivations. According to (3.15), it can be noted that: (1) The linearized model presented in (3.15) is an open-loop system with Δs, ΔEd and ΔEq as its control variables, since the internal dynamics of WPIGs is not included. Hence, (3.15) is also named systemside linearized model; (2) Only the state variables of induction generator (Δs, ΔEd

3.2 System Modeling for Damping Torque Analysis of a Power System. . .

69

K1

⎡ Δs ⎤ ⎥ ⎢ ⎢Δ Ed ⎥ ⎢Δ Eq ⎥ ⎦ ⎣

⎡ Kω 1⎤ ⎢K ⎥ ⎢ ω 2⎥ ⎢⎣ Kω 3⎥⎦

T



( pM + D)−1

Δω

Δδ

ω 0I p

K2 ΔE

' q

K4



∑ ΔE

( pTd 0 + K3 )

−1



( pTA + I )

−1

fd

KA

K5



⎡ K fd 1⎤ ⎢ ⎥ ⎢K fd 2⎥ ⎢K fd 3⎥ ⎣ ⎦

K6

T

⎡ Δs ⎤ ⎢ ⎥ ⎢Δ Ed ⎥ ⎢Δ Eq ⎥ ⎣ ⎦

[KE' 1, KE' 2, KE' 3 ] q

q

q

[Δs, ΔE , ΔE ]

T

d

q

Fig. 3.2 System-side linearized model diagram of a power system integrated with WPIGs Fig. 3.3 Representation of WPIG internal dynamics in frequency domain (WPIGside linearized model)

Δs ΔVw

GWPIG (p)

ΔEd ΔEq

and ΔEq) have a direct impact on the system damping and other state variables (e.g., state variables of DFIG converter controllers) affect system via Δs, ΔEd and ΔEq. For FSIG, Δs does not directly contribute to the system damping either since FSIG rotor is a closed circuit and thus physically separate from the grid. However, to keep a consistent form for the demonstration of WPIGs, Δs can be retained in (3.15) but with Kω1 ¼ KE0q 1 ¼ KE fd 1 ¼ 0. The linearized model in (3.15) is illustrated by Fig. 3.2 in frequency domain and p is the frequency domain operator.

3.2.2

Aggregate Model and Transfer Function of Wind Power Induction Generators

The internal dynamics of WPIGs includes dynamics of induction generators and converter controllers (if DFIG), which can be described by a set of first-order differential equations. In frequency domain, these equations can be converted and presented in the form of a SIMO controller as shown in Fig. 3.3, which is explained in details in Sect. 3.4. The input of the controller (ΔVw) is terminal voltage

70

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

associated with WPIG bus, and the outputs of the controller (Δs, ΔEd and ΔEq) are three state variables of WPIGs as mentioned in (3.15). Without losing generality, the transfer function is written in a general format as GWPIG ðpÞ ¼  T Gs ðpÞ GEd ðpÞ GEq ðpÞ , which can represent any type of WPIG. The full representation of GWPIG(p) is derived in Sect. 3.4 based on the internal dynamics of different types of WPIG. Figures 3.2 and 3.3 form a closed-loop linearized model with a clear physical insight for damping torque analysis of a power system affected by grid-connected WPIGs. If any system disturbance happens (represented by ΔVw), there should be a dynamic response from WPIG (reflected by (Δs, ΔEd and ΔEq). ThenΔs, ΔEd and ΔEq will in turn impact SGs and hence the system according to Fig. 3.2. It can be seen that the internal dynamics (i.e., GWPIG(p)) of WPIGs determines their dynamic response and plays a critical role in the dynamic interaction. Therefore, different internal dynamics is actually considered to be the major cause for different types of WPIG to have different damping mechanisms, which is carefully compared and investigated in this chapter.

3.3

Implementation Framework for Damping Torque Analysis of a Power System Affected by GridConnected Wind Power Induction Generators

Based on the models given in Figs. 3.2 and 3.3, a generic implementation framework of damping torque analysis to evaluate the damping torque contributions from different internal dynamic components of WPIGs and their impact on the system critical oscillation mode is established as follows. The forward path from Δs, ΔEd and ΔEq to the electromechanical oscillation loop of SGs can be obtained from Fig. 3.2 8 h i > Fws ðpÞ ¼ KF ðpÞ ðpTA þ IÞ1 KA K fd1 þ KE0q1  Kω1 > > > > < h i ð3:16Þ FwEd ðpÞ ¼ KF ðpÞ ðpTA þ IÞ1 KA K fd2 þ KE0q2  Kω2 > > > h i > > : FwE ðpÞ ¼ KF ðpÞ ðpTA þ IÞ1 KA K fd3 þ KE0  Kω3 q q3 where KF(p) ¼ K2[(pTd0 + K3) þ (pTA + I)1 + KAK6]1, and Fws(p), FwEd ðpÞ and FwEq ðpÞ are three m  1 matrices, assuming there are totally m SGs and l WPIGs in the system. Hence, the electric torque provided by the main dynamic components of WPIGs to electromechanical oscillation loop of SGs is 8 ΔTws ¼ Fws ðpÞGs ðpÞΔVw > < ΔTwEd ¼ FwEd ðpÞGEd ðpÞΔVw ð3:17Þ > : ΔTwEq ¼ FwEq ðpÞGEq ðpÞΔVw

3.3 Implementation Framework for Damping Torque Analysis of a Power. . .

71

where ΔTws, ΔTwEd and ΔTwEq include the electric torque contribution of WPIGs to all SGs and thus are m-dimention vectors, and ΔTw ¼ ΔTws þ ΔTwEd þ ΔTwEq . It can be revealed that all three dynamic components have their independent system channels to contribute to the system damping torque, which together forms the total damping impact ΔTw of WPIGs. Assume that the ith eigenvalue λi is the critical oscillation mode in the system. According to the algebraic equations of the linearized model of a multi-machine power system with WPIGs, it can obtain ΔVw ¼ CVw Xg ΔXg

ð3:18Þ

where ΔXg is the vector of state variables associated with SGs. If λi and vi is the ith eigenvalue and associated right eigenvector of state matrix in (3.15), it can have ΔXg ¼

n n X X vig ai vik ai , Δωk ¼ p  λ p  λi i i¼1 i¼1

ð3:19Þ

where vig is the vector inside vi corresponding to ΔXg, and vik is the element of vi corresponding to Δωk (angular speed of the kth SG). Based on (3.18) and (3.19), the relationship between ΔVw and Δωk can be derived. !

X n n X vig ai vik ai ΔVw ¼ CVw Xg ð3:20Þ Δωk ¼ γik Δωk p  λi p  λi i¼1 i¼1 Equation (3.17) can be further factorized to torque contribution of dynamic components of each WPIG to each SG, and hence the electric torque provided by different dynamics of the jth WPIG to the kth SG can be written as 8 ¼ Fwskj ðλi ÞGsj ðλi Þγijk Δωk ΔT > < wskj ΔTwEd kj ¼ FwEd kj ðλi ÞGEd j ðλi Þγijk Δωk ð3:21Þ > : ΔTwEq kj ¼ FwEq kj ðλi ÞGEq j ðλi Þγijk Δωk where subscript k and j denote the kth row and jth column element of corresponding matrices for Fwskj(λi), FwEd kj ðλi Þ and FwEq kj ðλi Þ, and the jth row of corresponding matrices for Gsj(λi), GEd j ðλi Þ, GEq j ðλi Þ and γijk. As the electric torque contribution from the jth WPIG to the kth SG is the linear superposition of each main dynamic component, ΔTwkj ¼ ΔTwskj þ ΔTwEd kj þ ΔTwEq kj . Similarly, considering the conl X ΔTwkj , tributions from all the WPIGs, the electric torque of the kth SG ΔTwk ¼ j¼1

which is the kth element of ΔTw.

72

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

The sensitivity of λi with respect to the electric torque coefficient of the kth SG can be computed to be Sik ¼

∂λi ¼ wik vik ∂TCwk

ð3:22Þ

where wik is the element of λi associated left eigenvector wi corresponding to Δωk. Thus, the variation of the ith eigenvalue λi caused by insertion of WPIG dynamics can be assessed by employing Sik Δλi ¼

m X

Sik TCwk ¼

k¼1

¼

m X k¼1

Sik

l  X

m l X X Sik TCwkj k¼1

j¼1

TCwskj þ TCwEd kj þ TCwEq kj



ð3:23Þ

j¼1

where TCwk and TCwkj are the electric torque coefficients of ΔTwk and ΔTwkj, and TCwskj, TCwEd kj and TCwEq kj are the coefficients of each main dynamic component in (3.21) respectively. On the basis of the derivations above, the generic implementation framework for damping torque analysis of multi-machine power system with different types of WPIGs is established, which is based in frequency domain but capable of providing deeper understandings about damping mechanisms than modal analysis. It can be seen that if closed-form solution of the WPIG transfer function is given, by substituting (3.16) into (3.21) and (3.23), both the damping torque contributions of WPIG dynamics components and eigenvalue variation should have an explicit expression, so that the damping mechanism of WPIGs can be easily examined and revealed. Therefore, a comprehensive comparison analysis is carried out in Sect. 3.4 based on the proposed framework.

3.4

Analytical Comparison on Damping Effectiveness And Robustness of Type 1 and 3 Wind Power Induction Generators

In the presented implementation framework of damping torque analysis, the external damping contribution channels for different types of WPIGs are quite similar as shown in (3.16) and (3.17), and hence their transfer functions (internal contribution channels) become a crucial part of comparison analysis, which are derived and investigated in this section. To facilitate the comparison and understanding of their damping mechanisms, a five-step model transformation from DFIG to FSIG are employed, in which FSIG is treated as a special case of DFIG. In addition to DFIG and FSIG, three transitional wound rotor generator models are proposed for comparison purposes and the corresponding linearized models are established.

3.4 Analytical Comparison on Damping Effectiveness And Robustness of Type. . .

3.4.1

73

Step 1: Detailed DFIG Model

Since the grid-side converter (GSC) controller of DFIG does not really affect the damping of system oscillation [10, 34, 35], its dynamics is ignored in this study. The linearized model of DFIGs considering the dynamics of both induction generator and rotor-side converter (RSC) controller is established 3 2 Δs_ 7 6 6 ΔE_ d 7 7 6 7 6 6 ΔE_ q 7 7 6 6 _ 7 6 ΔX ps1 7 7 6 6 ΔX_ 7 ps2 7 6 7 6 6 ΔX_ 7 qs1 5 4 ΔX_ qs2 3 2 M1 M1 0 0 0 0 0 w Dw w K1w 7 6 6 0 K3w K4w 0 0 7 0 K2w 7 6 7 6 6 0 0 K6w K7w 7 0 0 K5w 7 6 7 6 ¼6 0 KpsI1 K8w 0 0 0 0 0 7 7 6 6 0 KpsI2 K10w 0 0 0 7 0 KpsI2 K9w 7 6 7 6 6 0 0 0 0 7 0 0 KqsI1 K11w 5 4 2

0

3

2

0

KqsI2 K12w 3

0

0

KqsI2 K13w

0

M1 w Ks

Δs 7 6 7 6 6 ΔEd 7 6 KEd 7 7 6 7 6 7 6 7 6 6 ΔEq 7 6 KEq 7 7 6 7 6 7 6 7 6  6 ΔXps1 7 þ 6 KpsI1 KXps1 7ΔVw 7 6 7 6 6 ΔX 7 6 K K 7 ps2 7 6 6 psI2 Xps2 7 7 6 7 6 6 ΔX 7 6 K K 7 qs1 5 4 4 qsI1 Xqs1 5 ΔXqs2 KqsI2 KXqs2

ð3:24Þ where Mw is the inertia time constant, Dw is the damping coefficient, Xps1, Xps2, Xqs1 and Xqs2 are the state variables of RSC Integral Controllers illustrated in Fig. 3.4, and KpsI1, KpsI2, KqsI1 and KqsI2 are relevant parameters of Integral Controllers. The explicit form of matrix elements (K1w~K13w, Ks, KEd , KEq , KXps1 , KXps2 , KXqs1 and KXqs2 ) in (3.24) is given below

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

74

Irq

Ps ref s

P

+



Kpsp1



KpsI1 p

Qsref

+

Kqsp1



KqsI1 p



Xqs1

I sqref

+



Xss Xm



+



I rdref +



X ps1

I sdref Xss − + Xm

+





Vw Xm

Qs

I rqref



⎛ X2 ⎞ X V − s ⎜⎜ X rr − m ⎟⎟ I rd + s m w − rrIrq X X ss ss ⎠ ⎝

Kpsp2 K psI2 p

Ird



Vrq

+



Vrd

Xps2

Kqsp2 KqsI2 p

+

+

Xqs2

+

⎛ X2 ⎞ s ⎜⎜ Xrr − m ⎟⎟ I rq − rrIrd X ss ⎠ ⎝

Fig. 3.4 Configuration diagram of RSC control system model of DFIGs

Xrr Eq0 Kpsp2 Xss  , K2w ¼ I þ Kpsp1 jVw0 j , 2  Xss Xrr Xss Xrr  Xm Kpsp2 Xss Kqsp2 Xss  Xm ¼ , K4w ¼  , K5w ¼ I þ Kqsp1 jVw0 j , 2 Xrr Xrr Xss Xrr  Xm

K1w ¼ Isd0 þ K3w

X2m

K6w ¼  K9w ¼  K11w ¼

X2m

Kqsp2 Xss Xm Xrr jVw0 j , K7w ¼ , K8w ¼ , Xrr Xrr Xss Xrr  X2m  Xss Xrr Xss I þ Kpsp1 jVw0 j , K10w ¼  , Xm  Xss Xrr Xm

 Xrr jVw0 j Xss Xrr , K12w ¼  I þ Kqsp1 jVw0 j , 2  Xss Xrr Xss Xrr  Xm Xm

X2m

K13w ¼ 

Xss Xss Ed0 ½ Vwx0 Vwy0  , Ks ¼  2 , Xm Xm  Xss Xrr jVw0 j

Kpsp1 Kpsp2 Xss Isq0 ½ Vwx0 Vwy0  , Xrr jVw0 j    Xrr jVw0 j Xrr jVw0 j Kqsp2 Xss  þ Kqsp1 Isd0  ½ Vwx0 Xss Xrr  X2m Xss Xrr  X2m ¼ Xrr jVw0 j KEd ¼ 

KEq

Isq0 ½ Vwx0 Vwy0  Kpsp1 Xss Isq0 ½ Vwx0 Vwy0  , KXps2 ¼ , Xm jVw0 j jVw0 j   Isd0 Xrr þ ½ Vwx0 Vwy0 , KXqs1 ¼  jVw0 j Xss Xrr  X2m     Xss Xrr jVw0 j Xrr Kqsp1 Isd0  ¼ ½ Vwx0 Vwy0 :  Xm jVw0 j Xss Xrr  X2m Xss Xrr  X2m KXps1 ¼ 

KXqs2

Vwy0 

3.4 Analytical Comparison on Damping Effectiveness And Robustness of Type. . .

75

where the subscript 0 denotes the steady-state value of variables, d-q denotes the WPIG reference frame and x-y denotes the system reference frame. As the study focus is on the induction generator and RSC basic control, there is no additional damping control imposed to the RSC controller. The linearized model of RSC control can also be illustrated in frequency domain as shown in Fig. 3.4. According to Fig. 3.4, the transfer functions mentioned in previous section can be extended to their full explicit representation as presented in (3.25). 8  391 8 2 > KpsI2 K10w KpsI1 K8w > > > > > > þ K K K > 4w psI2 9w < > 7= K K K > p 3w psI1 8w 1 6 > 6 7 > þ G ð p Þ ¼ I  ð pI  K Þ 2w > Ed 4 5> > > p p > > > > : ; > > > >   2 3 > > K K K K psI2 10w psI1 Xps1 > > þ KpsI2 KXps2 K4w > > 6 7 K3w KpsI1 KXps1 > p 1 > 6 7 > þ  ð pI  K Þ K þ 2w E > d 4 5 > p p > > < 391 8 2 KqsI2 K13w KqsI1 K11w > > < = K þ K K > 7w qsI2 12w K K K p > 6w qsI1 11w > 5 > þ GEq ðpÞ ¼ I  ðpI  K5w Þ1 4 > > : ; p p > > >  3 > 2 > > KqsI2 K13w KqsI1 KXqs1 > > þ KqsI2 KXqs2 K7w > > 6 7 K K K p > 6w qsI1 X qs1 1 > 6 7 > þ  ð pI  K Þ K þ 5w > 4 Eq 5 > p p > > > > > > : Gs ðpÞ ¼ ðpMw þ Dw Þ1 ½Ks þ K1w GEd ðpÞ

ð3:25Þ It can be noted from Fig. 3.5 that: (1) The damping contributions of Δs, ΔEd and ΔEq to the system mainly consist of two parts, i.e., the dynamics of induction generator and RSC Integral Controller, which can be easily differentiated and split. In (3.25), the induction generator dynamics part of transfer functions includes the items associated with KEd and KEq , and the RSC controller dynamics part of transfer functions is associated with RSC controller parameter KpsI1, KpsI2, KqsI1 and KqsI2. That is to say, by setting these RSC controller parameters to zero, the damping effect of RSC dynamics is removed; (2) ΔEd and ΔEq have completely separate internal damping contribution channels due to the adoption of decoupled power control of DFIG. ΔEd is related to the P-Control and ΔEq is related to the Q-Control; (3) The damping contribution of Δs is only affected by its own dynamics and ΔEd as the rotor motion is mainly determined by the active power control.

3.4.2

Step 2: DFIG Model Without RSC Dynamics

Normally, the damping effect of RSC Integral Controllers (i.e., converter-side dynamics) is limited compared with the RSC Proportional Controllers which contribute to the induction generator-side dynamics. In Step 2, converter-side dynamics

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

76

Induction Generator-side Dynamics

ΔVw Ks

( pM w + Dw)−1



Converter-side Dynamics

Δs

Rotor Spin Dynamics

ΔX qs1

K6w

K1w

KqsI2K13w

Δ Eq

( pI

− K 5 w)

−1

ΔVw



K7w

ΔXqs2

I p

∑ KqsI2KXqs2

KEq

ΔX ps1

K3w



I p

(pI − K 2 w )−1



K4w

ΔXps2

I p



K Ed

KpsI2KX ps2

ΔVw

ΔVw

ΔVw

KqsI1K11w

KqsI2K12w

ΔVw

Q-Control



I p

KpsI1KX ps1

ΔVw

KpsI1K8w

KpsI2K10w

ΔE d

KqsI1KXqs1

K psI2K9w

P-Control

Fig. 3.5 WPIG-side linearized model diagram (internal dynamics of DFIG)

is not taken into account for the damping analysis and thus the first transitional generator model is obtained. Hence, (3.24) is reduced to three first-order equations associated with Δs, ΔEd and ΔEq only. 2 3 2 32 3 2 1 3 Δs_ Δs M w Ks M1 M1 0 w Dw w K1w 7 6 _ 7 6 76 7 6 6 0 54 ΔEd 5 þ 4 KEd 7 0 K2w 4 ΔE d 5 ¼ 4 5ΔVw ð3:26Þ _ ΔE q ΔEq KE q 0 0 K5w The diagram of the linearized model in (3.26) only includes the left part of Fig. 3.5 (induction generator-side dynamics). However, the explicit form and value of elements in (3.26) remain the same as in (3.24). As discussed in Step 1, the transfer functions of this transitional generator model should become 8 G ðpÞ ¼ ðpI  K2w Þ1 KEd > > < Ed ð3:27Þ GEq ðpÞ ¼ ðpI  K5w Þ1 KEq > > : Gs ðpÞ ¼ ðpMw  Dw Þ1 ½Ks þ K1w GEd ðpÞ Equation (3.27) can be also simply obtained by setting KpsI1 ¼ KpsI2 ¼ KqsI1 ¼ KqsI2 ¼ 0 in (3.25). It can be revealed from Fig. 3.5 that although dynamics model (Integral Controllers) of RSC controller is removed and only the RSC algebraic model (Proportional

3.4 Analytical Comparison on Damping Effectiveness And Robustness of Type. . .

77

Controllers) is retained in this step, the basic configuration of linearized model and damping contribution channels is not changed. In other words, the decoupled structure of damping contribution of ΔEd and ΔEq is not determined by the controller dynamics of RSC but actually by offset items of rotor voltage (or named offset rotor voltage) in RSC controller, which is intensively studied in the next step.

3.4.3

Step 3: DFIG Model with Offset Rotor Voltage Only

On the basis of Step 2, by setting all the parameters of PI controllers of RSC to zero, the second transitional generator model is established. Due to the only existence of offset rotor voltage in RSC, both the rotor dynamic equations associated with ΔEd and ΔEq are eliminated ( KEd ¼ KEq ¼ K2w ¼ K5w ¼ 0 ), and only rotor spin dynamics is retained in this linearized model. Equation (3.26) becomes 1 Δs_ ¼ M1 w Dw Δs þ Mw Ks ΔVw

ð3:28Þ

The diagram of the linearized model in (3.28) only includes the left upper corner of Fig. 3.5 (rotor spin dynamics). Based on the diagram, the transfer function of the second transitional generator model is derived to be Gs ðpÞ ¼ ðpMw þ Dw Þ1 Ks

ð3:29Þ

It can be seen from (3.28) that the elimination effect of the offset rotor voltage has removed the ‘original’ dynamics of induction generator associated with ΔEd and ΔEq and the ‘new’ generator-side dynamics of DFIG presented in Steps 1 and 2 are actually brought by the RSC algebraic model (Proportional Controllers) in a decoupled manner. This finding can be also proved by the explicit elements of (3.24) in Sect. 3.4.1. The key elements of generator-side dynamics K2w and KEd are mainly affected by the RSC P-Control algebraic model and K5w and KEq are mainly affected by the RSC Q-Control algebraic model, after the introduction of offset rotor voltage. In contrast, Ks and K1w are determined by the induction generator parameter as well as the steady-state operating status, which are not changed by the introduction of either RSC controller or offset rotor voltage.

3.4.4

Step 4: DFIG Model with Constant Rotor Voltage

Similar to Step 3, the effect of RSC PI controllers is ignored and rotor circuit is still wound in Step 4. The third transitional wound rotor generator model with constant rotor voltage output (sometimes also called open-loop control of RSC converter) is proposed in this step. As the rotor voltage output remains constant, the offset items of rotor voltage are removed and hence the ‘original’ generator-side dynamics is

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

78

reflected. As a result, the configuration of the linearized model is changed to Fig. 3.6 and the linearized equations become 2 3 2 32 3 2 1 3 Δs_ Δs Mw K s M1 M1 0 w Dw w K1w 7 6 _ 7 6 76 7 6 7 K3w K4w 54 ΔEd 5 þ 6 4 ΔE d 5 ¼ 4 K2w 4 0 5ΔVw ð3:30Þ ΔE_ q ΔEq KEq K5w K6w K7w where the elements of state matrix have been renumbered due to the change of state matrix configuration in this case and their explicit forms are given below K1w ¼ Isd0 þ

Xrr Eq0 Rr Xss , K2w ¼ Eq0 , K3w ¼ K7w ¼ , X2m  Xss Xrr X2m  Xss Xrr

Xss Ed0 ½ Vwx0 Vwy0  K4w ¼ s0 , K5w ¼ Ed0 , K6w ¼ s0 , Ks ¼  2 , Xm  Xss Xrr jVw0 j Rr X2 ½ Vwx0 Vwy0  KEq ¼  2 m Xm  Xss Xrr jVw0 j where the subscript 0 denotes the steady-state value of variables, d-q denotes the WPIG reference frame and x-y denotes the system reference frame. Compared with the explicit elements of (3.24) in Sect. 3.4.1, most elements in (3.30) are changed and determined by steady-state value of variables (e.g., K2w, K4w, K5w and K6w). Ks and K1w remains the same as indicated previously. It can be demonstrated from Fig. 3.6 that the ‘original’ induction generator-side dynamics has three significant features: (1) re-coupling of ΔEd and ΔEq represented by K4w and K6w; (2) Feedback support from Δs to ΔEd and ΔEq represented by and K5w; (3) The removal of KEd owing to DFIG d-q reference frame setting. Therefore,

ΔVw

Ks

( pMw + Dw)−1 Δ s



K 5w

K1w

Δ Eq

( pI

− K 7 w)

−1

K 4w



KEq

ΔVw

K2w

K6w

Δ Ed

( pI − K 3 w)−1



Fig. 3.6 WPIG-side linearized model diagram (internal dynamics of DFIG with constant rotor voltage or FSIG)

3.4 Analytical Comparison on Damping Effectiveness And Robustness of Type. . .

79

the damping mechanism of the third transitional generator model has been dramatically changed, which is also reflected by the transfer functions in (3.31), where all three transfer functions include Ks and KEq , and hence are coupled with each other. 8 91 8 < = > ðpI  K3w Þ1  K2w ðpMw þ Dw Þ1 K1w  > > > h i GEd ðpÞ ¼ > > 1 1 > : K4w ðpI  K7w Þ  K5w ðpMw þ Dw Þ K1w þ K6w ; > > > < n h io  K2w ðpMw þ Dw Þ1 Ks þ K4w ðpI þ K7w Þ1  K5w ðpMw þ Dw Þ1 Ks þ KEq > > > > > > G ðpÞ ¼ ðpM þ D Þ1 ½K þ K G ðpÞ > s w w s 1w Ed > > >  : 1  GEq ðpÞ ¼ ðpI  K7w Þ K5w Gs ðpÞ þ K6w GEd ðpÞ þ KEq ð3:31Þ

3.4.5

Step 5: FSIG Model

In particular condition, if constant rotor voltage in Step 4 is equal to zero, the third transitional wound rotor generator model becomes FSIG with the rotor circuit closed. As a result, the same linearized model ((3.30) and Fig. 3.6), transfer functions (3.31) and damping mechanisms can be applied to FSIG. However, it is worthy to mention that Δs of FSIG does not have a direct impact on the system damping as stated in Sect. 3.2.1 due to the closed physical structure of rotor, although the damping contribution from Δs is normally very small.

3.4.6

Main Findings in Analytical Comparison on Damping Mechanism of Type 1 and 3 Wind Power Induction Generators

By implementing the step-by-step model transformation analysis demonstrated above, essential difference and inner connection between DFIG and FSIG in their damping mechanisms of oscillation stability have been clearly revealed. The main points from the comparison analysis are summarized as follows: 1. The damping contributions of DFIG dynamics are essentially from the RSC control, which can be divided into two parts, i.e., the ‘new’ induction generator dynamics (related to RSC Proportional Controllers) and RSC controller dynamics (related to RSC Integral Controllers). The former represented by the RSC algebraic model normally accounts for the major part of the total damping contributions of DFIG.

80

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

2. The existence of the offset rotor voltage in RSC not only enables the decoupled PQ control of DFIG, but also establishes the decoupled structure of internal damping contribution channels of DFIG. 3. FSIG featured by the ‘original’ induction generator dynamics can be treated as a special case of DFIG model only when the open-loop control of RSC is applied. In this case, the damping contributions of FSIG only comes from the ‘original’ generator dynamics and are mainly determined by generator parameters and changeable steady-state value of variables as proved by the explicit elements of (3.30) in Sect. 3.4.4. The damping contributions channels of internal dynamics are coupled with each other. Therefore, based on the main points from the comparison of damping mechanisms of DFIG and FSIG, it can be rigorously concluded that the damping effect of DFIG is more robust since it is mainly determined by generator parameters and RSC controller parameters as proved by the explicit elements of (3.24) in Sect. 3.4.1. On the contrary, the damping impact of FSIG is comparatively limited and less controllable especially in the changing conditions due to the characteristic of ‘original’ induction generator dynamics. These significant empirical conclusions regarding damping effectiveness and robustness of the two typical WPIGs have been rigorously proved in an analytical manner. By substituting the detailed format of transfer functions of DFIG and FSIG derived in (3.25, 3.27, 3.29, 3.31) into (3.21) and (3.23), the damping torque analysis can be carried out, the numerical comparison of which is designed and presented in the next section.

3.5

Numerical Comparison

To demonstrate the proposed implementation framework and validate the findings from the analytical comparison, 16-machine and 68-node NYPS-NETS power system is employed and shown in Fig. 3.7. The models and parameters of the test system are given in Appendix 3.2. For demonstration purposes, a WPIG-based wind farm is planned to be connected to node 15 of the system. The connecting location could be any other nodes in the system, which does not affect the demonstration and validation. A typical set of WPIG parameters is provided in Appendix 3.2.

3.5.1

Example 3.1 (Base Case Comparison Study)

There are totally four inter-area oscillation modes in this test system and the selection of critical mode would not actually affect the validation of the proposed method and general results from the analytical comparison. Hence, the 31st eigenvalue λ31 is selected to be the system critical inter-area oscillation mode in the case study here for

3.5 Numerical Comparison

81

NYPS 14

8

A3

66 41

1

40

25

26

53

47

48

60

28

2

29

61

27

18

9

1

42

17

38

67

31

32

15

30

62

46

A4

63

36

6 7

54

22

12

35

A5

64 8

45

50

12

20

44

16

10

57

56

55 4

3

23

6

59 7

5

37

39

52 68

2

58

19

11 13

51

2

21

14

5

34

24 69

4

33 49

15

9

11

10

16

3

43

65 13

A2

A1

NETS

Fig. 3.7 Diagram of 16-machine 68-bus NYPS-NETS test system integrated with a WPIG

demonstration purposes. Before the internal dynamics of the WPIG is taken into account, the initial value of this critical inter-area oscillation mode is ð0Þ λ31 ¼ 0:1558 þ j3:3940, which is calculated from state matrix of the open-loop system presented in (3.15). Two WPIG models (i.e., detailed DFIG model and FSIG model) are demonstrated here. By adopting (3.16), the forward paths from the main dynamic components (Δs, ΔEd and ΔEq) of the DFIG or FSIG to the electromechanical oscillation loop of SGs are obtained and compared in Table 3.1. It can be seen from Table 3.1 that the DFIG has one more external damping contribution channel than FSIG due to the introduction of different algebraic models in (3.16) brought by different rotor structures, but the sum of the weightings of the external damping contribution channels for the DFIG and FSIG are approximately equal as a whole. The internal dynamics (transfer functions) of the DFIG and FSIG is computed by using (3.25) and (3.31) respectively and compared in Table 3.2. Table 3.2 clearly demonstrates that the main difference of damping mechanisms between the DFIG and FSIG actually lies in their internal damping contribution channels especially in GEd ðλ31 Þ and GEq ðλ31 Þ. Also, since the transfer functions of RSC integral controllers are less than 1% of each corresponding transfer function of DFIG, they are not listed in Table 3.2 separately. Then the eigenvalue variation caused by the introduction of the DFIG or FSIG main dynamics components are estimated by (3.23) and shown in Table 3.3. Based on Table 3.3, the estimation of the critical eigenvalue after the insertion of the DFIG or FSIG dynamics can be obtained

82

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

Table 3.1 Comparison on norm of forward paths (external damping contribution channels) of DFIG and FSIG to electromechanical oscillation loop of SGs DFIG model G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16

|Fws(λ31)| 0.0031 0.0034 0.0043 0.0067 0.0041 0.0057 0.0062 0.0038 0.0014 0.0011 0.0020 0.0005 0.0003 0.0001 0.0000 0.0000

jFwEd ðλ31 Þj 0.0104 0.0116 0.0147 0.0228 0.0142 0.0193 0.0209 0.0131 0.0052 0.0036 0.0070 0.0017 0.0010 0.0002 0.0000 0.0001

Table 3.2 Comparison on transfer functions (Internal Damping Contribution Channels) of DFIG and FSIG

Table 3.3 Eigenvalue variation brought by main dynamic components of DFIG and FSIG



FwE ðλ31 Þ q 0.0151 0.0245 0.0283 0.0502 0.0341 0.0376 0.0380 0.0285 0.0298 0.0081 0.0194 0.0046 0.0020 0.0011 0.0004 0.0011

FSIG model jFwEd ðλ31 Þj 0.0117 0.0130 0.0165 0.0256 0.0160 0.0217 0.0234 0.0147 0.0058 0.0041 0.0079 0.0020 0.0012 0.0003 0.0000 0.0001



FwE ðλ31 Þ q 0.0154 0.0248 0.0287 0.0508 0.0345 0.0382 0.0385 0.0289 0.0301 0.0082 0.0196 0.0047 0.0021 0.0011 0.0004 0.0011

DFIG model Gs(λ31) GEd ðλ31 Þ GEq ðλ31 Þ

ΔVWX 0.0037  j0.1560 0.0454  j0.0049 1.1412  j0.1217

ΔVWY 0.0009 – j0.0381 0.0111  j0.0012 0.2786 – j0.0297

FSIG model Gs(λ31) GEd ðλ31 Þ GEq ðλ31 Þ

ΔVWX 0.0074  j0.0072 0.2208 þ j0.0098 0.0907  j0.2901

ΔVWY 0.0018  j0.0018 0.0539 þ j0.0024 0.0221  j0.0708

Δs ΔEd ΔEq Total

ΔλDFIG 31 0.0008 þ j0.0005 0.0005  j0.0007 0.0248  j0.0443 0.0261  j0.0446

ΔλFSIG 31 0 0.0016  j0.0029 0.0067 þ j0.0061 0.0082 þ j0.0032

ð0Þ

ð3:32aÞ

ð0Þ

ð3:32bÞ

DFIG λDFIG ¼ 0:1819 þ j3:3494 31est ¼ λ31 þ Δλ31 FSIG λFSIG ¼ 0:1640 þ j3:3972 31est ¼ λ31 þ Δλ31

3.5 Numerical Comparison

83

To verify the results in (3.32a, 3.32b), the eigenvalue of the closed-loop system including the DFIG or FSIG dynamics is calculated to be λDFIG ¼ 0:1822 þ j3:3496 31

ð3:33aÞ

¼ 0:1640 þ j3:3975 λFSIG 31

ð3:33bÞ

Therefore, by comparing the results of (3.32a, 3.32b) and (3.33a, 3.33b), the implementation framework of damping torque analysis is validated. It can be summarized from the above comparison analysis that: (1) Both DFIG and FSIG play a positive role in damping power system oscillation for this study case owing to the positive damping impact from each dynamic component; (2) Given the same system network and loading condition, the external damping contribution channels of the DFIG and FSIG tend to be roughly equal. The difference in the total damping effectiveness is essentially brought by their different internal damping contribution channels. Time domain simulation is also carried out for verification of the proposed damping torque analysis. A three-phase short-circuit fault is applied to node 1 at 0.2s and cleared at 0.3s. G5 and G15 are the main generators related to the critical oscillation mode. Hence, the power angle difference between G5 and G15 is plotted in Fig. 3.8.

Fig. 3.8 Observation of G5-G15 power angle curve with different types of WPIG models

84

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

Fig. 3.8 shows that the non-linear simulation results align with the eigenvalue estimation from frequency domain analysis in Table 3.3 and (3.32a, 3.32b). The damping contribution from the FSIG is quite limited, which does not make a notable change to the power oscillation curve compared with the case of open-loop system. The damping effect of the DFIG is better although it has a slightly bigger initial oscillating amplitude.

3.5.2

Example 3.2 (Comparison Study Under Different Wind Penetration Conditions)

The damping torque analysis to compare damping effectiveness in a single wind speed condition is demonstrated above. In this section, in order to provide useful guidance to system operator for the real-time operation of WPIGs, the damping robustness of DFIG and FSIG is further assessed and compared under different wind penetration conditions. To simulate the intermittence of the wind power in the real case, the power output of DFIG and FSIG is set in a range from the cut-in power0.2 p. u to the rating power 2.0 p. u. As G5 is the main related synchronous generator for the critical oscillation mode, the total damping torque provided by the dynamic components of DFIG and FSIG to G5 is computed by (3.21) respectively under different wind power levels in Fig. 3.9. Then the real part of critical eigenvalue of the closed-loop system considering DFIG and FSIG dynamics and open-loop system without any WPIG dynamics are calculated and displayed in Fig. 3.10. It can be demonstrated by results in Fig. 3.9 and Fig. 3.10 that: 1. Although the damping of open-loop system (calculated from (3.15)) decreases with the increasing wind power, both DFIG and FSIG dynamics contribute a positive damping torque to the critical oscillation mode in this case. Generally, the damping effectiveness of FSIG is limited when compared with DFIG in the same loading condition. 2. The damping torque contribution of DFIG and FSIG in Fig. 3.9 is in exact proportion to their impact on critical eigenvalue variation in Fig. 3.10 (i.e., the difference between ‘open-loop system’ and two ‘closed-loop systems’). 3. DFIG shows more robustness in damping contribution than FSIG under different wind penetration conditions as it is indicated in Sect. 3.4 that the damping effect of DFIG is mainly determined by generator parameters and RSC control while that of FSIG is affected by generator parameters and changing FSIG operating point. Hence, once the generator is designed, FSIG shows less robustness of damping effect.

3.5 Numerical Comparison

85

Fig. 3.9 Comparison on total damping torque provided by DFIG and FSIG under different wind power output levels

Fig. 3.10 Comparison on the real part of critical eigenvalue of DFIG and FSIG connected systems under different wind power output levels

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

86

3.6

Summary of Analytical and Numerical Comparison Analysis

The main conclusions drawn from the analytical comparison in Sect. 3.4 and validated by the numerical comparison in Sect. 3.5 are summarized in the following: 1. The damping effect of DFIG mainly determined by induction generator parameters and RSC control (‘new’ induction generator dynamics plus RSC controller dynamics) is more robust under different wind conditions and also more effective if the RSC control parameters are properly tuned; 2. The damping effect of FSIG determined by induction generator parameters and changeable FSIG operating point (‘original’ induction generator dynamics) is comparatively limited and less robust in the changing wind conditions; 3. The inner connection between DFIG and FSIG is that DFIG will degenerate to FSIG and lose its advantageous properties in damping mechanism if open control of RSC is applied. The presented work can effectively facilitate the system operator’s understanding of different dynamics of DFIG and FSIG in a complex operational environment.

Appendix 3.1: A Typical Example of a SMIB System with Interface Equations of a WPIG From Fig. 3.11, it can obtain XtL

SG

XLb

VL

Vt

Vb ILb

ItL Iw

XwL

Vw

WPIG

Fig. 3.11 Diagram of a SMIB power system connected with a WPIG

Appendix 3.1: A Typical Example of a SMIB System with Interface. . .

     ILb ¼ ItL þ Iw ¼ ItL þ Vw  VL ¼ ItL þ Vw  Vt þ jXtL ItL jXwL jXwL t ¼ jXtL ItL þ jXLb ILb þ V b V

87

ð3:34Þ ð3:35Þ

Substituting (3.34) into (3.35) gives    jXwL XLb  XLb  b Vt  Vw  V 1þ XtL XLb þ XLb XwL þ XwL XtL XwL XwL    jXwL XLb  XLb  Vt  Vw  1 ¼ 1þ XtL XLb þ XLb XwL þ XwL XtL XwL XwL  L ¼ V w  jXwL Iw , it can have As  Iw ¼  ItL þ ILb and V ItL ¼ 

w þ Y t þ Iw ¼ Y 1 V 2 V

jXtL XtL XLb þ XLb XwL þ XwL XtL

ð3:36Þ

ð3:37Þ

jðXtL þ XLb Þ jXLb 2 ¼  ,Y . XtL XLb þ XLb XwL þ XwL XtL XtL XLb þ XLb XwL þ XwL XtL (3.37) can be linearized to be

1 ¼  where Y

ΔIw ¼ Y1 ΔVw þ Y2 ΔVt

ð3:38Þ

As the standard algebraic linearized model of a WPIG can be written as ΔIw ¼ CwΔXw + DwΔVw, by eliminating ΔIw, (3.38) becomes ΔVw ¼ ðY1  Dw Þ1 Cw ΔXw  ðY1  Dw Þ1 Y2 ΔVt

ð3:39Þ

Substituting (3.39) into the linearized equation of (3.36) to eliminate ΔVw gives ΔItL ¼ R1IB ΔVt þ R2IB ΔXw

ð3:40Þ

Transforming (3.40) from the Infinite Bus reference frame to d-q reference frame by introducing Δδ, (3.40) becomes ΔItL ¼ R1 ΔVt þ R2 ΔXw þ R3 Δδ

ð3:41Þ

Substituting the linearized SG equation ΔVtd ¼ XqΔItLq and ΔVtq ¼ ΔE0q  X0d ΔItLq into (3.41) and decomposing ΔItL to ΔItLd and ΔItLq gives ΔItLd ¼ Rδd Δδ þ RE0q d ΔE0q þ RIq d ΔItLq þ RXw d ΔXw ΔItLq ¼ Rδq Δδ þ RE0q q ΔE0q þ RIq q ΔItLq þ RXw q ΔXw

ð3:42Þ

Then (3.42) is substituted into the SMIB system linearized model in (3.43) and the Phillips-Heffron model of a SMIB system with interface equations of a WPIG is derived.

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

88

Δδ_ ¼ ω0 Δω Δω_ ¼ M1 ðΔTE  DΔωÞ ΔE_0q ¼ T1 d0 ðΔEQ þ ΔE fd Þ ΔE_ fd ¼ ðΔE fd  KA ΔVt ÞΔT1 A

ð3:43Þ

where   ΔTE ¼ ΔItLq E0q0 þ Iq0 ΔE0q þ ΔItLq Xq  X0d ItLd0 þ ItLq0 Xq  X0d ΔItLd ,  ΔEQ ¼ ΔE0q  Xd  X0d ΔItLd , ΔVtd ¼ XqΔItLq and ΔVtq ¼ ΔE0q  X0d ΔItLd .

Appendix 3.2: Data of Examples 3.1 and 3.2 Example 16-Machine 68-Bus New York and New England Power System [36] (Tables 3.4, 3.5 and 3.6) All the synchronous generators employ sixth-order detailed model with damping D ¼ 0.0. The structure of first-order excitation system model is shown by Fig. 3.12. The parameters of excitation system model are KA ¼ 3.95, TA ¼ 0.1s, efdmax ¼ 10.0, efdmin ¼  10.0.

Data of DFIG and FSIG Induction Generator Parameters Mw ¼ 3.4s, Dw ¼ 0, Rr ¼ 0.0007, Xs ¼ 0.0878, Xr ¼ 0.0373, Xm ¼ 1.3246, Xr3 ¼ 0.05, Xss ¼ Xs þ Xm, Xrr ¼ Xr þ Xm, Pw ¼ 2.0 p. u., Vw ¼ 1.015 p. u.

Control Parameters of RSC Kpsp1 ¼ Kqsp1 ¼ 0.2, Kpsp2 ¼ Kqsp2 ¼ 1, KpsI1 ¼ KqsI1 ¼ 12.56 s1, KpsI2 ¼ KqsI2 ¼ 62.5 s1.

Appendix 3.2: Data of Examples 3.1 and 3.2

89

Table 3.4 Bus data Bus no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Voltage – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Angle – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Active power consumption 2.5270 0.0000 3.2200 5.0000 0.0000 0.0000 2.3400 5.2200 1.0400 0.0000 0.0000 0.0900 0.0000 0.0000 3.2000 3.2900 0.0000 1.5800 0.0000 6.8000 2.7400 0.0000 2.4800 3.0900 2.2400 1.3900 2.8100 2.0600 2.8400 0.0000 0.0000 0.0000 1.1200 0.0000 0.0000 1.0200 60.0000 0.0000 2.6700 0.6563 10.0000

Reactive power consumption 1.1856 0.0000 0.0200 1.8400 0.0000 0.0000 0.8400 1.7700 1.2500 0.0000 0.0000 0.8800 0.0000 0.0000 1.5300 0.3200 0.0000 0.3000 0.0000 1.0300 1.1500 0.0000 0.8500 0.9200 0.4700 0.1700 0.7600 0.2800 0.2700 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1946 3.0000 0.0000 0.1260 0.2353 2.5000

Active power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Reactive power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – (continued)

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

90

Table 3.4 (continued) Bus no. 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

Voltage – – – – – – – – – – – 1.0450 0.9800 0.9830 0.9970 1.0110 1.0500 1.0630 1.0300 1.0250 1.0100 1.0000 1.0156 1.0110 1.0000 1.0000 1.0000

Angle – – – – – – – – – – – – – – – – – – – – – – – 0.0000 – – –

Active power consumption 11.5000 0.0000 2.6755 2.0800 1.5070 2.0312 2.4120 1.6400 1.0000 3.3700 24.7000 – – – – – – – – – – – – – – – –

Reactive power consumption 2.5000 0.0000 0.0484 0.2100 0.2850 0.3259 0.0220 0.2900 1.4700 1.2200 1.2300 – – – – – – – – – – – – – – – –

Active power generation – – – – – – – – – – – 2.5000 5.4500 6.5000 6.3200 5.0520 7.0000 5.6000 5.4000 8.0000 5.0000 10.0000 13.5000 – 17.8500 10.0000 40.0000

Reactive power generation – – – – – – – – – – – – – – – – – – – – – – – – – – –

Appendix 3.2: Data of Examples 3.1 and 3.2

91

Table 3.5 Line data Branch No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

From bus 1 1 2 2 2 3 3 4 4 5 5 6 6 6 7 8 9 10 10 10 12 12 13 14 15 16 16 16 16 17 17 19 19 20 21 22 22 23 23 25 25 26

To bus 2 30 3 25 53 4 18 5 14 6 8 7 11 54 8 9 30 11 13 55 11 13 14 15 16 17 19 21 24 18 27 20 56 57 22 23 58 24 59 26 60 27

R 0.0035 0.0008 0.0013 0.0070 0.0000 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0.0000 0.0004 0.0023 0.0019 0.0004 0.0004 0.0000 0.0016 0.0016 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008 0.0003 0.0007 0.0013 0.0007 0.0007 0.0009 0.0008 0.0006 0.0000 0.0022 0.0005 0.0032 0.0006 0.0014

X 0.0411 0.0074 0.0151 0.0086 0.0181 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.0250 0.0046 0.0363 0.0183 0.0043 0.0043 0.0200 0.0435 0.0435 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135 0.0059 0.0082 0.0173 0.0138 0.0142 0.0180 0.0140 0.0096 0.0143 0.0350 0.0272 0.0323 0.0232 0.0147

B 0.6987 0.4800 0.2572 0.1460 0.0000 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.1130 0.1389 0.0000 0.0780 0.3804 0.2900 0.0729 0.0729 0.0000 0.0000 0.0000 0.1723 0.3660 0.1710 0.1342 0.3040 0.2548 0.0680 0.1319 0.3216 0.0000 0.0000 0.0000 0.2565 0.1846 0.0000 0.3610 0.0000 0.5310 0.0000 0.2396

Transformer ratio 1 1 1 1 1.025 1 1 1 1 1 1 1 1 1.07 1 1 1 1 1 1.07 1.06 1.06 1 1 1 1 1 1 1 1 1 1.06 1.07 1.009 1 1 1.025 1 1 1 1.025 1 (continued)

92

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

Table 3.5 (continued) Branch No. 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

From bus 26 26 28 29 9 9 9 36 34 35 33 32 30 30 1 31 33 38 46 1 47 47 48 35 37 43 44 39 39 45 50 50 49 52 42 41 31 32 36 37 41 42

To bus 28 29 29 61 30 36 36 37 36 34 34 33 31 32 31 38 38 46 49 47 48 48 40 45 43 44 45 44 45 51 52 51 52 42 41 40 62 63 64 65 66 67

R 0.0043 0.0057 0.0014 0.0008 0.0019 0.0022 0.0022 0.0005 0.0033 0.0001 0.0011 0.0008 0.0013 0.0024 0.0016 0.0011 0.0036 0.0022 0.0018 0.0013 0.0025 0.0025 0.0020 0.0007 0.0005 0.0001 0.0025 0.0000 0.0000 0.0004 0.0012 0.0009 0.0076 0.0040 0.0040 0.0060 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

X 0.0474 0.0625 0.0151 0.0156 0.0183 0.0196 0.0196 0.0045 0.0111 0.0074 0.0157 0.0099 0.0187 0.0288 0.0163 0.0147 0.0444 0.0284 0.0274 0.0188 0.0268 0.0268 0.0220 0.0175 0.0276 0.0011 0.0730 0.0411 0.0839 0.0105 0.0288 0.0221 0.1141 0.0600 0.0600 0.0840 0.0260 0.0130 0.0075 0.0033 0.0015 0.0015

B 0.7802 1.0290 0.2490 0.0000 0.2900 0.3400 0.3400 0.3200 1.4500 0.0000 0.2020 0.1680 0.3330 0.4880 0.2500 0.2470 0.6930 0.4300 0.2700 1.3100 0.4000 0.4000 1.2800 1.3900 0.0000 0.0000 0.0000 0.0000 0.0000 0.7200 2.0600 1.6200 1.1600 2.2500 2.2500 3.1500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Transformer ratio 1 1 1 1.025 1 1 1 1 1 0.946 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.04 1.04 1.04 1.04 1 1 (continued)

Appendix 3.2: Data of Examples 3.1 and 3.2

93

Table 3.5 (continued) Branch No. 85 86

From bus 52 1

To bus 68 27

R 0.0000 0.0320

X 0.0030 0.3200

B 0.0000 0.4100

Transformer ratio 1 1

Table 3.6 Machine data Generator no. G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16

Bus name B53 B54 B55 B56 B57 B58 B59 B60 B61 B62 B63 B64 B65 B66 B67 B68 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

Generator base (MVA) 100 100 100 100 100 100 100 100 100 100 100 100 200 100 100 200 0.0690 0.2820 0.2370 0.2580 0.3100 0.2410 0.2920 0.2800 0.2050 0.1150 0.1230 0.0950 0.0286 0.0173 0.0173 0.0334

xa 0.0125 0.0350 0.0304 0.0295 0.0270 0.0224 0.0322 0.0280 0.0298 0.0199 0.0103 0.0220 0.0030 0.0017 0.0017 0.0041 0.0280 0.0600 0.0500 0.0400 0.0600 0.0450 0.0450 0.0500 0.0500 0.0450 0.0150 0.0280 0.0050 0.0025 0.0025 0.0060

xd 0.1000 0.2950 0.2495 0.2620 0.3300 0.2540 0.2950 0.2900 0.2106 0.1690 0.1280 0.1010 0.0296 0.0180 0.0180 0.0356 0.0250 0.0500 0.0450 0.0350 0.0500 0.0400 0.0400 0.0450 0.0450 0.0400 0.0120 0.0250 0.0040 0.0023 0.0023 0.0055

0

xd 0.0310 0.0697 0.0531 0.0436 0.0660 0.0500 0.0490 0.0570 0.0570 0.0457 0.0180 0.0310 0.0055 0.00285 0.00285 0.0071 1.50 1.50 1.50 1.50 0.44 0.40 1.50 0.41 1.96 1.50 1.50 1.50 1.50 1.50 1.50 1.50

00

xd 0.0250 0.0500 0.0450 0.0350 0.0500 0.0400 0.0400 0.0450 0.0450 0.0400 0.0120 0.0250 0.0040 0.0023 0.0023 0.0055 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035

0

Td0 ðsÞ 10.20 6.56 5.70 5.69 5.40 7.30 5.66 6.70 4.79 9.37 4.10 7.40 5.90 4.10 4.10 7.80 42.0 30.2 35.8 28.6 26.0 34.8 26.4 24.3 34.5 31.0 28.2 92.3 248.0 300.0 300.0 225.0

3 Damping Torque Analysis of Small-Signal Angular Stability of a Power. . .

94

efd0

Vtref + Vt −

Σ

KA 1+TAs

Δefd +

+ Σ

efd max

efd

efd min Fig. 3.12 The first-order excitation system model of synchronous generator

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17. Tsourakis G, Nomikos BM, Vournas CD (2009) Contribution of doubly fed wind generators to oscillation damping. IEEE Trans Energy Conver 24(3):783–791 18. Kunjumuhammed LP, Pal BC, Anaparthi KK, Thornhill NF (2013) Effect of wind penetration on power system stability. In: IEEE power and energy Soc general meeting (PES), Vancouver, BC, pp 1–5. 19. Gautam D, Goel L, Ayyanar R, Vittal V, Harbour T (2011) Control strategy to mitigate the impact of reduced inertia due to doubly fed induction generators on large power systems. IEEE Trans Power Syst 26(1):214–224 20. Arani MFM, El-Saadany EF (2013) Implementing virtual inertia in DFIG-based wind power generation. IEEE Trans Power Syst 28(2):214–224 21. Ma J, Qiu Y, Zhang WB, Song ZX, Thorp JS (2016) Research on the impact of DFIG virtual inertia control on power system small-signal stability considering the phase-locked loop. IEEE Trans Power Syst 32(3):2094–2105 22. Dominguez-Garcia J, Bianchi FD, Gomis-Bellmunt O (2013) Control signal selection for damping oscillations with wind power plants based on fundamental limitations. IEEE Trans Power Syst 28(4):4274–4281 23. Arani MFM, Mohamed Y (2015) Analysis and impacts of implementing droop control in DFIG-based wind turbines on microgrid/weak-grid stability. IEEE Trans Power Syst 30 (1):385–396 24. Surinkaew T, Ngamroo I (2014) Coordinated robust control of DFIG wind turbine and PSS for stabilization of power oscillations considering system uncertainties. IEEE Trans Sustain Energ 5(3):823–833 25. Surinkaew T, Ngamroo I (2016) Hierarchical co-ordinated wide area and local controls of DFIG wind turbine and PSS for robust power oscillation damping. IEEE Trans Sustainable Energ 7 (3):943–955 26. Liu Y, Gracia JR, King TJ, Liu YL (2015) Frequency regulation and oscillation damping contributions of variable-speed wind generators in the U.S. Eastern Interconnection (EI). IEEE Trans Sustain Energ 6(3):951–958 27. Zeni L, Eriksson R, Goumalatsos S, Altin M, Sorensen P, Hansen A, Kjar P, Hesselbak B (2016) Power oscillation damping from VSC-HVDC connected offshore wind power plants. IEEE Trans Power Deliver 31(2):829–838 28. Singh M, Allen AJ, Muljadi E, Gevorgian V, Zhang YC, Santoso S (2015) Interarea oscillation damping controls for wind power plants. IEEE Trans Sustain Energ 6(3):967–975 29. Mishra Y, Mishra S, Li FX, Dong ZY, Bansal RC (2009) Small-signal stability analysis of a DFIG-based wind power system under different modes of operation. IEEE Trans Energy Conver 24(4):972–982 30. Mokhtari M, Aminifar F (2014) Toward wide-area oscillation control through doubly-fed induction generator wind farms. IEEE Trans Power Syst 29(6):2985–2992 31. Chandra S, Gayme DF, Chakrabortty A (2014) Coordinating wind farms and battery management systems for inter-area oscillation damping: a frequency-domain approach. IEEE Trans Power Syst 29(3):1454–1462 32. Jamehbozorg A, Radman G (2015) Small-signal analysis of power systems with wind and energy storage units. IEEE Trans Power Syst 30(1):298–305 33. Yu YN (1983) Electric power system dynamics. Academic Press, New York 34. CIGRE Technical Brochure on Modeling and Dynamic Behavior of Wind Generation as it Relates to Power System Control and Dynamic Performance, Working Group 01 (2006) Advisory Group 6, Study Committee C4 Draft Rep 35. Rodriguez JM, Fernandez JL, Beato D, Iturbe R, Usaola J, Ledesma P, Wilhelmi JR (2002) Incidence on power system dynamics of high penetration of fixed speed and doubly fed wind energy systems: study of the Spanish case. IEEE Trans Power Syst 17(4):1089–1095 36. Rogers G (2000) Power system oscillations. Kluwer, Norwell, MA

Chapter 4

Modal Analysis of Small-Signal Angular Stability of a Power System Affected by Grid-Connected DFIG

Linearized model of a power system with a grid-connected variable speed wind generator (VSWG) is derived in this chapter in three steps. First, the linearized model of the VSWG is established. That includes the establishment of linearized model of a DFIG, a PMSG and a PLL. Second, the linearized model of the power system is derived. Finally, the linearized model of the power system with the VSWG is established by combining the model of the VSWG and the power system.

4.1

Decomposed Modal Analysis of the Impact of a GridConnected DFIG on Power System Small-Signal Angular Stability

Grid connection of a DFIG for the wind power generation brings about the change of system load flow and introduces the dynamic interactions. Those are two main factors that the grid-connected DFIG affects power system small-signal angular stability. In this section, a method of decomposed modal analysis is presented which can separately examine the impact of those two affecting factors introduced by the DFIG. Separate examination provides a potential way to gain better understanding on and deeper insight into the mechanism about how the DFIG affects power system small-signal angular stability.

4.1.1

Method of Decomposed Modal Analysis

For a multi-machine power system connected with a DFIG shown by Fig. 4.1, the following linearized state-space model of the power system can be established © Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_4

97

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

98

A multimachine power system

Pw + jQw

Vw

qw

DFIG

Ps+ jQs Pr + jQr GSC

RSC C

GSC control system

RSC control system

Fig. 4.1 A DFIG connected to a multi-machine power system

   d ΔXg Agg ¼ Adg dt ΔXd

Agd Awd



ΔXg ΔXd



  ΔXg ¼A ΔXd

ð4:1Þ

where ΔXd is the vector of all the state variables associated with the DFIG, ΔXg is the vector of all the state variables of the synchronous generators. By computing the electromechanical oscillation modes (EOMs) of the power system from the state matrix A, the effect of the DFIG on system small-signal angular stability can be examined. Obviously, the computation gives the overall effect of the DFIG, which includes that of change of system load flow brought about by the DFIG and that of the dynamic interaction between the DFIG and the synchronous generators. Normally, separate examination of two affecting factors of the DFIG, i.e., the change of system load flow and dynamic interaction, cannot be obtained from the conventional modal analysis based on the state-space model of Eq. (4.1). In order to examine the two affecting factors separately so as to understand how the DFIG affects the EOMs of the power system, the closed-loop interconnected model derived in 2.4 is used, where the DFIG is the open-loop subsystem in the feedback loop and the rest of the power system (ROPS) is the open-loop subsystem in the forward path. The closed-loop interconnected model is shown by Fig. 2.11 and presented as Fig. 4.2. The effect of the dynamic interactions introduced by the DFIG is individually estimated by applying the damping torque analysis (DTA). Thus, two affecting factors of the DFIG can be examined separately.

4.1.1.1

Electric Torque Contributed by the Grid-Connected DFIG

In the power system of Fig. 4.1, the power output from the DFIG, Pw þ jQw, is the physical exhibition of dynamic interactions between the DFIG and the synchronous

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . . Fig. 4.2 Closed-loop interconnected model of the power system with the DFIG

99

Subsystem of rest of power system ΔPw

Gvp (s) +

ΔVw

Gvq (s)

ΔQw

Hdq (s) Hdp (s) DFIG subsystem

generators. Denote Pw0 þ jQw0 as the power output from the DFIG at the steady-state operation of the power system. It can have Pw þ jQw ¼ Pw0 þ jQw0 þ ΔPw þ jΔQw

ð4:2Þ

Obviously ΔPw þ jΔQw characterizes the dynamic interactions between the DFIG subsystem and the ROPS subsystem, i.e., the synchronous generators. The state-space model of the ROPS subsystem is derived in Sect. 2.3 to be (2.126) and given below 8 < d ΔX ¼ A ΔX þ b ΔP þ b ΔQ g g g p w q w dt ð4:3Þ : T ΔVw ¼ cgv ΔXg þ dvp ΔPw þ dvq ΔQw Hence, the state-space model of the ROPS subsystem can be rewritten as 2

Δδ

3

2

0

ω0 I

0

32

Δδ

3

2

0

3

2

0

3

d6 76 7 6 7 6 7 7 6 4 Δω 5 ¼ 4 A21 A22 A23 54 Δω 5 þ 4 bP2 5ΔPw þ 4 bQ2 5ΔQw dt bQ3 A31 A32 A33 bP3 Δz Δz   ΔVw ¼ Cg1 Cg2 Cg3 ΔXg þ dvp ΔPw þ dvq ΔQw

ð4:4Þ

where Δδ and Δω is the vector of deviation of angular positions and speed variables of the synchronous generators respectively, Δz is the vector of all the other state variables of the synchronous generators in the power system, ΔVw is the deviation of the magnitude of voltage at the terminal of the DFIG. The transfer function model of the DFIG subsystem is derived in Sect. 2.4.2 to be (see (2.130)), ( ΔPw ¼ Hdp ðsÞΔVw ð4:5Þ ΔQw ¼ Hdq ðsÞΔVw

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

100

It is interesting to note that the DFIG only responds to the variation of the magnitude of the terminal voltage, ΔVw, of the DFIG. In the power system, normally dynamic variation of ΔVw is limited and so the dynamic response of the DFIG is very small. This explains why normally the dynamic interactions introduced by the DFIG are weak and the DFIG is considered to be “inertia-less”. The closed-loop interconnected model shown by Fig. 4.4 can be redrawn in more details from (4.4) and (4.5). The redrawn model is shown by Fig. 4.3. Figure 4.3 clearly depicts the two-ways dynamic interactions between the DFIG and the synchronous generators. One of the ways is the response of the DFIG to ΔVw to inject a variable power ΔPw þ jΔQw into the power system as described by (4.5). The other way is the response of the power system to ΔPw þ jΔQw to generate ΔVw as described by (4.4). From Fig. 4.3, the effect of the dynamic interactions introduced by the DFIG can be estimated by applying the damping torque analysis (DTA) as follows. Transfer function matrix from the DFIG output power, ΔPw þ jΔQw, to its electric torque contribution to the electromechanical oscillation loops of the synchronous generators can be obtained from (4.4) or Fig. 4.3 to be

ROPS subsystem

ΔTQ

A21

Δω

( sI - A22 )-1

+

A32 Δz

bP2

( sI - A33 )-1

bQ2

C g3

D Pw DQw

Δδ

s

A23

ΔTP

w0 I

H dp ( s ) H dq ( s )

C g2

DVw

+

A31

bP3

bQ3

d vp

C g1

+ d vq

DFIG subsystem Fig. 4.3 Closed-loop interconnected model of the multi-machine power system with the gridconnected DFIG

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . .

ΔTP ¼ GP ðsÞ ¼ bP2 þ A23 ðsI  A33 Þ1 bP3 ΔPw ΔTQ ¼ GQ ðsÞ ¼ bQ2 þ A23 ðsI  A33 Þ1 bQ3 ΔQw

101

ð4:6Þ

Hence from (4.5) and (4.6), the electric torque contribution from the DFIG is obtained to be   ΔTw ¼ ΔTP þ ΔTQ ¼ GP ðsÞHdp ðsÞ þ GQ ðsÞHdq ðsÞ ΔVw ð4:7Þ Thus the electric torque provided by the DFIG to the kth synchronous generator in the power system is   ΔTwk ¼ gPk ðsÞHdp ðsÞ þ gQk ðsÞHdq ðsÞ ΔVw

ð4:8Þ

where gPk(s) and gQk(s)is the kth element of GP(s) and GQ(s) respectively.

4.1.1.2

Method of Decomposed Modal Analysis

Consider an assumed case that ΔPw þ jΔQw ¼ 0. This is when the DFIG is completely decoupled dynamically with the ROPS. Thus, the only factor that the DFIG affects the system small-signal angular stability is the change of system load flow it introduces. Obviously, when ΔPw þ jΔQw ¼ 0, the DFIG is degraded into a constant power source, Pw0 þ jQw0, as shown by (4.2), and the state-space model of the ROPS of (4.4) becomes 2 3 2 32 3 2 3 Δδ 0 ω0 I Δδ 0 Δδ d4 Δω 5 ¼ 4 A21 A22 A23 54 Δω 5 ¼ Ag 4 Δω 5 ð4:9Þ dt Δz Δz Δz A31 A32 A33 Therefore, by modelling the DFIG as the constant power source and computing the EOMs of the power system from state matrix Ag, the impact of the change of the load flow brought about by the DFIG on system small-signal angular stability can be determined. Denote λi ¼ – ξi þ jωi as the ith EOM of concern computed from the ω i as the EOM of concern of open-loop state matrix Ag in (4.9). Denote ^λ i ¼ ^ξ i þ j^ the closed-loop interconnected system of Fig. 4.3 corresponding to λi ¼ – ξi þ jωi. The difference between the closed-loop and open-loop EOM of concern is Δλi ¼ ^λ i λi which is caused by ΔPW þ jΔQw 6¼ 0 Δλi ¼ ^λ i  λi can be estimated by the DTA as follows. Denote λi ¼  ξi þ jωi and vj, j ¼ 1, 2,   M as the eigenvalues and associated right eigenvectors of open-loop state matrix Ag in (4.9). Solution of (4.9) can be expressed as ΔXg ¼

M M X X v ja j v jk a j , Δωk ¼ s  λ s  λj j j¼1 j¼1

ð4:10Þ

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

102

where Δωk is the deviation of rotor speed of the kth synchronous generator and a state variable, vjk is the element of vj corresponding to Δωk. From the second equation of (4.4) and (4.5), it can have ΔVw ¼

1  dvp

Cg ΔXg Hdp ðsÞ  dvq Hdq ðsÞ

ð4:11Þ

From (4.10) and (4.11),

ΔVw ¼ Δωk

¼

M X v ja j 1 Cg 1  dvp Hdp ðsÞ  dvq Hdq ðsÞ j¼1 s  λ j M X v jk a j s  λj j¼1

M X v ja j 1 ðs  λi Þ Cg 1  dvp Hdp ðsÞ  dvq Hdq ðsÞ s  λj j¼1

ð s  λi Þ

ð4:12Þ

M X v jk a j s  λj j¼1

Let s ¼ λi in the above equation, ΔVw ¼

Cg vig 1 Δωk ¼ γik ðλi ÞΔωk 1  dvp Hdp ðλi Þ  dvq Hdq ðλi Þ vik

ð4:13Þ

Hence, at the complex oscillation frequency λi ¼  ξi þ jωi, the damping torque provided by the DFIG to the kth synchronous generator can be obtained from (4.8) and (4.13) to be    ΔTwDk ¼ Re gPk ðλi ÞHdp ðλi Þ þ gQk ðλi Þ Hdq ðλi Þ γik ðλi Þ Δωk ¼ Dwk Δωk ð4:14Þ where Re{} denotes the real part of a complex number. Let the sensitivity of the EOM of concern to the damping torque coefficient of the kth synchronous generator be Sik ¼

∂λi ∂Dwk

ð4:15Þ

From (4.14) and (4.15), Δλi ¼

N X k¼1

Sik Dwk

ð4:16Þ

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . .

103

Thus, Δλi can be quantitatively calculated to estimate the impact of dynamic interactions introduced by the DFIG on power system small-signal angular stability. In addition, Eq. (4.16) clearly indicates that the DFIG contribute the damping torque to the synchronous generators to affect the EOMs. Based on the discussions above, the method to separately examine the impact of change of system load flow and dynamic interactions brought about by the DFIG on the EOMs can be summarized as follows. 1. Model the DFIG as a constant power source and derive the state matrix Ag. Compute the EOM of concern from Ag, λi, to examine the effect of the change of load flow brought about by the DFIG on the system small-signal angular stability. 2. Compute the impact of the dynamic interactions of the DFIG with the synchronous generators on the oscillation mode to be Δλi by use of (4.16) with the full dynamics of the DFIG being included. 3. The total impact of the DFIG is ^λ i  λi þ Δλi

ð4:17Þ

Integration of large number of wind farms, each of which may consist of tens even hundreds of the DFIGs, will increase the dimension of state matrix, A, in (4.1) dramatically. Numerical complexity and difficulty of modal analysis for A may grow considerably in the examination of the impact of the integration of wind farms. The proposed method of decomposed modal analysis provides a way to avoid the problem. λi is computed from the state matrix Ag, dimension of which does not change with the integration of the DFIGs because the DFIG’s dynamics are not included. Thus the estimation of the impact of the dynamics of the DFIG, Δλi, involves no modal analysis of state matrix with increased dimension. If Δλi is found to be small, the DFIG can be modelled as a constant power source in the examination. This will reduce the computational burden significantly if the integration of large number of wind farms needs to be examined. Examination of dynamic interactions of the DFIG with the synchronous generators is very important to understand the involvement and behavior of the DFIG in power system dynamic transient. The estimation can be carried out by calculating the participation factors (PFs) of the state variables of the DFIG for the EOMs of concern. Calculation of Δλi as proposed above provides an alternative way to assess the dynamic interactions. The assessment does not need to carry out the modal analysis of the closed-loop state matrix A with increased dimension. While normally, computation of the PFs is based on the modal analysis of A. In addition, computation of Δλi can provide the estimation of not only the scale but also the direction (positive to improve or negative to reduce the damping of the EOM of concern) of the dynamic interactions.

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

104

4.1.2

Example 4.1

4.1.2.1

The Example Power System

Configuration of an example 16-machine 68-bus power system is shown by Fig. 4.4. In this example, the simplified third-order model of the synchronous generator and the first-order model of the AVR recommended in [1–3] were adopted. The bus data, line data, machine data are given in Tables 4.10, 4.11, and 4.12 in Appendix 4.1 respectively. No PSS was installed on any synchronous generator. There are four inter-area EOMs in the power system, which are crucial to the small-signal angular stability of the example system. They are ð0Þ

ð0Þ

ð0Þ

ð0Þ

λ1 ¼ 0:1000 þ j5:3419, λ2 ¼ 0:1353 þ j4:6374, λ3 ¼ 0:0960 þ j3:9617, λ4 ¼ 0:2639 þ j3:0428 where superscript (0) indicates the value of the modes when no DFIG is connected to the example power system. In order to demonstrate the application of the method introduced in the previous section, it is assumed that a DFIG is to be connected to the example power system. The candidate locations of connection are node 8 and 16. Though the proposed method is applicable for any type of the model of the synchronous generators and the DFIG, in the example power system, a third-order model of the synchronous generator and a simple first-order model of the automatic voltage regulator (AVR) were used [1]. All the synchronous generators were not equipped with the power NYPS 14

8

A3

66

1

40

41

48

53

47

38

31 62

15

63

36

A5

50 52 68 16

8

45 37

39 44

6 7

64

35 51

21 22

5

34

24

4

33 49

16 15 14

11

10

46

12

54 2

12 11

20 56

55 4

3

65 13

A2

NETS

Fig. 4.4 Configuration of example 16-machine 68-bus power system

58

19

13

10

A1 43

61 9

17 3

9

29

28 27

18

30

32

26

25

2 1

42 67

A4

60

57 5

23 59

6 7

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . .

105

system stabilizers (PSS). The loads were modelled as constant impedance. Normally, for the study of power system small-signal angular stability, such models of the synchronous generator, the AVR and the loads are sufficient if the focus of the study is not the impact of the synchronous generator, the AVR or the loads [1]. Load flow change introduced by the DFIG was balanced by G13 at slack bus 65. A detailed fourteenth-order model of the DFIG was used [4]. The DFIG adopted the reactive power control with a fixed power factor (0.95). Parameters of the DFIG, PI gains of RSC and GSC of DFIG are listed in Tables 4.13, 4.14, and 4.15 of Appendix 4.1 respectively.

4.1.2.2

Decomposed Modal Analysis

Firstly, the DFIG was modelled as a constant power injection, Pw0, to examine the effect of change of load flow introduced by the DFIG at the candidate connecting locations. Pw0 increased gradually from Pw0 ¼ 0 to Pw0 ¼ 2 p. u. Qw0 changed accordingly with a fixed power factor of 0.95. Changes of four inter-area oscillation modes, λi, i ¼ 1, 2, 3, 4, were calculated from the open-loop system state matrix Ag and are displayed in Figs. 4.5 and 4.6 as solid curves respectively, where the arrows show the direction of change of λi, i ¼ 1, 2, 3, 4 when Pw0 increased. Secondly, based on the closed-loop interconnected model of the example power system shown by Fig. 4.2, impact of dynamic interactions introduced by the DFIG was estimated by use of (4.16) to give Δλi, i ¼ 1, 2, 3, 4 when Pw0 increased. Results of λi þ Δλi, i ¼ 1, 2, 3, 4 are presented in Figs. 4.5 and 4.6 as dashed curves respectively. In order to confirm the results obtained above by use of the method of decomposed modal analysis, normal linearized model of the example power system with the DFIG’s dynamics being included shown by (4.1) was established. The interarea EOMs were calculated from state matrix A to be ^λ i , i ¼ 1, 2, 3, 4 and displayed as dashed curves with crosses in Figs. 4.5 and 4.6, which overlap the dashed curves (λi þ Δλi, i ¼ 1, 2, 3, 4). Results of Figs. 4.5 and 4.6 confirm the correctness of (4.16), i.e., the total effect of the DFIG, ^λ i , i ¼ 1, 2, 3, 4, is separated successfully by the method of decomposed modal analysis into two parts, the effect of the change of load flow brought about by the DFIG, λi, i ¼ 1, 2, 3, 4, and that of dynamic interactions introduced by the DFIG, Δλi, i ¼ 1, 2, 3, 4.

4.1.2.3

The Impact of Dynamic Interactions

From Figs. 4.5 and 4.6, it can be seen that the effect of dynamic interactions introduced by the DFIG on system small-signal angular stability can be negative or positive. For example, Fig. 4.5(1) shows that the dash curve is on the right hand side of the solid one for the first EOM from Pw0 ¼ 0 to Pw0 ¼ 2 p. u. This means that the effect of the dynamic interaction on the mode (Δλ1) is negative, being

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

106 Imaginary axis

Imaginary axis

5.342

4.65

(0)

l1

4.64

(0)

l2

4.63 5.3415

4.62

Pw0 increased

4.61

lˆ 1 l1 + Δl1

l1 5.341

Pw0 increased

4.6

l2

lˆ 2

4.59 4.58

Δl1 -0.1

-0.1001

4.57 -0.0999

-0.0998

Δl2

l2 + Δl2

-0.145

-0.135

-0.125

-0.115

Imaginary axis

-0.105

Real axis

Real axis Imaginary axis

3.9625

3.05 (0)

(0)

3.962 3.9615 3.961 3.9605

l3

3.9595

-0.0995

3.035 3.03

l3 + Δl3

3.025 3.02

l3

3.015

Δl3 -0.0985

l4

3.04

Pw0 increased lˆ 3

3.96

3.045

Pw0 increased l4 + Δl4

3.01 -0.0975

-0.0965

-0.0955

Real axis

-0.265

lˆ 4

l4 Δl4

-0.26

-0.255

-0.25

Real axis

Fig. 4.5 Trajectories of inter-area EOMs when the level of wind penetration increases with the DFIG being connected at node 8

detrimental to the small-signal angular stability of the example power system. Whilst for other three EOMs, the effect of dynamic interactions is beneficial because adding Δλi, i ¼ 2, 3, 4 on λi, i ¼ 2, 3, 4 makes the modes moving towards the left on the complex plane. Table 4.1 presents the computational results of the damping torque contribution from the DFIG to the synchronous generators when it is connected at node 8 and Pw0 ¼ 2 p. u.From Table 4.1, it can be seen that the DFIG contributed negative damping torque to the majority of the synchronous generators for the first EOM. It provided the positive damping torque to the majority of the synchronous generators for other three EOMs. This explains why the effect of dynamic interactions introduced by the DFIG was negative for the first EOM (Δλ1) and positive for other three EOMs (Δλi, i ¼ 2, 3, 4). From Figs. 4.5 and 4.6, it can be observed that the shift of Δλi, i ¼ 1, 2, 3, 4 in the horizontal direction is much smaller than that of λi þ Δλi, i ¼ 1, 2, 3, 4 or ^λ i , i ¼ 1, 2, 3, 4. Hence, when the DFIG was connected at the candidate locations, its impact on system small-signal angular stability was mainly from the change of

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . . Imaginary axis

Imaginary axis

5.342

4.64

(0)

l2

(0)

l1

5.3418

107

4.62

5.3416

Pw0 increased

5.3414

Pw0 increased

4.6 4.58

5.3412

lˆ 1

5.341

l1 + Δl1

l1

5.3408

l2

4.56

l2 + Δl2

4.54

lˆ 2

4.52

5.3406

Δl2

Δl1

-0.1001

-0.1

-0.0999

-0.0998

-0.14

-0.13

-0.12

-0.11

-0.1

-0.09

Real axis Imaginary axis

Imaginary axis 3.05

3.963 (0)

l3

3.962

3.96 3.959

3.04

3.02 3.01

l3 + Δl3

3

3.958

-0.103

Pw0 increased

2.98

l3

Δl3 -0.101

l4

lˆ 4

2.99

3.957 3.956

(0)

l4

3.03

Pw0 increased lˆ 3

3.961

-0.08

Real axis

l4 + Δl4

Δl4

2.97 -0.099

-0.097

-0.095

-0.27

-0.26

-0.25

-0.24

Real axis

-0.23

-0.22

Real axis

Fig. 4.6 Trajectory of inter-area oscillation modes when the level of wind penetration increases with the DFIG being connected at node 16

Table 4.1 Computational results of the damping torque contribution from the DFIG to the synchronous generators

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16

Mode 1 1.0971 2.8677 2.8583 1.0519 0.6180 0.9118 0.8674 1.2503 0.7538 2.8498 16.1222 1.5291 1.7722 0.0729 0.0160 0.2304

Mode 2 4.5036 11.2763 9.8262 2.4555 1.4108 2.1746 1.9086 3.3788 1.6448 2.3268 3.8338 1.9379 3.3472 6.1747 3.2344 8.0689

Mode 3 2.2922 5.2071 4.8684 1.5013 0.8927 1.3770 1.1927 1.7510 1.0196 4.7636 4.9318 2.8577 5.4747 0.2198 0.3234 0.3815

Mode 4 0.1205 0.2861 0.2965 0.1212 0.0740 0.1095 0.1002 0.1238 0.0816 0.1153 0.2951 0.2051 0.4192 0.2728 0.0578 0.3747

108

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Table 4.2 Computational results of summation of the participation factors of all the state variables of the DFIG Mode PFi107

1 0.39594

2 1.5378

3 9.6013

4 105.10

load flow it introduced. Small Δλi, i ¼ 1, 2, 3, 4 indicated weak dynamic coupling between the DFIG and the synchronous generators. To confirm this, Table 4.2 gives the computational results of summation of the participation factors of all the state variables of the DFIG, PFi, i ¼ 1, 2, 3, 4, when it was connected at node 16 and Pw0 ¼ 2.0 p. u. Obviously, very small numerical results in Table 4.2 confirm the nearly decoupling of the DFIG with the synchronous generators. Though the calculation of PFi, i ¼ 1, 2, 3, 4 can give an estimation of the dynamic interactions between the DFIG and the synchronous generators, estimation by use of Δλi, i ¼ 1, 2, 3, 4 is better in two aspects. (1) Δλi, i ¼ 1, 2, 3, 4 does not need to perform the modal analysis of state matrix A with increased dimension to include the DFIG’s dynamics as PFi, i ¼ 1, 2, 3, 4 does; (2) Δλi, i ¼ 1, 2, 3, 4 can not only provide the estimation of the scale of dynamic interactions, but also indicate whether the interactions are positive to improve or negative to reduce the small-signal angular stability of the example power system.

4.1.2.4

The Impact of Load Flow Change

From Figs. 4.5 and 4.6, it can be observed that the effect of change of load flow due to the DFIG (λi, i ¼ 1, 2, 3, 4, solid curves) on system small-signal angular stability can be positive or negative. For example, from Fig. 4.5 it can be seen that when the DFIG was connected at node 8, with the increase of the penetration level of the wind, λ1 and λ3 moved towards left on the complex plane hence benefiting the small-signal angular stability of the example power system. While λ2 and λ4 move towards right which was detrimental to the small-signal angular stability of the example power system. With the increase of penetration level of wind power, shift of the EOMs towards right indicated the potential threat to system stability. Hence it is important to identify the biggest movement towards right when the wind power is dispatched in power system operation or when a wind farm is planned to be connected at various candidate locations. The method of decomposed modal analysis suggests a simpler way to identify the most dangerous scenarios of wind power dispatching and connections as to be explained as follows. By comparing the solid curves (λi) and the dash ones with crossed (^λ i) in Figs. 4.5 and 4.6, it can be seen that when the range of horizontal shift of ^λ i is relatively big, the scale and direction of change of λi are very close to those of ^λ i . Thus, the biggest negative impact of the DFIG on the system small-signal angular stability can be estimated approximately from λi when the DFIG is modelled simply as a constant power injection. For example, from Figs. 4.5 and 4.6 it can be seen that the range of horizontal variation of λ2 and λ4 was bigger. The biggest shift of the modes to the

4.1 Decomposed Modal Analysis of the Impact of a Grid-Connected DFIG on. . .

109

right can be identified from the solid curves (λ2 and λ4) when the DFIG was modelled as a constant power injection. Therefore, it can be proposed that in dispatching wind power or planning the connection of wind farms, the wind generations are modelled firstly as constant power injections. Thus, modal computation from state matrix Ag (λi) can provide a primary and fast scanning to find the most dangerous cases of wind power dispatching or connection. Secondly, the effect of dynamic interactions due to the DFIGs’ dynamics is estimated for those most dangerous cases identified. By doing so, the computational cost involved in the examination would be significantly reduced as the dimension of state matrix Ag does not increase with the wind generations modelled as constant power sources. However, prescreening the most dangerous cases by modelling the wind power as constant power source may miss the case when the dynamic interactions introduced by the DFIG are strong. This special case is possible and will be discusses in Chap. 5 of the book.

4.1.2.5

The Impact of Reactive Power Control/Terminal Voltage Control

Study has shown that the impact of reactive power/voltage control adopted by the DFIG on the damping of power system EOMs was different [5–9]. A more detailed examination on the difference can also be carried out by use of the proposed method as to be demonstrated as follows. Table 4.3 present the computational results when the DFIG was connected at node 16 (Pw0¼3.0 p.u.) in the example power system. Two types of control adopted by the DFIG are compared in Table 4.3: (1) the reactive power control with a fixed power factor of 0.95; (2) the terminal voltage control with a reference value of voltage to be 1.0 p.u. When the DFIG adopts reactive power or terminal voltage control, not only the dynamic interactions introduced by the DFIG but also the Table 4.3 Representative computational results of the DFIG with the reactive power or voltage control

Modes λf1 λ10 Δλ1 λf2 λ20 Δλ2 λf3 λ30 Δλ3 λf4 λ40 Δλ4

Reactive power control 0.0997 þ j5.3396 0.0998 þ j5.3396 0.0001  j0.0000 0.1332 þ j4.4231 0.0313 þ j4.3761 0.1091 þ j0.0506 0.1098 þ j3.9578 0.1114 þ j3.9472 0.0037 þ j0.0099 0.2437 þ j2.8888 0.1684 þ j2.8761 0.0771 þ j0.0206

Voltage control 0.0998 þ 5.3396i 0.0998 þ 5.3396i 0.0000 þ 0.0000i 0.0936 þ 4.4426i 0.0795 þ 4.4504i 0.0141  0.0078i 0.1071 þ 3.9545i 0.1063 þ 3.9537i 0.0008 þ 0.0009i 0.2131 þ 2.9231i 0.2077 þ 2.9260i 0.0054  0.0029i

110

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Table 4.4 Representative computational results of the DFIG with the parameters of controllers being changed

Modes λf1 λ10 Δλ1 λf2 λ20 Δλ2 λf3 λ30 Δλ3 λf4 λ40 Δλ4

Reactive power control 0.0997 þ j5.3396 0.0998 þ j5.3396 0.0001  j0.0000 0.1424 þ j4.4194 0.0313 þ j4.3761 0.1199 þ j0.0464 0.1103 þ j3.9587 0.1114 þ j3.9472 0.0036 þ j0.0110 0.2537 þ j2.8779 0.1684 þ j2.8761 0.0905 þ j0.0102

Voltage control 0.0997 þ j5.3396 0.0998 þ j5.3396 0.0000 þ j0.0000 0.1414 þ j4.4164 0.0795 þ j4.4504 0.0623  j0.0374 0.1106 þ j3.9587 0.1063 þ j3.9537 0.0043 þ j0.0059 0.2529 þ j2.8735 0.2077 þ j2.9260 0.0490  j0.0584

change of load flow brought about by the DFIG is different. By using the method of decomposed modal analysis, a mor.3-3, ^λ i , i ¼ 1, 2, 3, 4 were calculated from the closed-loop state matrix A in (4.1) in order to confirm the correctness of the computation provided by use of the method of decomposed modal analysis. It is interesting to note that the impact of dynamic interactions introduced by the DFIG with the terminal voltage control was always smaller than that with the reactive power control (Δλi, i ¼ 1, 2, 3, 4). The same results have been reported in [8, 9]. The reason may be that the terminal voltage control applied by the DFIG constrained the variation of “feedback signal”, ΔVw, in Fig. 4.3, resulting in less dynamic interactions between the DFIG and the synchronous generators and hence less damping torque contribution from the DFIG. To further validate the results presented in Table 4.3, parameters of both PI reactive power controller and PI terminal voltage controller of the DFIG were reduced by 100 times. Computational results are given in Table 4.4. From Tables 4.3 and 4.4, it can be seen that with the smaller PI gains, the control ability of controllers were weakened, causing bigger dynamic interactions between the DFIG and synchronous generators, which was beneficial to the system small-signal angular stability.

4.2

Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on Power System Small-Signal Angular Stability

When impact of grid-connected wind farms on power system small-signal angular stability is examined, a commonly-adopted strategy has been to represent a gridconnected wind farm by a single variable speed wind generator (VSWG), such as a DFIG. This, in fact, has assumed that an equivalent dynamic model of single VSWG

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

111

can be used to examine the impact of wind farm on power system small-signal angular stability. In this section, the conditions of equivalence under this assumption are examined. Firstly, the sensitivity of a power system EOM of concern to the dynamic interactions introduced by a grid-connected DFIG for wind power generation is derived. The derived modal sensitivity is proposed as an index to indicate the impact of dynamic interactions (index of dynamic interactions, IDI) introduced by the DFIG on the EOM of concern. The analytical expression of the IDI explicitly connects the impact with the frequency response of the DFIG at the frequency of the EOM of concern. From the analytical expression of the IDI, it can be clearly seen that when the impact on the EOM of concern is examined, an equivalent dynamic model of single DFIG can be used to represent a wind farm, as long as the frequency response of this equivalent DFIG model around the frequency of the EOM of concern approximately matches the frequency response of the wind farm. Thus, the condition to represent the dynamics of the wind farm by the dynamic model of a single DFIG is established. Based on the proposed index, analysis is carried out in the section with following generally-applicable conclusion obtained: Impact of dynamic interactions introduced by the DFIG on power system EOM of concern increases when the level of wind power penetration increases. Furthermore, an indicator of the DFIG control (IDC) is derived for comparing the impact of the terminal voltage control and reactive power control implemented by the DFIG. The derivation provides some insight into the difference between the impact of terminal voltage and reactive power control on power system small-signal angular stability. Main merit of the IDC is the computational simplicity to determine that either the terminal voltage or reactive power control is more beneficial to the power system small-signal angular stability, as no knowledge of DFIG dynamic model is needed for computing the IDC.

4.2.1

The Index of Dynamic Interactions (IDI)

4.2.1.1

The Impact of The Grid-Connected DFIG

For a multi-machine power system connected with a DFIG shown by Fig. 4.1, the closed-loop interconnected model was derived in Sect. 2.4.2 of Chap. 2 which consists of the open-loop DFIG subsystem in the feedback loop and the ROPS subsystem in the forward path. The closed-loop interconnected model is shown by Fig. 4.2 and was used in the previous section to introduce the method of decomposed modal analysis. The state-space model of ROPS subsystem is given by (4.3). The transfer function model of the ROPS subsystem can be obtained from (4.3) to be ΔVw ¼ Gvp ðsÞΔPw þ Gvq ðsÞΔQw

ð4:18Þ

112

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

The transfer function model of the DFIG subsystem in Fig. 4.2 is (4.5). The statespace representation of the DFIG subsystem is (2.124) and rewritten below 8 d > > > ΔXd ¼ Ad ΔXd þ bdq ΔVw < dt ð4:19Þ ΔPw ¼ cdp T ΔXd þ ddp1 ΔVw > > > : ΔQw ¼ cdq T ΔXd þ ddq1 ΔVw From (4.3) and (4.19), the closed-loop state-space model of the power system with the DFIG can be obtained to be d ΔX ¼ AΔX dt

ð4:20Þ

where the vector of state variables, ΔX, and the closed-loop state matrix, A, are given in (2.131). Denote ^λ i as an EOM of concern in the power system with the DFIG. ^λ i is an eigenvalue of closed-loop state matrix, A, in Eq. (4.20). Denote λni as the corresponding EOM of concern when the DFIG is not connected to the power system. Impact of grid connection of the DFIG on the EOM of concern is ^λ i  λni . Examination in the previous section shows that ^λ i  λni is caused by change of load flow and dynamic interactions brought about by the DFIG. The method of decomposed modal analysis introduced in the previous section can examine those two affecting factors separately in the following two steps. First, consider the case that there are no dynamic interactions between the DFIG and the ROPS such that ΔPw þ jΔQw ¼ 0. In the closed-loop interconnected system shown by Fig. 4.2, the feedback loop is open. In this case, the DFIG becomes a constant power source, Pw0 þ jQw0, and power system model degrades from (4.3) to d ΔXg ¼ Ag ΔXg dt

ð4:21Þ

Denote λi as the EOM of concern corresponding to ^λ i when the DFIG is modelled as the constant power source, Pw0 þ jQw0. λi is an eigenvalue of open-loop state matrix Ag and hence is the open-loop EOM. Injection of constant power, Pw0 þ jQw0, into the power system changes the system load flow to affect the open-loop EOM of concern. Hence, the impact of load flow change introduced by the DFIG on the EOM of concern is λi  λni. Second, consider the case that ΔPw þ jΔQw 6¼ 0 and the feedback loop in Fig. 4.2 is closed. The EOM of concern is ^λ i and hence is the closed-loop EOM. The difference between the closed-loop and open-loop EOM of concern, Δλi ¼ ^λ i  λi , is caused by the dynamic interactions brought about by the DFIG. Hence, it measures the impact of dynamic interactions introduced by DFIG on the EOM of concern, λi. Thus, the total impact of the DFIG is separated as

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

^λ i  λi |fflfflffl{zfflfflffl}

^λ i  λni ¼ |fflfflfflffl{zfflfflfflffl} Total impact

Impact of dynamic interactions

þ

λi  λni |fflfflfflffl{zfflfflfflffl}

113

ð4:22Þ

Impact of load flow change

In the previous section, the DTA is applied to estimate the impact of dynamic interactions on the EOM of concern, Δλi ¼ ^λ i  λi . In fact, Δλi ¼ ^λ i  λi can be calculated directly from eigensolution of open-loop and closed-loop state matrix, Ag and A, to determine the exact impact of dynamic interactions introduced by the DFIG. However, direct calculation can only examine the impact on a case-by-case basis and would be difficult to gain deeper insight into and better understanding on the impact. In order to carry out analytical examination on the impact of dynamic interactions brought about by the DFIG to draw generallyapplied conclusions, an index of dynamic interactions (IDI) is derived in the next subsection.

4.2.1.2

Derivation of the IDI

Denote λk, k ¼ 1, 2,   n as the eigenvalues of open-loop state matrix Ag; pk and rk are the corresponding left and right eigenvectors. According to the theory of modal control [1], the transfer functions of the open-loop ROPS subsystem, i.e., the power system, can be written as 8 n X Rvpk > > G þ dvp ð s Þ ¼ > > < vp ð s  λk Þ k¼1 ð4:23Þ n > X Rvqk > > G ðsÞ ¼ > þ dvq : vq ð s  λk Þ k¼1 where Rvpk ¼ pkT bP cgT rk and Rvqk ¼ pkT bQ cgT rk are the residues of input-output pair ΔPw  ΔVw and ΔQw  ΔVw of the open-loop state-space model of the power system respectively, which is given by (4.3). From Fig. 4.2, the characteristic equation of closed-loop power system with the DFIG can be obtained to be 1 ¼ Hdp ðλÞGvp ðλÞ þ Hdq ðλÞGvq ðλÞ

ð4:24Þ

For the closed-loop EOM of concern, ^λ i , from Eqs. (4.23) and Eq. (4.24), it can have n P k¼1

n P

Rvpk Hdp ^λ i þ Rvqk Hdq ^λ i ^λ i λk ^ k¼1 λ i λk ¼1



^ 1  dvp Hdp λ i  dvq Hdq ^λ i

ð4:25Þ

114

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Multiplying both sides of (4.25) by ^λ i  λi and taking the limit ^λ i ! λi , it can have [1].

Rvpi Hdp ðλi Þ þ Rvqi Hdq ðλi Þ Δλi  lim ^λ i  λi ¼ ^ 1  dvp Hdp ðλi Þ  dvq Hdq ðλi Þ λ i !λi

ð4:26Þ

where Rvpi and Rvqi are the residues corresponding to λi for the input-output pair ΔPw  ΔVw and ΔQw  ΔVw in the open-loop state-space model of the power system respectively. For the open-loop EOM of concern, λi ¼  ξi þ jωi, normally |ωi| >> |ξi|. Thus, Hdp(λi)  Hdp(jωi) and Hdq(λi)  Hdq(jωi). Hence from Eq. (4.26), following IDI is proposed to indicate the impact of dynamic interactions introduced by the DFIG on the EOM of concern Δλi  IDIðjωi Þ ¼

Rvpi Hdp ðjωi Þ þ Rvqi Hdq ðjωi Þ 1  dvp Hdp ðjωi Þ  dvq Hdq ðjωi Þ

ð4:27Þ

The IDI derived in (4.27), in fact, is the modal sensitivity to the variations of dynamic interactions between the DFIG and power system because of the derivation of Eq. (4.26). Magnitude of IDI(jωi) indicates the level of impact of dynamic interactions on the EOM of concern. Phase of IDI(jωi) specifies the direction of impact of dynamic interactions, i.e., positive to improve or negative to reduce the damping of EOM of concern. For example, if |phase of IDI(jωi)| < 900, the IDI indicates that ^λ i is on the right hand side of λi on the complex plane. The dynamic interactions introduced by the DFIG degrade the damping of the EOM of concern. As the modal sensitivity, the IDI can be used as a pre-screening indicator when large numbers of scenarios of integration and operation of the DFIGs are examined. After potentially detrimental cases to power system stability are indicated by the IDI, more detailed and in-depth examination can be carried out to determine the impact of the DFIG more accurately. Like any index of modal sensitivity, the IDI can be an approximation of the actual impact, Δλi ¼ ^λ i  λi , as long as the dynamic interactions between the DFIG and power system, i.e., ΔPw þ jΔQw, are small. Normally in a power system, dynamic variations of ΔVw is very small. Hence, from Eq. (4.5), it can be seen that ΔPw and ΔQw should usually be small. This explains why the DFIG usually exhibits very limited dynamic response to the small disturbances in the power system due to the fast converter control implemented by the DFIG. Small dynamic interactions between the DFIG and power system ensures that usually, the IDI is a good approximation of the actual impact of dynamic interactions, i.e., Δλi ¼ ^λ i  λi . As a modal sensitivity, estimation error by the IDI may possibly become considerable when the dynamic interactions introduced by the DFIG are not small and when the IDI is not a constant. It is important to identify such case that the estimation error of the IDI on Δλi ¼ ^λ i  λi may be considerable when the IDI is applied. The identification establishes the boundary about the reliability of the IDI. Beyond the

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

115

boundary, other means of more detailed examination can be conducted in addition to the application of the IDI. A way to identify the case that the IDI might be unreliable is introduced as follows. Mathematical derivation of Eq. (4.26) implicitly means that



Hdp ^λ i ! Hdp ðλi Þ, Hdq ^λ i ! Hdq ðλi Þ, when ^λ i ! λi

ð4:28Þ

Hence, when the IDI is used as an approximation of the impact, i.e., IDIðjωi Þ  Δλi ¼ ^ λ i  λi , it should have



Hdp ^λ i  Hdp ðλi Þ, Hdq ^λ i  Hdq ðλi Þ

ð4:29Þ

Thus, if Hdp(jω) or/and Hdq(jω) changes dramatically around ωi, the approximation of (4.29) may yield considerable error such that IDIðjωi Þ  Δλi ¼ ^λ i  λi may be of considerable error of estimation. Such unusual case of radical changes of Hdp(jω) or/and Hdq(jω) around ωi is possible and can be elaborated as follows by using Hdp(jω) as an example. Denote λd ¼  ξd þ jωd as a complex pole of Hdp(s). It is a complex eigenvalue of open-loop state matrix of the DFIG, Ad in Eq. (4.19). Since |Hdp(λd)| ¼ | Hdp(ξd þ jωd)| ! 1, it is possible that |Hdp(jωi)| may increase significantly when ωi is close to ωd. In this case, from Eq. (4.5) it can be seen that ΔPw may become considerable around the oscillation frequency ωi, indicating substantial ^ dynamic interactions introduced by the DFIG. Thus, jΔλi j ¼ λ i  λi may possibly increase radically, indicating considerable impact of dynamic interactions introduced by the DFIG on the EOM of concern. This is the case that the IDI may be of considerable estimation error and even become unreliable. The case can be identified from the magnitude of frequency response of the DFIG, |Hdp(jω)|, around the oscillation frequency ωi, as being illustrated by Fig. 4.7. In Fig. 4.7, |Hdp(jω)| changes radically within the interval [ω1, ω2] where ωd locates. According to the analysis made above, if |Hdp(jω)| is at point A, the IDI should be a good approximation of the impact. If |Hdp(jω)| is at point B, estimation error by the IDI may be considerable. Hence, [ω1, ω2] defines the boundary. Inside Fig. 4.7 Identification of estimation error of the IDI

Hdp ( jw )

B

A

w w1

wd

w2

116

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

[ω1, ω2], the IDI may become unreliable with considerable estimation error and more detailed examination by other means is required. Therefore, the IDI can be used jointly with |Hdp(jω)| in the pre-screening to find out potential cases that the DFIG may threaten the system stability.

4.2.1.3

Further Discussions About the IDI

The IDI introduced in the previous subsection can provide an indication of the impact of dynamic interactions introduced by the DFIG, both the scale and direction, on the EOM of concern. Dynamic interaction of the DFIG with power system EOMs can be estimated by calculating the participation factors of state variables of the DFIG for the EOM of concern. However, the participation factors are a type of dimensionless index and hence can only provide the estimation of scale of dynamic interactions. Impact (both level and direction) of dynamic interactions on the EOM of concern cannot be assessed directly from computational results of the participation factors. Modal sensitivity to a certain parameter of the DFIG can indicate how variation of the parameter affects the EOM of concern in both level and direction. Thus, an index of modal-to-parameter sensitivity can show how a factor in the DFIG affects the EOM of concern in both level and direction. However, like the participation factors, index of modal-to-parameters sensitivity relies on the computational results and is applicable for case-by-case study. Normally, it would be difficult to draw generallyapplicable conclusions by applying the index of modal-to-parameters sensitivity and the participation factors. In the analytical expression of the proposed IDI of Eq. (4.27), the frequency response of the DFIG at the frequency of the EOM of concern, i.e., Hdp(jωi) and Hdq(jωi), is explicitly linked with the impact of dynamic interactions introduced by the DFIG. This is the main merit of the IDI as compared with the participation factors and modal-to-parameters sensitivity. The other main merits of the IDI are further elaborated as follows. Firstly, the analytical expression makes theoretical analysis possible to draw generally-applicable conclusion. This will be demonstrated in the next subsection. Secondly, when the impact of a wind farm on power system small-signal angular stability is examined, so far the equivalent dynamic model of a single DFIG has been used to represent dynamics of the wind farm. Adequacy of representing dynamics of the wind farm by dynamic model of the single DFIG has not been carefully investigated. The analytical expression of the IDI establishes the condition under which the equivalence of dynamic model of single DFIG and the wind farm for studying the impact of the wind farm. According to (4.27), it can be concluded that an equivalent dynamic model of single DFIG can be used to examine the impact of the wind farm on the EOMs of concern, as long as the frequency response of this equivalent DFIG model around the frequency of EOMs of concern approximately matches the frequency response of wind farm. Study cases are presented in Sect. 4.2.3 to demonstrate this aspect of the merit of the IDI. It is shown that when a single

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

117

DFIG with increased capacity is simply used to represent a wind farm, wrong result of assessment on the impact of the wind farm occurs. After the dynamic model of the single DFIG is adjusted to meet the condition of equivalence established above, correct assessment is obtained. Thirdly, for a practical large-scale power system, dimension of state matrix could be very high. Errors of system parameters caused by various factors may result in a state-space model of the power system with errors. Those can cause considerable problem in applying the modal analysis to assess the impact of the DFIG on the EOM of concern. In the IDI shown by (4.27), line parameters, dvp and dvq, and the residues, Rvpi and Rvqi, can be obtained from measuring data in field. Hdp(jw) and Hdq(jω) can also be obtained from frequency-response test of the DFIG. Hence, the IDI can become model-independent and be calculated from measuring data. Thus, the problems of model errors and computational complexity due to high-dimensional state matrix in modal analysis can be avoided. The IDI is of two computational advantages as follows. First, from (4.27) it can be seen that when the impact of changes of some key parameters of the DFIG is examined, proposed index, IDI(jωi), can be calculated from changes of Hdp(jωi) and Hdq(jωi) without need to perform eigensolution of closed-loop state matrix. This is very useful when the control parameters of the DFIG are tuned and the effect of tuning on the EOMs of concern need to be checked. In practical application, it is usually much easier to obtain the frequency response of the DFIG at given oscillation frequencies of the EOMs than to derive parametric dynamic model, which is normally needed for the calculation of the modal-to-parameter sensitivity and the participation factors. Second, compared with the method of direct calculation to find Δλi ¼ ^λ i  λi from eigensolution of closed-loop and open-loop state matrix, computational burden of the IDI is significantly reduced when the scale of power system is large and there are large numbers of the DFIGs to be examined. For each of the DFIGs being examined, direct calculation needs to perform eigensolution of closedloop state matrix and to select correct ^λ i corresponding to λi from the results of eigensolution. Hence, for N DFIGs to be examined, eigensolution of closed-loop state matrix has to be carried out for N times. Especially, selection of correct ^λ i from the results of eigensolution of closed-loop state matrix often is not a straightforward work when the scale of power system is large. Whilst computation of the IDI needs only to perform eigensolution of open-loop state matrix for once, no matter how many DFIGs needs to be examined.

4.2.2

The Impact of Dynamic Interactions Introduced by the DFIG

When the wind power penetration varies, impact of both the load flow change and dynamic interactions introduced by the DFIG may vary accordingly. In (4.22), λni is the EOM of concern when the DFIG is not connected; thus the active power output

118

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

from the DFIG, Pw0, is zero. λi is the EOM of concern when the DFIG is represented by a constant power source, Pw0. Change of Pw0 is balanced by the change of the active power output from the synchronous generators at the slack nodes in a power system if the total system load keeps unchanged. Hence, the impact of load flow change brought about by the DFIG on the EOM of concern, λi  λni in Eq. (4.22), is equivalent to that of change of active power output from the synchronous generators at slack nodes in the power system from Pg0 to Pg0  Pw0. Normally, the more the active power output from the synchronous generators varies, the more the EOMs are affected. Hence, it is easy to understand that when the level of wind power penetration is higher (bigger Pw0), the impact of load flow change introduced by the DFIG on the EOM of concern, i.e., λi  λni in Eq. (4.22), is bigger. Of course, if the modal sensitivity to the variation of active power output from the synchronous generators at slack nodes in the power system is always very small, |λi  λni| will be small, implying the limited impact of load flow change introduced by the DFIG on the EOM of concern, even if the wind power penetration is high. In this section, impact of dynamic interactions introduced by the DFIG, i.e., Δλi ¼ ^λ i  λi in Eq. (4.22), is examined. In order to draw generally-applicable conclusions, a simplified transfer function model of the DFIG is derived from the detailed 14th-order dynamic model of the DFIG introduced in Chap. 2. The examination then is carried out by using the IDI and the simplified transfer function model of the DFIG.

4.2.2.1

Simplified Transfer Function Model of the DFIG

Direction of terminal voltage of the DFIG, Vw, is taken as that of q axis of d  q coordinate of the DFIG as shown by Fig. 2.10. Voltage and flux equations of the rotor windings of the DFIG are given as (2.56) and (2.57) in Chap. 2. By omitting the first subscript d in (2.56) and (2.57), those equations are rewritten below.

dψrd ¼ ω0 Vrd þ sw ψrq þ Rr Ird dt

dψrq ð4:30Þ ¼ ω0 Vrq  sw ψrd þ Rr Irq dt ψrd ¼ Xm Isd  Xrr Ird , ψrq ¼ Xm Isq  Xrr Irq where sw is the rotor slip, Rr and Xrr are the resistance and self-inductance of the rotor windings, Xm is the magnetizing inductance, ψrd, ψrq, Vrd, Vrq, Ird and Irq are the flux, voltage and current of the rotor windings expressed in the d  q coordinate of the DFIG respectively, ω0 is the synchronous speed. Ignoring the transient and resistance of stator windings and from (2.55) and (2.57) (note: the first subscript d in (2.55) and (2.57) is omitted), Vwd ¼ 0 ¼ ψsq ¼ Xss Isq þ Xm Irq Vwq ¼ Vw ¼ ψsd ¼ Xss Isd  Xm Ird

ð4:31Þ

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

119

where Xss is the self-inductance of the stator windings, Irq Isd, Isq, Vwd and Vwq are the d and q component of stator current and terminal voltage of the DFIG respectively. From (4.30) and (4.31)       Xrr X2m d X2m Vrd ¼  Rr þ  Ird þ sw Xrr  Irq ω0 ω0 Xss dt Xss ð4:32Þ       Xrr X2m d X2m Xm Vrq ¼  Rr þ  Vw Irq  sw Xrr  Ird þ sw ω0 ω0 Xss dt Xss Xss Configuration of vector control system of the RSC of the DFIG is shown by Fig. 2.5, which is displayed as Fig. 4.8. From (4.32) and Fig. 4.8, an equivalent system to that shown by Fig. 4.8 can be obtained and shown by Fig. 4.9. From Fig. 4.9, it can have Xss ref Gq ðsÞΔIsq Xm   Xss ref Vw ΔIrd ¼ Gd ðsÞ  ΔIsd  Xm Xm ΔIrq ¼ 

ð4:33Þ

where Gq(s) and Gd(s) are the transfer function from ΔIrqref and ΔIrdref to ΔIrq and ΔIrd respectively in Fig. 4.9. Linearization of (4.31) is Xm 1 ΔIrd  ΔjVw j Xss Xss Xm ΔIsq ¼  ΔIrq Xss

ΔIsd ¼ 

active power control outer loop

Psref

I sqref

+ – Ps

Qsref

q-axis current control inner loop

K pq ( s)

I sdref

+ – Qs

K pd ( s)

reactive power control outer loop

I rqref

X – ss Xm

X – ss Xm

– sw ( X rr –

+ – I rq

+

I rdref

– Vw Xm

ð4:34Þ

+ – I rd

K iq ( s )

K id ( s)

d-axis current control inner loop

Fig. 4.8 Configuration of the RSC vector control system of the DFIG

X m2 X ) I rd + sw m Vw X ss X ss



+

– + sw ( X rr –

X m2 ) I rq X ss

Vrq

Vrd

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

120

Gq ( s) I sqref

Psref +

K pq ( s)

– Pe I sdref

Qsref +

K pd ( s)

– Qs

I rqref

X – ss Xm

X – ss Xm

+

+ –

I rdref

– Vw Xm

1

I rq

K iq ( s)

X2 Rr + ( – m )s w0 w0 X ss

Kid ( s)

X m2 Rr + ( )s – w0 w0 X ss

X rr

1

+ – I rd

I rd

X rr

Gd ( s)

Fig. 4.9 Equivalent system of RSC vector control system of the DFIG

From Fig. 4.9, (4.33) and (4.34), it can have ref ¼ Gq ðsÞKpq ðsÞðΔPs Þ ΔIsq ¼ Gq ðsÞΔIsq

1 ½Gd ðsÞ  1ΔVw Xss 1 ¼ Gd ðsÞKpd ðsÞðΔQs Þ þ ½Gd ðsÞ  1ΔVw Xss ref þ ΔIsd ¼ Gd ðsÞΔIsd

ð4:35Þ

where Ps and Qs are the active and reactive power output from the stator side of the DFIG respectively (see Fig. 4.1) and they are Ps ¼ Vwd Isd þ Vwq Isq ¼ Vw Isq Qs ¼ Vwq Isd  Vwd Isq ¼ Vw Isd

ð4:36Þ

Linearization of above equations is ΔPs ¼ Vw0 ΔIsq þ Isq0 ΔVw ΔQs ¼ Vw0 ΔIsd þ Isd0 ΔVw

ð4:37Þ

Active and reactive power output from the GSC of the DFIG is Pr (see Fig. 4.1) and Pr   swPs, Qr  0 [10–12], i.e., ΔPr  sw0 ΔPs  Ps0 Δsw , ΔQr  0

ð4:38Þ

From (4.35), (4.37) and (4.38), it can have   Vw0 Vw0 ΔQw ¼ ΔQs ¼ Hdq ðsÞΔVw ¼ Isd0  Vw0 Gd ðsÞKpd ðsÞ þ G d ðsÞ  ΔVw Xss Xss ð4:39Þ

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

121

Rotor motion equations of the DFIG are (2.58) when the rotor is represented by a two-mass shaft rotational system. The rotor can be simply modelled as a one-mass shaft. The torque can be considered to be same to the power. Thus (2.58) is simplified to obtain the following linearized rotor motion equation of the DFIG [10–12]. 1 Δsw ¼ ðΔPs þ ΔPr Þ ð4:40Þ Js where J is the constant of rotor inertia. From (4.38) and (4.40),    Ps0 Ps0 ΔPr ¼  sw0 þ 1þ ΔPs ¼ JðsÞΔPs Js Js

ð4:41Þ

From (4.35), (4.37), (4.38) and (4.41), it can have ΔPw ¼ ΔPs þ ΔPr ¼

Isq0 ½1 þ JðsÞ ΔVw ¼ Hdp ðsÞΔVw 1 þ Vw0 Gq ðsÞKpq ðsÞ

ð4:42Þ

Thus, the simplified transfer function model of the DFIG, Hdq(s) and Hdp(s), are derived as shown by Eqs. (4.39) and (4.42) respectively.

4.2.2.2

Impact of Dynamic Interactions Affected by the Level of Wind Power Penetration

Both the q-axis and d-axis current control inner loop in the RSC control system of the DFIG is derived from the electromagnetic transient of the rotor windings of the DFIG. The transient is electromagnetic and much faster than the electromechanical dynamic interactions between the DFIG and power system, which is the focus of discussions in this section. If this fast electromagnetic transient is ignored, Ird ¼ Irdref and Irq ¼ Irqref , which is equivalent to Gd(s) ¼ 1 and Gq(s) ¼ 1 as it can be seen from Fig. 4.9. Thus, by ignoring the electromagnetic transient of the rotor windings of the DFIG with Gd(s) ¼ 1 and Gq(s) ¼ 1, transfer function model of the DFIG shown by (4.39) and Eq. (4.42) are simplified to be Hdp ðsÞ 

Isq0 ½1 þ JðsÞ 1 þ Vw0 Kpq ðsÞ

ð4:43Þ

Hdq ðsÞ 

Isd0 1 þ Vw0 Kpd ðsÞ

ð4:44Þ

In the output equation of (4.3), dvp should be very small as the magnitude of busbar voltage is usually affected little by variation of line active power. Hence, the IDI given by (4.27) can be simplified to be

122

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Δλi  IDIðjωi Þ ¼

Rvpi Hdp ðjωi Þ þ Rvqi Hdq ðjωi Þ 1  dvq Hdq ðjωi Þ

ð4:45Þ

At steady state, reactive power output from the DFIG is Qw0 ¼ Vw0Isd0. Normally, the DFIG operates with high power factor such that Qw0  0. Hence, it can have Isd0  0. From (4.44), it can be seen that Hdq(jωi)  0. Hence, the IDI given in (4.45) can be further simplified) to be IDIðjωi Þ  Rvpi Hdp ðjωi Þ

ð4:46Þ

From Pr   swPs, Qr  0 [10–12], it can have Pw0 ¼ Ps0 þ Pr0  ð1  sw0 ÞPs0 ¼ ð1  sw0 ÞVw0 Isq0

ð4:47Þ

Thus, when the amount of active power output from the DFIG at steady state increases, Isq0 increases. From (4.43), it can be seen that |Hdp(jωi)| increases accordingly with the increase of Pw0. Hence, it can be concluded that increase of wind power penetration will lead to the increase of scale of the impact of dynamic interactions introduced by the DFIG, which is ^λ i  λi .

4.2.2.3

Impact of DFIG’s Terminal Voltage Control Vs. Reactive Power Control

A DFIG can be equipped with the terminal voltage control instead of reactive power control. In this case, the input signal in the reactive power control outer loop of Fig. 4.8, Qsref  Qs , is replaced by Vwref  Vw . In this case, from the reactive power control outer loop in Fig. 4.8, it can have ref ΔIsd ¼ Kpd ðsÞΔVw

ð4:48Þ

From (4.33), (4.34) and (4.48), it can have ΔIsd ¼ Gd ðsÞKpd ðsÞðΔVw Þ þ

1 ½Gd ðsÞ  1ΔVw Xss

ð4:49Þ

From (4.37), (4.38) and (4.49) ΔQw ¼ ΔQs ¼ Hdq ðsÞΔVw  Vw0 Vw0 ¼ Isd0  Vw0 Gd ðsÞKpd ðsÞ þ Gd ðsÞ  ΔVw Xss Xss

ð4:50Þ

By ignoring the electromagnetic transient of the rotor windings of the DFIG with Gd(s) ¼ 1, from (4.50) it can have

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

Hdq ðsÞ ¼ Isd0  Vw0 Kpd ðsÞ

123

ð4:51Þ

Impact of the DFIG’s terminal voltage control as compared with the reactive power control can be estimated by examining the changes of Kpd(s) as follows. Firstly, consider the extreme case that the gain value of outer loop PI controller is ref zero to have Kpd(s) ¼ 0. In this case, terminal voltage control results in Isd ¼ 0 as shown by Fig. 4.8. A well-designed reactive power control with the unity power ref ¼ 0 if the controller’s transient is ignored. Hence, factor (Qsref ¼ 0) also provides Isd it is expected that the impact of the DFIG with a normal reactive power control (with high power factor and well-designed PI controller) on the EOM of concern should be approximately equal to that of the DFIG with the terminal voltage control with gain value of PI controller being equal to zero. Denote λR as the EOM of concern in this case. Secondly, consider the case of a well-designed outer loop PI controller for the DFIG’s terminal voltage control with the EOM of concern being λV. The difference between the impact of dynamic interactions as caused by the reactive power and terminal voltage control can be determined by comparing λR and λV. The comparison can be carried out by increasing the gain value of outer loop PI controller of the DFIG’s terminal voltage control from zero and to estimate the change of the impact by use of (4.45). From (4.51), it can be seen that |Hdq(jωi)| will increase when the gain value of outer loop PI controller increases from zero to result in the EOM of concern to move from λR towards λV on the complex plane. The trend of motion from λR towards λV thus can be estimated from (4.45) to be H ðjω Þ

lim λV  λ R  lim Δλi ¼ lim jHdq ðjωi Þj!1 jHdq ðjωi Þj!1 jHdq ðjωi Þj!1

Rvpi Hdpdq ðjωi Þ þ Rvqi i

1dvp Hdp ðjωi Þ Hdq ðjωi Þ

 dvq

¼

Rvqi dvq

ð4:52Þ From (4.52), it can be seen that if (Real part of Rvqi/dvq) > 0, λV is on the left hand side of λR on the complex plane. Thus, damping of the EOM of concern is improved if the DFIG is equipped with terminal voltage control instead of reactive power control, and vice versa. Therefore, Rvqi/dvq can be used as an indicator of DFIG control (IDC) to approximately determine whether the terminal voltage control implemented by the DFIG is more beneficial to power system small-signal angular stability than the reactive power control or not. Calculation of the IDC is simple as no knowledge of DFIG’s dynamic model is required. Hence, it is useful at the stage of planning the integration of a wind farm when the details of the wind farm are unknown and the guidance on the selection of types of DFIG’s control is needed. The IDC is derived from the IDI and hence also a modal sensitivity. The IDC is for the comparison of the effect of |Hdq(jωi)| on the IDI between the reactive power and terminal voltage control implemented by the DFIG. It is derived on the basis that |Hdq(jωi)| of the terminal voltage control is much greater than |Hdq(jωi)| of the reactive power control in an extreme case that |Hdq(jωi)| ! 1. Hence, the IDC is

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

124

an indication of the trend of the modal sensitivity. It is not an approximation of the impact of either reactive power control or terminal voltage control implemented by the DFIG. The greater |Hdq(jωi)| of the terminal voltage control is than |Hdq(jωi)| of the reactive power control, the more accurate the IDC is.

4.2.3

Example 4.2

4.2.3.1

Impact of Dynamic Interactions as Affected by Level of Wind Power Penetration

Figure 4.10 shows the configuration of the example New England power system. Parameters and models of the system and synchronous generators given in [13] were used. In all the study cases presented in this section, detailed 14th-order dynamic model of the DFIG including its control systems was used [10–12]. The DFIGs operated with unity power factor. Parameters of the example are given in Appendix 4.2. The EOM of concern was an inter-area EOM between the tenth synchronous generator and the rest of the synchronous generators in the example New England power system as the inter-area EOM of concern was most lightly damped. 8 37

1 30

28

26

25

29

2 27

18 1

24

17

9

16

10

6 3

35

15

39

21

14 4 DFIG

38

5

6

DFIG

20 11

31 2

23

12

9

7

22

19

33

13 4 34

10

5

8 32 3

Fig. 4.10 Configuration of New England power system

36 7

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

125

Two DFIGs were connected to the example New England power system at node 22 and 39 respectively. Table 4.5 presents the computational results of the proposed index, IDI(jωi), when the active power output from the DFIGs Pw0 ¼ 5 p. u. and Pw0 ¼ 10 p. u. respectively for both DFIGs. Firstly, it can be seen from Table 4.5 that with the increase of active power output from both DFIGs, level of the impact of dynamic interactions on the inter-area EOM of concern increased as indicated by IDI (jωi). Correctness of IDI(jωi) was confirmed by direct calculation of ^λ i  λi from eigensolution of closed-loop and open-loop state matrix. Secondly, it can be seen from Table 4.5 that |Hdp(jωi)| increased with the increase of wind power penetration. |Hdq(jωi)| was approximately equal to zero. Hence, computational results of |Hdp(jωi)| and |Hdq(jωi)| demonstrate the correctness of analysis carried out in Sect. 4.2.2.2 to draw the generally-applicable conclusion from the proposed index that level of impact of dynamic interactions increases with increase of wind power penetration. Thirdly, IDI(jωi) presented in Table 4.5 indicated that the impact of the DFIG at node 22 was positive as the damping of inter-area EOM of concern was improved ( ^λ i  λi þ IDIðjωi Þ). Whilst the impact of the DFIG at node 39 was negative to reduce the damping of inter-area EOM of concern. In Table 4.5, PFdfig is the sum of the participation factors of all the state variables of the DFIGs. Computational results of PFdfig confirmed the increase of dynamic interactions between the DFIGs and power system when the wind power penetration increased, though PFdfig cannot indicate the direction of the impact of dynamic interactions.

Fourthly, the modal-to-Pw0 sensitivity, ∂ ^λ i  λi =∂Pw0 , can also be used as an index to estimate the impact of dynamic interactions introduced by the DFIGs on the inter-area EOM of concern when the level of wind power penetration varied. Computational results of this modal-to-Pw0 sensitivity are given in Table 4.5. From Table 4.5, it can be seen that both level and direction of impact of DFIGs indicated by the modal-to-Pw0 sensitivity were as same as those by IDI(jωi), confirming the correctness of proposed index, IDI(jωi). However,

it is very difficult to draw the generally-applicable conclusion by use of ∂ ^λ i  λi =∂Pw0 that impact of dynamic interactions increases with increase of wind power penetration which was obtained from proposed index IDI(jωi). Finally, IDI(jωi) was evaluated for the DFIG at node 22 and 39 when the level of wind power penetration changed from no wind (Pw0 ¼ 0 p. u.) to the maximum wind capacity allowed by the system operation (Pw0 ¼ 15 p. u., beyond which the load flow calculation of the example New England power system did not converge). Figures 4.11 and 4.12 show the results of evaluation with confirmation from the direct calculation of ^λ i  λi . From Figs. 4.11 and 4.12, it can be seen that the indication by IDI(jωi) on the impact of dynamic interactions introduced by the DFIGs at node 22 and 39 is correct. Estimation error of IDI(jωi) on the actual impact, ^λ i  λi , is small. Figure 4.13 presents the magnitude of frequency-response, |Hdp(jω)|, for the DFIG at node 22 and 39 respectively. From Fig. 4.13, it can be seen that |Hdp(jω)| is outside the boundary[ω1, ω2], within which the IDI may become unreliable with

Node 22 5 0.02437 þ 0.018577j 0.0286 þ 0.0133j

0.010632 2.8245 2.1552e05 0.00576 þ 0.0022j

DFIG location Pw0 IDI(jωi) b λ i  λi

PFdfig |Hdp(jωi)| |Hdq(jωi)|

∂ b λ i  λi

∂ Kpp þ Kpi 0.029117 6.0514 2.202e-05 0.01312 þ 0.0097j

10 0.0448 þ 0.0462j 0.06521 þ 0.0324j

Table 4.5 Impact of dynamic interactions when levels of wind power penetration varied

0.00048045 3.056 2.7174e05 0.00014 þ 0.0001j

Node 39 5 0.001652 þ 0.000526j 0.00152 þ 0.0009j

0.0016741 6.8517 3.0923e-05 0.0012 þ 0.0013j

10 0.004894 þ 0.00223j 0.00395 þ 0.004j

126 4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . . 160

0.3

IDI ( jωi ) ( lˆ – l )

Magnitude 0.25

i

Phase angle

155 150

i

0.2

Pw0 = 5

145

Pw0 =10

0.15

140

Pw0 = 5

0.1

Pw0 =10

135

IDI ( jωi ) (lˆ – l )

130 0.05 0

127

125 Active power output from the DFIG 7

5

3

1

11

9

13

120

15

i

i

Active power output from the DFIG 5

3

1

7

9

13

11

15

Fig. 4.11 Evaluation of the IDI for the DFIG at node 22 when wind power penetration varied

0.018

70

Magnitude

IDI ( jωi ) (lˆ – l )

0.015

i

0.012

Phase angle

60 50

i

Pw0 =10

Pw0 =10

40

0.009

30

Pw0 = 5

20

0.006

10

0.003 0

IDI ( jωi ) (lˆ – l )

Pw0 =5

0

i

Active power output from the DFIG 7

5

3

1

9

11

13

i

Active power output from the DFIG

15

-10

1

3

5

7

9

11

13

15

Fig. 4.12 Evaluation of the IDI for the DFIG at node 39 when wind power penetration varied

70

70

ωd

ω1

60

ω2

A

30 20

10

10 0

1

2

3 4 5 6 Frequency (Hz)

7

DFIG at node 22

8

Pw0 =10 Pw0 = 5

40

20

0

ω2

50

40 30

ωd

ω1

60 Magnitude

Magnitude

50

Pw0 =10 Pw0 = 5

0

A

0

1

2

6 5 4 3 Frequency (Hz)

7

8

DFIG at node 39

Fig. 4.13 Frequency response of |Hdp(jω)|

considerable error. Around ωi, |Hdp(jω)| does not change radically. Results presented in Figs. 4.11, 4.12 and 4.13 confirm the analysis made in Sect. 4.2.1.2 about the estimation error of the IDI.

128

4.2.3.2

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Dynamic Interactions as Affected by Parameters of RSC Active Power Controller

Table 4.6 presents the computational results of proposed index, IDI(jωi), when setting of PI parameters, Kpp and Kip, of the active power control outer loop of the RSC shown by Fig. 4.8 changed. From Table 4.6, it can be seen that with increased values of PI parameters, |Hdp(jωi)| decreased. Impact of dynamic interactions introduced by both DFIGs at node 22 and 39 on the inter-area EOM of concern decreased accordingly as indicated by IDI(jωi), though directions of impact did not change. Indication by IDI(jωi) in Table 4.6 was confirmed being correct by the results of direct calculation of ^λ i  λi and computation of the participation factors, PFdfig. Computational results of the index of modal-to-parameters sensitivity are also presented in Table 4.6 for confirmation and comparison. Impact of changes of Kpp and Kip on the inter-area EOM of concern presented in Table 4.6 can be elaborated as follows. In Fig. 4.8 and (4.43), Kpq(jωi) ¼ Kpp þ Kpi/jω. Hence, |Kpq(jωi)| increased when Kpp and Kip increased. Accordingly, |Hdp(jωi)| decreased as it can be seen from (4.43). Subsequently, |IDI(jωi)| decreased as indicated by (4.46). Hence, increase of the gain values of active power PI controller of the RSC led to decreased impact of dynamic interactions introduced by the DFIG on the inter-area EOM of concern. Elaboration above demonstrates the advantage of proposed index in the establishment of explicit connection between impact and frequency response of the DFIG, which makes analysis possible. Hence, computational results obtained in Table 4.6 can be explained and understood. Obviously, it would be difficult to carry out the same analysis as demonstrated above by use of index of participation factors and the modal-to-parameters sensitivity. In addition, the proposed index is of certain computational advantage: When Kpp and Kip varied, values of Hdp(jωi) and Hdq(jωi) were calculated from the DFIG’s dynamic model such that IDI(jωi) was calculated directly by use of (4.27). Hence, there was no need to carry out the eigensolution of power system state matrix every time when Kpp and Kip varied as the open-loop state space model of power system shown by (4.3) did not change with the variations of Kpp and Kip. Whilst computation of the participation factors and the modal-to-parameters sensitivity needs to perform eigensolution of power system state matrix every time when Kpp and Kip varied.

4.2.3.3

Considerable Dynamic Interactions Introduced by the DFIG

In order to demonstrate and evaluate the proposed index, IDI(jωi), when the dynamic interactions introduced by the DFIG are unusually considerable, a DFIG was connected at node 19 in the example New England power system. The active power output from the DFIG at steady state was 10 p.u. The DFIG operated with unity power factor. When the DFIG is modelled as a constant power injection, Pw0 ¼ 10 p. u., the open-loop inter-area EOM of the example New England

Node 22 1.15 0.019826 þ 0.013702j 0.02897 þ 0.0029j

1.6469 2.2021 0.0172–0.00433j

DFIG at Kpp, Kip IDI(jωi) b λ i  λi

PFdfig |Hdp(jωi)|

∂ λbi  λi

∂ Kpp þ Kpi 0.029117 6.0514 0.0676–0.096j

0.1.5 0.0448 þ 0.0462j 0.06521 þ 0.0324j

Table 4.6 Impact of dynamic interactions as affected by control parameters of the RSC

0.091813 2.5821 0.00143  0.0006668j

Node 39 1.15 0.0016709 þ 0.0012639j 0.00106 þ 0.0021j

0.0016741 6.8517 0.008  0.002j

0.1.5 0.004894 þ 0.00223j 0.00395 þ 0.004j

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . . 129

130

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Fig. 4.14 Frequency response of |Hdp(jω)| to assess the reliability of the IDI

40 35

Magnietud

30 25

ω1

20

H dp ( jωi )

15 10 5 0 0.2

ωd ω2 1.2

2.2

3.2 4.2 5.2 Frequency (Hz)

6.2

7.2 8

power system was λi ¼  0.33834 þ j3.5342; thus ωi ¼ 0.56. Computational results of the IDI was IDI(jωi) ¼ 0.046267  j0.37745, indicating that the dynamic interactions introduced by the DFIG may degrade the damping of the inter-area EOM of concern. For examining the estimation error by the IDI, magnitude of frequency response of the DFIG was obtained and is shown by Fig. 4.14. From Fig. 4.14, |Hdp(jωi)| was identified to be at point B within the boundary, [ω1, ω2]. Hence, it is possible that the IDI may have considerable estimation error on the impact of dynamic interactions introduced by the DFIG at node 19. To confirm the assessment above using the IDI, the closed-loop inter-area EOM was calculate to be ^λ i ¼ 0:22662 þ j3:2592. Hence, ^λ i  λi ¼ 0:11172  j0:275. The actual damping degradation estimation error by the IDI was indeed considerable, though the IDI indicated correctly that the dynamic interactions introduced by the DFIG at node 19 degraded the damping of the inter-area EOM of concern. To eliminate the detrimental effect of dynamic interactions introduced by the DFIG, control parameters of the DFIG were tuned such that the complex pole of Hdp(s), λd ¼ ξd þ jωd, moved upwards on the complex plane and ωd increased. After the control parameters were tuned, magnitude of frequency response of the DFIG was obtained and is shown by Fig. 4.15. From Fig. 4.15, it can be seen that |Hdp(jωi)| decreased and is at point A outside the boundary, [ω1, ω2]. This implies that firstly, the scale of dynamic interactions introduced by the DFIG decreased. Secondly, the IDI should be more accurate. Computational results of the IDI for the case of point A in Fig. 4.15 was IDI (jωi) ¼ 0.013282  0.036996j, indicating reduced impact of dynamic interactions. Confirmation from the computation of closed-loop inter-area EOM of concern was ^λ i ¼ 0:33834 þ 3:5333j. Hence, ^λ i  λi ¼ 0:0162  0:0347j. Indeed, the dynamic interactions introduced by the DFIG decreased and the effect of damping degradation was reduced. In addition, the IDI was a good approximation of the actual impact ^λ i  λi .

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . . Fig. 4.15 Frequency response of |Hdp(jω)| with increased ωd

131

70 60

Magnitude

50 40

A: H dp ( jωi )

30 20 ω1

ωd

10 0

Fig. 4.16 Participation factors for the closed-loop inter-area EOM of concern

0

1

ω2

2

5 4 3 Frequecy (Hz)

6

7

8

Participation factors

0.6 0.5

Case A Case B

0.4 0.3 0.2 0.1 0

sg1 sg2 sg3 sg4 sg5 sg6 sg7 sg8 sg9 sg10 dfig Generators

To further confirm the examination carried out above, the participation factors for the closed-loop inter-area EOM were calculated and computational results are displayed in Fig. 4.16. In Fig. 4.16, sgi, i ¼ 1, 2, . . .10 indicates the participation factors of the ith synchronous generator and dfig is the sum of the participation factors of all the state variables of the DFIG. Figure 4.17 presents the results of non-linear simulation. At 1 s of simulation, mechanical torque of the DFIG dropped by 10% and the drop was recovered in 100 ms. In Figs. 4.16 and 4.17, case B is the first case discussed above when the IDI was of considerable error, indicating strong dynamic interactions introduced by the DFIG. Case A is the second case when the control parameters of the DFIG were tuned such that the dynamic interactions introduced by the DFIG decreased. From Fig. 4.16, it can be observed that in case B, the inter-area EOM of concern was participated considerably by the DFIG, indicating strong dynamic interactions introduced by the DFIG. From Fig. 4.17, it can be seen that damping degradation caused by the strong dynamic interactions is noticeable in case B when the active power output from the DFIG varies visibly.

132

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Fig. 4.17 Confirmation by non-linear simulation

-1

Case A Case B

d10 – d1 (degree)

-2 -3 -4 -5 -6 -7 -8

0

1

2

3

4

5 6 Time(s)

7

8

9

10

Active power output of DFIG

10.4 10.2 10 9.8 9.6

9.2

4.2.3.4

Case A Case B

9.4 0

1

2

3

4

5 6 Time(s)

7

8

9

10

Representation of Wind Farm by a Single DFIG

In Sect. 4.2.1.3, following analytical conclusion is made from the proposed index, IDI(jωi): An equivalent dynamic model of single DFIG can be used to examine the impact of wind farm on EOMs of concern, as long as frequency response of this equivalent DFIG model around frequency of electromechanical oscillations of concern approximately matches the frequency response of wind farm. In this sub-section, tests were carried out to demonstrate and validate this analytical conclusion. The tests considered that node 22 and 39 in the example New England power system were connected with a wind farm respectively instead of a DFIG. Each of wind farms consisted of 20 DFIGs, among which 13 DFIGs used the reactive power control and 7 DFIGs adopted terminal voltage control. Parameters and operating conditions of those 20 DFIGs were different and grouped into three clusters. Parameters and operating conditions of DFIGs in each cluster were same. Thus transfer function models of DFIGs in each cluster were same and denoted as Hdpk(s) and Hdqk(s), k ¼ 1, 2, 3.

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

133

Table 4.7 Impact of dynamic interactions introduced by the wind farms Locations of wind farms b λ i  λi (first test: model of wind farms included all 20 DFIGs) b λ i  λi (second test: model of single DFIG was used) b λ i  λi (third test: model of single DFIG with tuned parameters was used) ID( jωi)

Node 22 0.0303 þ 0.037j

Node 39 0.0028 þ 0.0035j

0.0130 þ 0.0023i

0.0017 þ 0.0024i

0.043635 þ 0.045579j

0.0037496 þ 0.0028666j

0.04288 þ 0.046001j

0.0032252 þ 0.0029898j

In the first test, dynamic model of each of wind farms was established with detailed 14th-order dynamic models of all 20 DFIGs being used. Impact of wind farms on the inter-area EOM of concern was assessed by direct calculation of Δλi ¼ ^λ i  λi from eigensolution of open-loop and closed-loop state matrix. Results of assessment are presented in the second row of Table 4.7. It can be seen that impact of the wind farm at node 22 was negative as real part of Δλi > 0 such that the dynamic interactions introduced by the wind farm caused the decrease of damping of interarea EOM of concern. Whilst impact of the wind farm at 39 was positive. In this test, frequency response of two wind farms at node 22 and 39, FP  k(jω) and FQ  k(jω), k ¼ 22, 39was obtained by simulation and are shown by solid curves in Figs. 4.18 and 4.19. In the second test, transfer function model of a single DFIG in the first cluster of wind farms, Hdp1(s) and Hdq1(s), with increased power output equal to that of wind farms was simply used to represent wind farms. Impact of wind farms was assessed by direct calculation of Δλi ¼ ^λ i  λ from eigensolutions of open-loop and closedloop state matrix. Results of assessment are presented in the third row of Table 4.7. Comparing results in the second and third row of Table 4.7, it can be seen that representation of the wind farms by dynamic model of single DFIG produced incorrect assessment of the impact of the wind farms. Frequency response of the single DFIG with increased capacity to represent the wind farms at node 22 and 39 is CkHdp1(jω) and CkHdq1(jω), k ¼ 22, 39, where Ck, k ¼ 22, 39 are constants to represent the increased capacity of the single DFIG. In Figs. 4.18 and 4.19, CkHdp1(jω), k ¼ 22, 39 are shown by dash-dotted curves. From Figs. 4.18 and 4.19, it can be seen that at ω ¼ ωi, both the magnitude and phase ofFP  k(jω) and FQ  k(jω), k ¼ 22, 39 are different with those of CkHdp1(jω) and CkHdq1(jω), k ¼ 22, 39. That is why wrong assessment on the impact of the DFIG at node 22 and 39 was made by using the model of single DFIG. In the third test, PI gains of the RSC active and reactive power control outer loop of the DFIGs, Kpp, Kip, Kpq and Kiq, were tuned. Transfer function model of the DFIGs, CkHdp1(jω) and CkHdq1(jω), k ¼ 22, 39, changed with the tuning of the gains. Objective of tuning PI gains is to make Ck Hdp1 ðjωÞ  FPk ðjωi Þ Ck Hdq1 ðjωÞ  FQk ðjωi Þ, k ¼ 22, 39

ð4:53Þ

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

134

8

Magnitude

7 FP–22 ( j ω) C22GP 0–22 ( jω) C22 H dp1 ( j ω)

6 5 4

ωi

3 2 25

Frequency(Hz)

0.4 0.6

0.8

1

Magnitude

15

1.6

ωi

5 0

1.4

FQ –22 ( jω) C22GQ 0–22 ( jω) C22 H dq1 ( jω)

20

10

1.2

Frequency(Hz)

0.4 0.6

0.8

1

1.2

1.4

1.6

65 60 55 50 45 40 35 30 25 110 100 90 80 70 60 50 40 30 20

Phase angle

FP–22 ( j ω) C22G P0 –22 ( jω) C22 H dp1 ( j ω)

ωi Frequency(Hz)

0.4 0.6

0.8

1

1.2

1.4

1.6

Phase angle FQ –22 ( j ω) C22GQ 0–22 ( jω) C22 H dq1 ( j ω)

ωi

Frequency(Hz)

0.4 0.6

0.8

1

1.2

1.4

1.6

Fig. 4.18 Frequency response of the wind farm at node 22

With (4.53), the frequency response of the equivalent DFIGs’ model around frequency of electromechanical oscillations of concern approximately matches the frequency response of the wind farms. Hence, following objective function was set up 

Obj ¼ min Ck Hdp1 ðjωi Þ  j F Pk ð jωi Þj þ ∠Ck Hdp1 ðjωi Þ  ∠F Pk ðjωi Þ þ Ck Hdq1 ðjωi Þ  FQk ðjωi Þ þ ∠Ck Hdq1 ðjωi Þ  ∠FQk ðjωi Þ , k ¼ 22, 39 ð4:54Þ The Hooke-Jeeves direct searching algorithm [14] was used to find the solution of the above objective function. Initial searching step was 0.01. Solutions of objective functions by the direct searching were Kpp ¼ 0.12, Kip ¼ 5.2, Kpq ¼ 3.2 and Kiq ¼ 58 for the DFIGs. Then, PI gains of the DFIGs were changed to the searching solutions. Transfer function models of the equivalent single DFIG for the wind farms at node 22 and 39 were changed to CkGP0  k(s), CkGQ0  k(s), k ¼ 22, 39. Using this equivalent model of single DFIG, computational results of Δλi ¼ ^λ i  λi are given in the fourth row of Table 4.7. In addition, computational results of IDI(jωi) by using CkGP0  k(jωi), k ¼ 22, 39 are presented in the fifth row of Table 4.7. Comparing results in the second, fourth and fifth row of Table 4.7, it can be seen that estimation of impact of wind farms by using the dynamic model of single DFIG with parameters being tuned according to (4.53) is very close to the correct results.

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

8

Magnitude

7

FP −39 ( j ω) C39G P0 −39 (j ω) C39 H dp1 ( j ω)

6 5 ωi

4 3 2

Frequency(Hz)

0.4

25

0.6

0.8

1

Magnitude

15 ωi

5 0

1.4 1.6

FQ −39 ( jω) C39GQ0 −39 (j ω) C39 H dq1 ( jω)

20

10

1.2

Frequency(Hz)

0.4

0.6 0.8

1

1.2 1.4

1.6

65 60 55 50 45 40 35 30 25

110 100 90 80 70 60 50 40 30 20

Phase angle

135

FP −39 ( j ω) C39G P0 −39 (j ω) C39 H dp1 ( j ω)

ωi

Frequency(Hz)

0.4

0.6

0.8

1

1.2

1.4 1.6

Phase angle FQ −39 ( jω) C39GQ 0−39 (jω) C39 H dq1 ( jω)

ωi

Frequency(Hz)

0.4 0.6

0.8

1

1.2

1.4 1.6

Fig. 4.19 Frequency response of the wind farm at node 39

Therefore, above tests confirmed the analytical conclusion drawn in Sect. 4.2.1.3 about how correctly the dynamics of a wind farm can be represented by the dynamic model of single DFIG.

4.2.3.5

The IDI Obtained by Using the Measuring Data

The IDI can be applied to assess the impact of wind farms from measuring data instead of using parametric model of power system. This merit of the IDI is demonstrated in this subsection. Measuring data were obtained from the simulation tests which were polluted with white noises. In the simulation, the wind farms at node 22 and 39 were constructed by 20 DFIGs respectively. Firstly, frequency-response tests at the terminals of the wind farms at node 22 and 39 were conducted. Thus, the values of frequency response of the wind farms at the frequency, ωi, were obtained and are given in the first and second row of Table 4.8. Secondly, a small step increment of active power and reactive power output from the wind farms at node 22 and node 39, ΔPw and ΔQw, was applied. Subsequent increments of the magnitude of terminal voltage at node 22 and 39, ΔVw, were measured. Thus, dvp and dvq were obtained from the ratio of ΔPw and ΔQw to ΔVw respectively. The results are presented in the third and fourth row of Table 4.8. To confirm the results obtained from the measuring data and given in Table 4.8,

FP  k(jωi) FQ  k(jωi) dvp dvq Rvpi Rvqi IDI(jωi)

Locations of wind farms

Node 22 Results from measuring data 2.9238 þ 4.0322j 1.1229 þ 17.838j 0.0210 0.0422 0.0013 þ 0.0079j 0.0009 ‑ 0.0044j 0.0274 þ 0.0318j Results from parametric model 2.9921 þ 4.0539j 1.1122 þ 17.545j 0.0160 0.0378 0.0018 þ 0.0112j 0.0014 ‑ 0.0064j 0.0429 þ 0.0460j

Table 4.8 Impact of dynamic interactions assessed by the measuring data Node 39 Results from measuring data 2.9825 þ 4.0809j 1.1018 þ 17.7j 0.0084 0.0301 0.0005–0.0031j 0.0009 þ 0.0011j 0.0076þ 0.0035j

Results from parametric model 3.6246 þ 4.2322j 0.99652 þ 17.284j 0.0082 0.0287 0.0006–0.00029j 0.0002 þ 0.0003j 0.0032 þ 0.0030j

136 4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

4.2 Impact of Dynamic Interactions Introduced by a Grid-Connected DFIG on. . .

137

Table 4.9 Computational results of index to estimate impact of DFIG’s terminal voltage control as compared with reactive power control Rvqi/dvq

Node 22 0.045899  0.23573j

Node 15 0.01666  0.082519j

Node 1 0.012428 þ 0.018207j

computational results from parametric model of the example New England power system with the wind farms are listed Table 4.8. Finally, residues were obtained from the measuring data by using the method proposed in [15] instead of parametric model of power system. Results are presented in the fifth and sixth row of Table 4.8. Hence from the measuring data, the IDI was calculated and given in the last row of Table 4.8. Comparing the results of the IDI obtained from the measuring data and parametric model given in Table 4.8, it can be seen that the IDI obtained from the measuring data provides correct assessment on the impact of the wind farms at node 22 and 39.

4.2.3.6

Terminal Voltage Control Vs. Reactive Power Control

In order to demonstrate and validate the IDC, Rvqi/dvq, to predict the impact of DFIG’s terminal voltage control as compared with the reactive power control, two more optional connecting locations for the DFIG at node 22 were examined. Those two optional locations were node 15 and node 1 in the example New England power system. Computational results of Rvqi/dvq are given in Table 4.9. From Table 4.9, it can be seen that (Real part of Rvqi/dvq) < 0 if the DFIG was connected at node 22 and 15, which indicated λR to be on the left hand side of λV on the complex plane. Hence, the impact of DFIG’s reactive power control was more beneficial to the damping of the inter-area EOM of concern as compared with the reactive power control. Whilst (Real part of Rvqi/dvq) > 0 for node 1. Hence, damping of inter-area EOM of concern was benefited if the terminal voltage control was adopted by the DFIG connected at node 1 instead of the reactive power control. To confirm the correctness of prediction from the IDC, as given in Table 4.9, firstly the DFIG was equipped with a well-designed reactive power controller. Gain value of PI reactive power controller in the outer loop (see Fig. 4.8) was set to be Kpq ¼ 5, Kiq ¼ 50. The DFIG was connected at node 1, 15 and 22 respectively. The inter-area EOM of concern was calculated and indicated as λR on the complex plane in Fig. 4.20. Secondly, the reactive power controller was replaced by the terminal voltage controller. Gain values of terminal voltage PI controller were set to be zero, i.e., Kpq ¼ 0, Kiq ¼ 0, such that Kpd(s) ¼ 0. The inter-area EOM of concern was calculated and indicated as λV(0) in Fig. 4.20. Finally, gain values of terminal voltage PI controller were increased from Kpq ¼ 0.0, Kiq ¼ 0 to Kpq ¼ 1000, Kiq ¼ 10000. Trajectories of inter-area EOM of concern, λV, starting from λV(0) are presented in Fig. 4.20. From Fig. 4.20, it can be seen that λV(0) almost overlaps λR and the direction and trend of the variation of λR towards λV agrees well with the prediction by the IDC

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

138

4.1

Bus 22

Bus 15 Bus 1

Imagiary axis

4.0 λR

3.9



Rvqi dvq

3.8 3.7 direction of λR moving

towards λV λR

3.6

trajectory of λV

λV (0)

3.5

-0.36

-0.34

-0.32

-0.3

-0.28

-0.26

Real axis Fig. 4.20 Trajectories of the EOM when PI parameters of voltage controller increased

presented in Table 4.9. Hence, Fig. 4.20 confirms the analysis carried out to derive the IDC in Sect. 4.2.2.3 and the correctness of relative position of λV in respect to λR as being predicted by the computational results given in Table 4.9.

Appendix 4.1: Data of Example 4.1 Example 16-Machine 68-Bus New York and New England Power System [3] (Tables 4.10, 4.11, 4.12)

Appendix 4.1: Data of Example 4.1

139

Table 4.10 Bus data Bus no. 1 3 4 7 8 9 12 15 16 18 20 21 23 24 25 26 27 28 29 33 36 37 39 40 41 42 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Active power consumption 2.527 3.22 5 2.34 5.22 1.04 0.09 3.2 3.29 1.58 6.8 1.74 1.48 3.09 2.24 1.39 2.81 2.06 2.84 1.12 1.02 60 2.67 0.6563 10 11.5 2.6755 2.08 1.507 2.0312 2.412 1.64 2 4.37 24.7 – – – – – – –

Reactive power consumption 1.1856 0.02 1.84 0.84 1.77 1.25 0.88 1.53 0.32 0.3 1.03 1.15 0.85 0.92 0.47 0.17 0.76 0.28 0.27 0 0.1946 3 0.126 0.2353 2.5 2.5 0.0484 0.21 0.285 0.3259 0.022 0.29 1.47 1.22 1.23 – – – – – – –

Active power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2.5 5.45 6.5 6.32 5.052 7 5.6

Reactive power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 0.4789 2.5134 2.7087 2.0328 1.6098 1.8488 0.4723 (continued)

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

140

Table 4.10 (continued) Bus no. 60 61 62 63 64 65 66 67 68

Active power consumption – – – – – – – – –

Reactive power consumption – – – – – – – – –

Active power generation 5.4 8 5 10 13.5 35.91 17.85 10 40

Reactive power generation 0.0897 0.2378 0.0543 0.2462 1.9941 8.9526 0.5714 0.70456 4.7864

Table 4.11 Line data Branch no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

From bus 1 1 2 2 2 3 3 4 4 5 5 6 6 6 7 8 9 10 10 10 12 12 13 14 15 16 16 16

To bus 2 30 3 25 53 4 18 5 14 6 8 7 11 54 8 9 30 11 13 55 11 13 14 15 16 17 19 21

R 0.007 0.0008 0.0013 0.007 0 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0 0.0004 0.0023 0.0019 0.0004 0.0004 0 0.0016 0.0016 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008

X 0.0822 0.0074 0.0151 0.0086 0.0181 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.025 0.0046 0.0363 0.0183 0.0043 0.0043 0.02 0.0435 0.0435 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135

B 0.3493 0.48 0.2572 0.146 0 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.113 0.1389 0 0.078 0.3804 0.29 0.0729 0.0729 0 0 0 0.1723 0.366 0.171 0.1342 0.304 0.2548

Transformer ratio 1 1 1 1 1.025 1 1 1 1 1 1 1 1 1.07 1 1 1 1 1 1.07 1.06 1.06 1 1 1 1 1 1 (continued)

Appendix 4.1: Data of Example 4.1

141

Table 4.11 (continued) Branch no. 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

From bus 16 17 17 19 19 20 21 22 22 23 23 25 25 26 26 26 28 29 9 9 9 36 34 35 33 32 30 30 1 31 33 38 46 1 47 47 48 35 37 43 44 39

To bus 24 18 27 20 56 57 22 23 58 24 59 26 60 27 28 29 29 61 30 36 36 37 36 34 34 33 31 32 31 38 38 46 49 47 48 48 40 45 43 44 45 44

R 0.0003 0.0007 0.0013 0.0007 0.0007 0.0009 0.0008 0.0006 0 0.0022 0.0005 0.0032 0.0006 0.0014 0.0043 0.0057 0.0014 0.0008 0.0019 0.0022 0.0022 0.0005 0.0033 0.0001 0.0011 0.0008 0.0013 0.0024 0.0016 0.0011 0.0036 0.0022 0.0018 0.0013 0.0025 0.0025 0.002 0.0007 0.0005 0.0001 0.0025 0

X 0.0059 0.0082 0.0173 0.0138 0.0142 0.018 0.014 0.0096 0.0143 0.035 0.0272 0.0323 0.0232 0.0147 0.0474 0.0625 0.0151 0.0156 0.0183 0.0196 0.0196 0.0045 0.0111 0.0074 0.0157 0.0099 0.0187 0.0288 0.0163 0.0147 0.0444 0.0284 0.0274 0.0188 0.0268 0.0268 0.022 0.0175 0.0276 0.0011 0.073 0.0411

B 0.068 0.1319 0.3216 0 0 0 0.2565 0.1846 0 0.361 0 0.531 0 0.2396 0.7802 1.029 0.249 0 0.29 0.34 0.34 0.32 1.45 0 0.202 0.168 0.333 0.488 0.25 0.247 0.693 0.43 0.27 1.31 0.4 0.4 1.28 1.39 0 0 0 0

Transformer ratio 1 1 1 1.06 1.07 1.009 1 1 1.025 1 1 1 1.025 1 1 1 1 1.025 1 1 1 1 1 0.946 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (continued)

142

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Table 4.11 (continued) Branch no. 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

From bus 39 45 50 50 49 52 42 41 31 32 36 37 41 42 52 1

To bus 45 51 52 51 52 42 41 40 62 63 64 65 66 67 68 27

R 0 0.0004 0.0012 0.0009 0.0076 0.004 0.004 0.006 0 0 0 0 0 0 0 0.032

X 0.0839 0.0105 0.0288 0.0221 0.1141 0.06 0.06 0.084 0.026 0.013 0.0075 0.0033 0.0015 0.0015 0.003 0.32

B 0 0.72 2.06 1.62 1.16 2.25 2.25 3.15 0 0 0 0 0 0 0 0.41

Transformer ratio 1 1 1 1 1 1 1 1 1.04 1.04 1.04 1.04 1 1 1 1

Table 4.12 Machine data Generator No. G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16

Xd 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8

X0d 0.3582 0.3252 0.283 0.2994 0.36 0.3543 0.2989 0.3537 0.2871 0.5867 0.6531 0.5524 0.4344 0.485 0.585 0.6589

Td0 10.2 6.56 5.7 5.69 5.4 7.3 5.66 6.7 4.79 9.37 4.1 7.4 5.9 4.1 4.1 7.8

Xq 1.242 1.7207 1.7098 1.7725 1.6909 1.7079 1.7817 1.7379 1.7521 1.2249 1.7297 1.6931 1.7392 1.73 1.73 1.6888

M 2.3333 3.9494 3.9623 6.5629 3.7667 2.9107 3.3267 2.915 2.0365 2.9106 2.0053 5.1791 9.0782 3 3 4.45

D 10 10 10 13 8 14 2 10 1.8 30 30 70 100 90 70 60

KA 10 10 10 10 10 10 10 10 0 0 0 0 0 0 0 0

TA 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Appendix 4.2: Data of Example 4.2

143

Data of DFIG [4] (Tables 4.13, 4.14, 4.15)

Table 4.13 Basic data of DFIG

Jdr1 8 Rdr 0.0145

Table 4.14 PI gains of RSC of DFIG

Kdp1 0.2

Table 4.15 PI gains of GSC of DFIG

Kdp5 0.2

Jdr2 8 Xdm 2.4012

Kdi1 12.56

Kdp2 1

Kdi5 12.56

Ddm 0 Xdss 0.1784

Kdi2 62.5

Kdp6 1

Ddr1 0 Xdrr 0.1225

Kdp3 0.2

Kdi6 62.5

Kdi3 12.56

Ddr2 0 Cd 13.29

Kdp4 1

Kdp7 1

Appendix 4.2: Data of Example 4.2 Data of New England Power System [13] (Tables 4.16, 4.17 and 4.18)

Rds 0 Xfd 5

Kdi4 62.5

Kdi7 62.5

144

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Table 4.16 Line data Branch no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

From bus 1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 10 10 13 14 15 16 16 16 16 17 17 21 22 23 25 26 26 26 28 12 12 2 6 10

To bus 2 39 3 25 4 18 5 14 6 8 7 11 8 9 39 11 13 14 15 16 17 19 21 24 18 27 22 23 24 26 27 28 29 29 11 13 30 31 32

R 0.0035 0.001 0.0013 0.007 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0.0004 0.0023 0.001 0.0004 0.0004 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008 0.0003 0.0007 0.0013 0.0008 0.0006 0.0022 0.0032 0.0014 0.0043 0.0057 0.0014 0.0016 0.0016 0 0 0

X 0.0411 0.025 0.0151 0.0086 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.0046 0.0363 0.025 0.0043 0.0043 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135 0.0059 0.0082 0.0173 0.014 0.0096 0.035 0.0323 0.0147 0.0474 0.0625 0.0151 0.0435 0.0435 0.0181 0.025 0.02

B 0.6987 0.75 0.2572 0.146 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.113 0.1389 0.078 0.3804 1.2 0.0729 0.0729 0.1723 0.366 0.171 0.1342 0.304 0.2548 0.068 0.1319 0.3216 0.2565 0.1846 0.361 0.513 0.2396 0.7802 1.029 0.249 0 0 0 0 0

Transformer ratio 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.006 1.006 1.025 1.07 1.07

Appendix 4.2: Data of Example 4.2

145

Table 4.17 Machine data Generator no. G1 G2 G3 G4 G5 G6 G7 G8 G9 G10

Xd 0.1 0.295 0.2495 0.262 0.67 0.254 0.295 0.29 0.2106 0.2

X0d 0.031 0.0647 0.0531 0.0436 0.132 0.05 0.049 0.057 0.057 0.006

Xq 0.069 0.282 0.237 0.258 0.62 0.241 0.292 0.28 0.205 0.019

J 84 60.6 71.6 57.2 52 52.8 69.6 48.6 69 1000

D 40 40 40 40 40 40 40 40 40 40

Tf 10.2 6.56 5.7 5.69 5.4 7.3 5.66 6.7 4.79 7

Kf 0.06 0.05 0.06 0.06 0.02 0.02 0.02 0.02 0.02 0.01

Table 4.18 Bus data Bus no. 3 4 7 8 13 15 16 18 20 21 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Active power consumption 3.22 5 2.338 5.22 0.085 3.2 3.294 1.58 6.8 2.74 2.475 3.086 2.24 1.39 2.81 2.06 2.835 – 0.092 – – – – – – – 11.04

Reactive power consumption 0.024 1.84 0.84 1.76 0.88 1.53 0.323 0.3 1.03 1.15 0.846 0.922 0.472 0.17 0.755 0.276 0.269 – 0.046 – – – – – – – 2.5

Active power generation – – – – – – – – – – – – – – – – – 2.5 – 5.5 3.32 5.08 3.5 5.6 5.4 5.3 10

Reactive power generation – – – – – – – – – – – – – – – – – 1.0475 – 0.9831 0.9972 1.0123 1.0493 1.0635 1.0278 1.0265 1.03

KA 10 10 10 10 10 10 10 10 10 10

146

4 Modal Analysis of Small-Signal Angular Stability of a Power System. . .

Data of DFIG [16] (Tables 4.19, 4.20 and 4.21)

Table 4.19 Basic data of DFIG

Xdss 2.58

Xdm 2.40

Table 4.20 PI gains of RSC of DFIG

Kdp1 0.2

Kdi1 10

Table 4.21 PI gains of GSC of DFIG

Kdp5 0.1

Xdrr 2.52

Kdi5 5

Kdp2 0.4

Jdr1 3s

Kdi2 15

Kdp6 10

Jdr2 3s

Rdr 0.015

Kdp3 0.2

Kdi6 62.5

Rds 0

Kdi3 10

Xdf 0.05

Kdp4 0.4

Kdp7 0.2

Cd 30

Kdi4 15

Kdi7 10

References 1. Rogers G (2000) Power system oscillations. Kluwer, Norwell, MA 2. Yu YN (1979) Power system dynamics. Academic Press, New York 3. Kundur P (1994) Power system stability and control. McGraw-Hill, New York 4. Abad G, Lopez J, Rodriguez M, Marroyo L, Iwanski G (2011) Doubly fed induction machine: modeling and control for wind energy generation. John Wiley & Sons, Hoboken 5. Tsourakis G, Nomikos BM, Vournas CD (2009) Effect of wind parks with doubly fed asynchronous generators on small-signal stability. Electr Power Syst Res 79(1):190–200 6. Fan L, Miao Z, Osborn D (2008) Impact of doubly fed wind turbine generation on inter-area oscillation damping. In: Proc IEEE Power Eng Soc Gen Meeting, Pittsburgh, PA, pp 1–8 7. Tsourakis G, Nomikos BM, Vournas CD (2009) Contribution of doubly fed wind generators to oscillation damping. IEEE Trans Energy Conver 24(3):783–791 8. Vittal E, O’Malley M, Keane A (2012) Rotor angle stability with high penetrations of wind generation. IEEE Trans Power Syst 27(1):353–362 9. Vittal E, Keane A (2013) Identification of critical wind farm locations for improved stability and system planning. IEEE Trans Power Syst 28(3):2950–2958 10. Ekanayake JB, Holdsworth L, Jenkins N (2003) Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines. Elect Power Syst Res 67 (3):207–215 11. Feijóo A, Cidrás J, Carrillo C (2000) A third order model for the doubly-fed induction machine. Elect Power Syst Res 56(2):121 12. Ko HS, Yoon GG, Kyung NH, Hong WP (2008) Modeling and control of DFIG-based variablespeed wind-turbine. Elect Power Syst Res 78(11):1841–1849 13. Padiyar KR (1996) Power system dynamics stability and control. Wiley, New York 14. Avriel M (1976) Nonlinear programming analysis and method. Prentice Hall, Englewood Cliffs 15. Nudelland TR, Chakrabortt A (2014) A graph-theoretic algorithm for localization of forced harmonic oscillation inputs in power system networks. In: 2014 American Control Conference (ACC), Portland, OR, pp 1334–1340 16. MATLAB Simulink wind farm-DFIG detailed model. Wind Farm (DFIG Phasor Model)

Chapter 5

Small-Signal Angular Stability of a Power System Affected by Strong Dynamic Interactions Introduced from a Grid-Connected VSWG

Linearized model of a power system with a grid-connected variable speed wind generator (VSWG) is derived in this chapter in three steps. First, the linearized model of the VSWG is established. That includes the establishment of linearized model of a DFIG, a PMSG and a PLL. Second, the linearized model of the power system is derived. Finally, the linearized model of the power system with the VSWG is established by combining the model of the VSWG and the power system.

5.1

Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

A PMSG for the wind power generation is connected to a power system through fully controlled voltage source converter (VSC). Dynamic interactions between such VSC-controlled generation system and AC grid have been an actively pursued research topic in recent years. Impact of dynamic interactions between VSC-controlled system and external power system is twofold. One is on the dynamic performance of VSC-controlled system itself, which is reviewed in Chap. 1. Another is on the dynamic performance of external power system. Electromechanical low-frequency oscillation is one of typical power system dynamic problems and characterized by damping of power system EOMs. In [1], the Western Electricity Coordinating Council (WECC) power system with PMSGs in the United States for years 2010, 2020 and 2022 was examined. Computational results of the participation factors in [1] indicated that state variables associated with reactive power control outer loops of some PMSGs took part in the power system EOMs considerably. Converter oscillation modes of those PMSGs were also participated by some synchronous generators noticeably. Phenomenon of such considerable dual participations of the PMSGs and synchronous generators in the converter oscillation modes and the power system EOMs were found to be © Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_5

147

148

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

related with the nearness of frequencies of involved converter oscillation modes and the power system and adoption of type 4 reactive power control by the PMSGs [1]. Grid connection of a PMSG for the wind power generation brings about changes of system load flow and introduces dynamic interactions as a grid-connected DFIG does, which has been discussed in the previous chapter. Those are two main factors that the PMSG affects power system small-signal angular stability. It has been recognized that the dynamic interactions between the PMSG and the power system are normally weak. Power system EOMs usually are not expected to be affected considerably by the dynamic interactions introduced by the grid-connected PMSG. This has been examined in the previous chapter for the grid-connected DFIG. However, the observable dual participations of the PMSGs and synchronous generators reported in [1] in fact meant possible strong dynamic interactions between the PMSG and power system. Such strong dynamic interactions introduced by the PMSG may significantly affect power system small-signal angular stability. In this section, the strong dynamic interactions between a grid-connected PMSG and power system and their impact on power system small-signal angular stability are examined. The examination is based on the closed-loop interconnected dynamic model of the power system with the PMSG, which is derived in Sect. 2.3 of Chap. 2. Then, the damping torque analysis is applied to examine the damping torque contributions from the PMSG to the synchronous generators in power system, which has been introduced and applied for the grid-connected DFIG in Chaps. 3 and 4. Analytical results indicate that under a special condition that an open-loop converter oscillation mode of the PMSG is close to an open-loop EOM of the power system on the complex plane, strong dynamic interactions between the PMSG and power system may occur to considerably affect power system small-signal angular stability. This special condition is referred to as the open-loop modal resonance (OLMR). The consequence of the OLMR is examined to indicate that the modal resonance may possibly lead to the closed-loop converter oscillation mode and power system EOM moving towards the opposite directions in respect to the positions of resonant open-loop converter oscillation mode and the EOM on the complex plane. Thus, damping of either closed-loop converter oscillation mode or the EOM may decrease such that power system small-signal angular stability is affected negatively by the strong dynamic interactions introduced from the PMSG under the condition of the OLMR. Further analysis is conducted in the section to indicate that the responsible open-loop converter oscillation mode to cause the OLMR is related to the grid-side converter (GSC) control systems of the PMSG. Both the reactive power control loop with type 4 model and the active power control loop of the GSC may provide responsible converter oscillation mode to cause the OLMR. Finally, an example multi-machine power system with the PMSGs is presented to demonstrate and validate the analysis and conclusions made in the section.

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

149

5.1.1

Strong Dynamic Interactions Introduced by the GridConnected PMSG and the Impact

5.1.1.1

Strong Dynamic Interactions Introduced by the GridConnected PMSG

In Sect. 2.4.1 of Chap. 2, a closed-loop interconnected model of a power system with a grid-connected PMSG is derived. The derived closed-loop interconnected model is shown by Fig. 2.9 and redrawn as Fig. 5.1. The closed-loop interconnected model is comprised of the PMSG subsystem and the subsystem of the rest of power system (ROPS). State-space and transfer function models of the PMSG subsystem are (2.124) and (2.125), which are re-presented as follows. 8 d > > > < dtΔXp ¼ Ap ΔXp þ bpd ΔVw ð5:1Þ ΔPw ¼ cpp T ΔXp þ dpp1 ΔVw > > > : ΔQw ¼ cpq T ΔXp þ dpq1 ΔVw (

ΔPw ¼ Hpp ðsÞΔVw

ð5:2Þ

ΔQw ¼ Hpq ðsÞΔVw where  1 Hpp ðsÞ ¼ cpp T sI  Ap bpd þ dpp1 :  1 Hpq ðsÞ ¼ cpq T sI  Ap bpd þ dpq1

Fig. 5.1 Closed-loop interconnected model of the power system with the PMSG

ROPS subsystem ΔPw

Gvp (s) +

ΔQw

Gvq (s)

Hpq (s) Hpp (s) PMSG subsystem

ΔVw

150

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

The state-space and transfer function models are (2.126) and (2.127), which are re-listed as follows. 8 < d ΔX ¼ A ΔX þ b ΔP þ b ΔQ g g g p w q w dt ð5:3Þ : T ΔVw ¼ cgv ΔXg þ dvp ΔPw þ dvq ΔQw ΔVw ¼ Gvp ðsÞΔPw þ Gvq ðsÞΔQw

ð5:4Þ

where  1 Gvp ðsÞ ¼ cgv T sI  Ag bp þ dvp :  1 Gvq ðsÞ ¼ cgv T sI  Ag bq þ dvq The state-space model of the closed-loop interconnected system is (2.128), i.e., d ΔXgp ¼ Agp ΔXgp dt

ð5:5Þ

where ΔXgp ¼ [ΔXgT ΔXpT]T. Denote ^λ i a power system EOM of concerns with dynamics of PMSG being included. Hence, ^λ i is an eigenvalue of closed-loop state matrix, Agp, in (5.5). If ΔPw þ jΔQw ¼ 0, there are no dynamic interactions between the PMSG and the power system. In this case, the PMSG degrades into a constant power source, Pw0 þ jQw0, and the linearized model of the power system of (5.3) becomes d ΔXg ¼ Ag ΔXg dt

ð5:6Þ

Denote λi as the EOM corresponding to ^λ i when the PMSG is modelled as constant power source, Pw0 þ jQw0, when ΔPw þ jΔQw ¼ 0. Obviously, λi is an eigenvalue of open-loop state matrix of system, Ag. Difference of closed-loop and open-loop EOM, Δλi ¼ ^λ i  λi , is caused by the dynamic interactions introduced by the PMSG. Hence, Δλi ¼ ^λ i  λi measures the impact of dynamic interactions on the EOM of concerns. The closed-loop interconnected model shown by Fig. 5.1 establishes the link between the impact of the PMSG on EOM of concerns and the dynamic interactions introduced by the PMSG which exhibit as ΔPw þ jΔQw. The damping torque analysis can be applied to estimate Δλi ¼ ^λ i  λi , as it is introduced in Chaps. 3 and 4 for the grid-connected DFIG. The estimation on the impact of dynamic interactions introduced by the PMSG by use of the damping torque analysis is N X Sik Tk , k¼1 nh i o Tk ¼ Re gpk ðλi ÞHpp ðλi Þ þ gqk ðλi ÞHpq ðλi Þ γk ðλi Þ

Δλi ¼

ð5:7Þ

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

151

where gpk(s) and gqk(s) is the transfer function of the forward path from ΔPw and ΔQw to the electromechanical oscillation loop of the kth synchronous generator in the power system respectively, γk(s) is the reconstructing transfer function of ΔVw by the rotor speed of the kth synchronous generator, Δωk, i.e., ΔVw ¼ γk(s)Δωk, Sik is the sensitivity of the EOM of concern to the addition of damping torque on the kth synchronous generator, Re{} denotes the real part of a complex number. In Eq. (5.7), Tk is the coefficient of the amount of damping torque provided by the PMSG to the kth synchronous generator, TkΔωk. Derivation of (5.7) is similar to that of (4.16) for the grid-connected DFIG from (4.6) to (4.16) presented in Chap. 4. The derived (5.7) indicates that under a special condition, ΔPw þ jΔQw can be significant such that PMSG may introduce significant dynamic interactions with power system, as to be elaborated as follows. Denote λvsc as an open-loop converter oscillation mode of the PMSG. It is a complex pole of transfer function Hpp(s) or/and Hpq(s) such that |Hpp(λvsc)| ¼ 1 or/and |Hpq(λvsc)| ¼ 1. Thus, if λvsc is close to the EOM of concerns, λi, on the complex plane, i.e., λvsc  λi, |Hpp(λi)| or/and |Hpq(λi)| can be very big to result in significant ΔPw þ jΔQw at the complex frequency λi as it can be seen from Fig. 5.1 and (5.2). Significant dynamic interactions between the PMSG and power system may occur under this special condition.   According to (5.7), under the special condition that λvsc  λi, jΔλi j ¼ ^λ i  λi  is no longer small due to the significant increase of amount of the damping torque contributions from the PMSG, TkΔωk. Hence, the EOM of concerns may be affected and participated by the PMSG considerably.

5.1.1.2

The Open-Loop Modal Resonance (OLMR)

For the convenience of discussion, the special condition that the open-loop converter oscillation mode of the PMSG, λvsc, is close to the open-loop EOM of concern, λi, on the complex plane, i.e. λvsc  λi, is referred to as the open-loop modal resonance (OLMR). More generally, for the closed-loop interconnected system shown by Fig. 5.1, closeness of two open-loop oscillation modes separately from two subsystems on the complex plane is named as the condition of the OLMR. Normally, when the dynamic interactions between the PMSG and power system are weak, closed-loop EOM, ^λ i , is close to λi on the complex plane. However, under the condition of the OLMR, ^λ i could move away   from the position where λvsc  λi on the complex plane because jΔλi j ¼ ^λ i  λi  may not be small any more. It might be very difficult to analytically identify the exact position of ^λ i on the complex plane under the condition that λvsc  λi to strictly determine the impact of strong dynamic interactions caused by the OLMR. However, how ^λ i moves away in the neighborhood of λvsc  λi (^λ i ! λi) on the complex plane when λvsc  λi can be estimated as follows.

152

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

From Fig. 5.1, the characteristic equation of closed-loop interconnected system can be obtained to be Gvp ðλÞHpp ðλÞ þ Gvq ðλÞHpq ðλÞ ¼ 1

ð5:8Þ

Without loss of generality, assume that λi and λvsc is the pole of Gvp(s) and Hpp(s) respectively. Express Gvp(s) and Hpp(s) as Gvp ðsÞ ¼

gp ð s Þ hp ð s Þ , Hpp ðsÞ ¼ ð s  λi Þ ðs  λvsc Þ

^λ i should satisfy (5.8). Thus from (5.8) and (5.9), it can have    ^λ i  λi ^λ i  λvsc            ¼ gp ^λ i hp ^λ i þ ^λ i  λi ^λ i  λvsc Gvq ^λ i Hpq ^λ i

ð5:9Þ

ð5:10Þ

Under the condition of the OLMR, i.e., λvsc  λi, from (5.10) it can have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   lim Δλi ¼ lim ^λ i  λi   gp ðλi Þhp ðλi Þ ð5:11Þ ^ λ i !λi

^ λ i !i

Denote ^λ vsc as the closed-loop converter oscillation mode corresponding to λvsc. ^λ vsc should also satisfy (5.8). Taking the derivation similar to that from (5.8) to (5.11), it can have   lim Δλvsc ¼ lim ^λ vsc  λvsc ^λ vsc !λvsc

^λ vsc !λvsc

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   gp ðλvsc Þhp ðλvsc Þ   gp ðλi Þhp ðλi Þ

ð5:12Þ

From (5.11) and (5.12), it can be seen that when λvsc  λi, Δλi, and Δλvsc are approximately of the same value but opposite signs. This means that on the complex plane, the phase difference between vector Δλi and Δλvsc is about 180 . Closed-loop oscillation modes, ^λ i and ^λ vsc , intend to move away from each other in the opposite directions from λvsc  λi on the complex plane. Hence, it is very likely that either ^λ i or ^λ vsc will move towards the right on the complex plane from λvsc  λi. The OLMR may cause power system small-signal angular stability to decrease. Mathematically, derivations of (5.11) and (5.12) only stand correct for ^λ i and ^λ vsc in the neighborhood of λvsc  λi. However, they indicate how ^λ i and ^λ vsc intend to move away from λvsc  λi on the complex plane when the OLMR occurs. In (5.10), denote              ð5:13Þ f ^λ i ¼ gp ^λ i hp ^λ i þ ^λ i  λi ^λ i  λvsc Gvq ^λ i Hpq ^λ i

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

  Taylor series expansion of f ^λ i at point λi is   f ^λ i ¼ f ðλi þ Δλi Þ ¼ 0

153

ð5:14Þ

00

f ðλi Þ þ f ðλi ÞΔλi þ f ðλi ÞΔλi 2 þ   

From (5.13), it can have f(λi) ¼ gp(λi)hp(λi). Hence, (5.11) and (5.12),  in fact, are derived by only taking the first item in Taylor series expansion of f ^λ i . If ^λ i and ^λ vsc are far away from the neighborhood of λvsc  λi, (5.11) and (5.12) may not stand strictly, i.e., the phase difference between vector Δλi and Δλvsc may not exactly be 180 on the complex plane. Nevertheless, when the OLMR occurs, decreases of damping of ^λ i or ^λ vsc is quite possible. This will be demonstrated by examples in Sect. 5.1.2. In addition, if |gp(λi)hp(λi)| is small, the OLMR may not result in considerable movement of ^λ i and ^λ vsc on the complex plane as it can be seen from (5.11) and (5.12).

5.1.1.3

Existence of Converter Oscillation Modes

To examine whether Hpp(s) or Hpq(s) is of any complex pole to be a converter oscillation mode, a simplified version of transfer functions, Hpp(s) or Hpq(s), is derived as follows. The configuration of a PMSG being connected to an external power system is shown by Fig. 5.2, where Xf is the reactance of the filter. From Fig. 5.2, the linearized voltage equation across Xf is obtained to be dΔId ¼ ω0 ðΔVcd  ΔVd Þ þ ω0 ΔIq dt   dΔIq ¼ ω0 ΔVcq  ΔVq þ ω0 ΔId Xf dt Xf

ð5:15Þ

Direction of terminal voltage of the PMSG, Vw ∠θw, is taken as that of d axis of !

d  q coordinate of the GSC (see Fig. 2.8). Thus Vq ¼0, Vd ¼ Vw . Variation of power injected from the PMSG to the external power system is ΔPw ¼ Id0 ΔVw þ Vd0 ΔId , ΔQw ¼ Iq0 ΔVw  Vd0 ΔIq

Pin

Pw Vw qw = Vd + jVq

Vcd + jVcq A PMSG

Cd

Vdc

Xf

GSC

I d + jI q

Fig. 5.2 A PMSG connected to an external power system

Pw +jQw

External power system

ð5:16Þ

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

154

1

2

K p_vdc Vdcref – +

Id

3

+

K p_q +

K i_q

xid

s



s

V ref + + cd –

X f Iq

PWM Vq

I qref+

K p_iq +

K i_iq s

Iq

Qw = -Vd I q

3

+

K i_id

4

Qwref –

1

+ –

Ki_vdc + xvdc s

Vdc

Vd

K p_id

I dref+

+

Vcqref + + +

X f Id

active power control outer loop 2 d-axis current control inner loop reactive power control outer loop 4 q-axis current control inner loop

Fig. 5.3 Configuration of the GSC control system of the PMSG

Linearized dynamic equation of DC voltage across the capacitor in Fig. 5.2 is (see Sect. 5.2.1) Cd Vdc0

dΔVdc ¼ ΔPin  ΔPw dt

ð5:17Þ

The configuration of the converter control system of the GSC of the PMSG is shown by Fig. 2.3 and is represented as Fig. 5.3 below. Ignore the fast transient of the pulse width modulation (PWM) such that (see 2.33)) ref ref Vcd ¼ Vcd , Vcq ¼ Vcq

ð5:18Þ

From Fig. 5.3, (5.15) and (5.18), following equations can be obtained ΔId ¼ ΔIdref

¼

Kp

vdc s

þ Ki s

vdc

s2

ð5:19Þ

ΔVdc

ω0 Kp iq s þ Ki iq ω0 ΔI ref ¼ Giq ðsÞΔIqref X f þ ω0 Kp iq s þ Ki iq ω0 q  Kp q s þ Ki q  Iq0 ΔVpcc þ Vd0 ΔIq ¼ s

ΔIq ¼ ΔIqref

ω0 Kp id s þ Ki id ω0 ΔI ref ¼ Gid ðsÞΔIdref X f s2 þ ω0 Kp id s þ Ki id ω0 d

ð5:20Þ

ΔPin in (5.17) is determined completely by the state variables of the permanent magnet synchronous generator which are affected by the wind speed only. Thus, ΔPin is irrelevant with power system electromechanical dynamics, neither taking

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

DV w

I d0

155

D Pw

+ + Vd0

DI d

Gid (s )

DI

ref d

K p_vdc +

K i_vdc s

-1 DVdc CdVdc0 s

Fig. 5.4 Active power control path of the GSC Fig. 5.5 Reactive control path of the GSC

DV w

I q0

D Qw

– – Vd0

DI q

Giq ( s)

DI

ref q

K p_q +

Ki_q s

part in nor being affected by the dynamic interactions between the PMSG and power system. Hence, in Eq. (5.17) ΔPin ¼ 0 and (5.17) becomes Cd Vdc0

dΔVdc ¼ ΔPw dt

ð5:21Þ

From the first equation of (5.16), (5.19) and (5.21), the active power control implemented by the GSC can be shown by Fig. 5.4. From Fig. 5.4 ΔPw ¼ Hpp ðsÞΔVw where Hpp ðsÞ ¼ C

d Vdc0 s

ð5:22Þ

Id0 Cd Vdc0 s2 . d0 Gid ðsÞðKp vdc sþKi vdc Þ

2 þV

Similarly, from the second equation of (5.16) and (5.20), the reactive power control implemented by the GSC can be shown by Fig. 5.5. From Fig. 5.5, ΔQw ¼ Hpq ðsÞΔVw where

Gqv ðsÞ¼Iq0



1þVd0 Giq ðsÞ Kp q þ

Kiq s

ð5:23Þ

.

A simplification can be made for the transfer function model of the PMSG by ignoring the electromagnetic transient of the output current from the PMSG. This simplification is appropriate for the study of electromechanical dynamics of power system. With the simplification, the current control inner loops of the GSC control system of the PMSG can be removed to have ΔId ¼ ΔIdref and ΔIq ¼ ΔIqref , that is, Gid(s) ¼ 1 and Giq(s) ¼ 1 in (5.19) and (5.20). Thus, from (5.22) and (5.23), the simplified transfer function model of the PMSG is obtained to be Hpp ðsÞ ¼

Cd Vdc0

s2

Id0 Cd Vdc0 s2 þ Vd0 Kp vdc s þ Vd0 Ki

ð5:24Þ vdc

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

156

Hpq ðsÞ ¼ 

Iq0 s  1 þ Vd0 Kp q s þ Vd0 Ki

ð5:25Þ q

From (5.24), it can be seen that Hpp(s) may have a pair of complex poles to be a converter oscillation mode. Setting of PI parameters of the active power control outer loop, Kp_vdc and Ki_vdc, affects the position of the converter oscillation mode on the complex plane. Hence, if strong dynamic interactions occur with considerable variations of active power output from PMSG, ΔPw, due to the OLMR, retuning of PI parameters of the active power control outer loop of the GSC can change the position of the converter oscillation mode on the complex plane to dismiss the strong dynamic interactions. Since Vw ∠ θw is on the d axis of d  q coordinate of the GSC, Pw0 ¼ Vd0Id0. With the increase of the PMSG active power output, Id0 will increase. From (5.2) and (5.24), it can be seen that |Hpp(λi)| and thus ΔPw will increase accordingly with the increase of active power output from the PMSG to lead to increased dynamic interactions between the PMSG and power system. From (5.25), it can be seen that Hpq(s) is first-order and hence will not provide a converter oscillation mode. In fact, since Qw0 ¼  Vd0Iq0 and normally the PMSG operates with high power factor, Qw0  0 such that Iq0  0. From (5.25), it can be seen that Hpq(s)  0. Hence, ΔQw should usually be very small. However, if type 4 model of reactive power control [1] rather than that shown in Fig. 5.3 is used, Hpq(s) may provide a converter oscillation mode. An equivalent reactive power control path of type 4 model can be derived as shown by Fig. 5.6. A similar simplified transfer function to (5.25) can be derived from Fig. 5.6 for type 4 model to be Hpq ðsÞ ¼

Iq0 s2 þ Vd0 Kvi s s2 þ Kqi Kvi Vd0

ð5:26Þ

Obviously in this case, Hpq(s) is of a pair of complex poles. Hence when type 4 mode is used, setting of PI parameters of reactive power control outer loop of the PMSG may cause the OLMR to lead to strong dynamic interactions between the PMSG and power system. In this case, strong dynamic interactions occur with considerable variations of reactive power output from the PMSG, ΔQw. Retuning of PI parameters of reactive power control outer loop of the PMSG can terminate the strong dynamic interactions. DVw

I q0 –

D Qw –

Vd0

D xv K vi Giq (s ) D I qref s DIq

D xq + K qi s –

Fig. 5.6 Reactive control path of the GSC with the generic type 4 model

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

5.1.2

Example 5.1

5.1.2.1

Example Power System

157

Figure 5.7 shows the configuration of example New England power system with the PMSGs. Parameters of network and synchronous generators given in [2] were used. Detailed seventh-order model of synchronous generator and a third-order model of the AVR were adopted [2, 3]. Loads were modelled as constant impedance. Two PMSGs for the wind power generation were already connected at node 6 and 25. Three more PMSGs (PMSG1, PMSG2 and PMSG3) were to be connected at node 22 and 10. There were total nine EOMs in the example New England power system. The inter-area EOM of concern was of the lowest oscillation frequency and the related inter-area power oscillation was between the tenth synchronous generator and the rest of synchronous generators. Impact of dynamic interactions introduced by the grid connections of PMSG1, PMSG2 and PMSG3 on power system EOMs, especially the inter-area EOM, was examined. Parameters of the example power system are given in Appendix 5.1.

8 1 37

PM SG

30

28

26

25

29

2 27

18 1

24

17

38 9

16

10

6 3

39

35

15 21

14 PM SG

4 5

6

PM SG1

12

9

20 11

31 7

2

22

19

33

13 4

34

10

23 PM SG2

5

8 PM SG3

32 3

Fig. 5.7 Configuration of example New England power system with the PMSGs

36 7

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

158

700 600

Imaginary axis

40 20

Kp_vdc increased

0 -20 -40

Detailed model

-600

(5.30) used

-700 -800 -700 -600 -500 -30

-25 -20 Real axis

-15

-10

-5

0

Fig. 5.8 Trajectories of converter oscillation modes of PMSG1 with variation of parameters of the GSC active power control outer loop

5.1.2.2

Converter Oscillation Modes

All the PMSGs were modelled by the detailed dynamic model presented in Chap. 2. Typical parameters of the PMSG given in [4] were used. Configuration of the GSC control system shown by Fig. 5.3 was adopted by PMSG1. PMSG2 used the type 4 model of reactive power control of the GSC and typical parameters of type 4 model given in [5] were used. State-space model described by (5.1) for both PMSG1 and PMSG2 was established. Converter oscillation modes were calculated from the open-loop state matrix Ap in (5.1). When the outer loop PI parameters of the GSC control systems of PMSG1, Kp_vdc, Ki_vdc, Kp_q, and Ki_q shown in Fig. 5.3, varied with Ki_vdc ¼ cKp_vdc, Ki_vdc ¼ cKp_vdc, Ki_q ¼ cKp_q and c ¼ 40, computational results of the converter oscillation modes are displayed in Figs. 5.8 and 5.9 as the solid curves. Dotted areas in Figs. 5.8 and 5.9 are where power system EOMs usually locate with the oscillation frequency between 0.1 and 2.5 Hz. In Figs. 5.8 and 5.9, trajectories of the converter oscillation modes of PMSG1, being estimated by using (5.24) and (5.25), are also presented as the dashed curves. Computational results presented in Figs. 5.8 and 5.9 confirm the following analysis and conclusions made in the previous Sect. 5.1.1.3 about the existence of converter oscillation modes.

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

159

700 650

Imaginary axis

600 0.3

Kp_q increased

0 -0.3 -600

Detailed model -650 -700 -1000 -900

(5.31) used -700 -500

-20 -15 Real axis

-10

-5

0

Fig. 5.9 Trajectories of converter oscillation modes of PMSG1 with variation of parameters of the GSC reactive power control outer loop

1. Simplification made in the analysis is appropriate since the estimated converter oscillation modes (dashed curves) were very close to those calculated from the detailed state-space model of PMSG1 (solid curves). 2. A pair of converter oscillation modes were from Hpp(s). Hence, they were related with GSC active power control outer loop. The converter oscillation modes can get close to power system EOMs with the variations of PI parameters of active power control outer loop (Fig. 5.8), which may cause the OLMR. 3. Hpq(s) did not have any converter oscillation modes (Fig. 5.9). Hence, the variations of PI parameters of reactive power control outer loop will not cause the OLMR when the control configuration of reactive power control shown by Fig. 5.3 is used. When parameters in type 4 model of reactive power control outer loop of the GSC of PMSG2 varied with Kiv ¼ cKiq and c ¼ 13.5, trajectories of converter oscillation modes of PMSG2 were calculated from (5.26) and are displayed in Fig. 5.10 (the dashed line on the imaginary axis). Trajectories of complex poles of PMSG2 by using the detailed state-space model of PMSG2 are shown as the solid curve. It can be clearly seen that a pair of converter oscillation modes entered the area where power system EOMs locate. Hence, it is confirmed that when type 4 model is used, the GSC reactive power control system of the PMSG may cause the OLMR. This was reported in [1] and is explained analytically in Sect. 5.1.1.3.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

160

30

Kiv increased

Imaginary axis

20 10 0 -10

Detailed model

-20

(5.32) used -30 -0.8

-0.7

-0.6

-0.5

-0.4 -0.3 Real axis

-0.2

-0.1

0

Fig. 5.10 Trajectories of converter oscillation modes of PMSG2 with variation of parameters of the GSC reactive power control outer loop (type 4 model used)

5.1.2.3

Dynamic Interactions of PMSG1 with Power System

In this sub-section, impact of grid connection of PMSG1 was examined. Table 5.1 presents the results of modal computation when the power output from PMSG1 was Pvsc0 ¼ 0 p. u. , Qvsc0 ¼ 0 p. u. The inter-area EOM of concern was λi ¼  0.2841 þ 3.3356j. In case A, PI parameters of active power control outer loop of the GSC given in [4] were used (Kp_vdc ¼ 0.7147, Ki_vdc ¼ 7.1515). In case B, settings of PI parameters were changed toKp_vdc ¼ 0.22, Ki_vdc ¼ 3. From Table 5.1, it can be seen that in case A when λvsc was away from λi ¼  0.2841 þ 3.3356j on the complex plane, dynamic interactions   between PMSG1 and power system were small as measured by jΔλi j ¼ ^λ i  λi . Impact of PMSG1 on the inter-area EOM of concerns was small because ^λ i only changed slightly from λi with the dynamics of PMSG1 being included. However, in case B when λvsc was close to λi ¼  0.2841 þ 3.3356j on the complex plane, the inter-area EOM of concern was affected significantly and its damping decreased considerably. In this case, dynamic interactions introduced by PMSG1 imposed a serious threat to power system small-signal angular stability. Participation factors of state variables for the EOMs of concerns are the indication of dynamic interactions between the PMSG and power system, which were used in [1] extensively to examine the involvement of the PMSG in power system EOMs. Last column in Table 5.1 gives the computational results of sum of participation factors of all state variables of PMSG1, PFPMSG, for the inter-area EOM of concern. It clearly shows that when λvsc was close to λi (case B), PFPMSG increased

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

161

Table 5.1 Impact of dynamic interactions introduced by PMSG1 Case A B

a

λvsc 4.2882 þ 8.2115j 0.37006 þ j3.5007j

b λi 0.29245 þ j3.2749j 0.18242 þ j2.481j

b

0.4

0.25 0.2 0.15 0.1

0.35

0.25 0.2 0.15 0.1 0.05

0.05 0

PFPMSG(%) 6.142 74.648

0.3

0.3

Participation factors

Participation factors

0.35

|Δλi| 0.061272 0.86063

DVdc

Dxvdc State variables

Dxid

0

1

2

3

4 5 6 7 Generator number

8

9

10

Fig. 5.11 Computational results of participation factors in case B. (a) Participation of PMSG1 in inter-area oscillation mode; (b) participation of SGs in the converter oscillation mode of PMSG1

significantly, indicating considerable participation of PMSG1 in the inter-area EOM of concerns. A further split of total participation of PMSG1, PFPMSG, to individual state variables can help identifying the cause of participation [1]. Figure 5.11a presents the participation of three state variables of PMSG1 which contributed most to PFPMSG in case B. From Fig. 5.11a, it can be seen that indeed those state variables were associated with the active power control system of the GSC shown by Fig. 5.3 where Δxvdc and Δxid are indicated. Figure 5.11b is the participation of synchronous generators in the converter oscillation mode in case B. It shows considerable participations of synchronous generators in the converter oscillation mode. Figures 5.12 and 5.13 present further confirmation from the results of non-linear simulation. In the simulation, a 5% drop of PMSG1 output active power took place which was recovered in 0.1 s. From Fig. 5.12, it can be seen that in case B, active power exchange between PMSG1 and power system was significant. This confirms the occurrence of strong dynamic interactions and analysis made in Sect. 5.1.1.3 about strong dynamic interactions caused by the OLMR when the converter oscillation mode is the complex pole of Hpp(s). Consequently, damping of inter-area power oscillation decreased as shown by Fig. 5.13. In order to more clearly demonstrate how the OLMR caused the strong dynamic interactions between PMSG1 and power system, setting of PI parameters of active power control outer loop of PMSG1 in case A were changed gradually to that in case B. Trajectories of open-loop and closed-loop COMs and inter-area EOM with changes of parameters setting were calculated and are displayed in Fig. 5.14. From Fig. 5.14, it can be seen that when the open-loop converter oscillation mode, λvsc, got closer to the open-loop EOM of concern,λi, on the complex plane, the

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

162 11

Case A Case B

Active power output of PMSG1

10.8 10.6 10.4 10.2 10 9.8 9.6 9.4 9.2 9

0

1

2

3

4

5

6

7 8 Time(s)

9

10 11 12 13 14 15

Fig. 5.12 Exchange of active power between PMSG1 and power system 0

Case A Case B -2

d10–d1

-4

-6

-8

-10 -12

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time(s)

Fig. 5.13 Relative rotor positions between the tenth and first SG

closed-loop EOM, ^λ i , moved away from λi towards the right and the damping of inter-area EOM decreased. In Fig. 5.14, estimated positions of ^λ i and closed-loop converter oscillation mode, ^λ vsc , by using (5.11) and (5.12) are indicated by crosses, showing their opposite positions in respect to λi  λvsc on the complex plane. Hence, results presented in Fig. 5.14 confirm the analytical conclusion made in Sect. 5.1.1.3 that the OLMR may cause the damping of the EOM of concern to decrease.

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

163

5

Case A

λi – g p ( λi )hp ( λi )

Imaginary axis

4.5

λˆvsc

4

λvsc

Case B

3.5

λi

3

λˆi

2.5

λi + g p ( λi )h p ( λi ) 2 -0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Real axis Fig. 5.14 The OLMR between PMSG1 and power system

Table 5.2 Impact of dynamic interactions at different levels of active power output from PMSG1 Pw0 1 5 10

5.1.2.4

λi 0.27396 þ 3.2786j 0.28184 þ 3.3858j 0.37006 þ j3.5007j

b λi 0.28222 þ 3.2194j 0.29196 þ 3.0798j 0.18242 þ 2.481j

|Δλi| 0.059773 0.30617 0.86065

PFPMSG(%) 13.87 47.686 74.648

Dynamic Interactions Affected by the Level of Wind Power Penetration

Analysis in Sect. 5.1.1.3 has concluded that when the active power output from the PMSG increases, dynamic interactions introduced by the PMSG are expected to become stronger. This conclusion is confirmed by the computational results presented in Table 5.2. Confirmation from the computational results of participation factors is given in the last column of Table 5.2, Figs. 5.15 and 5.16. Figures 5.17 and 5.18 are the confirmation from the results of non-linear simulation. From Table 5.2 and Figs. 5.15, 5.16, 5.17 and 5.18, it can be seen that when the active power output from PMSG1 increased, dynamic interactions introduced by PMSG1 were stronger and the impact on the inter-area EOM of concern was larger. Consequently, damping of the inter-area oscillation decreased.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

164

0.35 Pw0=1 p.u. Pw0=10 p.u.

Participation factors

0.3 0.25 0.2 0.15 0.1 0.05 0

1

2

3

4 5 6 7 Generator number

8

9

10

Fig. 5.15 Participation of SGs in converter oscillation mode of PMSG1

0.4 Pw0=1 p.u. Pw0=10 p.u.

Participation factors

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

DVdc

Dxvdc

Dxid

State variables Fig. 5.16 Participation of PMSG1 in the inter-area EOM

5.1.2.5

The OLMR of PMSG1 with Local EOMs

It was demonstrated in Sect. 5.1.2.3 that the OLMR of PMSG1 with the inter-area EOM of concern occurred when setting of PI parameters of active power control outer loop of PMSG1 changed. In practice, parameters tuning is often carried out for a controller in order to obtain satisfactory control performance. Figure 5.19 shows the results of a test that when PI parameters of active power control outer loop of

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

165

Active power output of PMSG1

1 Pw0=1 p.u. Pw0=10 p.u.

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

3

2

1

0

4

5

6

7 8 9 10 11 12 13 14 15 Time(s)

Fig. 5.17 Variation of active power exchange between PMSG1 and power system at different levels of wind power penetration 0

Pw0=1 p.u. Pw0=10 p.u.

-2

d10 – d1

-4

-6

-8

-10

-12

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time(s) Fig. 5.18 Variation of relative rotor positions between the tenth and first SG at different levels of wind power penetration

PMSG1 were tuned, the OLMR of PMSG1 with five local EOMs of example New England power system occurred. In the test, PI parameters of active power control outer loop of PMSG1 were tuned gradually from Kp_vdc ¼ 0.255, Ki_vdc ¼ 16.9 to Kp_vdc ¼ 0.224, Ki_vdc ¼ 7.5 such that the open-loop converter oscillation mode of PMSG1, λvsc, moved accordingly on the complex plane. Trajectory of movement of λvsc is displayed as the dash-

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

166

9 8.5

l1 l vsc

Imaginary axis

8

l2

7.5

l3

lˆ vsc

7

lˆ i 6.5

l4 l5

6 5.5 5 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5

-0.48 -0.46 -0.44 -0.42 -0.4

-0.38

Real axis Fig. 5.19 The OLMR of PMSG1 with five local EOMs

dotted curve in Fig. 5.19. Trajectory of movement of closed-loop converter oscillation mode, ^λ vsc , along with λvsc is the dashed curve in Fig. 5.19. On the way of movement, λvsc was close to five open-loop local EOMs of the example New England power system, λi, i ¼ 1, 2, 3, 4, 5, which are indicated by triangles in Fig. 5.19. Trajectories of closed-loop local EOMs are shown by the solid curves in Fig. 5.19. From Fig. 5.19, it can be seen that the OLMR of converter oscillation mode with local EOMs occurred when λvsc was close to λi, i ¼ 1, 2, 3, 4, 5. Points of the OLMR are indicated by squares and triangles in Fig. 5.19. When the OLMR occurred, closed-loop local EOMs, ^λ i , i ¼ 1, 2, 3, 4, 5, were “driven” away from λi, i ¼ 1, 2, 3, 4, 5 to the points indicated by the filled circles. Closed-loop converter oscillation mode, ^λ vsc , moved away from λvsc to the points indicated by the hollow circles with corresponding numbers. Figures 5.19 shows that four closed-loop local EOMs, ^λ i , i ¼ 1, 2, 3, 4, 5, located on the right hand side of corresponding open-loop local EOMs, λi, i ¼ 1, 2, 3, 5. Hence, grid connection of PMSG1 caused the decrease of damping of those four local EOMs when the OLMR happened. The fourth closed-loop local EOM, ^λ 4 , moved to the left hand side of λ4, indicating improvement of the damping of the fourth local EOM when the OLMR occurred. However, the corresponding closedloop converter oscillation mode, ^λ vsc , was on the right hand side of λvsc, indicating the decrease of damping as caused by the OLMR.

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

167

Table 5.3 Estimation of the OLMR of PMSG1 with five local EOMs i 1 2 3 4 5

λi 0.55012 þ 8.4517j 0.47535 þ 7.8201j 0.52929 þ 7.1658j 0.4902 þ 5.8855j 0.51978 þ 6.3095j

gp(λi)hp(λi) 0.017556 þ 0.0088645j 0.063834 þ 0.02335j 0.022357–0.11377j 0.092222–0.22458j 0.0057484 þ 0.42818j

|gp(λi)hp(λi)| 0.019667 0.067971 0.11595 0.24278 0.42822

Results presented in Fig. 5.19 demonstrate that when the parameters of active power controller of the GSC of PMSG1 were tuned to cause the OLMR, power system small-signal angular stability decreased. In real power systems with the PMSGs, detrimental effect of the OLMR may not be noticed if such detailed analysis of the OLMR was not carried out. Figure 5.19 demonstrates that when the OLMR occurred, either ^λ i , i ¼ 1, 2, 3, 4 , 5 or ^λ vsc was on the right hand side of point of the OLMR where λvsc  λi. This demonstrates the correctness of analytical conclusion made from (5.11) and (5.12) in Sect. 5.1.1.3. A further confirmation is presented in Table 5.3 where computational ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi results of gp ðλi Þhp ðλi Þ at five points of the OLMR are given. From Table 5.3, estimated positions of ^λ i , i ¼ 1, 2, 3, 4, 5 and ^λ vsc by using (5.11) and (5.12) were calculated and are indicated by crosses in Fig. 5.19. From Fig. 5.19, it can be seen that the estimation from (5.11) and (5.12) was close to the actual positions of ^λ i , i ¼ 1, 2 , 3, 4, 5 and ^λ vsc indicated by hollow and filled circles. In addition, by comparing the third column of Table 5.3 and Fig. 5.19, it can be seen that when |gp(λi)hp(λi)| was small for the first two local EOMs, the impact of the OLMR was small.

5.1.2.6

Dynamic Interactions of PMSG2 with Power System

In [1], type 4 model for the GSC reactive power control outer loop of the PMSG was used instead of that shown in Fig. 5.3. Noticeable dynamic interactions were found being related to the state variables in the GSC reactive power control outer loop of the PMSG. Analysis in Sect. 5.1.1.3 has indicated that when type 4 model is used instead of that shown by Fig. 5.3, Hpq(s) could have a pair of converter oscillation modes to cause the OLMR. Reactive power control outer loop of PMSG2 in the example New England power system adopted the type 4 model. Modal computation presented in Fig. 5.10 has confirmed the analytical conclusion that there existed a pair of converter oscillation modes due to the type 4 model used by PMSG2. In this subsection, the impact of PMSG2 on the inter-area EOM of concern was examined as follow. Firstly, PMSG1 was connected at node 22 and parameters of active power control outer loop of PMSG1 were carefully tuned to avoid causing any OLMR. The

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

168

Table 5.4 Impact of dynamic interactions introduced by PMSG3 λvsc 0.032042 þ 0.31104i 0.29338 þ 3.3135i

Case C Case D

a

b λi 0.29245 þ 3.2749j 0.13382 þ 3.5714j

b

0.3

PFPMSG(%) 6.142 47.194

0.25

0.2

Participation factors

Participation factors

0.25

|Δλi| 0.034188 0.34609

0.2

0.15

0.1

0.15

0.1

0.05

0.05

0

0

DVdc

Dxvdc Dxv State variables

Dxq

1

2

3

4 5 6 7 Generator number

8

9

10

Fig. 5.20 Computational results of participation factors in Case D. (a) Participation of PMSG2 in inter-area oscillation mode; (b) participation of synchronous generators in converter oscillation mode of PMSG2

open-loop inter-area EOM of concern without dynamics of PMSG2 being included was calculated to be λi ¼  0.32559 þ 3.2833j. Secondly, dynamics of PMSG2 were included and typical parameters of type 4 model given in [5] were used (Kiv ¼ 5, Kiq ¼ 0.02). Results of modal computation are presented in Table 5.4 as case C. From the second row of Table 5.4, it can be seen that in case C, open-loop converter oscillation mode of PMSG2, λvsc, was away from open-loop EOM of concern, λi. Impact of dynamic interactions introduced by  PMSG2 was small because jΔλi j ¼ ^λ  λi  was small, indicating weak dynamic interactions brought about by PMSG2. Thirdly, parameters of type 4 model were changed to be Kiv ¼ 11.5, Kiq ¼ 1. Results of modal computation are presented in the third row of Table 5.4 as case D. It can be seen that in case D, open-loop converter oscillation mode of PMSG2, λvsc, was close to open-loop inter-area EOM of concern, λi. Damping of inter-area EOM of concern decreased significantly due to the inclusion of PMSG2 dynamics, indicating strong dynamic interactions between PMSG2 and power system. This was the effect of the OLMR between PMSG2 and power system. Fourthly, participation factors of sum of state variables of PMSG2, PFPMSG, were calculated and are presented in the last column of Table 5.4. Figure 5.20 shows the participation of synchronous generators in the converter oscillation mode and participation of state variables related to the reactive power control outer loop (type 4 model) of PMSG2 in the inter-area EOM of concern. They confirm the occurrence of strong dynamic interactions between PMSG2 and power system as caused by the OLMR.

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

169

Reactive power output of PMSG2

0.2

Case C Case D

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

2

4

6

8

10

12

14

16

18

20

Time(s) Fig. 5.21 Variation of reactive power output from PMSG2 -1

Case C Case D

-2 -3

d10 – d1

-4 -5 -6 -7 -8 -9 -10

0

2

4

6

8

10

12

14

16

18

20

Time(s) Fig. 5.22 Relative rotor positions between the tenth and first SG

Fifthly, non-linear simulation was conducted to provide further confirmation. Figures 5.21 and 5.22 present the results of simulation. From Figs. 5.21 and 5.22, it can be seen that significant dynamic interactions were due to the considerable variations of reactive power exchange between PMSG2 and power system. This caused decrease of damping of inter-area electromechanical oscillation. Finally, in order to demonstrate the OLMR introduced by PMSG2 more clearly, setting of PI parameters of type 4 reactive power controller of PMSG2 in case C were changed gradually to that in case D. Trajectories of open-loop and closed-loop

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

170

4 3.5

λi – g q ( λi )hq ( λi )

Imaginary axis

3 2.5

λi

Case D

λˆi

λi + g q ( λi )hq ( λi )

2

λvsc

1.5 1

λˆ

vsc

0.5 Case C 0 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Real axis Fig. 5.23 The OLMR between PMSG2 and power system

converter oscillation mode and the EOM involved in the OLMR with the change of parameters setting were calculated and are displayed in Fig. 5.23. From Fig. 5.23, it can be seen that with the change of PI parameters, open-loop converter oscillation mode, λvsc, got close to the open-loop EOM, λi, on the complex plane. Closed-loop inter-area EOM, ^λ i , moved away from λi towards the right on the complex plane and the damping of inter-area power oscillation decreased. In Fig. 5.23, estimated positions of closed-loop converter oscillation mode and the inter-area EOM, ^λ vsc and ^λ i , by using the derived equations similar to (5.11) and (5.12) are indicated by crosses. They are very close to the actual positions of ^λ vsc and ^λ i . This confirms the correctness of analysis and conclusions made in Sect. 5.1.1.3 about the OLMR. In Fig. 5.23, gq(s) ¼ (s  λi)Gvq(s), hq(s) ¼ (s  λvsc) Hpq(s).

5.1.2.7

Impact of Grid Connection of PMSG3

PMSG3 adopted the configuration of the GSC control system shown by Fig. 5.3 and used typical parameters given in [4]. Firstly, a converter oscillation mode of PMSG3 was calculated from the open-loop state matrix Ap in (5.1) to be λvsc ¼  4.2882 þ 8.2115j. With PMSG1 and PMSG2 being connected at node 22, the EOMs of example New England power system were calculated from the open-loop state matrix of power system Ag in (5.3). A local EOM was identified to

5.1 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected PMSG

171

9

λi – g p ( λi )hp ( λi )

8.8

Imaginary axis

8.6 8.4

λi

λˆ vsc

8.2 8

The OLMR

λvsc

7.8

λˆ i

7.6

λi + g p ( λi )h p ( λi ) 7.4 7.2 -4.6

-4.4

-4.2

-4

-3.8

-3.6

-3.4

Real axis Fig. 5.24 The OLMR between PMSG3 and power system

be close to λvsc ¼  4.2882 þ 8.2115j on the complex plane. The local EOM was λi ¼  4.1952 þ 8.3573j. Secondly, from the open-loop transfer functions of PMSG3 and power system, approximate positions of closed-loop converter oscillation mode and local EOM were estimated. The estimated positions are indicated by two crosses in Fig. 5.24. Thirdly, PMSG3 was connected at node 10 and closed-loop state matrix Agp in (5.5) was established. Closed-loop converter oscillation mode, ^λ vsc , and closed-loop EOM, ^λ i , were calculated from Agp. They are displayed in Fig. 5.24. From Fig. 5.24, it can be seen that the OLMR happened between the open-loop converter oscillation mode and the local EOM such that the damping of closed-loop local EOM decreased. The OLMR due to the grid connection of PMSG3 did not cause lightly or negatively damped local power oscillations, because open-loop local EOM was well damped before PMSG3 was connected. In a real power system, this may often be the case that the damping of the EOMs decreased as caused by the OLMR. However, grid connection of the PMSGs would not be identified as the cause of change of the damping of the EOMs if analysis of the OLMR was not carried out. Figure 5.25 presents the computational results of participation factors of synchronous generators and PMSG3 in the converter oscillation mode and the EOM. They clearly indicate the participations of SG2 and SG3 in the converter oscillation mode and the state variables associated with the active power control outer loop of PMSG3 in local EOM, thus confirming strong dynamic interactions introduced by PMSG3 as caused by the OLMR.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

172

a

b

0.45

1.2

0.4

1

Participation factors

Participation factors

0.35 0.3 0.25 0.2 0.15

0.8

0.6

0.4

0.1

0.2 0.05 0

0

DVdc

Dxvdc State variables

Dxid

1

2

3

7 6 5 4 Generator number

8

9

10

Fig. 5.25 Computational results of participation factors with PMSG3 being connected. (a) Participation of PMSG3 in the local EOM; (b) participation of SGs in the converter oscillation mode of PMSG3

5.2

Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

Grid connection of a DFIG for wind power generation changes the power system’s load flow conditions and introduces dynamic interactions. Changes in the load flow conditions and the dynamic interactions introduced by the DFIG can both affect the power system’s small-signal angular stability. In Sect. 4.1 of Chap. 4, a method was introduced to separately assess the impact of those two affecting factors, i.e., the change in the load flow conditions and the dynamic interactions introduced by the DFIG on the power system’s small-signal angular stability. Owing to the fast control speed of the VSC, it has been recognized that the dynamic interactions between the grid-connected DFIG and the power system are usually weak because of the dynamic “decoupling” effect of the VSC control. Using damping torque analysis, the reason for the low impact of the weak dynamic interactions introduced by the DFIG on the power system’s small-signal angular stability was explained in Sect. 4.1. Based on this, it was suggested to model the DFIG as a constant power source for pre-screening its impact on the power system’s stability. However, the computational results of the participation factors for a real largescale power system with grid-connected PMSGs in [1] reported an unusual phenomenon, wherein, there was a considerable participation of the PMSGs in the system EOMs. In the previous section, strong dynamic interactions introduced by the PMSGs were examined and are attributed to the OLMR between the converter oscillation modes of the PMSGs and power system EOMs. In this section, the strong dynamic interactions introduced by the DFIGs are investigated, because in case of strong dynamic interactions, the DFIG cannot be modeled as a constant power source for pre-screening its impact on the power system small-signal stability. Investigation on the strong dynamic interactions introduced by a grid-connected DFIG presented in this section is based on the establishment of a closed-loop

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

173

interconnected dynamic model of the power system with the DFIG, which was introduced in Sect. 2.4 of Chap. 2. In the established model, the power system and DFIG are modeled as two open-loop interconnected subsystems. The impact of the dynamic interactions introduced by the DFIG on the power system’s small-signal angular stability is assessed as the difference between the closed-loop and open-loop EOMs of the closed-loop interconnected model. Analysis is carried using the damping torque analysis to show that under the condition of the OLMR when an open-loop DFIG oscillation mode is close to an open-loop EOM on the complex plane, strong dynamic interactions between the DFIG and power system may occur, considerably affecting the power system’s small-signal angular stability. Further analysis reveals that the existence of the DFIG oscillation mode, causing the OLMR with the EOM, is related to the configuration and parameter setting of the RSC control system of the DFIG.

5.2.1

Strong Dynamic Interactions Introduced by the Grid-Connected DFIG and the Impact

5.2.1.1

Multivariable Closed-Loop Interconnected Model of the Power System with a Grid-Connected DFIG

Consider a multi-machine power system with a DFIG, as shown in Fig. 5.26. Pw þ jQw is the complex power output from the DFIG and Vw ∠θw is the terminal voltage of the DFIG. State-space model of the DFIG was derived in Sect. 2.2 as (2.108), which is re-written below:

A multimachine power system

Pw + jQw

DFIG

Ps + jQs

Rotor

GSC

GSC (grid side converter) control system

RSC

RSC (rotor side converter) control system

Fig. 5.26 Configuration of a multi-machine power system with a DFIG

Gear box

174

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

d ΔXd ¼ Ad ΔXd þ bv ΔVw þ b f Δθw dt ΔPw ¼ cp T ΔXd þ dd1 ΔVw þ dd2 Δθw

ð5:27Þ

ΔQw ¼ cq T ΔXd þ dd3 ΔVw þ dd4 Δθw The state-space model of the multi-machine power system was derived in Sect. 2.3 as (2.123), which is re-written as (5.28) below. d ΔXg ¼ Ag ΔXg þ bP ΔPw þ bQ ΔQw dt ΔVw ¼ cv T ΔXg þ dg1 ΔPw þ dg2 ΔQw

ð5:28Þ

Δθw ¼ c f T ΔXg þ dg3 ΔPw þ dg4 ΔQw From (5.27) and (5.28), the linearized model of the power system with the DFIG, shown in Fig. 5.1 is obtained to be, d ΔX ¼ Agd ΔX dt

ð5:29Þ

 T where ΔX ¼ ΔXg T ΔXd T . The dynamic models of the power system and the DFIG can be expressed by transfer function matrices. From (5.27) and (5.28), they are obtained to be Power system : " # " g11 ðsÞ ΔVw ¼ Δθw g21 ðsÞ DFIG : " # ΔPw ΔQw

" ¼

g12 ðsÞ

#"

g22 ðsÞ

d11 ðsÞ

d12 ðsÞ

d21 ðsÞ

d22 ðsÞ

ΔPw

# ¼ GðsÞ

ΔQw #"

ΔVw Δθw

"

#

" ¼ DðsÞ

ΔPw

#

ΔQw ΔVw

ð5:30Þ #

Δθw

where  1  1 g11 ðsÞ ¼ cv T sI  Ag bP þ dg1 , g12 ðsÞ ¼ cv T sI  Ag bQ þ dg2  1  1 g21 ðsÞ ¼ c f T sI  Ag bP þ dg3 , g22 ðsÞ ¼ c f T sI  Ag bQ þ dg4 d11 ðsÞ ¼ cp T ðsI  Ad Þ1 bv þ dd1 , d12 ðsÞ ¼ cp T ðsI  Ad Þ1 b f þ dd2

:

d21 ðsÞ ¼ cq T ðsI  Ad Þ1 bv þ dd3 , d22 ðsÞ ¼ cq T ðsI  Ad Þ1 b f þ dd4 The established model shown by (5.30) can be depicted by Fig. 5.27. It is a multivariable closed-loop interconnected dynamic model, where the dynamics of the power system and the DFIG are modeled separately as two open-loop interconnected subsystems.

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

g11 ( s )

DPw

175

DVw

+ g 21 ( s ) g12 ( s )

DQw

Dqw

+

g 22 ( s )

Power system

d12 ( s )

DFIG

+ d11 ( s ) d 22 ( s )

+ d 21 ( s ) Fig. 5.27 Multivariable closed-loop interconnected of the power system with the DFIG

5.2.1.2

Strong Dynamic Interactions Introduced by the DFIG

Denote ^λ i as the power system EOM of concern, of the closed-loop system shown in Fig. 5.27. Thus, ^λ i is a complex eigenvalue of the closed-loop state matrix, Agd, in (5.29). If ΔPw þ jΔQw ¼ 0, there are no dynamic interactions between the DFIG and the power system. The DFIG is degraded into a constant power source, Pw0 þ jQw0. In this case, the dynamic model of the power system shown by (5.28) becomes, d ΔXg ¼ Ag ΔXg dt

ð5:31Þ

Denote λi as the oscillation mode corresponding to ^λ i , when the DFIG is modeled as a constant power source. Obviously, λi is an eigenvalue of the open-loop state matrix, Ag, in (5.28) and (5.31). Hence, the difference between the closed-loop and open-loop EOMs, Δλi ¼ ^λ i  λi , measures the impact of the dynamic interactions of the DFIG with the power system, ΔPw þ jΔQw, on the concerned EOMs. Normally, the dynamic variations of ΔPw and ΔQw are small, indicating weak dynamic interactions between the DFIG and power system. The impact of the dynamic interactions on the power system’s small-signal angular stability,    ^ jΔλi j ¼ λ i  λi , should generally not be significant. A derivation of the damping

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

176

torque analysis similar to that presented in Sect. 4.1.1.1 of Chap. 4 will give the following estimation of Δλi as, Δλi ¼

N i X h Sik Re gpk ðλi Þd11 ðλi Þ þ gqk ðλi Þd21 ðλi Þ γkv ðλi Þ k¼1 h i þ gpk ðλi Þd12 ðλi Þ þ gqk ðλi Þd22 ðλi Þ γkf ðλi Þ

ð5:32Þ

where gpk(s) and gqk(s) are the transfer functions of the forward paths from ΔPw and ΔQw to the electromechanical oscillation loop of the kth synchronous generator, respectively; γkv(s) and γkf(s) are the reconstructing transfer functions of ΔVw and Δθw, respectively, by the rotor speed of the kth synchronous generator, Δωk, i.e., ΔVw ¼ γkv(s)Δωk and ΔVw ¼ γkf(s)Δωk; Sik is the sensitivity of the oscillation mode to the addition of damping torque on the kth synchronous generator; Re{} denotes the real part of a complex number. Denote λd as a complex eigenvalue of the open-loop state matrix of the DFIG, Ad, in (5.27), referred to as a DFIG oscillation mode, if its frequency is between 0.1 and 2.5 Hz. Thus, λd should be a pole of the transfer function matrix of the DFIG, D(s), in (5.30). It should satisfy|d11(λd)| ¼ 1 , |d12(λd)| ¼ 1 , |d21(λd)| ¼ 1or/and | d22(λd)| ¼ 1. If λd is close to the power system open-loop EOM of concern on the complex plane, i.e., λd  λi, then |d11(λi)|, |d12(λi)|, |d21(λi)|or/and |d22(λi)| should be large. In this case, it can be seen from (5.30) that the dynamic variations of either ΔPw or/and ΔQw can be significant at the complex oscillation frequency, λi, indicating strong dynamic interactions between the DFIG and the power system. From (5.32), it can be seen that in this case, the damping torque contribution from the DFIG to the kth synchronous generator can increase significantly at the complex oscillation frequency, λi. Hence, the EOM of concern may be affected considerably. This special condition of the OLMR between the open-loop power system and the DFIG, i.e., λd  λi, may lead to possible strong dynamic interactions between the DFIG and the power system. The impact of the OLMR introduced by the DFIG on the small-signal stability of the power system with the DFIG is examined in the following subsection.

5.2.1.3

The Open-Loop Modal Resonance (OLMR)

The characteristic equation of the multivariable closed-loop interconnected system shown by Fig. 5.27 is jI  GðsÞDðsÞj ¼ 0

ð5:33Þ

As λd and λi are the poles of open-loop transfer function matrices, G(s) and D(s), respectively, it can denote GðsÞ1 ¼ ðs  λi ÞG1 ðsÞ, DðsÞ1 ¼ ðs  λd ÞD1 ðsÞ

ð5:34Þ

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

177

Thus, characteristic equation of (5.33) becomes jðs  λi Þðs  λd ÞG1 ðsÞD1 ðsÞ  Ij ¼ 0

ð5:35Þ

Denote " KðsÞ ¼ G1 ðsÞD1 ðsÞ ¼ From (5.35) and (5.36),   ðs  λd Þðs  λi Þk11 ðsÞ  1    ðs  λd Þðs  λi Þk21 ðsÞ  ¼ 0

k11 ðsÞ

k12 ðsÞ

k21 ðsÞ

k22 ðsÞ

#

    ðs  λd Þðs  λi Þk22 ðsÞ  1   

ð5:36Þ

ðs  λd Þðs  λi Þk12 ðsÞ

When λd  λi, replacing s in (5.37) by ^λ i ¼ λi þ Δλi          k11 ^λ i k22 ^λ i  k12 ^λ i k21 ^λ i Δλ4i       k11 ^λ i þ k22 ^λ i Δλ2i þ 1  0

ð5:37Þ

ð5:38Þ

Substitute the following first-order Taylor series expansions at λi into (5.38)   k11 ^λ i ¼ k11 ðλi Þ þ k011 ðλi ÞΔλi ð5:39Þ   k22 ^λ i ¼ k22 ðλi Þ þ k022 ðλi ÞΔλi By ignoring the items higher than Δλ2i , it can have ½k11 ðλi Þ þ k22 ðλi ÞΔλ2i þ 1  0

ð5:40Þ

Hence sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Δλi   ¼ k ð λi Þ k11 ðλi Þ þ k22 ðλi Þ

ð5:41Þ

Denote ^λ d as the closed-loop oscillation mode of the DFIG corresponding to λd. ^λ d is also a solution of characteristic equation of (5.37). Hence, taking the derivation similar to that from (5.37) to (5.41), it can be obtained for ^λ d that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð5:42Þ Δλd ¼ ^λ d  λd   ¼ k ð λd Þ k11 ðλd Þ þ k22 ðλd Þ From (5.41) and (5.42), it can have ^λ i  λi  kðλi Þ ^λ d  λd  kðλd Þ  λi  kðλi Þ

ð5:43Þ

178

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

The above equations indicate that when the OLMR occurs, i.e., λd  λi, the corresponding closed-loop oscillation modes locate at approximately opposite positions on the complex plane in respect to the positions of coupled open-loop oscillation modes, λd  λi. Hence, there is always a closed-loop oscillation mode being on the right-hand side of λd  λi. This implies that the OLMR degrades the small-signal stability of the power system. When the EOM of concern, ^λ i , is on the right-hand side of λd  λi, the strong dynamic interactions caused by the OLMR degrade the damping of the EOM of concern. Since the impact of the OLMR is detrimental to power system small-signal stability, it is essentially important to carefully examine the condition under which the OLMR may possibly happen. The examination can be carried out by computing the oscillation modes of the open-loop subsystems in Fig. 5.27. In order to obtain some general guidelines in the examination instead of completely relying on the modal computation, in the following subsection, existence and sources of the openloop DFIG oscillation modes which may cause the OLMR are investigated. The investigation is conducted by deriving a simplified open-loop transfer function matrix model of the DFIG.

5.2.1.4

Non-existence of the DFIG Oscillation Mode

The assumptions for deriving the simplified transfer function matrix model of the DFIG are as follows: 1. The electromagnetic transient of the rotor windings of the DFIG is much faster than the electromechanical dynamic transient of the power system. Hence, the electromagnetic transient of the rotor windings can be ignored. 2. The active and reactive power outputs from the grid side converter (GSC) of the DFIG are Pr ¼  swPs, Qr ¼ 0, respectively, [6–9], where sw is the slip of the rotor motion of the DFIG. Hence, Pw ¼ Ps þ Pr ¼ (1  sw)Ps ¼ ωdPs, Qr ¼ 0, where ωd is the rotor speed of the DFIG. Configuration of the RSC control system of the DFIG is shown by Fig. 2.5, which is redrawn as Fig. 5.28. As per assumption (1) above, the current control inner loops shown in Fig. 5.28 can be removed to obtain [6–9]. Ird ¼ Irdref , Irq ¼ Irqref

ð5:44Þ

Direction of Vw is that of the q axis of the d-q coordinate system of the DFIG. Thus, the flux equations of the stator windings of the DFIG are (4.31) below, Vwd ¼ 0 ¼ ψsq ¼ Xss Isq þ Xm Irq Vwq ¼ Vw ¼ ψsd ¼ Xss Isd  Xm Ird

ð5:45Þ

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

q-axis current control inner loop

active power control outer loop Psref

+ – Ps

Qsref

+ – Qs

K pp +

K pq +

Kip

I sqref

s

Kiq

I rqref

X – ss Xm

I sdref



s

X ss Xm

– sw ( X rr –

+

Kiq ( s)

179

X m2 X ) I rd + sw m Vw X ss X ss

-

+

Vrq

– I rq

+

I rdref

+



– I rd

Kid ( s)

-

+

Vrd

Vw X2 d-axis current sw ( X rr – m ) I rq X m control inner loop X ss

reactive power control outer loop

Fig. 5.28 Configuration of the RSC control system of the DFIG

With assumption (2) above and (5.45) above, Ps ¼ Vwq Isq ¼ Vw Isq Pw ¼ ωd Ps

ð5:46Þ

Qw ¼ Qs ¼ Vwq Isd ¼ Vw Isd where ωd is the rotational speed of the DFIG. The shaft dynamics of the DFIG described by a two-mass model are (2.58). When a simple one-mass model is used, the rotor motion equation of the DFIG is dωd 1 ¼ ðPm  Ps Þ J dt

ð5:47Þ

where ωd ¼ 12 ðωd1 þ ωd2 Þ and J ¼ 12 ðJd1 þ Jd2 Þ is the constant of inertia of the rotor. Variation of mechanical power input is mainly affected by the wind speed and can be ignored to have ΔPm ¼ 0 . Thus, linearization of (5.47) is J

dΔωd ¼ ΔPs dt

ð5:48Þ

Linearizing (5.46) and then substituting (5.48), ΔPs ¼ Vw0 ΔIsq þ Isq0 ΔVw ΔPw ¼ Ps0 Δωd þ ωd0 ΔPs ¼

Ps0 ΔPs þ ωd0 ΔPs ¼ Js

ΔQs ¼ Vw0 ΔIsd þ Isd0 ΔVw ΔQw ¼ ΔQs

Ps0 ωd0  ΔPs Js

ð5:49Þ

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

180

Linearization of (5.45) is Xm Xm 1 ΔVw ΔIsq ¼  ΔIrq , ΔIsd ¼  ΔIrd  Xss Xss Xss

ð5:50Þ

Using linearized (5.44) and (5.50), from Fig. 5.28

Kip Xss ref Xss ΔIrq ¼ ΔIrqref ¼  ΔIsq ¼ Kpp þ ΔPs Xm Xm s Xss ref 1 ΔIrd ¼ ΔIrdref ¼  ΔIsd  ΔVw Xm Xm

Kiq Xss 1 ¼ Kpq þ ΔVw ΔQs  Xm Xm s

ð5:51Þ

From (5.49), (5.50) and (5.51),

Kip ΔPs ¼ Vw0 Kpp þ ΔPs þ Isq0 ΔVw s

Kiq ΔQs ¼ Vw0 Kpq þ ΔQs  Isd0 ΔVw s

ð5:52Þ

From (5.49) and (5.52), the elements in the transfer function matrix model of the DFIG are obtained to be

Ps0 ωd0  Isq0 s ΔPw Js  d11 ðsÞ ¼ ¼ , d12 ðsÞ ¼ 0 ΔVw 1 þ Vw0 Kpp s þ Kip d21 ðsÞ ¼

ð5:53Þ

ΔQs ΔQw Isd0 s  ¼ ¼ , d22 ðsÞ ¼ 0 ΔVw ΔVw 1 þ Vw0 Kpq s þ Kiq

The transfer function matrix model of the DFIG is 2

Ps0 ωd0  Isq0 s 6 Js 6  6 DðsÞ ¼ 6 1 þ Vw0 Kpp s þ Kip 6 4 Isd0 s   1 þ Vw0 Kpq s þ Kiq

3 7 07 7 7 7 5 0

ð5:54Þ

From (5.53) or (5.54), it can be seen that the DFIG does not have any oscillation mode, when the configuration of the RSC control system shown in Fig. 5.28 is used. Hence, it can be concluded that when the DFIG adopts the configuration of the RSC control system shown in Fig. 5.28, The OLMR should normally not occur and the dynamic interactions between the DFIG and power system should be weak. This conclusion is based on the simplified transfer function matrix model of the DFIG with assumptions (1) and (2) presented in the beginning of this subsection. It is to be

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

181

noted that the simplification ignores the faster transient of the rotor windings of the DFIG and the slower dynamics of the wind turbine compared to the electromechanical transient of the power system. In addition, the dynamic function of the GSC control system is represented by Pr ¼  swPs, Qr ¼ 0 [6–9]. Thus, the order of the dynamic model of the DFIG is considerably reduced.

5.2.1.5

Existence of the DFIG Oscillation Mode

The active power control outer loop in Fig. 5.28 can adopt the rotor speed of the DFIG as the input signal, instead of the active power [9]. In this case, the first equation in (5.51) becomes

Kip Xss ref Xss ref ΔIrq ¼ ΔIrq ¼  ΔIsq ¼  Kpp  ð5:55Þ Δωd Xm Xm s From the first equation in (5.49), (5.50) and (5.55),

Kip 1 ΔPs ¼ Vw0 Kpp þ ΔPs þ Isq0 ΔVw s Js

ð5:56Þ

Thus, ΔPs ¼

JIsq0 s2 ΔVw Js2 þ Vw0 Kpp s þ Vw0 Kip

ð5:57Þ

d11(s) can be obtained from (5.49) and (5.57) to be d11 ðsÞ ¼

ðωd0 Js  Ps0 ÞsIsq0 ΔPw ¼ ΔVw Js2 þ Vw0 Kpp s þ Vw0 Kip

ð5:58Þ

From (5.58), it can be seen that the DFIG now may have a pair of complex poles, i.e., oscillation modes. The existence of the DFIG oscillation modes may possibly cause the OLMR with the EOM of concern. In this case, the strong dynamic interactions between the DFIG and the power system may be accompanied by considerable variations of the active power output of the DFIG, ΔPw, during the electromechanical dynamic transient of the power system. The open-loop DFIG oscillation modes can be estimated from (5.58) to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vw0 Kpp  j Vw0 2 Kpp 2  4JKip ð5:59Þ λd  2J If the open-loop power system’s EOM of concern is known to be λi , a quick method to determine the range of the PI parameters of the active power control outer loop for identifying and avoiding modal coupling in this case, is to solve the following equation:

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

182

Qsref +

– Qs

K qi X rscQ s + – Vw

I sdref X ss K qv – s X Xm rscV

+

I rdref



+ – I rd

Kid ( s)

-

Vrd

+

Vw X2 d-axis current sw ( X rr – m ) I rq X ss X m control inner loop

reactive power control outer loop

Fig. 5.29 Configuration of the type 3 model for the RSC reactive power control system of the DFIG

Vw0 Kpp þ j

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vw0 2 Kpp 2  4JKip

ð5:60Þ 2J   ^ pp ; K ^ ip , setting the PI parameters of If the solution of the above equation is K   ^ pp  α; K ^ pp þ α and K the active power control outer loop outside the range,   ^ ip  β; K ^ ip þ β , (α and β are two appropriate positive numbers), may effectively K avoid and dismiss the OLMR. Figure 5.29 shows the configuration of the generic type 3 model which the RSC can use. From Fig. 5.29, the second equation in (5.51) becomes λi ¼

Xss ref 1  ΔVw ΔIrd ¼ ΔIrdref ¼  ΔIsd X Xm

m Xss Kqv Kqi 1 ΔQs  ΔVw  ¼ ΔVw Xm Xm s s

ð5:61Þ

Using (5.61), and the third equation in (5.49) and (5.50), respectively, d21 ðsÞ ¼

ΔQw Vw0 Kqv s  Isd0 s2 ¼ 2 ΔVw s þ Vw0 Kqv Kqi

ð5:62Þ

From (5.62), it can be seen that when the type 3 reactive power control model is used, the DFIG oscillation modes can be owing to the reactive power control outer loop in the RSC control system also. The parameter settings of the type 3 model may possibly lead to the OLMR. In this case, the OLMR may result in significant dynamic variations in the reactive power output of the DFIG, ΔQw.

5.2.2

Example 5.2

5.2.2.1

Evaluation of the Conclusions Regarding the Existence of the DFIG Oscillation Mode

The example New England power system shown by Fig. 5.7 is used in this section for demonstrating and evaluating the analysis and conclusions made in the previous

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

183

700 500

300

Imaginary axis

100 0.2 0 -0.2

-100 -300

Detailed model

-500

(5.58) used

-700 -610

-480

-330

-180

-30 -15 Real axis

-10

-5

0

Fig. 5.30 Trajectories of the open-loop eigenvalues of the DFIG, when Kpp and Kipare varied

section. A wind farm represented by a DFIG was connected to bus 22. A detailed dynamic model of the DFIG introduced in Chap. 2 is used. First, the RSC control system depicted in Fig. 5.28 was adopted by the DFIG connected at node 22. The state-space model of the DFIG as shown in (5.27) was established and the eigenvalues of the open-loop state matrix of the DFIG, Ad, were calculated. The PI parameters, Kpp and Kip, of the active power control outer loop were varied with Kip ¼ 20 Kpp. The trajectories of the complex eigenvalues of Ad are displayed the solid curves in Fig. 5.30. The PI parameters, Kpq and Kiq, of the reactive power control outer loop were varied with Kiq ¼ 20Kpq. The trajectories of the complex eigenvalues of Ad are displayed and highlighted in blue in Fig. 5.31. The shadow areas in Figs. 5.30 and 5.31 indicate the location of the power system EOMs with the oscillation frequency between 0.1 and 2.5 Hz. The entries of the trajectories of the eigenvalues of Ad into the shadow area are also displayed in Figs. 5.30 and 5.31. From these figures, it can be seen that there were no DFIG oscillation modes. In Figs. 5.30 and 5.31, the trajectories of the poles of d11(s) and d21(s) as calculated from (5.52), are also displayed as the dashed curves which overlaid the solid curves completely. They confirm the correctness of the assumptions made for deriving the elements,d11(s) and d21(s), in the simplified transfer function matrix in Sect. 5.2.1.4. Therefore, these figures confirm the analytical conclusion made in Sect. 5.2.1.4 that a DFIG oscillation mode does not occur, when the configuration of the RSC control system shown in Fig. 5.28 is used by the DFIG. In this case, there was no OLMR to cause the strong dynamic interactions between the DFIG and the power system.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

184

700 500

300

Imaginary axis

100 0.2 0 -0.2

-100 -300

Detailed model

-500 -700 -610

(5.58) used -480 -330

-180

-15 -30 Real axis

-10

-5

0

Fig. 5.31 Trajectories of the open-loop eigenvalues of the DFIG, when Kpq and Kiq are varied

Next, the configuration of the RSC active power control system in Fig. 5.28 adopted by the DFIG was changed by replacing the input signal of the active power by the rotor speed of the DFIG [9]. The PI parameters,Kpp and Kip, of the active power control outer loop of the RSC were varied with Kip ¼ 20Kpp. The trajectories of the complex eigenvalues of Ad are displayed as the solid curves in Fig. 5.32. In addition, the trajectories of the poles of d11(s) were calculated from (5.58), and displayed as the dashed curves in Fig. 5.32, which almost overlaid the solid curves; it can be seen that when the PI parameters of the RSC control system are varied, a pair of complex open-loop eigenvalues enter the shadow area, wherein, the EOMs are located. This confirms the analytical conclusion made in Sect. 5.2.1.5 that when the rotor speed input signal is used for the RSC active power control, a DFIG oscillation mode could exist, causing the OLMR with the power system EOMs. Finally, the configuration of the RSC reactive power control system shown in Fig. 5.29 was used by the DFIG. The PI parameters of the RSC reactive power control system were changed. The complex open-loop eigenvalues and poles of d21(s) were calculated from Ad and (5.62), respectively. When Kqi is varied from 0.1–6 with Kqv ¼ 13.5Kqi, the trajectories of the open-loop eigenvalues and the poles of d21(s) are displayed as the solid and dashed curve, respectively, in Fig. 5.33. From Fig. 5.33, it can be seen that a pair of DFIG oscillation modes existed owing to the adoption of the RSC reactive power control system shown in Fig. 5.29 that could possibly cause the OLMR.

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

185

700 400

Imaginary axis

100 20 0

-20 -100

Detailed model

-400 -700 -610 -480 -330 -180

(5.68) used -30 -20 Real axis

-15

-10

-5

0

Fig. 5.32 Trajectories of the open-loop eigenvalues of the DFIG, when the rotor speed input signal is used with Kpp and Kip varied

700 400 100

Imaginary axis

30 20 10 0 -10

-20 -30 -100

Detailed model

-400

(5.68) used

-700 -610 -480 -330 -180 -30

-1 -0.8 -0.6 -0.4 -0.2 Real axis

0

Fig. 5.33 Trajectories of the open-loop eigenvalues of the DFIG, when the RSC reactive power control system in Fig. 5.4 is used with Kqi and Kqv varied

186

5.2.2.2

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

The OLMR Owing to the RSC Active Power Control System of the DFIG

In this subsection, tests are carried out to demonstrate that the DFIG oscillation mode from the RSC active power control system of the DFIG connected at node 22 causes the OLMR with an EOM of the example New England power system. The example New England power system has a total of nine EOMs; the one with the lowest oscillation frequency at around 0.5 Hz is the EOM of concern. It is related to the inter-area oscillation of the tenth synchronous generator against the other synchronous generators. The DFIG connected at node 22 was operated with a fixed unity power factor and its active power output was 10 p.u. The configuration of the RSC active power control system shown in Fig. 5.28 was used. The following three cases were examined. Case A: The configuration of the RSC active power control system shown in Fig. 5.28 was used. Typical PI control parameters recommended in SIMULINK [10] were used with Kpp ¼ 1, Kip ¼ 100. Case B: The rotor speed input signal was used by the DFIG, replacing the active power input signal in Fig. 5.28. The PI parameters were retained at Kpp ¼ 1, Kip ¼ 100. Case C: Same as Case B but the values of PI gains were increased to Kpp ¼ 50, Kip ¼ 1000. Table 5.5 presents the results of the modal computations for the above three cases. In Table 4.5, λd is the open-loop DFIG oscillation mode associated with the RSC active power control system of the DFIG and was calculated from the open-loop state matrix, Ad; λi is the open-loop inter-area EOM of concern and was calculated from the open-loop state matrix, Ag, in (5.28); ^λ i is the closed-loop inter-area EOM of concern corresponding to λi and  was calculated from the closed-loop state matrix, Agd , in (5.29); jΔλi j ¼ ^λ i  λi  measures the impact of the dynamic interactions between the DFIG and the power system. From Table 5.5, it can be seen that in case A, the open-loop eigenvalue of the DFIG related to the RSC active power control is a real number. Hence, no DFIG oscillation mode existed. The effect of the DFIG’s dynamics on the inter-area EOM of concern was negligible. In case B, λd was close to λi and the OLMR between the DFIG and the power system occurred. The inter-area EOM of concern was affected significantly with a considerable decrease in the damping. In case C, λd was away from λi because of the increase in the PI control parameters. The OLMR disappeared

Table 5.5 Computational results of the DFIG’s dynamic impact owing to the RSC active power control system Case A B C

λd 16.667 0.3125 þ j3.5217 3.125 þ j10.735

λi 0.3144 þ j3.6009

b λi 0.31677 þ j3.6008 0.16193 þ j3.222 0.31191 þ j3.5947

|Δλi| 0.002372 0.40843 0.006681

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

187

4.6

Case C 4.4

lˆd

Imaginary axis

4.2 4 3.8

li

ld

Case B

3.6

lˆi

3.4 3.2 -0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

Real axis Fig. 5.34 The OLMR in case B

and the effect on the inter-area EOM of concern introduced by the DFIG’s dynamics was negligible. In order to clearly show the effect of the OLMR, as discussed in Sect. 5.2.1.4, the PI gains of the RSC active power controller of the DFIG were decreased such that the DFIG oscillation mode, λd, moved on the complex plane from the position in case C, towards λi. When the PI gains were decreased to the typical values recommended in [10], λd arrived at the position in case B such that λd  λi. The trajectory of the movement of λd is displayed as the solid curve in Fig. 5.34. The corresponding movement of the closed-loop EOM, ^λ i , and the closed-loop DFIG oscillation mode, ^λ d , (see Sect. 5.2.1.4) are also displayed as the dashed curves in Fig. 5.34, respectively. From Fig. 5.34, it can be seen that when λd was away from λi , ^λ i was close to λi. Thus, the dynamic interactions between the DFIG and the system were weak. However, when λd moved towards λi , ^λ d , moved along with λd such that ^λ i was “driven” away from λi towards the right on the complex plane for “avoiding” ^λ d . Thus, in case B, the damping of ^λ i decreased considerably, when the OLMR occurred. The participation factors of the DFIG’s state variables for the EOMs are indications of the dynamic interactions between the DFIG and the power system. Table 5.6 presents the computational results of the sum of the participation factors, PFdfig, of all the state variables of the DFIG connected at bus 22 for the inter-area EOM of concern. From Table 5.6, it can be clearly seen that in case B, when the OLMR

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

188

Table 5.6 Participation factors of the sum of all the state variables of the DFIG for the inter-area EOM of concern Case PFdfig(%)

A 0.40718

B 62.807

C 0.73363

0.25 Case C Case B

Participation factors

0.2

0.15

0.1

0.05

0

1

2

3

4 5 6 7 Generator number

8

9

10

Fig. 5.35 Participation of the synchronous generators in the DFIG oscillation mode

occurred, the DFIG’s state variables show considerably higher participations in the EOM of concern. The participation factors of the sum of the rotor position and speed of the synchronous generators in the DFIG oscillation mode were also computed for cases B and C. The computational results are presented in Fig. 5.35; it can be seen that when the OLMR occurred in case B, the synchronous generators participated considerably in the DFIG oscillation mode. Table 5.6, Fig. 5.35 indicate the significant dynamic interactions between the DFIG and the power system, when the OLMR occurred. Finally, to identify the source of the DFIG oscillation mode that caused the OLMR, the participation factors of the main state variables of the DFIG in cases B and C, respectively, are displayed in Fig. 5.36; it can be seen that when the OLMR occurred, the state variables with the maximum participation in the EOM of concern were Δωd and Δxd. These two state variables are related to the rotor motion equation of (5.48) and the PI controller of the RSC active power control outer loop. Hence, this confirms the analytical conclusion in Sect. 5.2.1.4 that the OLMR was due to the RSC active power control of the DFIG. Figures 5.37 and 5.38 present the results of non-linear simulation for cases B and C. In the simulation, a 20% drop in the wind power input to the DFIG occurred at 1 s and lasted for 100 ms. From Fig. 5.37, it can be seen that there were considerable

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

189

0.35 Case C Case B

Participation factors

0.3 0.25 0.2 0.15 0.1 0.05 0

Δωd

Δxd State variables

Fig. 5.36 Participation of the main state variables of the DFIG in the EOM of concern

11 Case C Case B

Active power outout of DFIG

10.8 10.6 10.4 10.2 10 9.8 9.6 9.4 9.2 9

0

1

2

3

4

5

6

7 8 Time(s)

9

10 11 12 13 14 15

Fig. 5.37 Active power output, Pw, from the DFIG. (a) Relative angular position between the tenth and first synchronous generator. (b) Relative angular position between the seventh and first synchronous generator.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

190

a

2 Case C Case B

1 0 -1

d 10 – d 1

-2 -3 -4 -5 -6 -7 -8 -9 -10

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time(s)

b

62

Case C Case B

61 60

d7 – d1

59 58 57 56 55 54

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Time(s) Fig. 5.38 Relative angular positions. (a) Relative angular position between the tenth and first synchronous generator; (b) relative angular position between the seventh and first synchronous generator

variations in the active power exchange between the DFIG and power system in case B, when there were strong dynamic interactions caused by the OLMR. Consequently, the damping of the inter-area low-frequency power oscillation decreased significantly, as shown in Fig. 5.38.

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

191

Table 5.7 Impact of the DFIG’s dynamics on the EOM of concern at different levels of wind power penetration Pw0 1 5 10

5.2.2.3

λi 0.30555 þ j3.43 0.30628 þ j3.5321 0.3125 þ j3.5217

b λi 0.29491 þ j3.3418 0.24155 þ j3.2682 0.16193 þ j3.222

|Δλi| 0.088839 0.27172 0.40843

PFdfig(%) 29.124 52.855 62.807

Impact of the DFIG’s Dynamics at Different Levels of Wind Power Penetration

At steady state, the active power output from the DFIG is, Pw0 ¼ ð1  sw0 ÞPs0 ¼ ωd0 Ps0 ¼ ωd0 Vw0 Isq0

ð5:63Þ

Hence, at a higher level of the wind power penetration, Pw0 and Isq0 are bigger. From (5.53) and (5.58), it can be seen that with an increased Isq0, |d11(λi)| increases accordingly. From (5.32), it can be observed that this will usually lead to the increase   of Δλi . Therefore, with the increase of the level of the wind power penetration, the impact of the DFIG’s dynamics on the power system’s EOMs is expected to increase. To demonstrate and confirm the conclusion made above regarding the impact of the DFIG’s dynamics at different levels of the wind power penetration, Table 5.7 presents the test results of the modal computations for the example New England power system with the DFIG at node 22, when the active power output, Pw0, was varied. The RSC control system of the DFIG was as same as that in case B in presented in Table 5.5. From Table 5.7, it can be seen that with the increase of the wind power penetration, the inter-area EOM of concern was affected more by the DFIG’s dynamics with an increased |Δλi|. The DFIG participated more in the EOM of concern with an increased participation factor, PFdfig. The computational confirmation regarding the participation factors of the synchronous generators in the DFIG oscillation mode is presented in Fig. 5.39, indicating the increased involvement of the synchronous generators in the DFIG oscillation mode, when the wind power penetration increased. Figure 5.40 depicts the participation factors of the state variables of the DFIG which had the highest participation in the EOM of concern, demonstrating a higher participation of the RSC active power control in the EOM of concern, when the wind penetration was higher. Further confirmation from the non-linear simulation is given in Figs. 5.41 and 5.42, showing the increased impact of the DFIG’s dynamics, when the wind power penetration increased.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

192 0.25

Pw0=1 p.u. Pw0=10 p.u.

Participation factors

0.2

0.15

0.1

0.05

0

1

2

3

4

5

6

7

8

9

10

Generator number

Fig. 5.39 Participation of the synchronous generators in the DFIG oscillation mode at different levels of wind power penetration

0.35

Participation factors

0.3

Pw0=1 p.u. Pw0=10 p.u.

0.25 0.2 0.15 0.1 0.05 0

Δωd

Δxd State variables

Fig. 5.40 Participation of the main state variables of the DFIG in the EOM of concern at different levels of wind power penetration

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG 0.6 0.5

Pw0=1 p.u. Pw0=10 p.u.

0.4

Active power variation of DFIG

193

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time(s)

Fig. 5.41 Active power output, ΔPw, from the DFIG at different levels of wind power penetration 6 Pw0=1 p.u. Pw0=10 p.u.

5 4 3

d10 – d1

2 1 0 -1 -2 -3 -4 -5 -6

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time(s)

Fig. 5.42 Relative angular position between the tenth and first synchronous generator at different levels of wind power penetration

5.2.2.4

The OLMR Owing to the RSC Reactive Power Control System of the DFIG

The analysis in Sect. 5.2.1.5 has concluded that when the type 3 model of the RSC reactive power control is used, the DFIG oscillation mode may originate from the

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

194

Table 5.8 Computational results of the DFIG’s dynamic impact owing to the RSC reactive power control system Case D E

a

λd 0.23355 þ j0.85 0.2823 þ j3.7341

b λi 0.31704 þ j3.5999 0.11453 þ j3.8062

|Δλi| 0.002823 0.28652

PFdfig (%) 0.43015 70.491

b

0.4

0.35

Participation factors

Participation factors

0.3

0.3

0.2

0.1

0.25 0.2 0.15 0.1 0.05

0

ΔxQ

ΔxV

State variables

0

1

2

3

4 5 6 7 8 Generator number

9

10

Fig. 5.43 Participation of the DFIG and SGs in case E. (a) Participation of the DFIG’s main state variables in the EOM; (b) participation of SGs in the DFIG oscillation mode

complex pole of d21(s) (see (5.62)), leading to the OLMR with the power system EOMs. This possibility has been confirmed by Fig. 5.33. In this subsection, the DFIG connected at bus 22 adopted the configuration of the RSC reactive power control system shown in Fig. 5.29 (type 3 model). The following two tests were carried out. Case D: Typical parameters of the type 3 model recommended in SIMULINK [11] were used with Kqi ¼ 0.05, Kqv ¼ 20. Case E: The values of the PI parameters of the type 3 model of the RSC reactive power control were changed to Kqi ¼ 1, Kqv ¼ 13.5. Table 5.8 presents the results of the modal computation for the above two cases. In case D, λd was away from λi ¼  0.3144 þ j3.6009. The dynamic interactions between the DFIG and power system were weak, imposing negligible impact on the inter-area EOM of concern. Hence ^λ i was close to λi and the damping of the interarea EOM did not change considerably, when the dynamics of the DFIG were included in power system. However, in case E, λd was close to λi ¼  0.3144 þ j3.6009. The OLMR occurred and the damping of ^λ i decreased considerably, indicating the occurrence of poorly damped inter-area oscillations in the example New England power system, caused by the grid connection of the DFIG at node 22. The participation factors of the sum of all the state variables of the DFIG, PFdfig, were calculated and the computational results are given in the last column of Table 5.8. Figure 5.43 presents the computational results of the participation factors of the main state variables of the DFIG for the EOM of concern and those of the synchronous generators for the DFIG oscillation mode in case E. The computational results reveal the significant participation of the synchronous generators and the

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

195

4 Case E

li

3.5

lfi

Imaginary axis

3 2.5 2 lfd

1.5

ld

1 Case D 0.5 -0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

Real axis Fig. 5.44 The OLMR in case E

DFIG in the DFIG oscillation mode and EOM of concern, respectively, confirming the considerable dynamic interactions between the DFIG and the synchronous generators, caused by the OLMR. For a clearer demonstration of the modal coupling, Fig. 5.44 presents the movement of the open-loop and closed-loop EOMs of concern and the DFIG oscillation mode on the complex plane, with the change of the parameters of the RSC reactive power controller of the DFIG from the typical values in case D to those in case E, when the OLMR occurred. From Fig. 5.44, it can be seen that when the open-loop DFIG oscillation mode, λd, (the solid curve) moves towards λi (hollow circle), the movement of ^λ d along with λd “drove” the closed-loop EOM, ^λ i , away from λi towards the right on the complex plane. Subsequently, the damping of the inter-area EOM of concern decreased owing to the OLMR. The pattern of the OLMR shown in Fig. 5.44 is as same as that shown in Fig. 5.34, validating the analysis made previously regarding the OLMR between the DFIG and the power system. Further validation from the non-linear simulation is presented in Figs. 5.45 and 5.46. In the simulation, an 80% load was lost at node 8 at 1 s of the simulation and the load loss was recovered in 100 ms. From Fig. 5.45, it can be seen that in case E, the dynamic variations of the reactive power output from the DFIG were obvious. This confirms that the OLMR was due to the RSC reactive power control, leading to the considerable increase of ΔQw. In this case, the damping of the inter-area EOM of concern was very small to be about 0.03, causing the reactive power output from the DFIG to oscillate longer.

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

196

0.2

Case D Case E

Reactive output of DFIG

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

1

2

3

4

5 6 Time(s)

7

8

9

10

Fig. 5.45 Variation of the reactive power output, ΔQw, from the DFIG

-4.2 Case D Case E

-4.3

d10 – d1

-4.4 -4.5 -4.6 -4.7 -4.8

0

1

2

3

4

5 6 Time(s)

7

8

9

10

Fig. 5.46 Relative angular position between the tenth and first synchronous generator

5.2.2.5

OLMR Between Multiple DFIGs

In this subsection, a test is carried out to demonstrate that in the OLMR, the openloop oscillation mode of the power system can be a DFIG oscillation mode of another DFIG in the power system. Thus, the OLMR occurs between two DFIGs. First, the test assumes that the wind farm connected at node 22 in the example New England power system was represented by two DFIGs. This is possible in practice, when the DFIGs in a wind farm distribute in two areas. The DFIGs in each area are aggregated and represented by a DFIG. Those two DFIGs connected at node

5.2 Impact of Strong Dynamic Interactions Introduced by a Grid-Connected DFIG

197

4.2 4

Imaginary axis

3.8

lfd2

3.6

ld1

3.4

ld2

3.2 3

lfd1

2.8 -1.25

-1

-0.75

-0.5

-0.25

0

Real axis Fig. 5.47 The OLMR between DFIG1 and DFIG2

22 were named as DFIG1 and DFIG2, respectively, for the convenience of discussion. The active power output from each of the DFIGs was 5 p.u. The configuration of the RSC active power control system shown in Fig. 5.28 with the rotor speed input signal being adopted by both the DFIGs. Typical PI control parameters recommended in SIMULINK [10] were used. The dynamic interactions of DFIG2 with the rest of the power system were examined. In this test, the open-loop statespace model of the power system described by (5.28) included the dynamics of DFIG1. The open-loop state-space model of the DFIG depicted by (5.27) included only the state variables of DFIG2. From the open-loop state matrix, Ag, in (5.28), a DFIG oscillation mode of DFIG1 was calculated and identified. This was the open-loop DFIG oscillation mode of the power system and indicated as λd1 in Fig. 5.47 by the hollow circle. From the open-loop state matrix, Ad, in (5.27), the open-loop DFIG oscillation mode of DFIG2 was calculated and indicated as λd2 in Fig. 5.47 by the hollow diamond. The corresponding closed-loop DFIG oscillation modes were calculated to be ^λ d1 and ^λ d2 , respectively. They are indicated by solid circle and diamond, respectively, in Fig. 5.47. Since the parameters of the RSC active power control systems of DFIG1 and DFIG2 were same, λd1 and λd2 were close to each other on the complex plane. Subsequently, the OLMR occurred between two DFIGs, causing both ^λ d1 and ^λ d2 to move away from λd1  λd2 on the complex plane, as shown in Fig. 5.47. The computational results of the participation factors are presented in Fig. 5.48, indicating considerable dynamic interactions between DFIG1 and DFIG2 when the OLMR occurred. SG10 in the example New England power system is an aggregated synchronous generator, representing an external large-scale power system [2]. With the

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

198

a

b

0.7

DFIG1 DFIG2

0.5 0.4 0.3 0.2

DFIG1 DFIG2

0.35

Participation factors

Participation factors

0.6

0.4

0.1

0.3 0.25 0.2 0.15 0.1 0.05

0

Δωdfig

0

ΔXω

Δωd

Δxd

State variables

State variables

Fig. 5.48 Participation factors of DFIG1 and DFIG2. (a) Participation of the DFIGs in ^λ d1 ; (b) Participation of the DFIGs in ^λ d2 5

lˆi

4.5

A

lˆ d2

Imaginary axis

4 3.5 3

ld1

ld2 2.5 2

lˆd1

1.5 1

-2

-1.5

-1

-0.5

0

0.5

Real axis

Fig. 5.49 The OLMR between DFIG1, DFIG2 and the power system

synchronous generators in the external power system withdrawing or participating in the operation, the constant of inertia of SG10 varies. Hence, in the second stage of the test, the inertia of SG10 was varied such that the inter-area EOM of concern moved on the complex plane. The movement of the EOM is displayed in Fig. 5.49 as the solid curve with the filled circles. Subsequently, the closed-loop DFIG oscillation modes, ^λ d1 and ^λ d2 , moved away from each other to “avoid” getting close to each other on the complex plane, as shown in Fig. 5.49. When the power system operated around point A, ^λ d1 moved into the right half of the complex plane. Thus, the power system lost the small-signal stability. The OLMR shown in Fig. 5.49 can also be explained as follows: The closeness of the open-loop DFIG oscillation modes of DFIG1 and DFIG2 cause both ^λ d1 and ^λ d2 to move away from the position, where λd1  λd2 on the complex plane. In order to

Appendix 5.1: Data of Examples 5.1 and 5.2

199

2 After DFIG2 was connected Before DFIG2 was connected

0

d10 – d1

-2 -4 -6 -8 -10 -12

0

2

4

6

8

10

12

14

16

18

20

Time(s) Fig. 5.50 Non-linear simulation when the OLMR occurred between the DFIGs

avoid ^λ d2 moving closer to ^λ d1 , ^λ d2 , “drove” ^λ d1 further towards the right. Eventually, the OLMR between DFIG1 and DFIG2 caused the loss of the system small-signal stability. Confirmation from the simulation results is presented in Fig. 5.50.

Appendix 5.1: Data of Examples 5.1 and 5.2 Data of New England Power System They are given in Appendix 4.2.1.

Data of DFIG They are given in Appendix 4.2.2.

Data of PMSG Basic data Xpd ¼ 0.25, Xpq ¼ 0.15, Xpf ¼ 0.05, Cp ¼ 30, Jpr ¼ 8s, ψpm ¼ 1.1

200

5 Small-Signal Angular Stability of a Power System Affected by Strong. . .

Control parameters of MSC Kpp1 ¼ 5, Kpi1 ¼ 20, Kpp2 ¼ 1, Kpi2 ¼ 100, Kpp3 ¼ 1, Kpi3 ¼ 100 Control parameters of GSC Kpp4 ¼ 0.8, Kpi4 ¼ 20, Kpp5 ¼ 0.15, Kpi5 ¼ 84.9, Kpp6 ¼ 0.28, Kpi6 ¼ 12, Kpp7 ¼ 0.15, Kpi7 ¼ 84.9

References 1. Quintero J, Vittal V, Heydt GT, Zhang H (2014) The impact of increased penetration of converter control-based generators on power system modes of oscillation. IEEE Trans Power Syst 29(5):2248–2256 2. Rogers G (2000) Power system oscillations. MA Kluwer, Norwell 3. Padiyar KR (1996) Power system dynamics stability and control. Wiley, New York 4. Kima HW, Kimb SS, Koa HS (2010) Modeling and control of PMSG-based variable-speed wind turbine. Electr Power Syst Res:46–52 5. MATLAB Simulink Wind Farm—Synchronous Generator and Full Scale Converter (Type 4) Detailed Model. Website: http://uk.mathworks.com/help/physmod/sps/examples/wind-farmsynchronous-generator-and-full-scale-converter-type-4-detailed-odel.html? requestedDomain¼www.mathworks.com 6. Ekanayake JB, Holdsworth L, Jenkins N (2003) Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines. Elect Power Syst Res 67 (3):207–215 7. Feijóo A, Cidrás J, Carrillo C (2000) A third order model for the doubly-fed induction machine. Elect Power Syst Res 56(2):121 8. Ko HS, Yoon GG, GG KNH, Hong WP (2008) Modeling and control of DFIG-based variablespeed wind-turbine. Elect Power Syst Res 78(11):1841–1849 9. Pena R, Clare JC, Asher GM (1996) Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation. IEEE Proc Electr Power Appl 143(3):231–241 10. MATLAB Simulink Wind Farm (DFIG Phasor Model) 11. MATLAB Simulink Wind Farm-DFIG Detailed Model

Chapter 6

Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Grid connection of a VSWG to a power system is realized by the converter control, which normally adopts the current vector control algorithm. This has been introduced in Chap. 2. Implementation of current vector control needs to track the relative positions between the d  q coordinate of the converter and the common x  y coordinate of the power system in order to determine the d  q coordinate of the converter. As being illustrated by Fig. 2.8, the direction of the terminal voltage (PCC voltage) of a PMSG in x  y coordinate of the power system is normally taken as that of d axis of d  q coordinate of the GSC of the PMSG. Figure 2.10 shows that the direction of the terminal voltage (PCC voltage) of a DFIG in x  y coordinate of the power system is often taken as that of d axis of d  q coordinate of the RSC of the DFIG. Hence, by tracking the phase of the terminal voltage of the VSWG, d  q coordinate of the converter can be determined for implementing the current vector control. This task of phase tracking is normally fulfilled by a phase locked loop (PLL). So far, various types of PLLs have been proposed. The core of majority of proposed PLLs have been a closed-loop control system for the phase tracking. In the PLL design, parameter setting for the closed-loop control is critical and needs to consider carefully the speed, accuracy, and robustness of the phase tracking, under variable power system operating conditions. For example, when high-bandwidth PLL parameters are set, a fast phase tracking performance can be obtained. However, a high-bandwidth may reduce the phase tracking ability of the PLL under unbalanced operating conditions. In addition to considering the phase tracking performance of the PLL itself, the impact of dynamic interactions brought about by the PLL on the power system stability should also be taken into account in the design. The PLL functions to link the dynamics of the VSWG and the power system. The impact of the dynamic interactions introduced by the PLL is in two folds: The firstfold impact caused by the dynamic interactions introduced from the PLL is on the stability of the VSWG. Investigation up-to-date has indicated that when a VSWG is

© Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_6

201

202

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

weakly connected to an external power system, the PLL may cause the small-signal instability in the VSWG system. In addition, when the bandwidth of the PLL is close to that of DC voltage control or current control inner loops of the GSC of the PMSG, dynamic interactions introduced by the PLL became strong which may affect the small-signal stability of the PMSG negatively. In the investigation, the PLL parameters (mainly the PI gain) were varied in order to demonstrate and identify the potential risk brought about by the PLL. The strategy of scanning the PLL parameter space by varying the PI gains of the PLL is effective in determining the constraints imposed by the stability requirement on the parameter setting of the PLL. In Sect. 1.2.1, this first-fold impact of the PLL has been reviewed. The second-fold impact of the PLL is on the power system stability. When the impact of a grid-connected VSWG on power system small-signal angular stability is investigated, often it was assumed that the PLL has a perfect phase tracking performance with zero phase-tracking error. Thus, the impact of the PLL was not considered in the investigation. However, results of examination in [1, 2] indicated that the PLL dynamics may affect the power system electromechanical oscillation modes (EOMs) considerably. In [1], the PI gains of the PLL were varied in a wide range for scanning the potential risk introduced by the PLL to the power system stability. With the variation of PI gains of the PLL, it was found from the EOM trajectories that, when the bandwidth of the PLL was reduced, the EOMs may be affected. Results of modal analysis by varying the PI gains of the PLL in [2] indicated that when a PLL oscillation mode (POM) was close to a power system EOM, the PLL affected the EOM considerably. However, it was not known whether the results of the case-by-case modal analysis in [1, 2] were special cases or not. In this chapter, the small-signal stability of a power system with a wind farm affected by the PLL is examined. The rest of the chapter is organized as follows. In the next section, a closed-loop interconnected linearized model of the power system with a PMSG is established, where the PLL for the PMSG and the rest of the power system (ROPS) are modeled as two separate open-loop interconnected subsystems. The established model clearly indicates that the effect of the PLL phase-tracking error on the power system small-signal stability can be assessed as the difference between the open-loop and closed-loop oscillation modes. Based on the established model, analysis is carried out to indicate that, when an open-loop oscillation mode of the PLL subsystem is close to an open-loop oscillation mode of the ROPS subsystem on the complex plane, the open-loop modal resonance may lead to the strong dynamic interactions between the PLL and the ROPS. It is very likely that openloop modal resonance may lead to a damping degradation of either the closed-loop oscillation mode of the PLL or the ROPS such that the small-signal stability of the power system with the VSWG decreases. In Sect. 6.2, the small-signal stability of example PMSG systems affected by the PLL is examined by using the theory of open-loop modal resonance presented in Sect. 6.1 of the chapter. In this example PMSG system, dynamics of the AC power system are not considered and hence the AC power system is modelled as an infinite

6.1 Impact of Open-Loop Modal Resonance Caused by the PLL. . .

Ppmsg

Pvsc GSC

V pcc q pcc

Vcd + jVcq PMSG

C

Vdc

Xf Pvsc +jQvsc

I d + jI q

Rest of power system

PLL

q pll

PLL

203

External power system

q pcc

Fig. 6.1 A PLL connected to an external power system

busbar. In Sect. 6.3, the small-signal angular stability of an example multi-machine power system with PMSGs is examined to demonstrate and validate the analysis and conclusions made about the open-loop modal resonance. Case studies show that open-loop modal resonance occurs when the power system operating conditions change or when the PLL parameters are varied. Subsequently, open-loop modal resonance causes poorly damped LEPOs in the example multi-machine power system.

6.1

Impact of Open-Loop Modal Resonance Caused by the PLL for a Grid-Connected PMSG

6.1.1

Closed-Loop Interconnected Model of a Power System with a PMSG

6.1.1.1

Function of a PLL

Figure 6.1 shows the configuration of a PMSG being connected to an external power system. The external power system may be a multi-machine power system with multiple grid-connected VSWGs. Function of the PLL is to track the phase of the terminal voltage of the PMSG at the point of common coupling (PCC), θpcc. The tracked phase, θpll, by the PLL is taken as the direction of the d-axis of the d  q coordinates of the grid side converter (GSC), as shown by Fig. 2.10, which is redrawn as Fig. 6.2. An ideal PLL has a perfect phase-tracking performance with zero phase-tracking error, θerror ¼ θpcc  θpll ¼ 0. In this case, the direction of the PMSG terminal voltage, Vpcc ∠ θpcc, coincides with that of daxis such that Vq  0. Strictly speaking, θerror ¼ θpcc  θpll 6¼ 0, although a well-designed PLL may have a negligible phase-tracking error, i.e.,θerror ¼ θpcc  θpll  0. When the negligible phase-tracking error is ignored, it can be assumed that the PLL is ideal

204

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Fig. 6.2 Phase tracking of the PMSG’s terminal voltage by the PLL to determine the relative position between d  qcoordinate of the GSC of the PMSG and common x  y coordinate of the power system

x q

V pcc

θ pcc d

θerror θ pll

y

θ pcc

with zero phase-tracking error. Consider the PLL as a closed-loop control system with the pair of input-output signals to be, θpcc  θpll. During the transient of the power system, the output, θpll, may not exactly follow the input, θpcc, at any instant of time such thatθpcc 6¼ θpll. Since the direction of d-axis direction of the d  q coordinate of the GSC is determined by θpll, as depicted in Fig. 6.2, Vd ¼ Vpcc cosθerror , Vq ¼ Vpcc sinθerror

ð6:1Þ

At steady state, θerror0 ¼ θpcc0  θpll0 ¼ 0; the linearization of (6.1) is, ΔVd ¼ ΔVpcc , ΔVq ¼ Vpcc0 Δθerror ¼ Vd0 Δθerror

ð6:2Þ

From (6.2), it can be seen that when the phase-tracking error of the PLL is considered such that θpcc 6¼ θpll, ΔVq 6¼ 0.

6.1.1.2

Closed-Loop Model

Denote hpll(s) as the transfer function of the PLL, i.e., Δθpll ¼ hpll(s)Δθpcc. Hence, the phase-tracking error of the PLL is,   θerror ¼ θpcc  θpll ¼ 1  hpll ðsÞ Δθpcc ¼ Hpll ðsÞΔθpcc ð6:3Þ Let the state-space realization of the transfer function, Hpll(s), be, d ΔXpll ¼ Apll ΔXpll þ bθ Δθpcc dt Δθerror ¼ cθ T ΔXpll þ dθ Δθpcc where ΔXpll is the vector of all the PLL state variables.

ð6:4Þ

6.1 Impact of Open-Loop Modal Resonance Caused by the PLL. . . Fig. 6.3 Closed-loop model of the power system with the PMSG

Dθerror Rest of the power system

205

Gg (s)

Dθpcc

PLL Hpll (s)

There are various schemes for building and designing a control system to fulfil a function for θpll to track θpcc. These schemes are often referred to as different PLLs. However, as long as the core of majority of the PLLs is a closed-loop control system with the pair of input-output signals and transfer function to be θpcc  θpll and hpll(s) respectively, the function of the PLL can be described by the state-space model of (6.4) with Hpll(s) ¼ cθT(sI  Apll)1bθ þ dθ. The pair of input-output signals for the PLL is converted to θpcc  θerror. This implies that for the rest of the power system excluding the PLL, the pair of input-output signals is θerror  θpcc. Hence, the following state-space model of the rest of the power system can be established. 8 < d ΔX ¼ A ΔX þ b Δθ g g g g error dt ð6:5Þ : T Δθpcc ¼ cg ΔXg þ dg Δθerror where ΔXg is the vector of all the state variables of the rest of the power system. The transfer function model of the rest of power system can be obtained from (6.5) as, Δθpcc ¼ Gg ðsÞΔθerror

ð6:6Þ

where Gg(s) ¼ cgT(sI  Ag)1bg þ dg. From (6.3) and (6.6), a closed-loop interconnected model of the power system with the PMSG is established, as displayed in Fig. 6.3. In the established model, the PLL and the rest of the power system are two separate open-loop interconnected subsystems, as illustrated in Fig. 6.1. The transfer functions of those two open-loop subsystems are Hpll(s) and Gg(s), respectively, as shown in Fig. 6.3. Their state-space representations are (6.4) and (6.5), respectively. The state-space model of the closedloop interconnected system shown by Fig. 6.3 is, d ΔX ¼ AΔX dt

ð6:7Þ

206

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

where  T ΔX ¼ ΔXg T ΔXpll T " # A11 A12 A¼ A21 A22 A11 ¼ Ag þ

d1 bP cg T þ d2 bQ cg T 1  d1 dP  d2 d Q

A12 ¼ bP cp1 T þ bQ cp2 T þ A21 ¼

ðd1 dP bP þ d2 dP bQ Þcp1 T þ ðd1 dQ bP þ d2 dQ bQ Þcp2 T 1  d1 dP  d2 dQ

bθ cg T 1  dP d1  dQ d2

A22 ¼ Apll þ

dP bθ cp1 T þ dQ bθ cp2 T 1  dP d1  dQ d2

6.1.2

Open-Loop Modal Resonance Caused by the PLL

6.1.2.1

The Condition of Open-Loop Modal Resonance (OLMR)

For an ideal PLL, the phase-tracking error is zero, i.e., Δθerror ¼ 0. In this case, the closed-loop interconnected model in Fig. 6.3 is open and the PLL does not introduce any dynamic interactions with the rest of power system. The state-space model of the rest of the power system (ROPS) becomes, d ΔXg ¼ Ag ΔXg dt

ð6:8Þ

Denote λi as an oscillation mode of the ROPS subsystem in Fig. 6.3. When an ideal PLL is considered (Δθerror ¼ 0), λi is an eigenvalue of the open-loop state matrix, Ag, in (6.5) or (6.8). Denote ^λ i as the oscillation mode corresponding to λi when the PLL is non-ideal with a phase-tracking error (Δθerror 6¼ 0). Obviously, ^λ i is an eigenvalue of the closed-loop state matrix, A, in (6.7). Hence, the impact of the PLL phase-tracking error on the oscillation mode of the power system is the difference between the open-loop and closed-loop oscillation mode Δλi ¼ ^λ i  λi . If ^λ i is on the right-hand side of λi on the complex plane, it implies that the PLL phase-tracking error degrades the damping of the oscillation mode and hence, is detrimental to the power system small-signal stability. A well-designed PLL should have a negligible phase-tracking error, i.e., Δθerror  0. Thus, the closed-loop system shown by Fig. 6.3 is “approximately” open. The difference between the open-loop and closed-loop EOMs of concern, Δλi ¼ ^λ i  λi , is negligible. In this case, the dynamic interactions between the PLL

6.1 Impact of Open-Loop Modal Resonance Caused by the PLL. . .

207

and the rest of power system are weak. This explains why an ideal PLL can normally be assumed, when the impact of the PMSG on the power system small-signal angular stability is examined. However, under the special condition of open-loop modal resonance (OLMR), this normality may change as to be elaborated as follow. Denote λpll as a complex pole of the PLL’s transfer function, Hpll(s), which is referred to as the PLL oscillation mode (POM). λpll is a complex eigenvalue of the open-loop state matrix, Apll, in (6.4). An open-loop modal resonance is the special condition that an open-loop POM is near the open-loop oscillation mode of the power system on the complex plane, i.e., λpll  λi. As |Hpll(λpll)| ¼ 1, |Hpll(λi)| is significant, when λpll  λi. From (6.3) and Fig. 6.3, it can be seen that under the condition of the OLMR, Δθerror may become significant around the complex frequency, λi. In this case, the dynamic interactions between the PLL and the rest of the power system may be strong. Considerable dynamic interactions are introduced by the PLL which exhibit as the considerable dynamic variations of Δθerror. It would be extremely difficult to establish the mathematical expression, Δλi ¼ ^λ i  λi , for determining the exact impact of the strong dynamic interactions introduced by the PLL under the condition of the OLMR. However, an estimate of Δλi ¼ ^λ i  λi in the neighborhood of λi ( ^λ i ! λi ) can be derived. The derived estimation indicates the location of ^λ i in respective from λi on the complex plane when the OLMR occurs. That is the impact of the OLMR caused by the PLL on the small-signal stability of the power system, which is discussed in the next subsection.

6.1.2.2

Impact of OLMR

From Fig. 6.3, the characteristic equation of the closed-loop interconnected system can be obtained as, 1 ¼ Gg ðλÞHpll ðλÞ

ð6:9Þ

Express the PLL open-loop transfer function and that of the rest of the power system in Fig. 6.3, respectively, as, G g ðsÞ ¼

n X

m X Rgk R  pllk  þ dθ þ dg , Hpll ðsÞ ¼ ð s  λk Þ s  λpllk k¼1 k¼1

ð6:10Þ

where λk, k ¼ 1, 2, . . .n are the eigenvalues of the open-loop state matrix, Ag, in (6.5); Rgk, k ¼ 1, 2, . . .n are the residues corresponding to λk, k ¼ 1, 2, . . .n; λpllk, k ¼ 1, 2, . . .m are the eigenvalues of the open-loop state matrix, Apll, in (6.4) and Rpllk, k ¼ 1, 2, . . .m are the residues corresponding to λpllk, k ¼ 1, 2, . . .m. The closed-loop oscillation mode of the power system, ^λ i , is a solution of (6.9). Hence, from (6.9) and (6.10), " #" # n m X X Rgk Rpllk   þ dθ þ dg 1¼ ð6:11Þ ðλ fi  λk Þ λ fi  λpllk k¼1 k¼1

208

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

2  Multiplying both sides of (6.11) by ^λ i  λi and under the condition of the OLMR, i.e., λpll  λi, 8 39 2 > > > > > > > 7> 6 n = 7   2 < 6 X Rgk ^λ i  λi  Rgi þ ^λ i  λi 6 7   þ dg 7 6 > > > 5> 4 k ¼ 1 ^λ i  λk > > > > ; : k ¼ 6 i 8 2 39 ð6:12Þ > > > > > > > > 6 7 < m 7=  6 X Rpllk 7 Rpll þ ^λ i  λi 6 þ d   θ 6 7> ^λ i  λpllk > > 4 5> > > k ¼ 1 > > ; : k 6¼ i where Rgi and Rpll are the residues corresponding to λi and λpll, respectively. Hence, in the neighborhood of λi (^λ i ! λi ), 2  2  Δλi 2 ¼ ^λ i  λi  lim ^λ i  λi  Rgi Rpll ^λ i !λi

ð6:13Þ

Denote ^λ pll as the closed-loop POM corresponding to λpll. The above derived equation, (6.13), will be more meaningful, when the estimation of ^λ pll is derived. ^ λ pll is also a solution of (6.9). With a derivation similar to that of (6.9) to (6.12) for ^λ pll , 2  2  Δλpll 2 ¼ ^λ pll  λpll  lim ^λ pll  λpll  Rgi Rpll ^ λ pll !λpll

ð6:14Þ

From (6.13) and (6.14), ^λ i  λi 

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Rgi Rpll , ^λ pll  λpll  Rgi Rpll

ð6:15Þ

The approximate estimation of the closed-loop POM and the oscillation mode of power system by (6.15) indicates that when the OLMR (λpll  λi) occurs, the closedloop POM and oscillation mode of power system intend to locate at the opposite positions with respect to the point of modal resonance, λpll  λi, on the complex plane. Hence, it is very likely that the OLMR may degrade the power system smallsignal stability. From (6.15), it can be seen that,   pffiffiffiffiffiffiffiffiffiffiffiffiffi  if Real part of Rgi Rpll  > Real part of λi or λpll  ð6:16Þ either the closed-loop POM or the closed-loop oscillation mode of power system may locate on the right half of the complex plane. In this case, considerable dynamic interactions are introduced by the PLL to cause the small-signal instability of the power system.

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL

209

The OLMR can be examined by calculating the complex eigenvalues of the PLL open-loop state matrix, Apll, in (6.4) and that of the rest of the power system, Ag, in (6.5). The proximity of an open-loop POM, λpll, to an open-loop EOM, λi, on the complex plane indicates a case of OLMR, i.e., λpll  λi. Then, further examination should be followed by the calculation of the open-loop residues, Rgi and Rpll, to estimate the impact of the OLMR on the power system small-signal stability using (6.16).

6.2

Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL

For the recent years, a great effort has been spent by many researchers to examine the small-signal stability of a grid-connected wind farm affected by the dynamic interactions introduced from the PLL. The investigation was to scan the degree of dynamic interactions by tuning the PI gains of the PLL. It was found that the dynamic interactions between the PLL and the DC voltage control of the converter control system of a PMSG were strongest when the bandwidth of the PLL was close to that of the DC voltage control system. When the bandwidth of the PLL got close to that of current control inner loops of the converter control systems, dynamic interactions between the PLL and current control inner loops increased. In Sect. 1.2.1 of Chap. 1, main results of the examination mentioned above have been reviewed. According to the control theory of a linear system, the bandwidth of a dynamic component in the system is normally close to the frequency of the open-loop oscillation mode of the dynamic component. Hence, if the bandwidth of PLL is ωp, an open-loop POM is λpll ¼  ξp  jωp. If the bandwidth of a dynamic component, such as the DC voltage control outer loop of the GSC of the PMSG is around ωg, it is possible that the frequency of an open-loop oscillation mode of the ROPS subsystem is around ωg. Hence, closeness of the bandwidth of PLL to the bandwidth of the dynamic component of the ROPS, such as the DC voltage control outer loop, implies that ωg  ωp. Then, it is possible that the OLMR may happen between λpll ¼  ξp  jωp and λg ¼  ξg  jωg. This explains why the closeness of bandwidths may result in strong dynamic interactions introduced by the PLL and increase the small-signal instability risk brought about by the PLL from the perspective of the OLMR. In this subsection, the first example power system with the grid-connected PMSG is presented to demonstrate how the OLMR explains the phenomenon of strong dynamic interactions introduced by the PLL when its bandwidth is close to that of converter control of the PMSG. The second example power system presented in the subsection demonstrates how the small-signal stability of a wind farm is affected by the PLL under the condition of OLMR.

210

6.2.1

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Example 6.1: A PMSG Connected to an External Power System

Configuration of an example power system is shown by Fig. 6.4, where a PMSG is connected to an infinite AC busbar via a transmission line represented by an equivalent lumped reactance, XL. Model and parameters of the PMSG and its converter control systems given in [3] were used. The PMSG was equipped with a synchronous reference frame (SRF) PLL [4–8]. Model and parameters of the SRF PLL given in [4–8] were adopted. Parameters of example power system are given in Appendix 6.1. The SRF PLL is the simplest and most commonly-used PLL scheme. Figure 6.5 shows the block diagram of linearized model of the SRF PLL [4–8]. The PI feedback control configuration in the SRF PLL is often the core of a more complicated PLL scheme. Hence, in order to demonstrate and evaluate the analysis and conclusions made in the previous section regarding the open-loop modal resonance, it considered that PMSG in Fig. 6.4 was equipped with a SRF PLL. Comparing Figs. 6.1 and 6.4, it can be seen that in this example power system, the external power system is represented by XL plus the infinite AC busbar. Hence, only the dynamics of the transmission line connecting the PMSG with the power system are considered. Other dynamics of the external power system are ignored. Therefore, in the closed-loop model shown by Fig. 6.3 established for the example power system, main dynamics of the ROPS subsystem include the control systems of the GSC of the PMSG and the transmission line represented by XL. Impact of dynamic interactions between the SRF PLL and those main dynamic components on the small-signal stability is examined under the condition of OLMR in this subsection as follows.

Fig. 6.4 A PMSG connected to an infinite AC busbar

Vpcc θpcc

Vb Infinite AC busbar

PMSG XL

Low-pass Filter

ΔVq

Δθ pcc +



Vd0

Fig. 6.5 Block diagram of an SRF PLL

F (s)

K pp K pi s

+ + Δx pll

1 s

Δθ pll

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL Fig. 6.6 Modal positions and trajectories when Kpi varied

211

90 70 Imaginary

lˆ pll

lpll lg1

50

C

B

lˆ g1

30 10

A -40

6.2.1.1

-30

Real -20

-10

0

The OLMR Between the PLL and the DC Voltage Control Outer Loop of the GSC

Firstly, an open-loop oscillation mode of the ROPS subsystem of example power system was calculated to be λg1 ¼  18.8716 þ j40.6902, which was found being associated with the DC voltage control outer loop of the GSC. PI gains of the PLL were initially set to be Kpp ¼ 0.18 and Kpi ¼ 1.2. The open-loop POM was calculated to be λpll ¼  24.3549 þ j8.1797. Position of λpll on the complex plane is indicated by hollow rectangle at point A in Fig. 6.6. Position of λg1 is indicated by hollow rectangle at point C. Secondly, in order to demonstrate and evaluate the small-signal instability risk caused by the OLMR between the PLL and the DC voltage control outer loop, the integral gain of the PLL was varied from the original value, Kpi ¼ 1.2 to Kpi ¼ 23. With the variation of Kpi, λpll moved from the initial position A towards λg1 on the complex plane as shown by the dashed curve in Fig. 6.6. At point B , λpll  λg1 where OLMR occurred and it was calculated that pffiffiffiffiffiffiffiffiffiffiffiffiffiffi the Rg1 Rpll ¼ 14:3527 þ j4:8796. Then, the closed-loop oscillation modes, ^λ pll and ^λ g1 , were estimated by using (6.15). The estimated positions are indicated by crosses in Fig. 6.6. It can be seen that under the condition of the OLMR, damping of ^λ g1 was poor; but the grid-connected PMSG system was still stable. Thirdly, to confirm the modal computation and estimation presented above, closed-loop oscillation modes, ^λ pll and ^λ g1 , were calculated from the closed-loop state matrix, A in (6.7). Trajectories of ^λ pll and ^λ g1 with the variation of Kpi are displayed in Fig. 6.6 by solid curves. Positions of ^λ pll and ^λ g1 when the OLMR occurred at point B are indicated by filled circles. They confirmed the correctness of the estimation made by using (6.15) that ^λ pll and ^λ g1 located at the opposite positions in respect to λpll  λg1. The PLL degraded the damping of ^λ g1 under the condition of the OLMR.

212

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Fourthly, the participation factors (PFs) for ^λ pll and ^λ g1 were computed to evaluate the dynamic interactions between the PLL and the DC voltage control outer loop. Computational results are presented in Fig. 6.7, where the dashed curves are the sum of the PFs of all the state variables of the PLL and the solid curves are the sum of the PFs of all the state variables of the DC voltage control outer loop. From Fig. 6.7, it can be seen that initially, the PLL and the DC voltage control outer loop took part in only its own oscillation modes, ^λ pll and ^λ g1 , separately. When λpll moved towards λg1, the PLL and the DC voltage control outer loop gradually participated more and more in both ^λ pll and ^λ g1 , indicating the increase of their dynamic interactions. Under the condition of the OLMR (point B) when Kpi ¼ 9.47, both ^λ pll and ^λ g1 were participated by the PLL and the DC voltage control outer loop considerably, confirming their strong dynamic interactions. Figure 6.8 shows the bandwidth of the PLL and the DC voltage control loop respectively. It can be seen that their bandwidths were close to each other when the OLMR happened, confirming the explanation in the previous section that the closeness of the bandwidths in fact indicated the condition of the OLMR. This explained the finding that when the bandwidth of the PLL was close to that of the Dc voltage control loop, their dynamic interactions increased.

b 0.8

0.8

Participation factor

Participation factor

a PLL 0.6 0.4 DC 0.2

DC

0.6 0.4

PLL 0.2

A

B

A

B

Fig. 6.7 Computational results of the PFs. (a) The PFs for ^λ p1 . (b) The PFs for ^λ g1 Fig. 6.8 Bandwidth of the PLL and DC voltage control outer loop

Bandwidth (Hz)

20

PLL

15 DC voltage control 10

5

A

B

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL Fig. 6.9 Modal positions and trajectories when Kpi varied

250

lpll lˆ pll

200 Imaginary

213

lg2 150

E lˆ g2

100 50

D Real

0.8 PLL 0.6 0.4

current control

0.2 D

E

-100

Participation factor

Participation factor

-120

-80

-60

-40

-20

0

0.8 current control 0.6 0.4 PLL

0.2 D

E

Fig. 6.10 Computational results of the PFs for b λ pll and b λ g1 when Kpi varied

6.2.1.2

The OLMR Between the PLL and the Current Control Inner Loops of the GSC

Another open-loop oscillation mode of the ROPS subsystem was λg2 ¼  44.5767 þ j147.6429, which was found being associated with the current inner loop of the GSC. The test similar to that presented in 6.2.1.1 above was carried out to demonstrate and evaluate the OLMR between the PLL and the current inner loops of the GSC by varying the PI gains of the PLL from Kpp ¼ 0.35, Kpi ¼ 13 to Kpp ¼ 0.35, Kpi ¼ 220. Modal positions and trajectories with the variations of the PI gains are shown in Fig. 6.9. It can be seen that the OLMR occurred at point E (Kpp ¼ 0.35, Kpi ¼ 91), where estimation on ^λ pll and ^λ g2 made by using (6.15) (crosses in Fig. 6.9) was correct that ^λ pll and ^λ g2 located at the opposite positions in respect to the resonant open-loop oscillation modes. Damping of ^λ g2 was degraded considerably by the PLL under the condition of the OLMR at point E. Figures 6.10 and 6.11 presents the computational results of the PFs and bandwidth of the PLL and the current control inner loop respectively. It can be seen that when λpll moved close to λg2 with the variations of the PI gains of the PLL, bandwidth of the PLL got close to that of the current control inner loops and their dynamic interactions increased. When the OLMR at point E happened, their

214

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Fig. 6.11 Bandwidth of the PLL and current control inner loops

80 Bandwidth (Hz)

PLL 60 current control 40

20

A

C

bandwidths were very close and dynamic interactions were strong. This explained the finding that when the bandwidth of the PLL was close to that of the current control inner loops, their dynamic interactions increased.

6.2.1.3

Weak Network Connection

As it was reviewed in 1.2, it has been found that weak network connection may cause the small-signal instability of the grid-connected PMSG system. The weakness of the network connection is defined by the short circuit ratio (SCR). When the SCR is below 2, the network connection is considered to be weak. In the examination in 6.2.1.1 and 6.2.1.2 above, SCR ¼ 2.1. To evaluate the condition of the OLMR as affected by the weak network connection, the equivalent transmission reactance in Fig. 6.4 was creased from XL ¼ 0.45 pu to XL ¼ 0.6 pu such that the SCR was reduced to SCR ¼ 1.7. Then, the tests as same as those conducted in 6.2.1.1 and 6.2.1.2 were carried out under the condition of reduced SCR (increased XL). Modal positions and trajectories were calculated and are presented in Fig. 6.12. From Fig. 6.12, it can be seen that when the OLMR occurred at point F and G, the closed-loop oscillation modes, ^λ g1 and ^λ g2 , moved into the right half of the complex plane. The grid-connected PMSG system became unstable. In Table 6.1, computapffiffiffiffiffiffiffiffiffiffiffiffiffi tional results of R are presented, confirming that gi Rpll , i ¼ 1, 2  pffiffiffiffiffiffiffiffiffiffiffiffiffi Real part of Rgi Rpll  increased considerably when the SCR decreased such that the grid-connected PMSG system lost the small-signal stability. pffiffiffiffiffiffiffiffiffiffiffiffiffi In fact, the magnitude of Rgi Rpll , i ¼ 1, 2 is proportional to XL. Hence, the bigger XL is, the bigger the impact of OLMR is. This is can be proved as follows. Dynamics of the ROPS subsystem of the example power system shown by Fig. 6.4 are comprised of the following components (For detailed linearized model of the PMSG, see Sect. 2.1 in Chap. 2).

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL

a

80

60 Imaginary

Fig. 6.12 Modal positions and trajectories when Kpi of PLL varied with reduced SCR (SCR ¼ 1.7). (a) The OLMR between the PLL and DC voltage control outer loop; (b) the OLMR between the PLL and current control inner loop

215

F

40

20 Real -40

-30

8

0

-10

-20

b 250

Imaginary

200 150 G 100 50 Real -120

-80

-40

0

20

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Table 6.1 Computational results of Rgi Rpll , i ¼ 1, 2 and closed-loop oscillation modes for study case (3) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^λ g1 ¼ 4:191 þ j41:804 The OLMR in Fig. 6.6 Rg1 Rpll ¼ 14:352  j3:979 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^λ g1 ¼ 1:528 þ j38:385 The OLMR in Fig. 6.12a Rg1 Rpll ¼ 18:361  j4:931 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^λ g2 ¼ 4:376 þ j153:561 The OLMR in Fig. 6.9 Rg2 Rpll ¼ 41:609 þ j29:081 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^λ g2 ¼ 5:163 þ j157:447 The OLMR in Fig. 6.12b Rg2 Rpll ¼ 58:675 þ j23:189

1. Filter reactance: 8 dΔId > > ¼ ω0 ðΔVcd  ΔVd Þ þ ω0 X f ΔIq dΔIq > :Xf ¼ ω0 ΔVcq  ΔVq  ω0 X f ΔId dt

ð6:17Þ

where Vd þ jVq is the PCC voltage of the PMSG, Vpcc ∠ θpcc, expressed in the d  q coordinate of the GSC, Id þ jIq is the output current of the GSC expressed in the d  q coordinate of the GSC.

216

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

2. DC capacitor (variation from the machine side Converter of the PMSG is ignored, as Ppmsg is not affected by the dynamic interactions between the PMSG and the power system; thus, it can be assumed that ΔPpmsg ¼ 0, see (2.28)) CVdc0

dΔVdc ¼ ΔPvsc dt

ð6:18Þ

3. Control systems of the GSC. Configuration of the control system of the GSC is shown by Fig. 2.3, which is redrawn as Fig. 6.13. Hence, linearized dynamic equations of DC voltage control and d-axis current control are 8 dΔxdc > > ¼ ki dc ΔVdc < dt > > dΔxid ¼ k ΔI þ k k : i d d i d p dt

ð6:19Þ dc ΔVdc

þ ki d Δxdc

Linearized equations of reactive power control and q-axis current control are 8 dΔxQ > > ¼ ki Q ΔQ < dt > dΔxiq > : ¼ ki q ΔIq þ ki q kp Q ΔQ þ ki q ΔxQ dt

ð6:20Þ

Ignoring the transient of the PMW algorithm such that in Fig. 6.13, Vcd ¼ Vcd∗ and Vcq ¼ Vcq∗. From Fig. 6.13, ( ΔVcd ¼ kp d ΔId þ kp d kp dc ΔVdc þ kp d Δxdc þ Δxid  X f ΔIq þ ΔVd ΔVcq ¼ kp q ΔIq þ kp q kp Q ΔQ þ kp q ΔxQ þ Δxiq þ X f ΔId þ ΔVq ð6:21Þ

K p_dc ref dc

V

+

– + Vdc

K p_Q

Qwref –

+ + Qw

Outer loop

Ki_Q + xQ s

+ – Id

Ki_dc + xdc s

Vd

K p_d

I dref+

Ki_d s

+ xid

K p_q

I qref +

+ – Iq

Inner loop

Ki_q s

+ xiq

Fig. 6.13 Configuration of control system of the GSC of the PMSG

+

*

+ Vcd – X f iq PWM

Vq + +

+ X f id

Vcq*

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL

217

4. Transmission line represented by XL. Denote Vx þ jVy as the PCC voltage of the PMSG, Vpcc ∠ θpcc, expressed in the common x  y coordinate of the power system. Denote Ix þ jIy as the output current of the GSC expressed in the common x  y coordinate of the power system. Voltage at the infinite busbar is constant. Hence, ( ΔVx ¼ XL ΔIy ð6:22Þ ΔVy ¼ XL ΔIx 5. Output power from the GSC is P þ jQ. Following linearized equations can be obtained (The phase-tracking error of the PLL at the steady state is zero. Hence, Vd0 ¼ Vpcc0 and Vq0 ¼ 0) (

ΔP ¼ Vpcc0 ΔId þ Id0 ΔVd þ Iq0 ΔVq

ð6:23Þ

ΔQ ¼ Vpcc0 ΔIq þ Id0 ΔVq  Iq0 ΔVd

6. Linearized equations of transformation between d  q and x  y coordinate. "

ΔIx

# ¼

ΔIy "

ΔVd ΔVq

"

#

" ¼

cos θpll0

 sin θpll0

sin θpll0

cos θpll0

cos θpll0

sin θpll0

 sin θpll0

cos θpll0

#"

#

ΔId

þ

ΔIq #"

"

ΔVx

#

Ix0

#

ΔVy

Iy0

" þ

Vq0 Vd0

Δθpll

ð6:24Þ

# Δθpll

ð6:25Þ

From (6.17) to (6.25), the state-space model of the ROPS subsystem of (6.5) is derived for the example power system shown by Fig. 6.4. It can have ΔXg ¼ ½ ΔId 2

ΔIq a11

6 6 a21 6 6 6 a31 6 6 Ag ¼ 6 0 6 6a 6 51 6 6a 4 61 a71

ΔVdc

Δxdc

Δxid

ΔxQ

0

a13

a14

a15

0

a22

a23

0

0

a26

a32

0

0

0

0

0

a43

0

0

0

0

a53

a54

0

0

a62

0

0

0

0

a72

0

0

0

a76

0

Δxiq T 3

7 a27 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 5 0

ð6:26Þ

218

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

2

0 2V2p0 kp q kp Q Id0 Iq0

3

7 6 7 6 6 X L 7 3 2 2 7 6 V þ Q X 0 0 L p0 7 6 7 6  7 6 Vp0 7 6 6 XL 7 7 6 V2p0 I2d0  I2q0 2 7 6 6 6 Vp0 þ Q0 XL 7  XL 7 7 6 7 6 7 6 CVdc0 V2 þ Q0 XL 7 6 p0 0 7 6 7 6 7 6 7 6 ¼ X bg ¼ 6 b , c ¼ 7 L g1 g 7 6 0 0 7 6 7 6 7 6 7 6 7 6 0 7 6 0 7 6 7 6 7 6 7 6 2V2p0 ki Q Id0 Iq0 7 6 0 5 4 6  XL 7 2 7 6 V þ Q X 0 L 7 6 p0 0 7 6 7 6 2V2 k k I I 5 4 p0 i q p Q d0 q0  XL 2 Vp0 þ Q0 XL ¼ XL cg1

ð6:27Þ dg ¼

Q0 XL þ Q0 XL

V2p0

where ω0 ω0 ω0 ω0 a11 ¼  kp d , a13 ¼ kp d kp dc , a14 ¼ kp d , a15 ¼ Xf Xf Xf Xf   ω0 a21 ¼  kp q kp Q Id0 XL , a22 ¼ ki Q Iq0 XL  Vg0 Xf    2Vp0 kp q kp Q Id0 Iq0 X2L ω0  a22 ¼ kp q kp Q Iq0 XL  Vp0  kp q þ Xf V2p0 þ Q0 XL  Vp0 X2L I2d0  I2q0 2V k I I X2 ω  , a26 ¼ 0 kp q þ p02 i Q d0 q0 L a23 ¼  Xf Vp0 þ Q0 XL CVdc0 V2 þ Q XL p0

a27 ¼

0

ω0 2Vp0 ki q kp Q Id0 Iq0 X2L þ Xf V2p0 þ Q0 XL

Vp0 þ Iq0 XL Id0 XL , a32 ¼ CVdc0 CVdc0 ¼ ki dc , a51 ¼ ki d , a53 ¼ ki d kp dc , a54 ¼ ki d ,

a31 ¼  a43

a61 ¼ ki Q Id0 XL , a71 ¼ ki q kp Q Id0 XL ,

a72 ¼ ki q kp

 Q

 Iq0 XL  Vp0  ki q , a76 ¼ ki

q

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL

0.03

Point A Point B Point F

0.01

PLL tracking error (pu)

PLL tracking error (pu)

0.02

0 -0.01 -0.02 0

0.2

0.4

0.6

time/s 0.8

Point D Point E Point G

0.02 0.01 0 -0.01 -0.02

time/s

-0.03 1

219

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 6.14 Confirmation of non-linear simulation—Example 6.1

Denote υgi and wgi the left and right eigenvectors of state matrix Ag in (6.26) for open-loop oscillation mode of the ROPS subsystem, λgi, i ¼ 1, 2. The residue Rgi for λgi can be calculated from the state-space model of the ROPS subsystem of (6.27) to be T υgi wgi bg1 ¼ X2L Rgi1 Rgi ¼ cg T υgi wgi bg ¼ X2L cg1

ð6:28Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From (6.28), Rgi Rpll ¼ XL Rgi1 Rpll . Hence, it is proved that the magnitude of pffiffiffiffiffiffiffiffiffiffiffiffiffi Rgi Rpll , i ¼ 1, 2 is proportional to XL. Figure 6.14 presents the confirmation of non-linear simulation. At 0.1 s of simulation, the reactive power output of the PMSG dropped by 0.02 p.u. which was recovered in 0.11 s. Two simulation results are shown in Fig. 6.13: (1) λpll was at point A in Fig. 6.6, point D in Fig. 6.9 (no OLMR occurred); (2) λpll was at point B in Fig. 6.6, point E in Fig. 6.9, point F and G in Fig. 6.12, which were under the condition of the OLMR. From Fig. 6.14, it can be seen that under the condition of the OLMR, considerable variations of phase-tracking error of the PLL were observed and poorly or negatively damped oscillations occurred.

6.2.2

Example 6.2: A PMSG Wind Farm Connected to an External Power System

Configuration of a PMSG wind farm connected to an external power system is shown by Fig. 6.15. Purpose of study is for the demonstration and evaluation of the analysis and conclusions made in Sect. 6.1, the example PMSG wind farm was simply comprised of only three PMSGs. They were aggregated models representing the PMSGs in three different areas in the wind farm. All the three PMSGs were equipped with the SRF PLLs. Model and parameters of the PMSG given in [6] were used. The parameters of the GSC control systems and the PLL of the second PMSG (PMSG2) and the third PMSG (PMSG3) were decreased and increased by 15% respectively to reflect the fact that they represented the PMSGs in different areas in the wind farm. Parameters of the example are given in Appendix 6.2. When a

220

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

GSC

Vcd + jVcq

PMSG1 C Vdc

Vpcc θpcc

Xf

I1

P1+jQ1

Inner loop Outer loop θerror PLL subsystem

PLL1

Vdc

Vb

θpcc

PMSG 2 ROPS subsystem

Xt1

XL

Infinite AC busbar

Xt2

PMSG 3

Xt3

Fig. 6.15 Configuration of a grid-connected PMSG wind farm system Fig. 6.16 Modal positions and trajectories when Kpi of PLL1 varied (SCR ¼ 2.15)

100

Imaginary

80 L 60 H

40 20 P -20

-15

Real -10

-5

0

practical grid-connected wind farm with more PMSGs is studied, the procedure to conduct the following demonstration and evaluation will be same. The PLL (PLL1) for the first PMSG (PMSG1) was considered. Firstly, a closedloop model shown by Fig. 6.3 was established, where the open-loop PLL subsystem in the feedback loop was PLL1; the open-loop ROPS subsystem included PMSG2, PMSG3, the rest of PMSG1 with PLL1 being excluded and the power network. The open-loop oscillation modes of the ROPS subsystem were calculated from the openloop state matrix, Ag in (6.5). Two open-loop oscillation modes, λg3 and λg4, were found being related with the DC voltage control loop (DC2) of PMSG2 and the PLL (PLL3) of PMSG3. Their positions on the complex plane are indicated by hollow rectangles at point H and L in Fig. 6.16. PI gains of PLL1 were Kpp ¼ 0.12, Kpi ¼ 3 initially. The open-loop oscillation mode of PLL1 was λpll ¼  13.6975 þ j17.8116 and its position is indicated by hollow circle at point P in Fig. 6.16.

6.2 Small-Signal Stability of a Grid-Connected PMSG Wind Farm Affected by the PLL Table 6.2 Computational results of study cases of wind farm system Point H in Fig. 6.16 when SCR ¼ 2.15 Point H in Fig. 6.17 when SCR ¼ 1.62 Point L in Fig. 6.16 when SCR ¼ 2.15 Point L in Fig. 6.17 when SCR ¼ 1.62

221

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rgi Rpll , i ¼ 3, 4 and closed-loop oscillation modes for the

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rg3 Rpll ¼ 4:8969  j3:9149 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rg3 Rpll ¼ 8:0309  j5:4493

^ λ g3 ¼ 5:9845 þ j45:9711 ^ λ g3 ¼ 1:2377 þ j45:1659

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rg4 Rpll , 1 ¼ 3, 4 ¼ 8:4176  j5:1637

^ λ g4 ¼ 2:4381 þ j73:8261

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rg4 Rpll ¼ 11:8275  j9:2929

^ λ g4 ¼ 1:9755 þ j68:1097

Secondly, the integral gain of PLL1 was varied from Kpi ¼ 3 to Kpi ¼ 40. Trajectory of λpll with the variations of Kpi is displayed in Fig. 6.16 by dashed curve. It can be seen that at point H, λpll  λg3 such that the OLMR between PLL1 and DC2 happened. At point L, λpll  λg4 where the OLMR between PLL1 and PLL3 occurred. pffiffiffiffiffiffiffiffiffiffiffiffiffi Thirdly, Rgi Rpll , i ¼ 3, 4 were computed for the OLMR at point H and L respectively. Computational results are given in the first and third row of Table 6.2. By using (6.15), the closed-loop oscillation modes, ^λ pll , ^λ g3 and ^λ g4 , under the condition of the OLMR at point H and L were estimated. The estimated positions of ^λ pll , ^λ g3 and ^λ g4 are indicated by crosses. It can be seen that the dynamic interactions between PLL1 and DC2 or PLL3 caused the stability degradation of the example grid-connected wind farm under the condition of the OLMR. However, the wind farm system was still stable. Finally, the equivalent transmission reactance of the example wind farm was increased from XL ¼ 0.31 pu to XL ¼ 0.42 pu such that the SCR decreased from SCR ¼ 2.15 to SCR ¼ 1.62. Under the condition of weak network connection, modal positions and trajectories were computed when Kpi varied. The computational pffiffiffiffiffiffiffiffiffiffiffiffiffi results are displayed in Fig. 6.17. Computational results of Rgi Rpll , i ¼ 3, 4 are given in the second and fourth row of Table 6.2. It can be seen that with reduced SCR, the wind farm system lost the small-signal stability under the condition of the pffiffiffiffiffiffiffiffiffiffiffiffiffi OLMR, because the magnitude of real part of Rgi Rpll , i ¼ 3, 4 increased considerably. Fourthly, ^λ pll , ^λ g3 and ^λ g4 were calculated from the closed-loop state matrix, A in (6.7). Trajectories of ^λ pll , ^λ g3 and ^λ g4 with the variation of Kpi are displayed by solid curves in Fig. 6.17, confirming the correctness of the estimation made by using (6.15). Confirmation of non-linear simulation is presented in Fig. 6.18. At 0.1 s of simulation, the reactive power output of the PMSG dropped by 0.02 p.u. which was recovered in 0.11 s. From Fig. 6.18, it can be seen that when the SCR was 2.15, the poorly damped oscillation happened as caused by the dynamic interactions introduced by PLL1 under the condition of the OLMR. When the network connection was weak (SCR ¼ 1.62), the dynamic interactions became stronger and the wind farm system lost the small-signal stability under the condition of the OLMR.

222

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Fig. 6.17 Modal positions and trajectories when Kpi of PLL1 varied (SCR ¼ 1.62)

100

lp1

lˆp1

lg4

Imaginary

80

L 60

lˆg4

lg3

40

H

lˆg3

20 P -20

-15

-10

-5

0

5

b PMSG1 PMSG2 PMSG3

0.014 0.007

PLL tracking error (pu)

PLL tracking error (pu)

a

Real

0 -0.007 -0.014 time/s 0

0.2

0.4

0.6

0.8

1

PMSG1 PMSG2 PMSG3

0.06 0.03 0 -0.03 -0.06

time/s 0

0.2

0.4

0.6

0.8

1

Fig. 6.18 Confirmation of non-linear simulation when the OLMR occurred. (a) SCR ¼ 2.15. (b) SCR ¼ 1.62

6.3

Example 6.3: Electromechanical Oscillation Modes of a Power System Affected by the PLL

Figure 6.19 shows the configuration of the New England test power system (NEPS) extensively used for studying the power system electromechanical oscillations [9– 11]. The parameters of the network and the synchronous generators (SGs) given in [9] were used. There were nine electromechanical oscillation modes (EOMs) in the NEPS. An inter-area EOM, between the tenth SG and the rest of SGs, was the EOM of main concern, with the smallest damping and lowest oscillation frequency. Two PMSGs were connected at node 22, in the NEPS and operated with unity power factors, each. The detailed 15th-order dynamic model of the PMSG, including its control systems recommended in [6], was used. Parameters of the test power system are given in Appendix 6.3. The first PMSG (PMSG1) connected at node 22 was equipped with an SRF PLL. The second PMSG was equipped with a more complicated PLL, the duel secondorder generalized integrator phase locked loop (DSOGI) PLL [12] designed to track

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

223

8 37

1 30

28

26

25

29

2 27

18 1

24

17

38 9

16

10

6 3

35

15

39

21

14 4

22

19

5

6

9

PMSG

20 11

31 7

23

12 33

13

36

4

2

7

34

10

5

8 32 3

Fig. 6.19 Configuration of the New England test power system (NEPS) with two PMSGs ω0 vα vabc

vα′

SOGI

v

v

+

qvα′ +

+ q

+ α



1/ 2 1/ 2

+

Park

v+β

vd+

PI

+ + ω ff



θ+

Clark vβ

SOGI

v′β qv′β v–α

+ –

+ +

1/ 2 1/ 2

v–β

Fig. 6.20 Configuration of a DSOGI PLL

the phase angle of the positive sequence voltage. The configuration of the DSOGI PLL is depicted in Fig. 6.20. The impact of the dynamic interactions introduced by the PMSGs connected at node 22, caused by the PLLs, on the inter-area EOM was examined. The PI gains of the PLLs were varied in a wide range to scan the possibility of OLMR between the PLLs and the NEPS. The PI gain values given in MATLAB [13] for the SRF PLL were used to start the scan; the PI gain of the DSOGI PLL was

224

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

initially set near the highest frequency EOM. This scanning strategy, by varying the PI gains of the PLLs, has been used to study the impact of the PLL on the stability of a VSC-based system. It was adopted in the following tests for investigating the potential danger posed by the OLMR. Thus, in PLL design, the identified danger can be avoided. When the PI gains of the PMSG PLLs were varied, the positions of the open-loop POMs on the complex plane changed. Hence, OLMR may be determined by varying the PI gains of the PLLs. When the operating conditions of the power system changed, the open-loop EOMs moved on the complex plane; this may also lead to open-loop modal resonance. In the following subsections, test results are presented to demonstrate the possibility that changes in the power system operating conditions and the PI gains of the PLLs may lead to OLMR.

6.3.1

Test 1: Changes of Power System Operating Conditions

For the PLL equipped by PMSG1, the closed-loop model in Fig. 6.3 was derived, where the open-loop transfer function of the SRF PLL is (see Fig. 6.5 of the linearized model of the SRF PLL and (6.3)), Hpll ðsÞ ¼

Δθerror s2 ¼ 2 Δθpcc s þ Vd0 FðsÞKpp s þ Vd0 FðsÞKpi

ð6:29Þ

The typical PI gains of the SRF PLL, given in [14], were Kpp ¼ 4.4 and Kpi ¼ 39.27. The open-loop and closed-loop POMs were calculated to be λpll ¼  2.1975 þ j5.8688 and ^λ pll ¼ 1:2655 þ j5:7057, respectively. The open-loop and closedloop EOMs of the power system were computed from the open-loop and closed-loop state matrices, Ag in (6.5) and A in (6.7), respectively. In Table 6.3, λi, i ¼ 1, 2, . . .9 were the open-loop EOMs of the NEPS, when the PLL phase-tracking error was assumed to be zero. ^λ i , i ¼ 1, 2, . . . 9 were the closed-loop EOMs, when the PLL phase-tracking error was considered. Hence, Δλi ¼ ^λ i  λi , i ¼ 1, 2, . . . 9 were caused by the dynamic interactions between the SRF PLL and the rest of the power system. In Table 6.3, λ9 and ^λ 9 were the open-loop and closed-loop interarea EOMs of main concern, respectively. From the first three columns of Table 6.3, it can be seen that the open-loop POM, λpll ¼  2.1975 þ j5.8688, was closer to λ6 and λ7, than to the other EOMs. Hence, the SRF PLL affected λ6 and λ7 more, and Δλ6 and Δλ7 were slightly larger. In order to examine whether a larger Δλ6 and Δλ7 were caused by the proximity of λpll to λ6 and λ7, the PI gains of the PLL were changed to Kpp ¼ 7.54, Kpi ¼ 377 such that the open-loop POM became λpll ¼  3.7699 þ j19.047 and was far from all the open-loop EOMs of the NEPS. The computational results of ^λ i , i ¼ 1, 2, . . . 9 and Δλi, i ¼ 1, 2, . . .9, when λpll ¼  3.7699 þ j19.047, are presented in the last two columns of Table 6.3. It can be seen that, when λpll moved far away from all the EOMs, the impact of the PLL on all the EOMs, Δλi, i ¼ 1, 2, . . .9, was less. This

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

225

Table 6.3 Computational results of the open-loop and closed-loop EOMs of the NEPS i

λi

1

0.55144þ 8.4521j 0.4689 þ7.8529j 0.32196 þ6.9978j 0.53018 þ7.1673j 0.45682 þ7.1209j 0.49977 þ5.9183j 0.48828 þ6.3816j 0.46338 þ6.1885j 0.30609 þ3.5525j

2 3 4 5 6 7 8 9

Typical PLL parameters were used ^λ i Δλi 0.5516þ 0.0002þ 8.4522j 0.0001j 0.47178 0.0029 þ7.8663j þ0.0134j 0.32533 0.0034 þ6.9988j þ0.0010j 0.53992 0.0097 þ7.1705j þ0.0032j 0.45623 0.0006 þ7.1211j þ0.0002j 0.69915 0.1994 þ6.5069j þ0.5886j 0.52733 0.0390 þ5.8967j 0.4849j 0.45379 0.0096 þ6.1997j þ0.0112j 0.31961 0.0135 þ3.2819j 0.2706j

PLL parameters were changed ^ λi Δλi 0.5514þ 8.4522j 0.46993 þ7.8515j 0.3226 þ6.9972j 0.53094 þ7.1649j 0.45686 þ7.121j 0.4975 þ5.9148j 0.49539 þ6.3584j 0.4665 þ 6.1894i 0.30175 þ3.525j

3e-05þ 0.0001j 0.00103 0.0014j 0.00064 0.0006j 0.00076 0.0024j 4e-05 þ0.0001j 0.00218 0.0035j 0.00711 0.0232j 0.00314 þ0.0009j 0.00434 0.0275j

confirmed that when the typical PI gains recommended in [14] were used, a slight OLMR of λpll with λ6 and λ7 occurred such that Δλ6 and Δλ7 were larger. This explained why the closed-loop POM, ^λ pll ¼ 1:2655 þ j5:7057, was on the right of the open-loop POM, λpll ¼  2.1975 þ j5.8688, on the complex plane. The OLMR of λpll with λ6 and λ7 caused the damping degradation in^λ pll ¼ 1:2655 þ j5:7057. In the test presented above, the OLMR was not strong and the closed-loop EOMs and POM were slightly affected. The PI gains were varied to demonstrate that, when the OLMR was eliminated, the impact of the PLL became negligible. In the following test, it is shown that when the operating conditions of the NEPS changed, the open-loop POM moved closer to the EOM, λ7. Subsequently, the OLMR became stronger, causing a greater PLL impact on the power system EOMs. In the test, the change in the operating conditions of the NEPS was the gradual increase in the seventh SG constant of inertia from 69.6–80s. This replicated the changes in the power system operating conditions, when there was more generation as the spinning reserve at the seventh SG site. With the gradual increase in inertia, the seventh open-loop EOM, λ7, moved closer to the open-loop POM, λpll ¼  2.1975 þ j5.8688, on the complex plane, as shown by the dashed curve in Fig. 6.21. The movement ended at λ7 ¼  2.0761 þ j5.6494, which was closer to λpll ¼  2.1975 þ j5.8688 than it was before the constant of inertia was changed. The OLMR points, λpll  λi, are indicated by a hollow rectangle and circle, respectively, in Fig. 6.21. The trajectories of the corresponding movement of the seventh closed-loop EOM, ^λ 7 and the closed-loop POM, ^λ pll , are also displayed in Fig. 6.21 as solid curves. Their positions corresponding to the open-loop modal

226

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL 6.6

Imaginary axis

6.4

λˆ7

6.2

λ7 6

Inertia gradually increased

λpll 5.8

λˆpll

A

5.6

5.4 -3

-2.75 -2.5

-2.25

-2

-1.75 -1.5

-1.25

-1

-0.75 -0.5

-0.25

Real axis Fig. 6.21 Modal resonance caused by the gradual increase in inertia of the seventh SG

resonance points are indicated by the filled rectangle and circle, respectively. From Fig. 6.21, it can be seen that, when λpll and λ7 moved closer to each other, ^λ pll and ^λ 7 moved away from the corresponding open-loop modes λpll and λ7, indicating a larger difference between the closed-loop and open-loop modes, i.e., a greater PLL impact. From Fig. 6.21, it can be seen that ^λ pll and ^λ 7 (indicated by filled rectangle and circle) located approximately at opposite positions, with respect to the open-loop modal resonance points, indicated by a hollow rectangle and circle, respectively. The positions of ^ λ pll and ^λ 7 were also estimated using (6.15) and are indicated by crosses in Fig. 6.21; it can be seen that the estimation is close to the actual positions indicated by the filled rectangle and circle. Hence, the analytical conclusion made in Sect. 6.1, from (6.15), was confirmed. In this case study, λpll was well damped initially and the movement of ^λ pll towards the right on the complex plane was not significant. Hence, although OLMR degraded the damping of the closed-loop POM, the power system small-signal stability was still maintained. At the OLMR points, the participation factors (PFs) were computed for the closed-loop POM, ^λ pll and the closed-loop EOM, ^λ 7 . The computational results are presented in Fig. 6.22, depicting a considerable participation of the SGs in ^λ pll and the PLL in ^λ 7 . This confirmed that, when OLMR occurred, the dynamic interactions between the PLL and the SGs were considerable.

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

227

0.16

0.25

0.14 Participation factors

Participation factors

0.2 0.15 0.1 0.05

0.12 0.1 0.08 0.06 0.04 0.02

0

1

2

3

4 5 6 7 8 Generator number

(a)

9

10

0 Δθpll

Δxpll-vq

(b)

Fig. 6.22 Computational results of the participation factors. (a) Participation of the SGs in the POM; (b) participation of the PLL in the EOM

6.3.2

Test 2: Variation of PLL Gains (SRF PLL for PMSG1)

The second test was to show that when the PI gains of the PLL were changed, the open-loop POM moved on the complex plane to cause OLMR with different EOMs consecutively. The variation of the PI gains, in the test, was to identify certain selections of the PLL PI gains, which introduced the detrimental effect of OLMR. Thus, these selections should be avoided, when setting the PI gains of the PLL. In the test, the PI gains of the PLL were changed gradually from Kpp ¼ 0.90, Kpi ¼ 64 to Kpp ¼ 0.57, Kpi ¼ 13.2. With this change, the trajectory of the open-loop POM, λpll, was depicted by dashed curves and the movement of the closed-loop POM, ^λ pll , by solid curves in Figs. 6.23 and 6.24; it can be seen that while moving, λpll came close to the four open-loop EOMs, λ2, λ4,λ6, and λ9. The OLMR points of λpll with these four open-loop EOMs are indicated by hollow rectangles and circles. In Figs. 6.23 and 6.24, the movement of the closed-loop EOMs is also displayed by solid curves. The filled rectangles and circles indicate the positions of the closedloop EOMs and POMs, when the OLMR occurred. First, from Figs. 6.23 and 6.24, it can be seen that with respect to the OLMR points indicated by hollow rectangles and circles, the closed-loop modes, ^λ pll and^λ i , indicated by filled rectangles and circles, located at opposite positions. This confirmed the analytical conclusion made in Sect. 6.1 from (6.15). The estimated positions of ^ λ pll and ^λ i obtained from (6.15) are depicted by crosses. It can be seen that the estimation approximately indicated the actual positions of ^λ pll and ^λ i . Next, from these figures, it can be seen that the level of the PLL impact caused by the OLMR was different. From (6.15), it can be when the OLMR  observed that, pffiffiffiffiffiffiffiffiffiffiffiffiffi Real part of Rgi Rpll . Table 6.4 occurs, the impact of the PLL is affected by  pffiffiffiffiffiffiffiffiffiffiffiffiffi Real part of Rgi Rpll  at four points of the presents the computational results of pffiffiffiffiffiffiffiffiffiffiffiffiffi OLMR; Real part of Rgi Rpll  for the OLMR between λpll and the open-loop interarea EOM of main concern, λ9, was the largest. From Fig. 6.23, it can be seen that the movement of the closed-loop inter-area EOM of main concern from the point of

228

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL 8

Enlarged view is presented by Fig.9

Imaginary axis

7

6

5

lpll lˆ pll

4

lˆ 9 l9

3

2 -0.6

-0.2

-0.3

-0.4

-0.5

-0.1

Real axis Fig. 6.23 Modal resonance caused by PLL parameter tuning 8

l2

7.5

Imaginary axis

lpll

lˆ2

lˆpll

7

lˆ4

l4

6.5

l6 6

lˆ6 5.5

5 -0.6

-0.55

-0.5

-0.45

-0.4

Real axis Fig. 6.24 Enlarged view of the modal resonance caused by PLL parameter tuning

modal resonance was the farthest towards the right, confirming the computational results presented in Table 6.4. Figure 6.23 also shows that the OLMR of the PLL with the inter-area EOM caused a considerable damping degradation of the inter-area EOM, posing a serious threat to the power system small-signal angular stability. Further examination of this

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

0.35

0.16

0.3

0.14

0.25 0.2 0.15 0.1 0.05 0

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Real part of Rgi Rpll  0.059689 0.017552 0.080128 0.12531

λi i¼2 i¼4 i¼6 i¼9

Participation factors

Participation factors

Table 6.4 Estimation of the impact of the OLMR for using (6.15)

229

0.12 0.1 0.08 0.06 0.04 0.02

ΔVdc

Δxvdc

Δθpll

State variables

(a)

Δxpll-vq

0

1

2

3

4

5

6

7

8

9 10

Generator number

(b)

Fig. 6.25 Computational results of the PFs, when the OLMR of the PLL with the inter-area EOM of concern occurred. (a) Participation of the PMSG in ^ λ 9 ; (b) participation of the SGs in ^ λ pll

particularly dangerous case of open-loop modal resonance, by calculating the PFs, is presented in Fig. 6.25; it can be seen that, when the OLMR of the PLL with the interarea EOM occurred, both the PMSG and the SGs participated in the inter-area EOM and POM considerably, indicating strong dynamic interactions between the PMSG and the SGs. Simulation tests were carried out for further confirmation. An 80% load drop occurred at node 8 at 1.0 s of simulation. The lost load was restored in 100 ms. For comparison, the simulation results for two cases are presented in Figs. 6.26 and 6.27, respectively. In case A, the typical PLL parameters recommended in [14] are used (Kpp ¼ 4.4 and Kpi ¼ 39.27). In case B, the OLMR of the PLL with the inter-area EOM occurred, as shown in Fig. 6.23. From Fig. 6.26, it can be seen that, when the OLMR occurred, there were considerable variations in the reactive power output of the first PMSG. The damping degradation of the inter-area EOM of main concern was confirmed by the simulation results in Fig. 6.27, where poorly damped power oscillations were observed. Figure 6.26 shows that considerable dynamic interactions between the first PMSG and the power system were caused by the OLMR. The dynamic interactions were exhibited as variations in the reactive power output from the first PMSG. This can be elaborated as follows.

230

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Reactive power output of PMSG1

1 Case A Case B

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

2

4

6

8

10 12 Time(s)

14

16

18

20

Fig. 6.26 Reactive power output from the PMSG1

-54 Case A Case B

-56

d10 – d7(deg.)

-58 -60 -62 -64 -66 -68 -70 -72

0

2

4

6

8

10

12

14

16

18

20

Time(s) Fig. 6.27 Relative rotor position between the tenth and seventh SGs

The active and reactive power outputs of a PMSG are (see Fig. 6.1), Pvsc ¼ Vd Id þ Vq Iq , Qvsc ¼ Vq Id  Vd Iq

ð6:30Þ

At steady state, the PLL phase-tracking error is zero. Hence, Vq0 ¼ 0. From (6.30), Qvsc0 ¼  Vd0Iq0. Usually, the PMSG operates with a high or even unity power factor such that Qvsc0 ¼  Vd0Iq0  0. Hence, Iq0  0. The linearization of (6.30) is,

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

Vd Voltage Forward Feedback

Vdcref – + Vdc

K dc ( s )

– Qp

K q ( s)

αf s+α f

I dref+

– Id

Voltage Forward Feedback

Q pref +

231

K id ( s )

αf

Vq

ref + + Vcd – X f Iq

PWM

s+α f

I qref + – Iq

K iq ( s )

+

+ Vcqref +

X f Id

Fig. 6.28 Configuration of the GSC vector control system

ΔPvsc ¼ Id0 ΔVd þ Vd0 ΔId , ΔQvsc ¼ Id0 ΔVq  Vd0 ΔIq

ð6:31Þ

The linearized dynamic equation of the DC voltage across the GSC capacitor in Fig. 6.1 is, (Ppmsg is not affected by the dynamic interactions between the PMSG and the power system; thus, it can be assumed that ΔPpmsg ¼ 0), CVdc0

dΔVdc ¼ ΔPvsc dt

ð6:32Þ

From Fig. 6.1 (see 6.17), dΔId ¼ ω0 ðΔVcd  ΔVd Þ þ ω0 X f ΔIq dt   dΔIq ¼ ω0 ΔVcq  ΔVq  ω0 X f ΔId Xf dt Xf

ð6:33Þ

Figure 6.28 shows the configuration of the current vector control system of the GSC (see Fig. 6.13). From Fig. 6.28, (6.32), and (6.33), the following equations are obtained: sω0 ΔVd , ðs þ α f Þ½X f s þ ω0 Kid ðsÞ sω0  ΔVq , ΔIq ¼ Gq ðsÞΔQvsc  ðs þ α f Þ X f s þ ω0 Kiq ðsÞ ΔId ¼ Gd ðsÞΔVdc 

where, G d ðsÞ ¼

ω0 Kq ðsÞKiq ðsÞ ω0 Kdc ðsÞKid ðsÞ , G q ðsÞ ¼ X f s þ ω0 Kiq ðsÞ X f s þ ω0 Kid ðsÞ

ð6:34Þ

232

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

From (6.31), (6.32), (6.33), and (6.34),

CVdc0 Id0 s ΔPvsc ¼ CVdc0 s þ Vd0 Gd ðsÞ

CVd0 ω0 Vdc0 s2  ΔVd , ðs þ α f Þ½CVdc0 s þ Vd0 Gd ðsÞ½X f s þ ω0 Kid ðsÞ

Id0 ΔQvsc ¼ 1  Vd0 Gq ðsÞ ω0 Vd0 s    ΔVq  ðs þ α f Þ 1  Vd0 Gq ðsÞ X f s þ ω0 Kiq ðsÞ

ð6:35Þ

Substituting (6.2) in (6.35),

CVdc0 Id0 s ΔPvsc ¼ CVdc0 s þ Vd0 Gd ðsÞ

CVd0 ω0 Vdc0 s2 ΔVpcc , ðs þ α f Þ½CVdc0 s þ Vd0 Gd ðsÞ½X f s þ ω0 Kid ðsÞ

CVdc0 Vd0 Id0 s ΔQvsc ¼ CVdc0 s þ Vd0 Gd ðsÞ CV2d0 ω0 Vdc0 s2  Δθerror ðs þ α f Þ½CVdc0 s þ Vd0 Gd ðsÞ½X f s þ ω0 Kid ðsÞ



ð6:36Þ

The derived equations of (6.36) indicate that the phase-tracking error of the PLL mainly exhibits as dynamic variations of the reactive power output of the PMSG, ΔQvsc. Hence, in the simulation results in Fig. 6.26, considerable variations in the reactive power output of the first PMSG are observed.

6.3.3

Test 3: Variation of SCR and Wind Power Penetration (SRF PLL for PMSG1)

It is reviewed and introduced in Sect. 1.1 of Chap. 1 that a short circuit ratio (SCR) measures the strength of the connection between a VSC-based system, such as the PMSG, and the power system. Normally, if the SCR is lower than 3.0, the connection is considered weak. Recent study has indicated that, when the grid connection of the VSC-based system is weak, the PLL may degrade its stability. The following test examined the effect of the SCR on the OLMR. This test follows test 2 in the previous subsection, when the OLMR occurred between the POM, λpll and the inter-area EOM of the power system, λ9. In the test, the impedance of the transmission line connected PMSG1 and node 22 in the NEPS,

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

233

6 λˆpll

Imaginary axis

5 4

λpll –

λpll

Rgi Rpll

λˆ9

3 λ9

2 SCR varied from 10 to 2.5 1

λ9 + 0 -0.8

-0.7

-0.6

-0.5

-0.4 -0.3 Real axis

Rgi Rpll 0

-0.1

-0.2

0.1

Fig. 6.29 Open-loop modal resonance affected by SCR variation 0.2 SCR =10 SCR =2.5

0.2 0.15 0.1 0.05 0

Δθpll Δxpll State variables of the PLL

SCR =10

Participation factors

Participation factors

0.25

SCR =2.5

0.15 0.1 0.05 0

1

2

4

3

5

6

7

8

9 10

Generator number

Fig. 6.30 PF computational results, when SCR ¼ 2.5 and SCR ¼ 10

XL, was varied such that the SCR varied between SCR ¼ 10and SCR ¼ 2.5. Figure 6.29 shows the trajectories of the closed-loop POM, ^λ pll and the inter-area EOM, ^λ 9 , on the complex plane; the arrows indicate the direction of the SCR ^λ pll and ^λ 9 , using (6.15), decrease. The trajectories of the estimated positionspof ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi are also displayed. They are indicated as λpll þ Rgi Rpll and λ9  Rgi Rpll , respectively, in Fig. 6.29. From Fig. 6.29, it can be seen that the OLMR degraded the damping of the interarea EOM for both a weak (SCR ¼ 2.5) and strong (SCR ¼ 10) grid connection of PMSG1. In both the cases, the dynamic interactions between PMSG1 and the power system were strong, as indicated by the PF computational results in Fig. 6.30. The weaker the grid connection, the greater was the damping degradation of the interarea EOM caused by the OLMR.

234

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

From (6.30) and the discussions following it in the previous subsection, with Iq0  0, Pvsc0  Vd0Id0. Hence, when the wind power penetration, i.e., Pvsc0, increased, Id0 increased accordingly. From (6.36) and the discussions following it, it can be seen that, when the OLMR occurred at a higher level of wind power penetration, the variations of ΔQvsc were more. This implied that at a higher level of wind power penetration, the OLMR may cause greater damping degradation of the power system EOMs. The following test evaluated the impact of the OLMR, when wind power penetration increased. This test followed test 2 in Sect. 6.3.2, when the OLMR occurred between λpll and λ9. The active power output from PMSG1 increased from 0.5 to 10 p. u. The trajectories of the closed-loop POM, ^λ pll and the inter-area EOM, ^λ 9 , on the complex plane are displayed in Fig. 6.31; the arrows indicate the direction of the increase in the PMSG1 active power output. In Fig. 6.31, the trajectories of p theffiffiffiffiffiffiffiffiffiffiffiffiffi estimated positions of ^λ pll and ^λ 9 , using (6.15), are also displayed as λpll þ Rgi Rpll and pffiffiffiffiffiffiffiffiffiffiffiffiffi λ9  Rgi Rpll , respectively. The PF computational results are displayed in Fig. 6.32. From Figs. 6.31 and 6.32, it can be seen that, when the wind power penetration from PMSG1 increased, the dynamic interactions between the PLL and the rest of the power system increased and the OLMR caused a greater damping degradation of the inter-area EOM.

5

λˆ pll

Imaginary axis

4.5 4

λpll

3.5 λpll – 3 2.5

Rgi Rpll

λˆ 9 λ9

The active power output of the PMSG

λ9 +

increased from 0.5 to 10 2 -0.5 -0.45 -0.4 -0.35 -0.3

Rgi Rpll

-0.25

-0.2

-0.15

Real axis Fig. 6.31 The OLMR when the wind power penetration from PMSG1 increased

-0.1

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

0.2

0.2 Ppmsg 0 =0.5

Ppmsg 0 =0.5

Participation factors

Participation factors

0.25

0.15 Ppmsg 0 =10 0.1 0.05 0

235

Δθpll

Ppmsg 0 =10

0.15 0.1 0.05

Δxvq-pll

State variables of the PLL

0

1

2

3

4 5 6 7 8 Generator number

9 10

Fig. 6.32 PF Computational results, when the wind power penetration from PMSG1 increased

6.3.4

Test 4: DSOGI PLL for PMSG

In the above three subsections, the SRF PLL for PMSG1 was examined. In this subsection, the DSOGI PLL for PMSG2 was considered for demonstrating and evaluating the OLMR. The configuration of the DSOGI PLL is shown in Fig. 6.20. Differing from the SRF PLL, the DSOGI PLL can track the phase angle of the positive sequence voltage more accurately. The open-loop oscillation modes of the DSOGI PLL were not in the frequency range of the electromechanical oscillations of the NEPS, when the parameters of the DSOGI PLL given in [12] were adopted. However, this does not imply that there is no possibility of modal resonance of the DSOGI PLL with the power system EOMs, in practice. For example, due to the existence of modeling errors, and variations in the power system and PMSG operating conditions, the PLL parameters often need to be tuned in the field. Tuning is generally a trial-and-error procedure, and guidance on the tuning is useful for avoiding detrimental occurrences. Hence, by varying the PI gains of the DSOGI PLL in a wide range to scan the potential dangers posed by open-loop modal resonance, on one hand, the analysis about the OLMR can be demonstrated and evaluated for the DSOGI PLL; on the other hand, the scanning results provide guidance for tuning the PI gains, as they approximately indicate the PI gain ranges that should be avoided. Therefore, in this test, the PI gains of the DSOGI PLL were varied for scanning, in the frequency range of the NEPS EOMs. The scanning results of the DSOGI PL, similar to Fig. 6.23 for the SRF PLL, are shown in Fig. 6.33; it can be seen that, when the open-loop POM of the DSOGI PLL, indicated by circles, was in the neighborhood of the open-loop EOMs, indicated by the squares, the OLMR occurred. The corresponding closed-loop POM and EOMs were located approximately on the opposite side of the open-loop oscillation modes. The predicted positions of the closed-loop POM and EOMs were calculated using (6.15) and indicated by the crosses. Obviously, the evaluation results of the OLMR for the DSOGI PLL depicted in Fig. 6.33 are as same as those for the SRF PLL depicted in Fig. 6.23.

236

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

9 8 Imaginary axis

7 6 5 λ9 4 3 2 -0.6

λˆ9

λˆpll λpll -0.5

-0.2

-0.3 -0.4 Real axis

-0.1

Fig. 6.33 Modal resonance caused by DSOGI PLL parameter tuning

a

b

0.25

0.16

Participation factors

Participation factors

0.14

0.2 0.15 0.1 0.05

0.12 0.1 0.08 0.06 0.04 0.02

0

Δθpll Δxvq-pll State variables of the PLL

0

1

2

3

4

5

6

7

8

9

10

Generator number

Fig. 6.34 PF Computational results for the DSOGI PLL. (a) Participation of the PMSG in ^λ 9 ; (b) participation of the SGs in ^λ pll

For the OLMR between λpll and λ9, the PF computational results for ^λ pll and ^λ 9 are presented in Fig. 6.34. Strong interactions between the DSOGI PLL and the SGs were confirmed.

6.3.5

Test 5: The Case of DFIG

In the NEPS shown by Fig. 6.19, a wind farm represented by a DFIG, instead of two PMSGs, was connected to bus 22. A detailed 15th-order dynamic model of the DFIG (see Chap. 3 for the model of the DFIG) was used that included the dynamics of the

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

237

RSC and GSC control systems (seventh-order), the rotor motion (first-order), a SRF PLL (second-order), the induction generator (second-order), a grid-side converter impedance, and a capacitor (third-order). The dynamics associated with the pitch control and wind variations were not considered. The multivariable closed-loop state-space model shown by Fig. 5.27 for the NEPS with the DFIG connected to bus 22 was established with the dynamics of the PLL being included. The DFIG adopted the configuration of the RSC control system shown in Fig. 5.28 and the active power output of the DFIG was 10 p.u. Typical PI gains recommended in SIMULINK for the PLL were used with Kpp ¼ 4.3929, Kpi ¼ 39.2712. First, from the open-loop state matrix, Ad, in (5.33), a DFIG oscillation mode (DOM) was calculated as λd ¼  2.1974 þ j5.8687. It was identified to be associated with the SRF PLL from the computation of the participation factors of the state variables of the DFIG. From the open-loop state matrix of the power system, Ag, in (5.34), an open-loop EOM was calculated to be λi ¼  2.1005 þ j5.9133. It was identified to be a local EOM associated with SG4 and SG5 in the NEPS. The OLMR occurred between λd ¼  2.1974 þ j5.8687 and λi ¼  2.1005 þ j5.9133, as shown in Fig. 6.35, where ^λ i and ^λ d are the closedloop local EOM and DOM corresponding to λi and λd, respectively. From Fig. 6.35, it can be seen that the OLMR caused a considerable decrease in the damping of the local EOM, when the DFIG was connected at node 22. The computational results of the participation factors are presented in Fig. 6.36, confirming significant dynamic interactions between SG5 and the DFIG. 6.4 λˆ i

6.3

Imaginary axis

6.2 6.1 6

ˆλd

5.9 λd

λi

5.8 5.7 -2.8

-2.6

-2.4

-2 -2.2 Real axis

Fig. 6.35 The OLMR between the local EOM and the PLL

-1.8

-1.6

-1.4

238

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

b 0.9

0.4

0.8

0.35

0.7

0.3

Participation factors

Participation factors

a

0.6 0.5 0.4 0.3 0.2

0.2 0.15 0.1 0.05

0.1 0

0.25

1

2

4

3

8 7 6 5 Genereator number

9

10

0

Δxp

Δφp State variables

Fig. 6.36 Participation factors of the DFIG and SGs. (a) Participation of SGs in the DOM; (b) participation of the PLL in the local EOM

5.5 lˆd

Imaginary axis

5 4.5 ld

4

li

3.5

lˆi

3 2.5 -0.7

-0.6

-0.5

-0.4 Real axis

-0.3

-0.2

-0.1

Fig. 6.37 The OLMR between the inter-area EOM and PLL

Next, the PI gains of the PLL were decreased to move λd away from λi for avoiding the OLMR. However, when the PI parameters were changed to Kpp ¼ 0.6409, Kpi ¼ 13.1947, λd moved close to the inter-area EOM of the NEPS. The OLMR between the PLL and the inter-area EOM occurred, as shown in Fig. 6.37. Consequently, the damping of the inter-area EOM decreased considerably. The OLMR between the PLL and the inter-area EOM was confirmed by the computational results of the participation factors shown in Fig. 6.38. Further confirmation from the non-linear simulation is presented in Fig. 6.39.

6.3 Example 6.3: Electromechanical Oscillation Modes of a Power System. . .

a

b

0.14

Participation factors

Participation factors

0.12 0.1 0.08 0.06 0.04 0.02 0

239

1

2

3

4

5

6

7

8

9

10

Generator number

0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Δθpll Δxpll State variables of the PLL

Fig. 6.38 Participation factors of the DFIG and SGs. (a) Participation of SGs in the DOM; (b) participation of the PLL in the inter-area EOM

-2.5 PLL was of typical parameters PLL parameters were decreased

-3 -3.5 d10 – d1

-4 -4.5 -5 -5.5 -6 -6.5

0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Time(s)

Fig. 6.39 Non-linear simulation, when the OLMR occurred between the DFIG and the inter-area EOM, caused by the PLL

240

6 Small-Signal Stability of a Power System with a VSWG Affected by the PLL

Appendix 6.1: Data of Example 6.1 (Tables 6.5, 6.6 and 6.7) Table 6.5 Basic data of PMSG [3, 6]

Table 6.6 PI gains of MSC of PMSG [3, 6]

Kpi1 20

Kpp1 5

Table 6.7 PI gains of GSC of PMSG [3, 6]

Kpp4 2

Table 6.8 Basic data of PMSG

Xpd 0.25

Table 6.9 PI gains of MSC of PMSG

Kpp1 5

Table 6.10 PI gains of GSC of PMSG

Kpp4 0.8

ψpm 1

Xpq 0.2

Xpd 0.2

Kpi4 20

Xpq 0.15

Kpi4 20

Kpp5 0.5

Kpi5 84.9

Appendix 6.2: Data of Example 6.2 Data of PMSG1 They are given in Appendix 6.1.

Line Data Xt2 ¼ 0.02, Xt3 ¼ 0.02, XL ¼ 0.02.

Appendix 6.3: Data of Example 6.3 Data of New England Power System They are given in Appendix 4.2.1.

Data of PMSG (Tables 6.8, 6.9 and 6.10) Data of SRF PLL Kppll ¼ 4.4,Kipll ¼ 39.27

Kpi5 100

ωpref 0.8 Kpp2 1

Kpp5 0.15

Jpr 8s

Kpp2 1

ψpm 1.1

Kpi1 20

ωpref 0.8

Vpdc 1

Kpi2 100 Kpp6 1 Jpr 8s

Kpp3 1 Kpi6 10

Kpi3 100 Kpi7 100

Xpf 0.05

Cp 30

Kpp3 1

Kpi6 12

Cp 30

Kpp7 0.5

Vpdc 1

Kpi2 100

Kpp6 0.28

Xpf 0.02

Kpp7 0.15

Kpi3 100

Kpi7 84.9

References

241

References 1. Wang YJ, Hu JB, Zhang DL, Ye C, Li Q (2015) DFIG WT Electromechanical transient behavior influenced by PLL: modelling and analysis. In: International conference of renewable power generation, Beijing, pp 1–5 2. Wang ZW, Shen C, Liu F (2014) Impact of DFIG with phase lock loop dynamics on power systems small signal stability. In: IEEE PES general meeting, National Harbor, MD, pp 1–5 3. Li S, Haskew TA, Swatloski RP, Gathings W (2012) Optimal and direct-current vector control of direct-driven PMSG wind turbines. IEEE Trans Power Electron 27(5):2325–2337 4. Kaura V, Blasko V (1997) Operation of a phase locked loop system under distorted utility conditions. IEEE Trans Ind Appl 33(1):58–63 5. Chung SK (2000) A phase tracking system for three phase utility interface inverters. IEEE Trans Power Electron 15(3):431–438 6. Shi L, Crow ML (2008) A novel PLL system based on adaptive resonant filter. In: 40th North American Power Symposium, Calgary, AB, pp 1–8 7. Timbus A, Liserre M, Teodorescu R, Blaabjerg F (2005) Synchronization methods for three phase distributed power generation systems—An overview and evaluation. In: IEEE 36th power electronics specialists conference, Recife, pp 2474–2481 8. Kesler M, Ozdemir E (2011) Synchronous-reference-frame-based control method for UPQC under unbalanced and distorted load conditions. IEEE Trans Ind Electron 58(9):3967–3975 9. Padiyar KR (1996) Power system dynamics stability and control. Wiley, New York 10. Rogers G (2000) Power system oscillations. MA Kluwer, Norwell 11. Wang HF, Du WJ (2016) Analysis and damping control of power system low-frequency oscillations. Springer, New York 12. Golestan S, Monfared M, Freijedo FD (2013) Design-oriented study of advanced synchronous reference frame phase-locked loops. IEEE Trans Power Electron 28(2):765–778 13. MATLAB Simulink wind farm-DFIG detailed model. https://uk.mathworks.com/help/ physmod/sps/examples/wind-farm-dfig-detailed-model.html 14. Autom R, Co AB (1997) Operation of a phase locked loop system under distorted utility conditions. IEEE Trans Ind Appl 33(1):58–63

Chapter 7

Small-Signal Stability of a Power System Integrated with an MTDC Network for the Wind Power Transmission

As it is introduced in Chap. 1, small-signal stability of an AC power system integrated with a multi-terminal DC (MTDC) network for the wind power transmission is determined by the dynamic interactions between the VSCs and synchronous generators (SGs). The dynamic interactions are through both the MTDC network and AC grid. Hence, the small-signal stability of an MTDC/AC power system is a complicated issue, which is addressed in this chapter by focusing on the following three particular aspects. The first is the small-signal angular stability of an AC power system affected by the dynamic interactions brought about by the MTDC network. The MTDC network is integrated with the AC power system by VSC control. Control speed of the VSCs is much faster than the electromechanical transient of the AC power system. Hence, it has been recognized that dynamic interactions between the MTDC network and the AC power system are normally weak. Usual inertia-less response of the VSCs implies very small impact of the MTDC’s dynamics on the small-signal angular stability of the AC power system. Can the small-signal stability of a MTDC/AC power system be simply examined by modelling the MTDC network as constant power injections into the AC power system? Obviously, if there is a possibility that the dynamic interactions between the MTDC and AC system are strong under a special condition, modelling the MTDC as constant power injections may possibly miss out the potential threat imposed by the strong dynamic interactions on power system small-signal angular stability. In the first subsection of this chapter, the damping torque analysis (DTA) is applied to examine the special condition—the open-loop modal resonance (OLMR), under which the MTDC network may introduce strong dynamic interactions with the AC power system to affect the smallsignal angular stability of the MTDC/AC power system considerably. The OLMR has been studied in Chap. 5 and 6 for the VSWGs and PLL respectively. The second aspect is the small-signal stability of an independent MTDC network affected by the dynamic interactions between the VSCs via the MTDC network. When this particular aspect of problem is examined, the dynamic interactions between the SGs through the AC grid are not considered. The examination can be © Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4_7

243

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

244

carried out for an MTDC power system, where each of the VSCs is connected to a SG and there is no AC grid connecting all the SGs. In the second subsection of this chapter, this aspect of small-signal stability problem is investigated. The third aspect is the impact of a particular type of AC/DC dynamic interactions between the VSCs and SGs—the dynamic interactions introduced by a selected control system of a VSC in the MTDC/AC power system. Selection of the control system of the VSC enables the examination to be focused and simplified. The smallsignal stability of MTDC/AC power system as affected by this particular type of AC/DC dynamic interactions is studied in the third subsection of the chapter.

7.1

Small-Signal Angular Stability of an AC Power System Affected by the Integration of an MTDC Network for the Wind Power Transmission

7.1.1

The Open-Loop Modal Resonance (OLMR) in the MTDC/AC Power System

7.1.1.1

Linearized Model of AC Power System

Figure 7.1 shows the configuration of an AC power system integrated with a VSC-based MTDC network for wind power transmission. In order to examine the small-signal angular stability of the AC power system affected by the dynamic interactions introduced by the MTDC network, a closed-loop linearized model of the MTDC/AC power system is established in this subsection.

P1 + jQ1

Wind farms

I MTDC1 I dc1

...

M T D C

C1

Vdc1

I d 1 + jI q1

VSC-1

SG-1

Pj + jQ j

I MTDCj I dcj Cj

VSC-j

I dj + jI qj

...

PM + jQM

Network

I MTDCM I dcM

CM

Vj

X fj

Vdcj

VdcM

Vg1

V1 X f1

VM

A C

Vgk

SG-k

Network

Vgn

X fM VSC-M

I dM + jI qM

SG-n

Fig. 7.1 An AC power system integrated with a MTDC network for the wind power transmission

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

245

Denote Pj þ jQj, j ¼ 1, 2, . . .M as the complex power output from the jth VSC to the AC power system. From Fig. 7.1, the active and reactive power injected from the jth VSC to the power system can be obtained to be P j ¼ Vxj Ixj þ Vyj Iyj , Q j ¼ Vyj Ixj  Vxj Iyj , j ¼ 1, 2, . . . M

ð7:1Þ

where subscript x and y denotes the x and y component of a phasor under system common reference coordinate respectively; Vj ¼ Vxj þ jVyj, j ¼ 1, 2, . . .M and Ij ¼ Ixj þ jIyj, j ¼ 1, 2, . . .M are the terminal voltage of and the output current from the jth VSC respectively. Linearization of (7.1) is ΔP j ¼ Ixj0 ΔVxj þ Iyj0 ΔVyj þ Vxj0 ΔIxj þ Vyj0 ΔIyj

ð7:2Þ

ΔQ j ¼ Iyj0 ΔVyj þ Ixj0 ΔVyj þ Vyj0 ΔIxj  Vxj0 ΔIyj

where subscript 0 denotes the value of a variable at the steady state. In matrix form, the above equations are " # ΔP ð7:3Þ ¼ F1 j ΔI þ F2 j ΔV ΔQ where ΔP ¼ ½ΔP1 ΔP2    ΔPM T , ΔQ ¼ ½ ΔQ1 ΔI ¼ ½ ΔI1

ΔI2

ΔV j ¼ ½ ΔVxj



ΔQ2

ΔIM T , ΔV ¼ ½ ΔV1

ΔVyj T , ΔI j ¼ ½ ΔIxj

   ΔQM T ,

ΔV2

   ΔVM T ,

ΔIyj T

Following linearized network equation with the PCC nodes of the VSCs and nodes of the synchronous generators (SGs) being kept can be established [1] #" " # " # Ygg Ygj ΔIg ΔVg ¼ ð7:4Þ Y jg Y jj ΔI ΔV where  ΔVg ¼ ΔVg1 T



ΔVgj ¼ ½ ΔVgxj

ΔVgyj T , ΔIgj ¼ ½ ΔIgxj

ΔVgn T

T

 , ΔIg ¼ ΔIg1 T



ΔIgn T

T

,

ΔIgyj T

Vgxj þ jVgyj and Igxj þ jIgyj is the terminal voltage and output current of the jth SG in the power system respectively. The following linearized model of the SGs can be established [1]. sΔXAC ¼Ag ΔXAC þBg ΔVg , ΔIg ¼Cg ΔXAC þDg ΔVg

ð7:5Þ

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

246

Substituting the second equation of (7.4) into (7.3) gives " #   2 1 ΔP  21 ΔV¼ F2 j þF1 j Y jj F1 j Y jg ΔVg  F2 j þF1 j Y jj ΔQ

ð7:6Þ

From the second equation of (7.5) and the first equation of (7.4), it can have   Ygg  Dg ΔVg þYgj ΔV¼Cg ΔXAC ð7:7Þ Substituting (7.6) into (7.7) " ΔVg ¼F3 ΔXAC þF4 where

ΔP

# ð7:8Þ

ΔQ

h i1  1 F3 ¼ Ygg  Dg  Ygj F2 j þF1 j Y jj F1 j Y jg Cg

h i1  1  1 F4 ¼  Ygg  Dg  Ygj F2 j þF1 j Y jj F1 j Y jg Y jg F2 j þF1 j Y jj Substituting (7.8) into the first equation of (7.5), following state equations are obtained. d ΔXAC ¼ AAC ΔXAC þ BAC ΔU dt

ð7:9Þ

where ΔU¼½ ΔP ΔP ¼ ½ ΔP1

ΔQ , ΔP2

   ΔPM T , ΔQ ¼ ½ ΔQ1

ΔQ2



ΔQM T ,

Substituting (7.8) into (7.6) " ΔV¼CAC ΔXAC þDAC where

ΔP ΔQ

# ¼CAC ΔXAC þDAC ΔU

ð7:10Þ

 1  1 CAC ¼  F2 j þF1 j Y jj F1 j Y jg F3 , DAC ¼ F2 j þF1 j Y jj  1  F2 j þF1 j Y jj F1 j Y jg F4

Writing (7.9) and (7.10) together, following state-space model of the AC power system is established: d ΔXAC ¼ AAC ΔXAC þ BAC ΔU dt ΔV ¼ CAC ΔXAC þ DAC ΔU

ð7:11Þ

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

247

where ΔXAC is the vector of all the state variables of the AC power system. From (7.11), the transfer function matrix model of the AC power system can be obtained to be GðsÞ¼CAC ðsI  AAC Þ1 BAC þDAC

ð7:12Þ

ΔV¼GðsÞΔU

ð7:13Þ

Hence,

7.1.1.2

Linearized Model of the MTDC Network

The pair of input-output signal vectors of the MTDC network are ΔV  ΔU. Thus, following transfer function matrix model of the MTDC network can be derived. ΔU¼HðsÞΔV

ð7:14Þ

The detailed derivation of (7.14) is as follows. Assume that L VSCs are connected to L wind farms and M VSCs are connected to the AC power system in the MTDC network of Fig. 7.1. For a VSC being connected to a wind farm, active wind power output from the wind farm is Pwj ¼ VdcwjIdcwj, j ¼ 1, 2, . . .L. Since the wind power output is only determined by the factors, such as wind speed, which are irrelevant with the disturbances on the side of AC power system, it can be assumed to be a constant. Thus, for the jth VSC being connected to the wind farm, it can have ΔPwj ¼ Vdcwj0 ΔIdcwj þ Idcwj0 ΔVdcwj ¼ 0, j ¼ 1, 2, . . . L

ð7:15Þ

Following network equation for the MTDC network can be established 2

ΔIMTDC1

3

2

0

y11 ðsÞ

0

y12 ðsÞ

0 6 7 6 0 4 ΔIMTDC 5 ¼ 6 4 y21 ðsÞ Y22 ðsÞ 0 0 ΔIdcw y31 ðsÞ Y32 ðsÞ

32 3 ΔVdc1 76 0 7 Y23 ðsÞ 7 54 ΔVdc 5 0 ΔVdcw Y33 ðsÞ

1st VSC

0

y13 ðsÞ

ðM  1Þ VSCs VSCs being connectedto wind farms ð7:16Þ

where ΔIMTDC ¼ ½ ΔIMTDC2

ΔIMTDC3

   ΔIMTDCM T , ΔVdc ¼ ½ ΔVdc2

ΔVdc3

   ΔVdcM T

In the following derivation, Laplace operator s in (7.16) is omitted for the simplicity of expressions. The jth element of ΔIdcw and ΔVdcw is ΔIdcwj and ΔVdcwj as given in (7.15). Using (7.15) to delete the last row in (7.16), it can have

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

248

"

ΔIMTDC1

#

ΔIMTDC

" ¼

y11

y12

y21

Y22

#"

ΔVdc1

# ð7:17Þ

ΔVdc

For the kth VSC being connected with the AC power system, it can have dΔIdk ¼ ω0 ðΔVcdk  ΔVdk Þ þ ω0 X fk ΔIqk dt   dΔIqk ¼ ω0 ΔVcqk  ΔVqk  ω0 X fk ΔIdk X fk dt X fk

ð7:18Þ

where Xfk is the reactance of the output filter of the kth VSC, ω0 is the synchronous frequency. Without the loss of generality, assume that the first VSC uses the DC voltage control and the rest of VSCs implement the active power control. Figures 7.2 and 7.3 show the configuration of the vector control system of the kth VSC being connected to the AC system. Ignoring the very fast transient of the PWM algorithm, ref ref Vcdk ¼ Vcdk , Vcqk ¼ Vcqk , k ¼ 1, 2, . . . M. Express the transfer function of various PI controllers in Figs. 7.2 and 7.3 as Kqi1 Kdci , Kq ðsÞ ¼ Kqp1 þ , s s Kiqi1 Kidi1 Kid ðsÞ ¼ Kidp1 þ , Kiq ðsÞ ¼ Kiqp1 þ s s Kpik Kqik , Kqk ðsÞ ¼ Kqpk þ , Kpk ðsÞ ¼ Kppk þ s s Kiqik Kidik Kidk ðsÞ ¼ Kidpk þ , Kiqk ðsÞ ¼ Kiqpk þ s s Kdc ðsÞ ¼ Kdcp þ

Current control inner loop

DC voltage control outer loop

Vd1

DC voltage control loop ref dc1 +

V

– Vdc1

Kdc (s )

I

ref d1

+ – Id1

Kid (s )

– Q1

ref I q1 + K q (s )



Iq1

ref + + Vcd1 +

X f1 Iq1 Vq1

Reactive power control loop

Q1ref +

ð7:19Þ

K iq (s )

ref + + Vcq1 +

X f1 Id1

Fig. 7.2 Converter control system of the first VSC with DC voltage control

PWM

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

249

Power control outer loop

Current control inner loop Active power control loop

ref k

P + – Pk

K pk (s )

I

ref dk

+ – I dk

Vdk ref + + Vcdk Kidk (s ) – X fk Iqk

Reactive power control loop

Qkref + –

K qk (s )

Qk

I qkref +

– I qk

PWM

Vqk ref + + Vcqk Kiqk (s ) + X fk Idk

Fig. 7.3 Converter control system of the kth VSC with active power control

From Figs. 7.2 and 7.3, (7.18) and (7.19), Kdcp s þ Kdci ΔVdc1 s Kppk s þ Kpik ref ΔPk , k ¼ 2, 3, . . . M ΔIdk ¼ s Kqpk s þ Kqik ref ΔQk , k ¼ 1, 2, . . . M ¼ ΔIqk s ω0 Kidpk s þ Kidik ω0 ΔI ref , k ¼ 1, 2, . . . M ΔIdk ¼ X fk s2 þ ω0 Kidpk s þ Kidik ω0 dk ω0 Kiqpk s þ Kiqik ω0 ΔI ref , k ¼ 1, 2, . . . M ΔIdk ¼ X fk s2 þ ω0 Kiqpk s þ Kiqik ω0 qk

ref ¼ ΔId1

ð7:20Þ

Matrix form of (7.20) is   ΔId ¼ diagGpk ðsÞΔP ¼ Gp ðsÞΔP ΔIq ¼ diag Gqk ðsÞ ΔQ ¼ Gq ðsÞΔQ

ð7:21Þ

where T ΔId ¼ ½ ΔId2 ΔId3    ΔIdM T , ΔIq ¼ ½ ΔIq1 ΔIq2    ΔIqM  , T T ΔP ¼ ½ ΔP2 ΔP3    ΔPM  , ΔQ ¼ ½ ΔQ1 ΔQ2    ΔQM 

Take the direction of PCC voltage, Vk, k ¼ 1, 2, . . .M, as that of d axis of the d  q coordinate of the kth VSC. It can have Vdk ¼ Vk, Vqk ¼ 0, k ¼ 1, 2, . . .M. Hence linearized active and reactive power output from the VSCs can be written as ΔP1 ¼ Id10 ΔV1 þ V10 ΔId1 ¼ Vdc10 ΔIdc1 þ Idc10 ΔVdc1 ΔP ¼ Id0 ΔVk þV0 ΔId ¼Vdc0 ΔIDC þIdc0 ΔVdc ΔQ ¼ Iq0 ΔV  V0 ΔIq

ð7:22Þ

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

250

where ΔVk ¼ ½ ΔV2



ΔVM T , ΔV ¼ ½ ΔV1

ΔV2

   ΔVM T ,

   ΔIdcM T , Id0 ¼ diag½Idk0 , V0 ¼ diag½Vk0 ,   ¼¼ diag½Vdck0 , Idc0 ¼ diag½Idck0 , Iq0 ¼ diag Iqk0

ΔIDC ¼ ½ ΔIdc2 Vdc0

ΔV3 ΔIdc3

From (7.21) and (7.22),

 1 ΔP¼ I  V0 Gp ðsÞ Id0 ΔVk  1 ΔQ¼  IþV0 Gq ðsÞ Iq0 ΔV

ð7:23Þ

On the DC side of the VSC, the linearized model is dΔVdc1 þ ΔIdc1 ¼ ΔIMTDC1 , dt   dΔVdc þ ΔIDC ¼ ΔIMTDC , C ¼ diag C j , j ¼ 2, 3, . . . M C dt C1

ð7:24Þ

From (7.17) and (7.24), ΔVdc ¼ ðsC  Y22 Þ1 y21 ΔVdc1  ðsC  Y22 Þ1 ΔIDC From (7.21), (7.22) and (7.25),  1 ΔP ¼ I  V0 Gp ðsÞ Id0 ΔVk ¼Vdc0 ΔIDC þ Idc0 ΔVdc ¼Vdc0 ΔIDC þIdc0 ðsC  Y22 Þ1 y21 ΔVdc1  Idc0 ðsC  Y22 Þ1 ΔIDC

ð7:25Þ

ð7:26Þ

Thus

h i1  1 ΔIDC ¼ Vdc0  Idc0 ðsC  Y22 Þ1 I  V0 Gp ðsÞ Id0 ΔVk h i1  Vdc0  Idc0 ðsC  Y22 Þ1 Idc0 ðsC  Y22 Þ1 y21 ΔVdc1

ð7:27Þ

From (7.17), (7.25) and (7.27), ΔIdc1 ¼ y11 ΔVdc1 þ y12 ΔVdc ¼ y11 ΔVdc1 þ y12 ðsC  Y22 Þ1 y21 ΔVdc1  y12 ðsC  Y22 Þ1 ΔIDC ¼ y11 ΔVdc1 þ y12 ðsC  Y22 Þ1 y21 ΔVdc1 h i1  1 y12 ðsC  Y22 Þ1 Vdc0  Idc0 ðsC  Y22 Þ1 I  V0 Gp ðsÞ Id0 ΔVk h i1 þy12 ðsC  Y22 Þ1 Vdc0  Idc0 ðsC  Y22 Þ1 Idc0 ðsC  Y22 Þ1 y21 ΔVdc1 ¼ y1 ðsÞΔVdc1 þ yðsÞΔVk ð7:28Þ

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

251

From (7.23) and (7.28), ΔIdc1 ¼ ½y1 ðsÞ  C1 sΔVdc1 þ yðsÞΔVk

ð7:29Þ

From (7.20), (7.22) and (7.28) ΔP1 ¼ Id10 ΔV1 þ V10 ΔId1 ¼ Vdc10 ΔIdc1 þ Idc10 ΔVdc1 ¼ Id10 ΔV1 þ V10 Gp1 ðsÞΔVdc1

ð7:30Þ

¼ Vdc10 ½y1 ðsÞ  C1 sΔVdc1 þ Vdc10 yðsÞΔVk þ Idc10 ΔVdc1 From the above equation,  1 ΔVdcl ¼ V10 Gp1 ðsÞ  Vdc10 y1 ðsÞ þ Vdc10 C1 S  Idc10 Vdc10 yðsÞΔVk  1 Id10 V10 Gp1 ðsÞ  Vdc10 y‘ 1 ðsÞ þ Vdc10 C1 s  Idc10 ΔV1

ð7:31Þ

Substituting (7.31) into (7.30), ΔP1 ¼ gp1 ðsÞΔV1 þ gpk ðsÞΔVk

ð7:32Þ

where gp1 ðsÞ ¼ fVdc10 ½y1 ðsÞ  C1 s þ Idc10 gId10  1 V10 Gp1 ðsÞ  Vdc10 y‘1 ðsÞ þ Vdc10 C1 s  Idc10 gpk ðsÞ ¼ fVdc10 ½y1 ðsÞ  C1 s þ Idc10 g  1 V10 Gp1 ðsÞ  Vdc10 y1 ðsÞ þ Vdc10 C1 s  Idc10 Vdc10 yðsÞ þ Vdc10 yðsÞ Writing (7.23) and (7.32) together, the transfer function matrix model of the MTDC network of (7.14) is obtained. Let the state-space realization of the MTDC network’s transfer function matrix model of (7.14) be d ΔXDC ¼ ADC ΔXDC þ BDC ΔV dt ΔU ¼ CDC ΔXDC þ DDC ΔV

ð7:33Þ

where ΔXDC is the vector of all the state variables of the MTDC network and HðsÞ¼CDC ðsI  ADC Þ1 BDC þDDC

7.1.1.3

ð7:34Þ

The Closed-Loop Model of the MTDC/AC Power System

From (7.13) and (7.14), the closed-loop model of the MTDC/AC power system is obtained and shown by Fig. 7.4. The state-space model of the MTDC/AC power system is obtained from (7.11) and (7.34) to be

252

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Fig. 7.4 Closed-loop model of the AC/MTDC power system

G(s)

ΔU

H(s)

ΔV

AC power system

MTDC network

d ΔX¼AΔX dt

ð7:35Þ

where  T ΔX ¼ ΔXAC T ΔXDC T , " AAC þ BAC ðI  DDC DAC Þ1 DDC CAC A¼ BDC ðI  DAC DDC Þ1 CAC

#

BAC ðI  DDC DAC Þ1 CDC ADC þ BDC ðI  DAC DDC Þ1 DAC CDC

Dynamic interactions between the MTDC network and the AC power system physically are the dynamic variations of complex power output from the MTDC, ΔPj þ jΔQj, j ¼ 1, 2, . . .M. If ΔPj þ jΔQj ¼ 0, j ¼ 1, 2, . . .M, there are no dynamic interactions between the MTDC network and the AC power system. In this case, small-signal angular stability of the AC power system is not affected by dynamics of the MTDC network at all. State-space model of the AC power system shown in (7.11) is degraded to d ΔXAC ¼AAC ΔXAC dt

ð7:36Þ

Denote λACi as an electromechanical oscillation mode (EOM) of concern in the open-loop AC power system when ΔPk þ jΔQk ¼ 0, k ¼ 1, 2, . . .M. λACi is a λ ACi as the EOM complex eigenvalue of open-loop state matrix, AAC. Denote b λ ACi is an eigenvalue of closedcorresponding to λACi when ΔPk þ jΔQk 6¼ 0. b loop state matrix, A in (7.35). Obviously, difference between b λ ACi and λACi, b ΔλACi ¼ λ ACi  λACi , is caused by the dynamic interactions between the MTDC network and the AC power system. This difference is the impact of dynamic interactions introduced by the MTDC network on the small-signal angular stability of AC power system. Based on the closed-loop model, dynamic interactions between the MTDC and the AC power system, ΔPk þ jΔQk, are linked with the difference of closed-loop and open-loop EOM of concern, which is a measurement of the impact of the dynamic interactions.

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

7.1.1.4

253

Damping Torque Analysis (DTA)

Re-write state equation in (7.11) as 32 3 2 3 2 3 2 3 2 0 Δδ 0 0 Δδ 0 ω0 I M M X6 d6 76 7 X6 7 7 7 6 4 bP2k 5ΔPk þ 4 bQ2k 5ΔQk 4 Δω 5¼4 A21 A22 A23 54 Δω 5þ dt k¼1 k¼1 bQ3k Δz A31 A32 A33 bP3k Δz ð7:37Þ where Δδ and Δω are the vectors of deviations of angular positions and speeds of the SGs respectively, Δz is the vector of all other state variables in the AC power system. Express the transfer function matrix model of the MTDC network of (7.14) as " # " # H P ðsÞ ΔP ΔV ð7:38Þ ¼ HðsÞΔV ¼ HQ ðsÞ ΔQ From (7.37) and (7.38), the electric torque provided from the MTDC network via M VSCs to the SGs is obtained ΔTDC ¼

M X gpk ðsÞΔPk þ gqk ðsÞΔQk k¼1

¼

M X

M M X X gpk ðsÞ hpkj ðsÞΔV j þ gqk ðsÞ hqkj ðsÞΔV j

k¼1

j¼1

ð7:39Þ

j¼1

where, gpk(s) ¼ bP2k + A23(sI  A33)1bP3k, gqk(s) ¼ bQ2k + A23(sI  A33)1bQ3k, hpkj(s) and hqkj(s), k, j ¼ 1, 2, . . .M are the elements of HP(s) and HQ(s) respectively. It is proved in [2] that in the AC power system, following equation can always be established: ΔV j ¼ γ jm ðλACi ÞΔωm

ð7:40Þ

From (7.39) and (7.40), damping torque contribution from the MTDC network to the electromechanical oscillation loop of the mth SG is obtained to be ΔTdm ¼ Re

M X

gpkm ðλACi Þ

k¼1 M X

þ

k¼1

gqkm ðλACi Þ

M X hpkj ðλACi Þγ jm ðλACi Þ j¼1

M X



ð7:41Þ

hqkj ðλACi Þγ jm ðλACi Þ Δωm ¼ dm Δωm

j¼1

where gpkm(s) and gqkm(s) is the mth element of gpk(s) and gqk(s) respectively, Re [] denotes the real part of a complex number.

254

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

ACi Denote Sim ¼ ∂λ as the sensitivity of the EOM of concern to the damping torque ∂dm coefficient of the mth SG. The impact of dynamic interactions between the MTDC network and the AC power system on the EOM of concern is obtained from (7.41) to be

λ ACi  λACi ¼ ΔλACi ¼ b

N X

Sim dm

ð7:42Þ

m¼1

where N is the total number of the SGs in the AC power system. Denote λDCi as an oscillation mode of the MTDC subsystem. It is a complex eigenvalue of open-loop state matrix of the MTDC subsystem, ADC in (7.33). λDCi should be at least a complex pole of one element of transfer function matrix H(s), hpkj(s) and hqkj(s), k, j ¼ 1, 2, . . .M, defined in (7.38). Without loss of generality, assume λDCi is a complex pole of hp11(s); thus |hp11(λDCi)| ¼ 1. Open-loop modal resonance (OLMR) is the special condition that open-loop oscillation mode of the MTDC subsystem, λDCi, is closed to the EOM of concern of the open-loop AC subsystem, λACi, on the complex plane, i.e., λDCi  λACi. Obviously, when λDCi  λACi, |hp11(λACi)| is significantly large. From (7.38), it can be seen that ΔP1 may be large at the complex frequency λACi. This means that the first VSC may respond considerably to disturbances in AC power system. Hence, strong dynamic interactions between the MTDC and the AC power system may occur when the OLMR, λDCi  λACi, happens. From (7.41), it can be seen that when λDCi  λACi such that |hp11(λACi)| is significantly large, it is possible that the damping torque contribution from the first VSC to some of the SGs in the AC power system is significant. Consequently, corresponding damping coefficients, dm, may be big; thus ΔλACi may not be small any more. This explains why the OLMR may cause strong dynamic interactions between the MTDC and the AC power system to affect the EOM of concern considerably.

7.1.1.5

Impact of Strong Dynamic Interactions Under the Condition of OLMR

Denote b λ DCi as the closed-loop oscillation mode corresponding to. It is a pair of conjugate complex eigenvalues of closed-loop state matrix, A in (7.35). Similarly, λ DCi  λDCi is also caused by dynamic interactions between the MTDC ΔλDCi ¼ b network and the AC power system. Hence, the relative positions of the closed-loop λ DCi , in respect to the positions of the open-loop modes, λDCi  λACi, modes, b λ ACi and b indicate how much the dynamic interactions between the MTDC network and the AC power system affect the small-signal stability of MTDC/AC power system under the condition of OLMR. Using the theory on modal resonance, the indication can be further examined as follows.

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

a

255

b

Imaginary axis

Imaginary axis

area A

area B

lˆDCi lDCi

area B

l ACi

lˆACi

lDCi

lˆACi

l ACi lˆDCi area A

Real axis

Real axis

Fig. 7.5 Illustration of near strong modal resonance around point of the OLMR

Modal resonance is the coincidence of eigenvalues of state matrix of a linear system. There are two types of modal resonance mathematically defined, strong modal resonance and weak modal resonance [3]. Only near strong modal resonance may occur in power systems when certain system parameters vary [4–6]. It has been mathematical proved that in the neighborhood of near strong modal resonance, two resonant eigenvalues intend to move towards opposite directions [3–6]. Near strong modal resonance of two power system EOMs with variations of some system parameters was investigated in [4–6]. Results of investigation showed that two EOMs moved close to each other and then towards opposite directions on the complex plane. Subsequently, one EOM moved towards the right on the complex plane to lead to damping degradation of power system low-frequency oscillations [4–6]. For the MTDC/AC power system shown in Fig. 7.1, when the dynamic interactions between the MTDC network and the AC power system are weak, b λ DCi is close λ ACi is close to λACi respectively on the complex plane. Hence, when λDCi to λDCi and b moves from a position away from λACi with variations of some parameters of openλ ACi should loop MTDC network, b λ DCi should move along with λDCi. Meanwhile, b remain close to λACi owing to the weak dynamic interactions between the MTDC network and the AC power system. This is illustrated by shadow areas A in Fig. 7.5a. However, when λDCi moves close to the point of the OLMR, λDCi  λACi, on the λ ACi if dynamic complex plane, b λ DCi moves along with λDCi to get close to b interactions between the MTDC network and the AC power system remain being weak. Closeness of two eigenvalues of closed-loop state matrix, A in (7.35), b λ DCi and b λ ACi , implies that a near strong modal resonance is destined to happen in the λ ACi and b λ DCi should neighborhood of point of the OLMR, λDCi  λACi. In this case, b move towards the opposite directions away from the point of the OLMR as illusλ ACi  λACi trated by circled areas B in Fig. 7.5a. This is the case that both ΔλACi ¼ b

256

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

and ΔλDCi ¼ b λ DCi  λDCi are not small any more, indicating increased dynamic interactions between the MTDC network and the AC power system. Hence, in the neighborhood of point of the OLMR, λDCi  λACi, strong dynamic interactions may λ DCi move occur and their impact may be considerable. In addition, since b λ ACi and b away towards the opposite directions from the point of the OLMR, λDCi  λACi, it is λ DCi may locate on the right hand side of λDCi  λACi, very likely that either b λ ACi or b indicating negative impact of dynamic interactions between the MTDC network and the AC power system on the small-signal stability of the MTDC/AC power system. On the other hand, when some parameters of open-loop AC subsystem change, λACi moves from a position away from λDCi towards the point of the OLMR, λDCi  λACi. A similar analysis to that presented above can be illustrated by Fig. 7.5b. It is likely that near strong modal resonance may occur in the neighborhood of point of the OLMR, λDCi  λACi. Consequently, the small-signal stability of the MTDC/AC power system may possibly decrease under the condition of the OLMR. From the examination presented above and illustration in Fig. 7.5, it can be seen that the point of the OLMR, in fact, indicates the location, around which the near strong modal resonance of closed-loop MTDC/AC power system may occur. It can be concluded that the impact of strong dynamic interactions caused by the OLMR may very likely reduce the small-signal stability of the MTDC/AC power system.

7.1.2

Example MTDC/AC Power Systems

7.1.2.1

Example 7.1—New England Power System Integrated with a MTDC Network for the Wind Power Transmission

The New England test system (NETS) has been used to study power system smallsignal angular stability in many occasions [7] and also as the example power systems in the previous chapters. Figure 7.6 shows the configuration of modified NETS. Modifications were the integration of a ten-terminal DC network for wind power transmission. Five VSCs were connected with the wind farms. Each of wind farms was modelled as a PMSG. Detailed dynamic model of the PMSGs and parameters given in [8] were used. Other five VSCs were connected to the NETS. Parameters of SGs, loads and network of the NETS given in [7] were used. Appendix 7.1 presents the data of the example power system with the MTDC network. There were in total nine EOMs in the NETS. How the EOMs were affected by the integration of the MTDC was examined. An inter-area EOM with lowest frequency is the mode of concern, λACi, in the examination. It was most lightly damped EOM. Low-frequency oscillations related to λACi is between G10 and rest of the SGs. Two types of MTDC control strategies were examined. The first one was the master-slave control [9] with the VSC1 adopting DC voltage control and other VSCs using active power control. The second type of control strategy was the DC voltage droop control [9, 10] implemented by the VSC1 and VSC2. Other VSCs adopted

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

26 25 2

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8

257

13

44

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42

41

VSC-1 VSC-5 VSC-4 VSC-3 VSC-2

10

7

21

r

MTDC NET

r

32 8

NETS

3

WIND FARMs

Fig. 7.6 Configuration of the NETS integrated with a ten-terminal DC network

active power control. Wind active power delivery to the NETS was shared equally between VSC1 and VSC2. The droop coefficient was 0.1%. Parameters of the ten-terminal DC network (MTDC) recommended in [9, 10], including control parameters of the VSCs, were used.

7.1.2.2

Strong Dynamic Interactions When the Master-Slave Control Was Implemented

First, total wind power penetration to the NETS via the MTDC network gradually increased from Pw0 ¼ 10 p.u to a higher level, Pw0 ¼ 16 p.u. It was found that a complex eigenvalue of open-loop state matrix of the MTDC subsystem, λDCi, moved close to the EOM of concern, λACi, as shown by dashed curves in Fig. 7.7, where arrows indicate the direction of increase of wind power penetration to the NETS via the MTDC network. In Fig. 7.7, trajectories of complex eigenvalues of state matrix of closed-loop MTDC/AC system corresponding to λACi and λDCi are shown by solid λ DCi . curves. They are b λ ACi and b From Fig. 7.7, it can be seen that when Pw0 ¼ 10 p.u, λDCi and λACi position away λ ACi is close to λACi from each other on the complex plane. b λ DCi is close to λDCi and b respectively. This was the case of weak dynamic interactions between the MTDC and the NETS. With increase of wind power penetration, λDCi and λACi moved close to each other on the complex plane. Around point A when Pw0 ¼ 16 p. u, λDCi  λACi

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

258

Fig. 7.7 Modal trajectories when the OLMR occurred (Master-slave control)

Imaginary axis (rad/s) 3.8 Master-slave control

λˆACi

λDCi 3.6

λˆDCi A

3.4 -0.5

1.0 Pwo=10 pu Pwo=16 pu

Pwo=10 pu Pwo=16 pu

0.8 Participation factors

0.4 Participation factors

0.1 Real axis

-0.1

-0.3

0.5

0.3

0.2

0.6

0.4

0.2

0.1

0

λACi

1 2 3 4 5 6 7 8 Generator number

9 10

1 2 3 4 5 VSC number

0

1

2

3 4 5 6 7 8 Generator number

9 10

1 2 3 4 5 VSC number

Fig. 7.8 Participation factors (PFs) for b λ ACi and b λ DCi (Master-slave control)

and b λ ACi moved away into the right half of complex plane, causing small-signal angular instability of the NETS. From Fig. 7.7, it can be seen that when λDCi  λACi, λ ACi  λACi was not small, indicating significant dynamic interactions ΔλACi ¼ b between the MTDC and the NETS. Computational results of participation factors (PFs) of the SGs and the VSCs for b λ ACi and b λ DCi are presented in Fig. 7.8 when Pw0 ¼ 10 p.u and Pw0 ¼ 16 p. u respectively. From Fig. 7.8, it can be seen that when Pw0 ¼ 10 p.u (λDCi was away λ ACi was mainly participated by the SGs and b λ DCi from λACi on the complex plane), b was dominantly participated by VSC1. This indicated weak dynamic interactions between the MTDC network and the NETS. However, when Pw0 ¼ 16 p. u λ ACi and b λ DCi were participated considerably by the SGs and (λDCi  λACi), both b VSC1, confirming strong dynamic interactions between the MTDC network and the

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

0.6 Active power output from VSC1 (pu)

Fig. 7.9 Non-linear simulation (Master-slave control)

259

Pwo=10 pu Pwo=16 pu

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

0

2

4

6

8

10

6

8

10

Time(s) 8 Pwo=10 pu Pwo=16 pu Dd1 – Dd10 (deg.)

4

0

-4

-8

0

2

4 Time(s)

NETS. That was why under the condition of the OLMR, λDCi  λACi, the inter-area EOM of concern was affected considerably by the dynamic interactions. To further confirm the results of modal computation presented above, Fig. 7.9 shows the results of non-linear simulation. Simulation was carried out with a sudden loss of 80% load at node 10 in the NETS at 0.1 s of simulation. The lost load was recovered in 0.1 s. From Fig. 7.9, it can be seen that when Pw0 ¼ 10 p.u, dynamic variations of active power output from the VSC1 were very small, indicating weak dynamic interactions between the MTDC network and the NETS. The integrated MTDC/NETS was stable. However, when the wind penetration increased to Pw0 ¼ 16 p. u, dynamic variations of active power output from VSC1 were

260

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Fig. 7.10 Modal trajectories to dismiss the OLMR (Master-slave control)

Imaginary axis (rad/s) 6 Master-slave control 5.5

5

4.5 λDCi 4

3.5 -0.6

λˆACi

λˆDCi λACi -0.5

-0.3

-0.1

0.1 Real axis

significant, indicating strong dynamic interactions. The integrated MTDC/NETS became unstable. Participation factors (PFs) of various state variables of VSC1 for b λ ACi were calculated under the condition of the OLMR. It was found that the state variables associated with the DC voltage control system of VSC1 participated in b λ ACi mostly. Hence, in order to examine if the strong dynamic interactions may be dismissed by tuning control parameters of the MTDC, parameters of the DC voltage controller of VSC1 were changed from Kdcp ¼ 0.3, Kdci ¼ 2.4 to Kdcp ¼ 0.5, Kdci ¼ 20. Trajectory of λDCi on the complex plane with parameters tuning of the DC voltage controller of VSC1 are displayed in Fig. 7.10, when the wind power penetration was Pw0 ¼ 16 p. u. In Fig. 7.10, hollow circles are the positions of open-loop EOMs of the NETS and solid curves are the trajectories of the corresponding closed-loop EOMs. From Fig. 7.10, it can be seen that when parameters of the DC voltage controller were tuned, λDCi moved away from the open-loop inter-area EOM of concern, λACi. Closed-loop EOM of concern, b λ ACi , moved back towards the open-loop EOM of concern, λACi. On the way that λDCi moved away from λACi, three closed-loop local EOMs were affected when λDCi was close to the corresponding open-loop local EOMs. It can be observed that the impact of dynamic interactions brought about by the MTDC network on those three local EOMs of the NETS was not very significant, though the closed-loop local EOMs were “driven” towards the right slightly by the closeness of λDCi. Results shown in Fig. 7.10 confirmed that the damping degradation of inter-area EOM of concern was caused by the OLMR, λDCi  λACi. In addition, condition of the OLMR can be dismissed by tuning the parameters of DC voltage controller of VSC1.

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . .

7.1.2.3

261

Numerical Evaluation of Connections Between the OLMR and Near Strong Modal Resonance

Nearness of two closed-loop oscillation modes can be described by the mode-toparameter sensitivity: When variation of a parameter causes two closed-loop modes moving towards each other, the mode-to-parameter sensitivity increases towards the infinity [3]. As infinite mode-to-parameter sensitivity is not practical, two closedloop modes cannot be identical in any practical system, rather than get close to each other, but never be in coincidence on the complex plane. Two closed-loop modes getting close to each other will move away from each other at a point of their proximity to each other. This is the point on the complex plane when the near strong modal resonance occurs [3]. Hence, at the point of near strong modal resonance, the mode-to-parameter sensitivity reaches the maximum. In order to numerically demonstrate and evaluate the connections between the near strong modal resonance and the OLMR, PI gains of the DC voltage controller of VSC1 were initially set to be Kdcp ¼ 0.21, Kdci ¼ 1.95. Then, integral gain was changed from Kdci ¼ 1.95 to Kdci ¼ 3.5. Trajectory of λDCi on the complex plane with changes of Kdci is displayed by dashed curve in Fig. 7.11, when the wind power penetration was Pw0 ¼ 16 p. u. In Fig. 7.11, solid curves are the trajectories of λ DCi . From Fig. 7.11, it can be seen that with closed-loop oscillation modes, b λ ACi and b the increase of Kdci, λDCi moved towards λACi firstly. Around the points indicated by hollow circles, λDCi almost overlapped λACi and then moved away from λACi. Between point A and E on the dashed trajectory of λDCi, closed-loop modes started moving away from open-loop modes, as being caused by the closeness of λDCi and λACi. Hence, it was identified that the OLMR occurred between point A and E. Strongest OLMR happened at point C on the dashed trajectory of λDCi when two open-loop modes almost overlapped. Corresponding positions of closed-loop λ DCi , are indicated by filled circles on the solid curves. They were modes, b λ ACi and b b λ DCi were furthest from each other on the complex plane as caused by when λ ACi and b Fig. 7.11 Trajectories of open-loop and closed-loop oscillation modes

Imaginary axis (rad/s) 4.5

E Ki_vdc1 =2.7

D 4 D

C λˆDCi

λDCi

3.5

λACi B

B 3 -0.6

A -0.5

-0.3

λˆACi

Ki_vdc1 =2.15 -0.1

0.1 Real axis

262

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Fig. 7.12 Variations of | hp11(λACi)|

⏐hpll(lACi)⏐

110

60

10

D 3

E

Kdci

4

0.5 Participation factors of MTDC

Fig. 7.13 Variations of the PFs of sum of state variables of the MTDC network

AB C

0.25

VSC-1

Others 0

A

B

C

D

3

Kdci

E

the OLMR. For convenience of discussion, point C is called the point of strongest OLMR. Figures 7.12 and 7.13 respectively show the variation of |hp11(λACi)| and the PFs of summation of all the state variables of VSC1 and rest of the MTDC network for the closed-loop EOM, b λ ACi . From Figs. 7.12 and 7.13, it can be seen that the closer the λDCi was to λACi, the bigger |hp11(λACi)| and the PFs were; thus the stronger the dynamic interactions between the MTDC and AC system were. At point C of strongest OLMR, |hp11(λACi)| and the PFs reached the peak, indicating strongest dynamic interactions.

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . . Fig. 7.14 Variations of mode-to-parameter sensitivity

263

⏐ΔlˆACi⏐⁄ΔKdci

1.5

1

0.5

0 A

B

Variations of mode-to-parameter sensitivity,

C

D

3

Kdci

   b  Δλ ACi 

,with increase of Kdci are ΔKdci displayed in Fig. 7.14. It can be observed that the sensitivity achieved the peaks at point B and D, which are the points when the closed-loop EOM, b λ ACi , was closest to b λ DCi . They were the points of near strong modal resonance as shown in Fig. 7.11.

7.1.2.4

Strong Dynamic Interactions When the DC Voltage Droop Control Was Implemented

When the DC droop voltage control was implemented by VSC1 and VSC2, increase of wind power penetration from Pw0 ¼ 10 p.u to Pw0 ¼ 16 p.u to cause strong dynamic interactions between the MTDC network and the NETS is shown by Fig. 7.15. In Fig. 7.15, arrows indicate the direction of increase of wind power penetration. From Fig. 7.15, it can be seen that the closeness of open-loop mode of the MTDC network, λDCi, to the open-loop inter-area EOM of concern, λACi, led to the small-signal angular instability of the NETS. λ DCi are presented in Computational results of participation factors for b λ ACi and b Fig. 7.16 when Pw0 ¼ 10 p.u and Pw0 ¼ 16 p.u respectively. From Fig. 7.16, it can be seen that strong dynamic interactions occurred between VSC1, VSC2 and the NETS when Pw0 ¼ 16 p.u. Further confirmation from non-linear simulation is presented in Fig. 7.17. Significant variations of active power output from VSC1 and VSC2 were observable. The integrated MTDC/NETS became unstable as caused by strong dynamic interactions between the MTDC network and the NETS. Participation factors of all the state variables of VSC1 and VSC2 for b λ ACi were calculated and compared. It was found that state variables of the DC voltage control systems of both VSC1 and VSC2 participated in b λ ACi mostly when the OLMR occurred. In order to dismiss the condition of the OLMR, parameters of the DC voltage controllers of VSC1 and VSC2 were tuned respectively when Pw0 ¼ 16 p.u.

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

264

Fig. 7.15 The OLMR occurred at a high level of wind penetration (DC voltage droop control)

Imaginary axis (rad/s) 3.7 DC voltage droop control

3.6

3.5

λDCi

λˆACi

λˆDCi B λACi

3.4 -0.6

-0.5

-0.3

Pwo=10 pu Pwo=16 pu

Pwo=10 pu Pwo=16 pu 0.8 Participation factors

0.4 Participation factors

0.1 Real axis

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3 4 5 6 7 8 Generator number

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ˆ ACi PFs for λ

1 2 3 4 5 VSC number

0

1 2 3 4 5 6 7 8 Generator number

9 10

1 2 3 4 5 VSC number

ˆ DCi PFs for λ

Fig. 7.16 Participation factors (PFs) for b λ ACi and b λ DCi (DC voltage droop control)

Movements of λDCi on the complex plane with the parameters tuning are shown in Fig. 7.18. From Fig. 7.18, it can be seen that when the parameters of the DC voltage controller of either VSC1 or VSC2 were tuned, λDCi moved away from the openloop inter-area EOM of concern, λACi. Subsequently, the closed-loop inter-area EOM moved back towards λACi and its damping was improved. This confirmed that damping degradation of inter-area EOM of concern caused by the OLMR can be dismissed by tuning parameters of the DC voltage controllers of the VSCs which took part in the droop voltage control. From Fig. 7.18, it can be seen that the OLMR occurred between λDCi and other local EOMs of the NETS when the parameters of the DC voltage controller of either VSC1 or VSC2 were tuned. However, impact of the OLMR between λACi and those local EOMs was not significant.

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . . 0.15

Pwo=10 pu Pwo=16 pu

Active power output from VSC2 (pu)

Active power output from VSC1 (pu)

0.4

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Fig. 7.17 Non-linear simulation (DC voltage droop control)

a

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DC voltage droop control

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DC voltage droop control

6

6 Trajectories of closedloop local EOMs

Trajectories of closedloop local EOMs

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Fig. 7.18 Modal trajectories when parameters of VSC1 and VSC2 were tuned (DC voltage droop control). (a) Parameters of the DC voltage controller of VSC1 were tuned. (b) Parameters of the DC voltage controller of VSC2 were tuned

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

266

7.1.2.5

Example 7.2—Test with the CIGRE Five–terminal DC Network

Figure 7.19 shows the configuration of the NETS integrated with the CIGRE fiveterminal DC network [11]. Data of the CIGRE MTDC network is presented in Appendix 7.2. The DC network adopted the master-slave control scheme with VSC1 using the DC voltage control, VSC2 and VSC3 using the active power control. VSC4 and VSC5 were connected separately to two wind farms. The total wind power penetration was 16 p.u. In this subsection, test results to numerically evaluate the connections between the near strong modal resonance and the OLMR are presented. In the test, PI gains of the DC voltage controller of VSC1 were varied with Kdcp ¼ KpiKdci, Kpi ¼ 0.15. The integral gain was changed from Kdci ¼ 2 to Kdci ¼ 5. Trajectory of λDCi on the complex plane is displayed by dashed curve in Fig. 7.20, where solid curves are the corresponding trajectories of closed-loop oscillation λ DCi . Figures 7.21 and 7.22 are the variations of |hp11(λACi)| and modes, b λ ACi and b the PFs of all the state variables of VSC1 and the rest of the MTDC network for the closed-loop EOM, b λ ACi . Figure 7.23 shows the variations of mode-to-parameter  b  Δλ ACi  . sensitivity, ΔKdci From Fig. 7.20, it can be seen that the OLMR occurred between point A and E on the trajectory of λDCi. Whilst at point F on the trajectory of λDCi, dynamic interactions between the CIGRE five-terminal DC network and AC system were obviously weak. The near strong modal resonance happened at point B and D (see Fig. 7.23). 26 25 2

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Fig. 7.19 New England power system integrated with the CIGRE five-terminal DC system

7.1 Small-Signal Angular Stability of an AC Power System Affected by the. . . Fig. 7.20 Trajectories of open-loop and closed-loop oscillation modes with variation of Kdci

267

Imaginary axis (rad/s) 6 F 5 E D

4

λˆDCi

λACi C

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3 -1.0

Fig. 7.21 Variations of |hp11(λACi)| with changes of Kdci

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⏐hpll(lACi)⏐

110

60

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0.5 Participation factors of MTDC

Fig. 7.22 Variations of the PFs of the MTDC with changes of Kdci

λˆACi

B A

-0.5

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VSC-1

Others 0 A

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D 3

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E

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

268

Fig. 7.23 Variations of mode-to-parameter sensitivity with changes of Kdci

⏐ΔlˆACi⏐⁄ΔKdci

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1

0.5

0 A

B

C

D

3

Kdci

E

The strongest point of OLMR was C, when the damping degradation caused by the OLMR was the most. The closer the λDCi was to λACi, the stronger the dynamic interactions between the CIGRE five-terminal DC network and AC system were. At point C, the dynamic interactions were strongest. Confirmation from non-linear simulation is presented in Fig. 7.24 when the CIGRE five-terminal DC network/NETS operated at point C of strongest OLMR and point F of weak modal coupling. At 0.1 s of simulation, 80% load at node 10 in the NETS was lost. The lost load was recovered in 0.1 s. From Fig. 7.24, it can be observed that when the OLMR occurred, the active power output from VSC1 varied considerably, indicating strong dynamic interactions. The strong dynamic interactions caused growing low-frequency inter-area power oscillations.

7.2

Small-Signal Stability of an MTDC Network for the Wind Power Transmission Affected by the Dynamic Interactions Between the VSCs

7.2.1

Small-Signal Stability Analysis

7.2.1.1

Linearized Model When a VSC Implementing Active Power Control Is Considered

Figure 7.25 shows the configuration of an MTDC network for wind power transmission being connected to multiple SGs. The configuration is different to that of the MTDC/AC power system shown by Fig. 7.1 because in Fig. 7.25, there is no AC transmission grid connecting M SGs. Each SG is connected to one of the VSCs which are connected by the MTDC network. Hence, there are two types of dynamic interactions in the MTDC/AC power system shown by Fig. 7.25: (1) AC/DC dynamic interactions between a VSC and a SG; (2) DC/DC dynamic interactions

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

1 Active power output from VSC1 (pu)

Fig. 7.24 Confirmation from non-linear simulation of the NETS with the CIGRE five-terminal DC system

269

Weak dynamic interactions at F Open-loop modal coupling at C

0.5

0

-0.5

-1

0

2

4

6

8

10

8

10

Time(s) Weak dynamic interactions at F Open-loop modal coupling at C

Dd1–Dd10 (deg.)

8

4

0

-4

-8

0

2

4

6 Time(s)

between the VSCs via the MTDC network. For the simplicity of discussion, the MTDC/AC power system shown by Fig. 7.25 is referred to as the MTDC power system. In this subsection, the small-signal stability of the MTDC power system affected by the dynamic interactions between the VSCs is examined. In order to conduct the examination, closed-loop model of the MTDC power system needs to be established firstly. Consider the kth VSC (VSC-k) connected to the SG in the MTDC power system. Denote Pk þ jQk and Vk as the power output from and the magnitude of terminal voltage of VSC-k. State-space model of the kth SG (SG-k) can be written as

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

270

P1 + jQ1

Wind farms

V1

I MTDC1 I dc1

...

M T D C

C1

X f1

Vdc1

I d 1 + jI q1

Pk + jQk IMTDCk I dck

Ck

VSC-k

I dk + jI qk

...

PM + jQM

VdcM

SG-k

VM

I MTDCM I dcM

CM

Vk

X fk

Vdck

Network

SG-1

VSC-1

X fM VSC-M

I dM + jI qM

SG-M

Fig. 7.25 Configuration of a MTDC/AC power system

d ΔXk ¼ Ak ΔXk þ bpk ΔPk þ bqk ΔQk dt ΔVk ¼ ck T ΔXk þ dpk ΔPk þ dqk ΔQk

ð7:43Þ

where ΔXk is the vector of all the state variables of SG-k. From (7.43), transfer function model of SG-k is obtained to be ΔVk ¼ Gpk ðsÞΔPk þ Gqk ðsÞΔQk

ð7:44Þ

where Gpk(s) ¼ ckT(sI  Ak)1bpk þ dpk, Gqk(s) ¼ ckT(sI  Ak)1bqk þ dqk. Take the direction of the terminal voltage of VSC-k as that of d axis of d  q coordinate of VSC-k. Thus, Vdk ¼ Vk , Vqk ¼ 0 and ΔVdk ¼ ΔVk , ΔVqk ¼ 0

ð7:45Þ

where subscript d and q respectively indicates d and q component of a variable in the d  q coordinate of VSC-k. From (7.44), linearized active and reactive power output from VSC-k can be obtained to be ΔPk ¼ Idk0 ΔVk þ Vk0 ΔIdk ¼ Vdck0 ΔIdck þ Idck0 ΔVdck ΔQk ¼ Iqk0 ΔVk  Vk0 ΔIqk

ð7:46Þ

where subscript 0 denotes the value of a variable at the steady state. Linearized line current equations at the terminal of VSC-k are (7.18). Consider that VSC-k adopts the active power control. Configuration of vector control system of VSC-k is shown by Fig. 7.3. From (7.18) and Fig. 7.3, ΔIdk ¼ hpk ðsÞΔPk , ΔIqk ¼ hqk ðsÞΔQk

ð7:47Þ

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

+

G pk (s)

2

D (s) ΔVk

H pk (s)

3

+ VSC control closed loop

Δ Pk 1

+

1 Vdck 0



I dck 0

H qk (s)

1/ Ck s 5

Gqk (s)

271

ΔVdck

C (s)

4 ΔQk

MTDC closed loop



Δ I dck

+ 6

ΔI MTDCk

M (s) Fig. 7.26 Closed-loop interconnected model when VSC-k uses the active power control ω K ðsÞK ðsÞ

ω K ðsÞK ðsÞ

idk pk iqk qk where hpk ðsÞ ¼ X0fk sþω , hqk ðsÞ ¼ X0fk sþω . 0 Kidk ðsÞ 0 Kiqk ðsÞ

Substituting (7.47) into (7.46) ΔPk ¼ Hpk ðsÞΔVk , ΔQk ¼ Hqk ðsÞΔVk

ð7:48Þ

I

where Hpk ðsÞ ¼ 1VIk0dk0hpk ðsÞ , Hqk ðsÞ ¼ 1þVk0qk0 hqk ðsÞ. Linearized DC voltage equation of VSC-k is Ck

dΔVdck ¼ ΔIMTDCk  ΔIdck dt

ð7:49Þ

From (7.44), (7.46), (7.48) and (7.49), the closed-loop model of the MTDC power system is obtained and shown by block diagram of Fig. 7.26. In Fig. 7.26, the VSC control closed loop on the left is depicted by (7.44) and (7.48). From (7.46), ΔPk  Idck0ΔVdck ¼ Vdck0ΔIdck, which together with (7.49) describes the upper loop D(s) on the right in Fig. 7.26; whilst dynamics of the rest of the MTDC power system excluding the dynamics of VSC-k and SG-k are expressed by transfer function M(s) as ΔIMTDCk ¼ M(s)ΔVdck. The closed-loop model of Fig. 7.26 is derived for the VSC which adopts the active power control. In the derivation, the entire MTDC power system is divided into two parts: (1) VSC-k and SG-k; (2) the rest of MTDC power system. The division is reflected at point ⑤ and ⑥ in Fig. 7.26. The part of rest of MTDC power system is modelled below point ⑤ and ⑥, above which is the part of SC-k and SG-k. From Fig. 7.26, it can be seen that there are two closed loops in the model. The VSC control closed loop reflects the AC/DC dynamic interactions between SG-k and the control system of VSC-k. Dynamic interactions between the dynamics of VSC-k and the rest of MTDC power system are through the MTDC closed loop in Fig. 7.26. It can be noted that the dynamic connection between those two closed loops is unidirectional at point ①.

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

272

7.2.1.2

Linearized Model When a VSC Implementing DC Voltage Control or DC Voltage Droop Control Is Considered

Consider that VSC-k adopts the DC voltage control. Configuration of DC voltage control is shown by Fig. 7.2. Comparing Fig. 7.2 and Fig. 7.3, it can be seen that in two types of converter control configurations, only the input to the DC voltage and active power control outer loop is different. However, this difference changes the configuration of closed-loop model as to be demonstrated as follow. From (7.18) and Fig. 7.2, ΔIdk ¼ hpk ðsÞΔVdck , ΔIqk ¼ hqk ðsÞΔQk

ð7:50Þ

Substituting (7.50) into (7.46), ΔPk ¼ Idk0 ΔVk þ Vk0 hpk ðsÞΔVdck , ΔQk ¼ Hqk ðsÞΔVk

ð7:51Þ

From (7.44), (7.46), (7.49) and (7.51), the closed-loop model of the MTDC power system can be derived and is shown by the block diagram of Fig. 7.27. Comparing Figs. 7.26 and 7.27, it can be seen that in Fig. 7.27, there is an additional VSC closed loop, which reflects the dynamic interactions between the control system of VSC-k and the rest of MTDC power system via the MTDC network. The additional VSC closed loop is formed by (7.51), which indicates that when the DC voltage control is implemented by VSC-k, variation of active power output from VSC-k is affected not only by ΔVk as it is in the case of active power control, but also by ΔVdck. The closed-loop model of Fig. 7.27 is derived for the VSC which adopts the DC voltage control. Figures 7.26 and 7.27 show two models for the same MTDC power system shown by Fig. 7.25. Up to the VSC being considered, the model can be that shown by either 7.26 or Fig. 7.27. For example, consider that the MTDC network implements the master-slave control where VSC-1 adopts the DC voltage control and all the other VSCs use the active power control. When VSC-1 is selected to derive the closed-loop model of the MTDC power system, the derived model is shown by Fig. 7.27. If any of other VSCs using the active power control is chosen, the derived model is shown by Fig. 7.26.

+

G pk (s)

2

ΔVk 3

+ VSC control closed loop

Gqk (s)

D (s) I dk 0

ΔPk 1

+

+

H qk (s)

I dck 0

VSC closed loop

Vk 0h pk (s) 4 ΔQk C (s)

1 Vdck 0



1/ Ck s 5

ΔVdck

MTDC closed loop

M (s) Fig. 7.27 Closed-loop model when VSC-k uses DC voltage control

Δ I dck – + 6

ΔI MTDCk

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

273

Power control outer loop

Current control inner loop DC voltage droop control

PWM Reactive power control loop

Fig. 7.28 Converter control system of VSC-k with DC voltage droop control

Finally, consider the case that the DC voltage droop control is implemented by VSC-k. In this case, input to the outer loop in Fig. 7.2 is changed  DC voltage  control ref ref from Vdck  Vdck to Kdpk Pkref  Pk þ Vdck  Vdck , where Kdpk is the droop coefficient. Configuration of the reactive power control of VSC-k remains unchanged. Figure 7.28 shows the configuration of the DC voltage droop control implemented by VSC-k. In case of DC voltage droop control, (7.50) becomes   ΔIdk ¼ hpk ðsÞ Kdpk ΔPk þ ΔVdck , ΔIqk ¼ hqk ðsÞ ΔQk ð7:52Þ Substituting (7.52) into (7.46) Idk0 ΔVk þ Vk0 hpk ðsÞΔVdck  1  Vk0 hpk ðsÞKdpk  ð Þ ¼ Dk s Idk0 ΔVk þ Vk0 hpk ðsÞΔVdck ΔQk ¼ Hqk ðsÞΔVk ΔPk ¼

ð7:53Þ

Similar to the case of VSC-k with the DC voltage control, the closed-loop model of the MTDC power system when VSC-k with the DC voltage droop control is obtained and shown by the block diagram of Fig. 7.29, where Dk(s) ¼ (1  KdpkVk0 gpk(s))1. The VSC closed loop is formed by (7.53).

7.2.1.3

Small-Signal Stability of the MTDC Network Affected by the OLMR Between the VSCs

A simpler representation of block diagram for Fig. 7.26 is shown by Fig. 7.30. From Fig. 7.30, it can be seen that there are three open-loop subsystems in the closed-loop model. Transfer functions of those three open-loop subsystems are C(s), D(s) and M

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

274

+

2

G pk (s)

D (s)

ΔVk 3

+ VSC control closed loop

I dk 0

Dk (s)

+

1

Δ Pk

I dck 0

VSC closed loop

H qk (s)

1 Vdck 0



1/ Ck s 5

Vk 0h pk (s)

Gqk (s)

+

MTDC closed loop

ΔVdck

4 ΔQk C (s)

Δ I dck – + 6

Δ I MTDCk

M (s) Fig. 7.29 Closed-loop interconnected model when VSC-k uses the DC voltage droop control

Fig. 7.30 Simpler representations of closedloop interconnected model for Fig. 7.26

C (s)

D (s) MTDC closed loop

5

ΔVdck

M (s)

a

C (s)

6

VM (s)

Δ I MTDCk

VM (s)

b C (s)

D (s)

V (s) 5

ΔVdck VSC closed loop

MTDC closed loop

M (s)

V (s) 6

ΔI MTDCk

5

ΔVdck

MTDC closed loop

M (s)

6

Δ I MTDCk

VM (s)

Fig. 7.31 Simple representation of closed-loop interconnected model for Fig. 7.27 and 7.29

(s) respectively. Open-loop subsystems, D(s) and M(s), form a closed-loop subsystem VM(s) as shown in Fig. 7.30b. For the convenience of discussion, D (s) is referred to as the open-loop VSC subsystem and M(s) is named as the openloop subsystem of the RDCN (rest of MTDC network). If and only if both the openloop subsystem C(s) and the closed-loop subsystem VM(s) in Fig. 7.30b are stable, the MTDC power system is stable. Obviously, stability of the closed-loop subsystem VM(s) is determined by the dynamic interactions between the VSCs through the MTDC closed loop. A simpler representation of the block diagram for Figs. 7.27 and 7.29 is shown by Fig. 7.31. In Fig. 7.31b, transfer function of the open-loop VSC subsystem in the forward loop is V(s), which consists of C(s) and D(s). The open-loop subsystem of the RDCN in the feedback loop is still M(s). For the convenience of discussions, the transfer function of closed-loop system in Fig. 7.31b is still denoted as VM(s), which consists of V(s) and M(s). Hence, in both cases of Fig. 7.30 and 2.31, the impact of

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

275

dynamic interactions between the VSCs on the small-signal stability of the MTDC power system can be studied by examining the stability of the closed-loop system VM(s). According to the theory of stability of a closed-loop system, if poles of two interconnected open-loop subsystems move close to each other on the complex plane, dynamic interactions between two open-loop subsystems increase and the stability of the closed-loop interconnected system decreases. This has been demonstrated in Chaps. 5 and 6 when the VSWGs and the PLL are examined. For the convenience of discussions, the closeness of complex poles of two open-loop subsystems in Fig. 7.30 and 7.31 is still referred to as open-loop modal resonance (OLMR). Impact of OLMR on the stability of closed-loop system VM(s) in Fig. 7.30 and 7.31, i.e., the small-signal stability of the power system with the MTDC network, is examined as follows. Denote λm as an oscillation mode of the open-loop subsystem of the RDCN. It is a complex pole of M(s) in Fig. 7.30 and 7.31. Denote λv as an oscillation mode of open-loop VSC subsystem. It is a complex pole of either D(s) in Fig. 7.30 or V(s) in Fig. 7.31. The OLMR is when λm is close to λv on the complex plane, i.e., λv  λm. Since λm is related with the dynamics of other VSCs in the MTDC power system, λ v and b λ m as the oscillation λv  λm may be the OLMR between the VSCs. Denote b modes of the closed-loop system VM(s) in Figs. 7.30 and 7.31 corresponding to λv and λm respectively. Impact of OLMR between the VSCs on the small-signal λ m when stability of MTDC power system can be examined by estimating b λ v and b the OLMR occurs. b λ m are the solutions of the following characteristic equations of closed-loop λ v and b system VM(s) shown by Figs. 7.30 or 7.31. 1 ¼ DðsÞMðsÞ or 1 ¼ VðsÞMðsÞ

ð7:54Þ

Denote D ðsÞ ¼

dð s Þ vð s Þ m ðsÞ or VðsÞ ¼ , M ðsÞ ¼ ðs  λ v Þ ðs  λ v Þ ð s  λm Þ

ð7:55Þ

Under the condition of open-loop modal coupling, i.e., λv  λm, for b λ v it can have         2 b λ v  λv b λ v  λm ¼ d b λv m b λv  b λ v  λv or         2 b λ v  λv b λ v  λm ¼ v b λv m b λv  b λ v  λv λ v ! λv Thus, in the neighborhood of λv, i.e., b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b λ v  λv   vðλv Þmðλv Þ λ v  λv   dðλv Þmðλv Þ or b

ð7:56Þ

ð7:57Þ

276

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Transfer functions of open-loop subsystems can be written as DðsÞ ¼ or VðsÞ ¼

X Ri Rv þ ðs  λv Þ ðs  λ i Þ i¼1 i6¼v

M ðsÞ ¼

X Rj Rm   þ ð s  λm Þ s  λj

ð7:58Þ

j¼1 i6¼v

where λi are the poles of open-loop subsystem D(s) in Fig. 7.30 or V(s) in Fig. 7.31, λj are the poles of the open-loop RDCN subsystem M(s), Ri and Rj are the residues corresponding to λi and λj; Rv and Rm are the residues corresponding to λv and λm respectively. From (7.55) and (7.58), it can have v(λv)m(λv) ¼ RvRm. Hence, (7.57) becomes pffiffiffiffiffiffiffiffiffiffiffiffi b ð7:59Þ λ v  λv  Rv Rm Taking the derivation similar to that from (7.56) to (7.59) for b λ m , it can have pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi b ð7:60Þ λ m  λm  Rv Rm  λv  Rv Rm The above derived equations of (7.59) and (7.60) indicate that when the OLMR λ m , locate at positions occurs, the corresponding closed-loop complex poles, b λ v and b approximately opposite to each other in respect to the positions where λv  λm. λm, Hence, it is very likely that one of the closed-loop oscillation modes, either b λ v or b is on the right hand side of corresponding open-loop oscillation mode. This implies that the OLMR between the VSCs degrades the closed-loop stability of the MTDC power system. From (7.59) and (7.60), it can be seen that either b λ v or b λ m may be in the right half of complex plane if  pffiffiffiffiffiffiffiffiffiffiffiffi  ð7:61Þ real part of Rv Rm  > jreal part of λv or λm j That is the condition under which the OLMR between the VSCs may lead to the instability of the MTDC network. From Fig. 7.26, it can be seen that for the VSC adopting the active power control, the open-loop VSC subsystem D(s) in the closed-loop system VM(s) in Fig. 7.30 is a first-order system and is not of any oscillation mode. Thus, there is no possibility that the VSC takes part in the OLMR to cause the instability of the MTDC power system. Hence, when the MTDC network adopts the master-slave control, one VSC uses the DC voltage control and all the other VSCs adopt the active power control. There is no OLMR between two or more VSCs to happen. Thus, it can be concluded that in the MTDC network with the master-slave control, no OLMR between the VSCs can occur to cause the instability of the MTDC power system.

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

277

If the DC voltage droop control is implemented by the MTDC network, there are at least two VSCs adopting the DC voltage droop control. From Fig. 7.31, it can be seen that the open-loop VSC subsystem, V(s), may have an oscillation mode because the VSC closed loop, C(s), is a part of subsystem V(s). Hence, it is possible that the OLMR may occur between the VSCs adopting the DC voltage droop control to lead to the instability of the MTDC network. Therefore, it can be concluded that when the DC voltage droop control is implemented by the MTDC network, instability risk of MTDC power system increases because of the possibility of OLMR between the VSCs.

7.2.2

Example MTDC Power Systems

7.2.2.1

Example 7.3—An Eight-Terminal DC Network

Fig. 7.32 shows the configuration of an example MTDC power system. Parameters of the MTDC and AC system given in [4, 12–14] were used. The MTDC network is of eight terminals. VSC-k (k ¼ 1,2,3,4) operated with unity power factor and their active power output at steady state was 3.43 p.u, 1 p.u, 0.5 p.u and 0.5 p.u respectively. Each of other four VSCs is connected to a wind farm represented by a PMSG. Detailed 14th-order model and parameters of the PMSG recommended in [8] were used. Data of the example MTDC network is given in Appendix 7.3. Tests were conducted to demonstrate and evaluate the impact of OLMR between the VSCs on the small-signal stability of example MTDC power system. In the tests, the master-slave control and the DC voltage droop control were implemented respectively by the MTDC network. When the master-slave control was implemented, VSC-1 adopted the DC voltage control and other three VSCs used r1 r8

x1

r7 200km 200kmr6 x6 x7

30km x 8

r9

x6 r8 8

r6

C2 40km VSC-2

r5

200km

x5 130km

P2 + jQ2

X f3

x7 200km

V1

C1 r2 P + jQ1 VSC-1 1 V2 30km x2 Xf2 A r3 x3

40km x 9

30km x

X f1

200km

V3

P3 + jQ3 r7 r4 C3 V4 x4 30km VSC-3

SG-1

SG-2

SG-3

Xf4

C4 VSC-4 Fig. 7.32 Configuration of an example AC/MTDC power system

P4 + jQ4

SG-4

278

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

the active power control. When the DC voltage droop control was implemented, VSC-1 and VSC-2 adopted the DC voltage droop control. Droop coefficient was 0.01. VSC-3 and VSC-4 still used the active power control. With a partition at point A in Fig. 7.30, the closed-loop models of the example MTDC power system were established. The open-loop VSC subsystem in the forward loop in Figs. 7.30 and 7.31 was comprised of VSC-2 and SG-2 with ΔIMTDC2 and ΔVdc2 being the input and output variable respectively. Hence, the open-loop state-space model of the open-loop VSC subsystem was d ΔXv ¼ Av ΔXv þ bv ΔIMTDC2 dt ΔVdc2 ¼ cv T ΔXv þ dv ΔIMTDC2

ð7:62Þ

where ΔXv is the vector of all the state variables of VSC-2 and SG-2. When the MTDC network implemented the master-slave control, VSC-2 used the active power control. Thus, transfer function of the VSC subsystem was Vs ðsÞ ¼ cv T ðsI  Av Þ1 bv þ dv

ð7:63Þ

When the MTDC network implemented the DC voltage droop control, VSC-2 used the DC voltage droop control. Hence, transfer function of the VSC subsystem was VðsÞ ¼ cv T ðsI  Av Þ1 bv þ dv

ð7:64Þ

The open-loop RDCN subsystem in the feedback loop in Figs. 7.30 and 7.31 was comprised of the rest of example MTDC power system with ΔVdc2 and ΔIMTDC2 being the input and output variable respectively. Following state-space model of the open-loop RDCN subsystem was established d ΔXm ¼ Am ΔXm þ bm ΔVdc2 dt ΔIMTDC2 ¼ cm T ΔXm þ dm ΔVdc2

ð7:65Þ

where ΔXm is the vector of all the state variables of the rest of example MTDC power system excluding those of VSC-2 and SG-2. Thus, transfer function of the open-loop RDCN subsystem is MðsÞ ¼ cm T ðsI  Am Þ1 bm þ dm

ð7:66Þ

Complex poles of the open-loop VSC and RDCN subsystems were calculated respectively from the open-loop state matrix As in (7.62) and Am in (7.65). They are the complex poles of transfer functions, Vs(s) in (7.63) or (7.64) and M(s) in (7.66). Tests were focused on the OLMR between λv and λm. λv was an open-loop oscillation mode of the VSC subsystem, Vs(s) or V(s). λm was an open-loop complex pole of RDCN subsystem M(s). The participation factors (PFs) for λv and λm were calculated from (7.62) and (7.65). Computational results of the PFs indicated that λm

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

279

and λv were associated with the current control inner loops of VSC-1 and VSC-2 respectively. Initially, parameters of control systems of VSC-1 and VSC-2 were all same except the integral gains of current control inner loop, Kiik, k ¼ 1, 2. For VSC-1, Kii1 ¼ 0.13 p. u; and for VSC-2, Kii2 ¼ 0.1p. u. To find the OLMR between VSC-1 and VSC-2, value of integral gain of current control inner loop of VSC-1 was varied from Kii1 ¼ 0.13 p. u to Kii1 ¼ 0.7 p. u. With the variation of Kii1, open-loop oscillation mode, λm, moved on the complex plane. Trajectories of movement are displayed in Figs. 7.33 and 7.34 as dashed curves. In Figs. 7.33 and 7.34, positions of λv on the complex plane are indicated by hollow circles. From Figs. 7.33 and 7.34, it can be seen that at point ① and ② on the dashed curves indicated by hollow circles when Kii1 ¼ 0.1 p. u, λm was close to λv. Thus, ① and ② may be the points of OLMR. 230

Imaginary axis (rad./s.)

Fig. 7.33 Modal positions and trajectories of the example MTDC system (Master-slave control)

225

λm

220

λv

215

210 -6.35

-6.31

-6.33 Real axis (rad./s.)

260

Imaginary axis (rad./s.)

Fig. 7.34 Modal positions and trajectories of the example MTDC system (DC voltage droop control)

1

240

220

ˆλm

λm

λv

ˆλv

2

200

180 -9.5

-7

-2 -4.5 Real axis (rad./s.)

0.5

280

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

To examine whether the OLMR occurred at point ① and ②, residues corresponding to λv and λm were computed fromp(7.62) ffiffiffiffiffiffiffiffiffiffiffiffi and (7.65). When the MTDC network used the master-slave control, Rv Rm ¼ 0. This means that although at point ① in Fig. 7.33, λv  λm, closed-loop oscillation modes were estimated being equal to the open-loop oscillation modes. The closeness of openloop modes, i.e., λv  λm, affected little the stability of the example MTDC power system. This implied that λv in fact was not a complex pole of open-loop subsystem D(s) in the closed-loop system VM(s) in Fig. 7.30 and thus, the OLMR did not occur. This confirmed the analysis made in Subsect. 7.2.1.3 that λv was a complex pole of C (s) because VSC-2 adopted the active power control. When the DC voltage droop control was implemented by the MTDCpnetwork, ffiffiffiffiffiffiffiffiffiffiffiffi computational results of residues from (7.62) and (7.65) were Rv Rm ¼   ffiffiffiffiffiffiffiffiffiffiffi ffi p 2:11  j0:73. Thus, at point ② in Fig. 7.34, real part of Rv Rm  ¼ 2:11 >|real part of λv| ¼ 1.81. This indicated that at point ②, the OLMR happened. According to the estimation by (7.61), the OLMR at point ② may cause the small-signal instability of the example/MTDC power system. In order to confirm the conclusion obtained above from the modal analysis for the open-loop subsystems of (7.62) and (7.65), the state-space model of closed-loop MTDC power system was derived from (7.62) and (7.65) to be d ΔX¼AΔX dt

ð7:67Þ

where  T ΔX ¼ ΔXv T ΔXm T 2 bv dm cv T 6 Av þ 1  d d v m 6 A¼6 T 4 b d m v dm c v bm cv T þ 1  dm dv

3 bv dm dv cm T bv cm þ 1  dv dm 7 7 7 bm dv cm T 5 Am þ 1  dm dv T

Closed-loop oscillation modes corresponding to λv and λm were calculated from λ m . When the master-slave the closed-loop state matrix, A in (7.67), to be b λ v and b control was implemented by the MTDC network, trajectory of b λ v with variation of λ m was also in Kii1 was found to coincide completely that of λv in Fig. 7.33. b λ v and position of b λ m are not coincidence with λm in Fig. 7.33. Thus, trajectory of b b b displayed in Fig. 7.33. Therefore, computational results of λ v and λ m confirmed that the closeness of λv to λm on the complex plane at point ① in Fig. 7.33 caused no λ m , not affecting the small-signal stability of the example MTDC change of b λ v and b power system at all. Hence, point ① in Fig. 7.33 was not a point of OLMR between VSC-1 and VSC-2 when the master-slave control was implemented by the MTDC network with VSC-2 adopting the active power control. When the MTDC network implemented the DC voltage droop control, trajectories of closed-loop poles, b λ v and b λ m , with variation of Kii1 were calculated from the closed-loop state matrix A in (7.67) and are displayed by solid curves in Fig. 7.34.

1.2

Master-slave Voltage droop

Participation factors

Participation factors

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

0.6

1.2

281

Master-slave Voltage droop

0.6

0

0 PFv

PFm

The PFs for ˆλv

PFv

PFm

The PFs for ˆλm

Fig. 7.35 Computational results of the PFs

Positions of b λ v and b λ m corresponding to λv and λm at point ② are indicated by solid circles. Obviously, the OLMR between VSC-1 and VSC-2 at point ② caused the small-signal instability of the example MTDC system, confirming the estimation made previously by using (7.61). Computational results of the PFs for b λ v and b λ m corresponding to λv and λm at point ① and ② in Figs. 7.33 and 7.34 are presented in Fig. 7.35. In Fig. 7.35, PFv is the sum of the PFs of all the state variables of VSC-2 and PFm is the sum of the PFs of all the state variables of VSC-1. From Fig. 7.35, it can be seen that when the MTDC network implemented the master-slave control, VSC-1 participated only in b λ m and VSC-2 took part in only b λ v . Hence, there were no dynamic interactions between VSC-1 and VSC-2. When the MTDC network adopted the DC voltage droop λ m . This indicated the control, both VSC-1 and VSC-2 participated in both b λ v and b strong dynamic interactions between VSC-1 and VSC-2 when the OLMR occurred. Results of non-linear simulation are displayed in Fig. 7.36. At 0.1 s of simulation, 80% of the load at the AC terminal of VSC-1 was lost. The lost load was recovered in 0.1 s. In Fig. 7.36, solid curves are the simulation results when the MTDC network implemented the master-slave control. Dashed curves are the simulation results when the DC voltage droop control was implemented by the MTDC network. Parameters of control systems of VSC-2 were set such that the example MTDC power system operated at point ① and ② indicated in Figs. 7.33 and 7.34. From Fig. 7.36, it can be seen that when the master-slave control was used by the MTDC network, the example MTDC power system was stable. However, when the DC voltage control was implemented by the MTDC network, the oscillations occurred and the example MTDC power system was unstable. The instability was caused by the OLMR between VSC-1 and VSC-2 which used the DC voltage droop control. The oscillation frequency was 34.7 Hz, This was in consistence with the results of open-loop modal analysis presented in Fig. 7.34, where the OLMR occurred at point ② and the negatively damped closed-loop SSO mode was b λ v ¼ 0:17 þ j217:6ðrad:=s:Þ.

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

282

Variation of IMTDC1 (p.u)

0.6

DC voltage droop control Master slave control

0

Time (second)

-0.6 0

0.5

1

1.5

2

Variation of Vdc1 (p.u)

0.1 DC voltage droop control Master slave control 0

Time (second) -0.1 0

0.5

1

1.5

2

Variation of IMTDC2 (p.u)

0.2 DC voltage droop control Master slave control 0

Time (second)

-0.2 0

0.5

1

1.5

2

Variation of Vdc2 (p.u)

0.1 DC voltage droop control Master slave control 0

Time (second) -0.1

0

Fig. 7.36 Non-linear simulation

0.5

1

1.5

2

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

283

Tests presented above demonstrated and confirmed the following analytical conclusions made in Subsect. 7.2.1: (1) When the MTDC network uses the master-slave control, no OLMR between the VSCs may occur to degrade the small-signal stability of MTDC power system; (2) When the DC voltage droop control is used by the MTDC network, it is possible that the OLMR between the VSCs may occur to lead to the stability degradation of MTDC power system. In the worst case, instability may occur as caused by the dynamic interactions between the VSCs. The instability risk can be identified by using (7.61). Study cases of the example MTDC power system presented above indicated that when two VSCs adopting the DC voltage droop control are of the same control system parameters, open-loop oscillation modes associated with the VSCs’ control systems are very possibly close to each other on the complex plane. This was the case demonstrated in Fig. 7.34. Hence, same settings of control system parameters of the VSCs in the MTDC network should be avoided if those VSCs take part in the DC voltage droop control, because it may likely bring about the risk of OLMR between the VSCs to degrade the small-signal stability of the MTDC power system.

7.2.2.2

Example 7.4—The CIGRE Five-Terminal DC Network

Figure 7.37 shows the configuration of the CIGRE five-terminal DC network [11] connecting the wind farms with the SGs. The master-slave control and the DC voltage droop control were implemented respectively by the five-terminal DC network. When the master-slave control was used, VSC-1 adopted the DC voltage control and other two VSCs connected with the SGs used the active power control. When the DC voltage droop control was implemented, VSC-1 and VSC-2 adopted the DC voltage droop control. Droop coefficient was 0.01. VSC-3 still used the active power control. Data of example MTDC network is given in Appendix 7.2. 200km

r1

x1

X f1

C1 VSC-1

r2 r6 500km

r4 400km

x6

x5

200km

P1 + jQ1

SG-1

500km

x2 Xf2

x4 r3

r5

V1

C2

x3

200km X f3

C3 VSC-3

P3 + jQ3

VSC-2

P2 + jQ2

V3

SG-3

Fig. 7.37 Configuration of power system with CIGRE five-terminal DC network

V2

SG-2

284

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Initially, parameters of control systems of VSC-1 and VSC-2 were set to be same except the integral gains of current control inner loops, Kiik, k ¼ 1, 2. For VSC-1, Kii1 ¼ 0.191 p. u; and for VSC-2, Kii2 ¼ 0.159 p. u. To examine the existence of the OLMR between VSC-1 and VSC-2, value of integral gain of current control inner loop of VSC-1 was varied from Kii1 ¼ 0.191 p. u to Kii1 ¼ 0.127 p. u. Modal trajectories similar to those presented in Figs. 7.33 and 7.34 were calculated and are shown by Figs. 7.38 and 7.39. From 7.38, it can be seen that when the master-slave control was used, there was no OLMR to cause the stability degradation of the CIGRE five-terminal DC power system. Figure 7.39 indicated the occurrence of OLMR when the DC voltage droop control was implemented by the CIGRE fiveterminal DC network. The OLMR led to the loss of the stability of the CIGRE fiveterminal DC power system. The negatively damped closed-loop SSO mode was b λ v ¼ 0:06 þ j213:7ðrad:=s:Þ. 230

Imaginary axis (rad./s.)

Fig. 7.38 Modal positions and trajectories of the CIGRE five-terminal DC power system (Master-slave control)

220 λv

λm 210

200

190 -9.27

230

Imaginary axis (rad./s.)

Fig. 7.39 Modal positions and trajectories of the AC/CIGRE five-terminal DC power system (DC voltage droop control)

-9.07

-9.17 Real axis (rad./s.)

220 λv

λm 210

ˆ λ v

ˆλ m

200

190 -11

-8

-5 Real axis (rad./s.)

-2

1

7.2 Small-Signal Stability of an MTDC Network for the Wind Power. . .

285

To confirm the results of modal computation presented in Figs. 7.38 and 7.39, non-linear simulation was conducted. Results of non-linear simulation are displayed in Fig. 7.40. At 0.1 s of simulation, 80% of the load at the AC terminal of VSC-1 was lost. The lost load was recovered in 0.1 s. It can be seen that growing oscillations occurred when the DC voltage droop control was implemented.

Variation of IMTDC1 (p.u)

0.2

DC voltage droop control Master slave control

0

Time(second) -0.2

0

0.75

1.5

1.75

3

Variation of Vdc1 (p.u)

0.1 DC voltage droop control Master slave control 0

-0.1

Time(second) 0

0.75

1.5

1.75

3

Variation of IMTDC2 (p.u)

0.3 DC voltage droop control Master slave control 0

Time(second) -0.3

0

0.75

1.5

1.75

3

Variation of Vdc2 (p.u)

0.1 DC voltage droop control Master slave control 0

Time(second) -0.1

0

0.75

1.5

1.75

Fig. 7.40 Non-linear simulation (AC/five-terminal DC power system)

3

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

286

7.3

Method of Open-loop Modal Analysis to Examine the Impact of Dynamic Interactions Introduced by a Selected VSC Control on the Small-Signal Stability of an MTDC/AC Power System

7.3.1

Method of Open-Loop Modal Analysis

7.3.1.1

Closed-Loop Model When the DC Voltage Control Is Selected

Consider the MTDC/AC power system shown by Fig. 7.1, where VSC-1 adopts the DC voltage control and VSC-k uses the active power control, with k ¼ 2, 3, . . .M. Figure 7.2 shows the configuration of the control system of VSC-1, which is comprised of the DC voltage control loop and reactive power control loop. In this subsection, the DC voltage control loop of VSC-1 is selected to derive a closed-loop model of the MTDC/AC power system. As indicated in Fig. 7.1, the power output from VSC-1 is P1 þ jQ1. The terminal voltage of VSC-1 expressed in its d  q coordinate is Vd1 þ jVq1. Normally, the direction of the terminal voltage of the VSC is chosen as that of d axis of the d  q coordinate, such that ΔVd1 ¼ ΔV1 and ΔVq1 ¼ 0, where V1 is the magnitude of the terminal voltage of VSC-1. The dynamic equations of the AC output current from VSC-k expressed in the d  q coordinate are given in (7.18). From Fig. 7.2,   ref ref Id1 ¼ Kdc ðsÞ Vdc1  Vdc1 ð7:68Þ  ref  ref  Vd1 þ X f1 Iq1 ¼ Kid ðsÞ Id1  Id1 Vcd1 where Vdc1 is the DC voltage of VSC-1, C1 is the DC capacitance of VSC-1, Idc1 is the DC current of VSC-1. Combining (7.68) and (7.18) and ignoring the very fast transient of the PWM ref algorithm such that Vcd1 ¼ Vcd1 , X f1

 ref  dId1 ¼ ω0 Kid ðsÞ Id1  Id1 dt

From (7.69) Id1 ¼

ω0 Kid ðsÞ I ref X f1 s þ ω0 Kid ðsÞ d1

ð7:69Þ

ð7:70Þ

Active power balance equation on the DC and AC side of VSC-1 is Vdc1 IMTDC1  C1 Vdc1

dVdc1 ¼ P1 dt

ð7:71Þ

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

287

Linearization of the first equation of (7.68), (7.70), and (7.71) is dΔVdc1 ¼ IMTDC10 ΔVdc1 þ Vdc10 ΔIMTDC1  ΔP1 dt ω0 Kid ðsÞ ΔI ref ¼ Kdc ðsÞΔVdc1 , ΔId1 ¼ X f1 s þ ω0 Kid ðsÞ d1

C1 Vdc10 ref ΔId1

ð7:72Þ

From (7.72) ΔId1 ¼ H0 ðsÞΔVdc1 ΔVdc1 ¼

Vdc10 1 ΔIMTDC1  ΔP1 C1 Vdc10 s  IMTDC10 C1 Vdc10 s  IMTDC10

ð7:73Þ

id ðsÞKdc ðsÞ where, H0 ðsÞ ¼ Xω0f1Ksþω . 0 Kid ðsÞ

Since Vq1 ¼ 0, Vd1 ¼ V1, ΔP1 ¼ Id10 ΔV1 þ Vd10 ΔId1

ð7:74Þ

ΔP1 ¼ Hv1 ðsÞΔV1 þ Hi1 ðsÞΔIMTDC1

ð7:75Þ

From (7.73) and (7.74),

where Hv1 ðsÞ ¼

Id10 ðsC1 Vdc10  IMTDC10 Þ , ðsC1 Vdc10  IMTDC10 Þ þ Vd10 H0 ðsÞ

Hi1 ðsÞ ¼

Vd10 Vdc10 H0 ðsÞ ðsC1 Vdc10  IMTDC10 Þ þ Vd10 H0 ðsÞ

From (7.75), it can be seen that ΔV1 and ΔIMTDC1 are the input variables to the selected DC voltage control of VSC-1 and ΔP1 is the output variable from the selected DC voltage control of VSC-1. This implies that to the rest of MTDC/AC power system, excluding the selected DC voltage control system of VSC-1, ΔV1 and ΔIMTDC1 are the output variables and ΔP1 is the input variable. From (7.75), the state-space model of the DC voltage control system of VSC-1 can be written as d ΔXp ¼ Ap ΔXp þ bd ΔIMTDC1 þ bv ΔV1 dt ΔP1 ¼ cp T ΔXp þ dd ΔIMTDC1 þ dv ΔV1

ð7:76Þ

where ΔXp is the vector of all the state variables of the DC voltage control system of VSC-1. Since the input variable is ΔP1 and output variables are ΔV1 and ΔIMTDC1 for the rest of the MTDC/AC power system, excluding the DC voltage control system of VSC-1, the state-space model of the rest of the MTDC/AC power system can be written as

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

288

d ΔXs ¼ As ΔXs þ bp ΔP1 dt ΔIMTDC1 ¼ cd T ΔXs þ d1 ΔP1 , ΔV1 ¼ cv T ΔXs þ d2 ΔP1

ð7:77Þ

where ΔXs is the vector of all the state variables of the rest of the MTDC/AC system, excluding those in the DC voltage control system of VSC-1. From (7.77), the following transfer function model of the rest of the MTDC/AC system can be obtained ΔIMTDC1 ¼ Gi1 ðsÞΔP1 , ΔV1 ¼ Gv1 ðsÞΔP1

ð7:78Þ

where Gi1(s) ¼ cdT(sI  As)‐1bp þ d1, Gv1(s) ¼ cvT(sI  As)1bp þ d2. From (7.75) and (7.78), a closed-loop model with the DC voltage control system of VSC-1 in the feedback loop can be established, and is shown in Fig. 7.41. The state-space model of the closed-loop interconnected system can be obtained from (7.76) and (7.77) as d ΔX ¼ AΔX ð7:79Þ dt where. 2

 ΔX ¼ ΔXp T

ðd1 bd þ d2 bv Þcp T 6 Ap þ 1  d d  d d d 1 v 2 6 A¼6 T 4 b p cp 1  dd d 1  dv d2

Fig. 7.41 Closed-loop interconnected model with the DC voltage control loop in the feedback loop

T ΔXs T 3 ðd1 bd þ d2 bv Þðdd cd T þ dv cv T Þ bd cd T þ bv cv T þ 7 1  dd d1  dv d2 7   7 5 dd bp cd T þ dv bp cv T As þ 1  dd d1  dv d2

Subsystem : rest of ΔP1 power system

Gi1 (s)

ΔIMTDC1

Gv1 (s)

ΔV1

Hv1 (s)

Subsystem: DC voltage control of VSC-1

Hi1 (s)

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

7.3.1.2

289

Closed-Loop Model When the Active or Reactive Power Control Is Selected

VSC-k, k ¼ 2, 3, . . .M use the active power control, configuration of which is shown by Fig. 7.3. The active power control system of VSC-k can be selected to derive a closed-loop model similar to that shown by Fig. 7.41 as follows. When the active power control system of VSC-k is considered, (7.68) becomes   ref Idk ¼ Kpk ðsÞ Pk  Pkref ð7:80Þ  ref  ref  Vdk þ X fk Iqk ¼ Kidk ðsÞ Idk  Idk Vcdk From Linearized (7.18) and (7.80), ΔIdk ¼

ω0 Kidk ðsÞ ref ΔI ref , ΔIdk ¼ Kpk ðsÞΔPk X fk s þ ω0 Kidk ðsÞ dk

ð7:81Þ

From (7.81), ΔIdk ¼ H0k ðsÞΔPk

ð7:82Þ

0 Kidk ðsÞ where H0k ðsÞ ¼ X fkωsþω Kpk ðsÞ. 0 Kidk ðsÞ

Since Vqk ¼ 0, Vdk ¼ Vk, ΔPk ¼ Idk0 ΔVk þ Vdk0 ΔIdk

ð7:83Þ

From (7.82) and (7.83), ΔPk ¼ Hpk ðsÞΔVk , Hpk ðsÞ ¼

Idk0 1  Vdk0 H0k ðsÞ

ð7:84Þ

Transfer function model of the selected active power control system of VSC-k is (7.84). From (7.84), it can be seen that ΔVk is the input signal of the selected active power control system of VSC-k and the output variable of the rest of the MTDC/AC system, and ΔPk is the output variable of the selected active power control system of VSC-k and the input variable of the rest of MTDC/AC system. Hence, following state-space model of the selected active control system of VSC-k can be obtained. d ΔXp ¼ Ap ΔXp þ bpv ΔVk dt ΔPk ¼ cp T ΔXp þ dpv ΔVk

ð7:85Þ

where ΔXp is the vector of all the state variables of the selected active control system of VSC-k. The state-space model for the rest of the MTDC/AC system in Fig. 7.1, excluding the active control system of VSC-k, can be written as

290

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

d ΔXsp ¼ Asp ΔXsp þ bsp ΔPk dt ΔVk ¼ cvp T ΔXsp þ dsp ΔPk

ð7:86Þ

where ΔXsp is the vector of all the state variables of the rest of the MTDC/AC system excluding those in the selected active control system of VSC-k. From (7.884), the following transfer function model of the rest of the MTDC/AC system can be obtained  1 ΔVk ¼ Gpk ðsÞΔPk , Gpk ðsÞ ¼ cvp T sI  Asp bsp þ dsp ð7:87Þ From (7.84) and (7.87), the closed-loop model is obtained, and shown in Fig. 7.42. The reactive power control of VSC-k, k ¼ 1, 2, 3, . . .M can also be selected to derive a closed-loop model of the MTDC/AC power system similar to that shown by Fig. 7.42. The derivation is as follows. From Fig. 7.3,   ref Iqk ¼ Kqk ðsÞ Qk  Qkref  ð7:88Þ ref ref  Vqk  X fk Idk ¼ Kiqk ðsÞ Iqk  Iqk Vcqk

From linearized (7.18) and (7.88), ref ΔIqk ¼ Kqk ðsÞΔQk , ΔIqk ¼

ω0 Kiqk ðsÞ ΔI ref X fk s þ ω0 Kiqk ðsÞ qk

ð7:89Þ

From (7.89), ΔIqk ¼ H1k ðsÞΔQk , H1k ðsÞ ¼

ω0 Kiqk ðsÞKqk ðsÞ X fk s þ ω0 Kiqk ðsÞ

ð7:90Þ

Since Vqk ¼ 0, Vdk ¼ Vk, linearized reactive power output equation from VSC-k can be obtained as Fig. 7.42 Closed-loop model with active power control system of VSC-k in the feedback loop

Subsystem: rest of the system ΔPk

Gpk (s)

ΔVk

Hpk (s) Subsystem: active power control of VSC-k

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

291

ΔQk ¼ Iqk0 ΔVk  Vdk0 ΔIqk

ð7:91Þ

From (7.90) and (7.91), ΔQk ¼ Hqk ðsÞΔVk , Hqk ðsÞ ¼

Iqk0 1 þ Vdk0 H1k ðsÞ

ð7:92Þ

The transfer function model of the selected reactive power control system of VSC-k is (7.92). From (7.92), it can be seen that ΔVk is the input variable of the selected reactive power control system of VSC-k and the output variable of the rest of the MTDC/AC system, ΔQk is the output variable of the selected reactive power control system of VSC-k and the input variable of the rest of the MTDC/AC system. Thus, the following state-space model of the selected reactive control system of VSC-k is obtained. d ΔXq ¼ Aq ΔXq þ bqv ΔVk dt ΔQk ¼ cq T ΔXq þ dqv ΔVk

ð7:93Þ

where ΔXq is the vector of all the state variables of the selected reactive control system of VSC-k. The state-space model for the rest of MTDC/AC system in Fig. 7.1, excluding the selected reactive control loop of VSC-k, can be written as d ΔXsq ¼ Asq ΔXsq þ bq ΔQk dt ΔVk ¼ cvq T ΔXsq þ dq2 ΔQk

ð7:94Þ

where ΔXsq is the vector of all the state variables of the rest of the MTDC/AC system, excluding those in the selected reactive control loop of VSC-k. From (7.94), the following transfer function model of the rest of the MTDC/AC system can be obtained  1 ΔVk ¼ Gqk ðsÞΔQk , Gqk ðsÞ ¼ cvq T sI  Asq bq þ dq2 ð7:95Þ From (7.92) and (7.95), the closed-loop model is derived, and is shown in Fig. 7.43. Fig. 7.43 Closed-loop model with reactive power control system of VSC-k in the feedback loop

Subsystem: rest of the system ΔQk

Gqk (s)

ΔVk

Hqk (s) Subsystem: reactive power control of VSC-k

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

292

7.3.1.3

Open-Loop Modal Analysis

Denote λti as an oscillation mode of the open-loop subsystem of the rest of the power system and λpk as a complex pole (oscillation mode) of the open-loop subsystem of the DC voltage control system of VSC-1. λti is a complex eigenvalue of the openloop state matrix As in (7.77). λpk is a complex eigenvalue of open-loop state matrix λ ti and b λ pk as the closed-loop oscillation modes corresponding to Ap in (7.76). Denote b λti and λpk, respectively. b λ ti and b λ pk are the complex eigenvalues of the closed-loop state matrix A in (7.79). When ΔP1 ¼ 0, the closed-loop system shown in Fig. 7.39 is open. Closed-loop and corresponding open-loop eigenvalues are the same. Hence, the differences λ ti between the closed-loop and corresponding open-loop oscillation modes, Δλti ¼ b λti and Δλpk ¼ b λ pk  λpk , are caused by ΔP1 6¼ 0, which is the physical exhibition of the AC/DC dynamic interactions between the DC voltage control system of VSC-1 and the rest of the MTDC/AC power system. Normally, the dynamic interactions are weak, such that ΔP1  0. The system shown in Fig. 7.41 should λ ti  λti  0 and Δλpk ¼ b λ pk  λpk  0. approximately be open such that Δλti ¼ b This implies that the weak dynamic interactions caused by the DC voltage control system of VSC-1 should normally affect little the oscillation modes in the MTDC/ AC system. However, under a special condition of open-loop modal resonance (OLMR), ΔP1 may become considerable, which is elaborated as follows. The special condition of OLMR is when λpk is close to λti on the complex plane, i.e., λpk  λti. Since λpk is a complex pole of Hi1(s) or/and Hv1(s) in Fig. 7.41, | Hi1(λpk)| ¼ 1 or/and |Hv1(λpk)| ¼ 1; thus |Hi1(λti)| or/and |Hv1(λti)| are significant when λpk  λti. From (7.75), it can be seen that ΔP1 may become considerable around the oscillation complex frequency λti. Hence, when λpk  λti, the dynamic interactions introduced by the DC voltage control system of VSC-1 may become λ ti strong. The impact of strong dynamic interactions caused by the OLMR is Δλti ¼ b λti and Δλpk ¼ b λ pk  λpk . The derivation of analytical expressions of Δλti and Δλpk may be very difficult to determine the exact impact of the dynamic interactions introduced by the DC voltage control system of VSC-1. However, an approximate estimation of Δλti and Δλpk can be derived as follows. From Fig. 7.41, the characteristic equation of the closed-loop system is obtained as 1 ¼ Hi1 ðsÞGi1 ðsÞ þ Hv1 ðsÞGv1 ðsÞ

ð7:96Þ

The transfer functions of open-loop subsystems in Fig. 7.41 can be written as Hi1 ðsÞ ¼ 

nh nh X X R1p R R R  þ dd þ  1pn , Hv1 ðsÞ ¼  2p  þ dv þ  2pn  s  λpk s  λ s  λ s  λpn pn pk n¼1 n¼1

g g X X R1ti R1k R2ti R2k þ dp1 þ , Gv1 ðsÞ ¼ þ dp2 þ ðs  λti Þ ð s  λk Þ ðs  λti Þ ðs  λ k Þ k¼1 k¼1

n

Gi1 ðsÞ ¼

n

ð7:97Þ

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

293

where λpk and λpn, n ¼ 1, 2, . . .nh are the eigenvalues of Ap in (7.76), Rmp and Rmpn (m ¼ 1, 2, n ¼ 1, 2, . . .nh) are the corresponding residues; λti and λk, ( k ¼ 1, 2, . . .ng) are the eigenvalues of As in (7.77), Rmti and Rmk, m ¼ 1, 2, k ¼ 1, 2, . . .ng are the corresponding residues. Substitute (7.97) into (7.96); replace s by b λ ti and then multiply both sides of the  2 b equation by λ ti  λti . With λpk  λti, it can have ( " #) nh X  2   R 1pn b   λ ti  λti  R1p þ b λ ti  λti dd þ s  λpn n¼1 ( " #) ng X   R1k b R1ti þ λ ti  λti dp1 þ þ ð s  λk Þ k¼1 ð7:98Þ ( " #) nh X   R2pn b   R2p þ λ ti  λti dv þ s  λpn k¼1 ( " #) ng X   R2k b R2ti þ λ ti  λti dp2 þ ð s  λk Þ k¼1 λ ti ! λti), the following estimation of Δλti ¼ b λ ti In the neighborhood of λpk  λti (b λti is obtained from (7.98) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð7:99Þ lim Δλti ¼ lim b λ ti  λti   R1p R1ti þ R2p R2ti bλ ti !λti bλ ti !λti The above estimation will become more meaningful when the estimation of Δλpk ¼b λ pk  λpk is established. b λ pk is also a solution of (7.96). By taking the derivation similar to that from (7.96) to (7.99) for b λ pk , it can have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð7:100Þ lim Δλpk ¼ lim b λ pk  λpk   R1p R1ti þ R2p R2ti bλ pk !λpk bλ pk !λpk From (7.99) and (7.100) b λ ti  λti  Rki , b λ pk  λpk  Rki  λti  Rki ð7:101Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where, for the convenience of discussion, Rki ¼ R1p R1ti þ R2p R2ti is called the open-loop modal residue. From (7.101), it can be seen that when the OLMR occurs, in the neighborhood of λ pk and b λ ti are located at the positions approximately opposite to each other λpk  λti, b on the complex plane with respect to the positions of λpk  λti. Thus, it is likely that λ ti may be on the right hand side of λpk  λti. Therefore, strong dynamic either b λ pk or b interactions caused by the OLMR may possibly degrade the damping of either b λ pk or b λ ti . In addition, if |Real part of Rki| >|Real part of λpk or λti|, it is possible that either

294

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

b λ ti may become negatively damped such that growing oscillations occur in the λ pk or b MTDC/AC power system. From the above discussions, it can be seen that to examine the risk caused by the AC/DC dynamic interactions introduced by the DC voltage control system of the VSC, the OLMR first needs to be identified. Secondly, the degree of stability degradation caused by the identified OLMR is estimated by computing the openloop modal residue Rki. The identification and estimation can be achieved from the results of modal analysis for the open-loop subsystems. The state-space model of the open-loop MTDC/AC power system can be obtained by assuming ΔP1 ¼ 0, i.e., assuming the active power output from VSC-1 to be a constant P10. Similarly, when the active power control system or reactive power control system is selected rather than the DC voltage control system, the active power and reactive power outputs from the VSC are assumed to be constant. With the assumption, the state-space model of the open-loop MTDC/AC power system can be derived. From the modal analysis of the derived open-loop subsystems, the OLMR can be identified and the degree of stability degradation caused by the identified OLMR can be estimated. Therefore, the following method of open-loop modal analysis to examine the impact of AC/DC dynamic interactions introduced from a selected VSC control system on the small-signal stability of the MTDC/AC power system can be proposed. 1. Modelling all the VSCs connecting the MTDC network and AC power system as constant power sources, Pk0 þ jQk0, k ¼ 1, 2, . . .M; the direction of terminal voltage of each VSC is taken as that of the d axis of the d  q coordinate of the VSC; establish the state-space model of the AC power system. From the established open-loop state matrix, calculate and identify the open-loop oscillation modes λti, i ¼ 1, 2, . . ., of the AC system. 2. Calculate the open-loop oscillation modes λpk, k ¼ 1, 2, . . ., of the control systems of the VSCs 3. If any λpk  λti is found, the corresponding open-loop modal residue Rki is calculated. By applying the criterion |Real part of Rki| > |Real part of λpk or λti|, the potential risk of growing oscillations is identified. In addition, the troublemaking control systems of the VSCs and the dynamic components, such as the SGs, in the AC power system responsible for the growing oscillations are identified. 4. The parameters of the identified control systems of the VSCs are tuned to move λpk away from λti on the complex plane. Thus, the identified OLMR is dismissed and the risk of growing oscillations is eliminated.

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

7.3.2

An Example MTDC/AC Power System for Applying the Open-Loop Modal Analysis

7.3.2.1

Example 7.5—Small-Signal Instability Risk Caused by the OLMR

295

Figure 7.44 shows the configuration of an example MTDC/AC power system. The part of AC power system is the well-known New England power system (NEPS). The parameters of loads and network of the NEPS given in [4] were used. The model of the SGs given in [15] for studying the sub-synchronous resonance (SSR) was used and their parameters were adjusted according to the capacity of the SGs. The MTDC network had eight terminals. Four terminals were connected by VSC-k, k ¼ 1, 2, 3, 4 to the nodes of the NEPS. VSC-1 used the DC voltage control and VSC-k (k ¼ 2, 3, 4) adopted the active power control. The active power output from VSC-k (k ¼ 1, 2, 3, 4) at the steady state was 3.43 p.u, 1 p.u, 0.5 p.u, and 0.5 p.u, respectively. These four VSCs operated with a unity power factor. Each of the other four VSCs was connected to a wind farm represented by a PMSG. In addition, a wind farm represented by a PMSG was connected at node 39. The detailed 14th-order model and parameters of the PMSG recommended in [8] were used. Model and parameters of the MTDC network given in [14] were used. Data of the example power system is given in Appendix 7.4. First, VSC-k (k ¼ 1, 2, 3, 4) were modeled as constant power sources. The openloop state-space model of the NEPS was established. Open-loop oscillation modes of 9

26

25

28

27 2 30 1

17

PMSG

15 14

5

GSC MSC

11

42 VSC-3

13

7 36

4

22 21

39

8

34

12 5

35

33 4

24

6

20

3

10

29

19

8

1

9

16

18

37

38

23

40 43 VSC-1 VSC-4

10

6

MTDC NET

r

7

2

31 NEPS

32 3

VSC-2 41

Fig. 7.44 Configuration of a test MTDC/AC power system

WIND FARMs

r

296

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

Table 7.1 Selected oscillation modes of the MTDC/NEPS Open-loop SSO modes λt1 ¼ 0:69 þj91:51

Related dynamic components Mass of the second low pressure cylinder (LPC) of SG1

λt2 ¼ 0:84 þj97:75

Mass of the second LPC of SG6

λt3 ¼ 0:71 þj100:8

Mass of the second LPC of SG5

λt4 ¼ 5:13 þj90:03

The grid side converter (GSC) of the PMSG at node 39

λp1 ¼ 1:98 þj90:98

DC voltage control system of VSC-1

Closed-loop SSO modes b λ t1 ¼ 0:02 þj91:12 b λ t2 ¼ 0:83 þj97:31 b λ t3 ¼ 0:71 þj100:8 b λ t4 ¼ 5:18 þj89:98 b λ p1 ¼ 2:58 þj90:12

the NEPS were calculated. In the first column of Table 7.1, four selected open-loop sub-synchronous oscillation (SSO) modes of the NEPS are listed as λti, i ¼ 1, 2, 3, 4. The dynamic components related to λti, i ¼ 1, 2, 3, 4 were identified by calculating the PFs from the open-loop state-space model of the NEPS. Results of identification are given in the second column of Table 7.1. Secondly, the open-loop oscillation modes of the MTDC network were calculated. λp1 related with the DC voltage control system of VSC-1 was found to be close to the torsional SSO mode of SG1, λt1 (see the last row in the first column of Table 7.1). Thirdly, the open-loop modal residue for the OLMR, λp1  λt1, was calculated and found to be R11 ¼ 1.96  j0.45. Hence, |Real part of R11| > |Real part of λt1 |, indicating that the OLMR may cause poorly or negatively damped torsional SSOs in the MTDC/NEPS. To confirm that the OLMR, λp1  λt1, may have caused the damping degradation of the SSOs in the MTDC/NEPS, the closed-loop oscillation modes were calculated, and are presented in the last column of Table 7.1. It can be seen that the closed-loop λ p1 , were located at the approximately opposite positions oscillation modes, b λ t1 and b on the complex plane with respect to λp1  λt1, confirming the conclusion made from the open-loop modal analysis. The closed-loop torsional SSO mode, b λ t1 , of SG1 was poorly damped, confirming that poorly damped torsional SSOs occur, caused by the DC voltage control of VSC-1. The OLMR is different from the closeness of frequency of the open-loop oscillation modes. For example, in Table 7.1, the frequency (imaginary part) of λt4 is close to that of λp1. However, on the complex plane, λt4 is not in the proximity of λp1 because the real parts of λt4 and λp1 are very different.

7.3 Method of Open-loop Modal Analysis to Examine the Impact of Dynamic. . .

7.3.2.2

297

Parameter Tuning of the VSC Control

Tuning parameters of the DC voltage control system of VSC-1 can move the openloop oscillation mode, λp1, away from the open-loop torsional SSO mode, λt1, to eliminate the OLMR, λp1  λt1. With the OLMR being eliminated, the damping of the closed-loop torsional SSO mode, b λ t1 , can be improved. Hence, the integral gain of the DC voltage control Kdci was tuned within the range from Kdci ¼ 0.93 to Kdci ¼ 1.54. With Kpi being tuned, the trajectory of λp1 on the complex plane is displayed as the dashed curve in Fig. 7.45. The positions of the open-loop torsional SSO modes, λti, i ¼ 1, 2, 3, listed in the first column of Table 7.1, are indicated by hollow circles in Fig. 7.45. It can be observed that when Kdci was tuned, λp1 was in the proximity of λti, i ¼ 1, 2, 3 consecutively at points ①, ②, and ③. Point ① was the case of OLMR, λp1  λt1, identified above, while points ② and ③ were the new cases of OLMR of the DC voltage control system of VSC-1 with the open-loop torsional SSO modes of SG5 and SG6, respectively. The computational results of the open-loop modal residues for the new cases of open-loop SSO modal proximity at points ② and ③ were R12 ¼ 1.76 þ j0.06 and R13 ¼ 1.61  j0.03, respectively. Since |Real part of R1k| > |Real part of λtk |, k ¼ 2, 3, it can be predicted that the OLMR at points ② and ③ may probably lead to the poorly or negatively damped SSOs in the MTDC/NEPS. From the open-loop modal residues, the estimated positions of the closed-loop oscillation modes were obtained by using (7.101) and indicated by crosses in Fig. 7.45. To confirm the above prediction made from the open-loop modal analysis, the trajectories of closed-loop oscillation modes, b λ ti , i ¼ 1, 2, 3 and b λ p1 , are displayed as solid curves in Fig. 7.45. The positions of closed-loop SSO modes, when the OLMR occurred at point ①, ②, and ③, are indicated by filled circles. They confirmed that the OLMR degraded the damping of closed-loop torsional SSO modes of the related SGs. When λp1 was away from λti, i ¼ 1, 2, 3 by tuning Kpi, the damping of the closed-loop torsional SSO modes was improved as they moved toward where the corresponding open-loop torsional SSO modes were located on the complex plane. Imag axis (rad/s) 102 3

lt3 2

lt2

92

1 lˆ p1

82 -4

lˆt3 lˆt2 lˆt1

lt1 lp1

Real axis -3

Fig. 7.45 OLMR caused by tuning Kpi

-2

-1

0

1

7 Small-Signal Stability of a Power System Integrated with an MTDC. . .

298

Imag axis (rad/s) 92

λˆt4

λˆp1

λˆt1

λt4 4

1

λp1

λt1 Real axis

87 -6.5

-4.5

-2.5

-0.5

Fig. 7.46 Trajectories of oscillation modes when Kpp is tuned

Three SGs involved in the open-loop modal analysis were at different geographical locations in the NEPS. Their electric distances to VSC-1 were different. However, the results presented in Fig. 7.45 indicated that the strong dynamic interactions can occur between VSC-1 and the torsional dynamics of those SGs, leading to poorly damped torsional oscillations. The impact of OLMR was determined by the open-loop modal residue defined in (7.101). Figure 7.45 shows that tuning Kpi may improve the damping of related closedloop torsional SSO modes. However, it may also cause new cases of OLMR. Hence, the proportional gain Kdcp of the DC voltage control system of VSC-1 was tuned within the range of Kdcp ¼ 0.16 to Kdcp ¼ 0.36. The trajectory of λp1 on the complex plane with Kdcp being tuned is displayed as the dashed curve in Fig. 7.46. The solid curves are the trajectories of the related closed-loop oscillation modes. From Fig. 7.46, it can be observed that when Kdcp was tuned, λp1 moved away from λt1 toward the left on the complex plane such that the damping of closed-loop torsional λ p1 was improved. However, the OLMR occurred between λp1 and λt4 modes b λ t1 and b at point ④, indicating the strong dynamic interactions between the VSC-1 and the PMSG at node 39. The open-loop modal residue was calculated and found to be R14 ¼ 2.25 þ j1.04. Since |Real part of R14| > > > > > for Pwi ¼ 0 > > > >  k0i -1 "  k0i # > > > k P  k P  k 0i wi 2i wi 2i > > exp  , > < k1i c0i k1i c0i k1i c0i ð8:2Þ f wpoweri ðPwi Þ ¼ > > for 0 < Pwi < Pri > > >   > > > Fwspeedi ðv fi Þ  Fwspeedi ðvri Þ δðPwi  Pri Þ, > > > > > for Pwi ¼ Pri > > > : 0, for Pwi < 0 or Pwi > Pri

Fwpoweri ðPwi Þ ¼

8

 k > Pwi k2i 0i > þ 1  Fwspeedi ðv fi Þ, 1  exp  > > k1i c0i > > > > > < for 0  Pwi < Pri > > > > > > > > > :

1,

for

Pwi ¼ Pri

1,

for

Pwi > Pri

0,

for

Pwi < 0

ð8:3Þ

where Pwi is the active power supplied by the ith wind power generation source (wind farm) connected to a multi-machine power system, fwpoweri() and Fwpoweri() is the PDF and CDF of the wind power, vci the cut-in wind speed, vri the rated wind speed, vfi the furling wind speed, Fwspeedi() the CDF of the Weibull distribution of wind

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

309

speed, δ() the impulse function and Pri the rated wind power. Parameters in (8.2) and (8.3) are given by the following equations k0i ¼ ðσi =μi Þ1:086 , c0i ¼ μi =Γð1 þ 1=k0i Þ, k1i ¼ Pri =ðvri  vci Þ, k2i ¼ k1i vci

ð8:4Þ

where Γ() is a Γ function, μi and σi is the mean and standard deviation of wind speed respectively. The wind power variation can be defined to be ΔPwi ¼ Pwi  Pw0i, where Pw0i is the deterministic wind power generation. According to the probability theory, the nth moment αnΔPwi of the wind power variation, ΔPwi, can be computed from (8.2) or (8.3) as follows Ð P P Ð P P αn ΔPwi ¼ Pri w0i w0i xn dFwpoweri ðxÞ ¼ Pri w0i w0i xn f wpoweri ðxÞdx   Ð Pw0i n  ¼ Pw0i x 1  Fwspeedi ðv fi Þ  Fwspeedi ðvci Þ δðx þ Pw0i Þdx

 k  k0i 1  xk2i þPw0i 0i k1i c0i Ð P P k0i x  k2i þ Pw0i þ Pri w0i w0i xn dx e k1i c0i k1i c0i  Ð P Pw0i n  ð8:5Þ þ PririPw0i x Fwspeedi ðv fi Þ  Fwspeedi ðvri Þ δ½x  ðPri  Pw0i Þdx    ¼ 1  Fwspeedi ðv fi Þ  Fwspeedi ðvci Þ ðPw0i Þn   þ Fwspeedi ðv fi Þ  Fwspeedi ðvri Þ ðPri  Pw0i Þn

 k  k0i 1  xk2i þPw0i 0i k1i c0i Ð P P k0i x  k2i þ Pw0i dx þ Pri w0i w0i xn e k1i c0i k1i c0i By performing the variable transformation twice, i.e. t ¼ x  k2i þ Pw0i and  k0i τ ¼ k1itc0i , the above equation becomes    ¼ 1  Fwspeedi ðv fi Þ  Fwspeedi ðvci Þ ðPw0i Þn   þ Fwspeedi ðv fi Þ  Fwspeedi ðvri Þ ðPri  Pw0i Þn  k0i P k h in Ð kri1i c0i2i 1 k0i k1i c0i τk0i þ ðk2i  Pw0i Þ eτ dτ þ  -k1ik2ic0i    ¼ 1  Fwspeedi ðv fi Þ  Fwspeedi ðvci Þ ðPw0i Þn   þ Fwspeedi ðv fi Þ  Fwspeedi ðvri Þ ðPri  Pw0i Þn  k0i ð Pri k2i n X k1i c0i k k0i τk0i eτ dτ þ Cnk ðk1i c0i Þk ðk2i  Pw0i Þnk 

αn

ΔPwi

k¼0

k

k2i 1i c0i

ð8:6Þ

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

310

where Cnk ¼

n! k!ðn  kÞ!

 k0i ð Pri k2i k1i c0i k k0i τk0i eτ dτ is an incomplete Γ function. and  k

k2i 1i c0i

The nth order cumulant, γnΔPwi , also known as the semi-invariant, is the polynomial in α1ΔPwi , α2ΔPwi , . . . , αnΔPwi , for example γ1

ΔPwi

¼ α1

ΔPwi

γ2

ΔPwi

¼ α2

ΔPwi

 α21

ΔPwi

¼ α3

ΔPwi

 3α1

γ3

ð8:7Þ

ΔPwi ΔPwi α2 ΔPwi

þ

2α31 ΔPwi

Hence by using (8.7), the nth order cumulant, γnΔPwi , can be computed from the various-order moment.

8.2.1.2

Construction of Distribution Function of Critical Eigenvalue

According to the probability theory in [1, 2], if the relationship between a random variable ρ and m other independent random variables ηj, j ¼ 1, 2, . . .m is linear, that is ρ ¼ a1η1 þ a2η2 þ    þ amηm, their nth order cumulants satisfy the following equation γn

ρ

¼ a1 n γ n

η1

þ a2 n γn

η2

þ    þ am n γn

ηm

ð8:8Þ

If there are m grid-connected wind generation sources (wind farms) in the multimachine power system and λk ¼ ξk þ jωk is the kth eigenvalue (critical eigenvalue) of the power system, the following relationship between the critical eigenvalue and the wind power generation can be established for power system small-signal stability analysis Δλk ¼ Δξk þ jΔωk ¼

m X ð∂λk =∂Pwi ÞΔPwi i¼1

¼

m X

ð8:9Þ

Re½ð∂λk =∂Pwi ÞΔPwi þ jIm½ð∂λk =∂Pwi ÞΔPwi

i¼1

where Re() and Im() denote the real and imaginary part of a complex variable respectively. The sensitivity of the critical eigenvalue with respect to m wind power sources at an equilibrium point in (8.9) can be computed in an analytical way [17] or conveniently in a numerical way given by the following equation ∂λk λk ðPwi þ ΔPwi Þ  λk ðPwi Þ ¼ , ΔPwi ∂Pwi

i ¼ 1, 2, . . . m

ð8:10Þ

The assumption of linearity is implied in deriving (8.9). For small-signal stability analysis, linearized model of power systems is used and the nonlinearities are not

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

311

considered. Hence, the assumption of linearity in (8.9) is tenable. From (8.8) and (8.9) it can have   n m

X ∂λk γn Δξk ¼ Re γn ΔPwi ð8:11Þ ∂Pwi i¼1 where γnΔξk is the nth order cumulant of the stochastic variation of the real part of the critical eigenvalue, Δξk. The mean of Δξk is μΔξk ¼ γ1Δξk . It is noted that (8.11) is derived under the assumption that all the wind power sources are independent. In practice this is true if wind farms locate far away to each other geographically. The spatial correlations between different wind farms will be considered in Sect. 8.2.1.4. The nth order central moment, βnΔξk , of Δξk is calculated from its cumulants by using the following equation β1

Δξk

¼0

β2

Δξk

¼ γ2

Δξk

β3

Δξk

¼ γ3

Δξk

β4

Δξk

¼ γ4

Δξk

¼ σΔξk 2 þ 3γ22

ð8:12Þ

Δξk

where σΔξk is the standard deviation of Δξk. From the cumulants and central moments of Δξk, the CDF and PDF of the Δξk  μΔξk standardized Δξk, Δξk ¼ , can be obtained by using the following wellσΔξk known Gram-Charlier expansion g1 ð1Þ g g Φ ðxÞ þ 2 Φð2Þ ðxÞ þ 3 Φð3Þ ðxÞ þ    1! 2! 3! g1 ð1Þ g2 ð2Þ g3 ð3Þ f _ ð x Þ ¼ g 0 φð x Þ þ φ ð x Þ þ φ ð x Þ þ φ ð x Þ þ    Δξ k 1! 2! 3!

F

_

Δξ k

where F

_

Δξk

ðxÞ ¼ g0 ΦðxÞ þ

ðxÞ and f

_

Δξk

ð8:13Þ

ðxÞ is the CDF and PDF of Δξk respectively, Φ(x) and φ(x) the

CDF and PDF of standard normal distribution respectively and the superscript number (n) denotes the nth order derivative of Φ(x) and φ(x). Coefficients in the Gram-Charlier expansion of (8.13) are the polynomial in the central moments of Δξk as given as follows g0 ¼ 1 g1 ¼ g2 ¼ 0 g3 ¼  g4 ¼

β3

β4

Δξk

σ3Δξk Δξk

σ4Δξk

3

ð8:14Þ

312

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

Obviously, the CDF of Δξk can easily be obtained from that of Δξk to be   x  ΔμΔξk FΔξk ðxÞ ¼ F _ ð8:15Þ Δξk σΔξk Because Δξk ¼ ξk  ξk0 (ξk0 is the deterministic value of the real part of λk), the CDF of ξk can be obtained to be Fξk ðxÞ ¼ FΔξk ðx  ξk0 Þ

ð8:16Þ

The PDF is the derivative of the CDF of ξk obtained in (8.16). Therefore, the probability of the real part of the kth eigenvalue in the power system with m grid-connected wind power sources to be negative can be computed by (8.1).

8.2.1.3

Treatment of Ending Values of Distribution Function of Critical Eigenvalue

The wind power distribution is not continuous as given by (8.2). Thus f ξk ðxÞ 6¼ 0 only exists over a certain interval of ξk, i.e., [ξk_left, ξk_right]. The CDF and PDF given by (8.16) is for ξk within [ξk_left, ξk_right] and their value at the left end (ξk ¼ ξk_left) and right end (ξk ¼ ξk_right) of the interval needs to be calculated separately. Hence (8.16) needs to be modified. This modification is carried out by dividing the wind generation sources into two groups firstly. Group A is of positive Re(∂λk/∂Pwi) and group B is of negative Re(∂λk/∂Pwi). In deterministic eigenvalue analysis, ξk is calculated as ξk_left, when the wind generation sources in group A is at the cut-in wind power (i.e., Pwi ¼ 0, i 2 A) and in group B is at the furling wind power (i.e., Pwi ¼ Pri, i 2 B). ξk_right is calculated similarly but when the wind generation sources in group A is at the furling wind power and group B the cut-in wind power. The modified CDF and PDF are ( 8   ) Y 1  Fwspeedi1 ðvfi1 Þ  Fwspeedi1 ðvci1 Þ  > > >   > > > Fwspeedi2 ðvfi2 Þ  Fwspeedi2 ðvri2 Þ > i 2A , i 2B 1 2 > > > > >  δðx  ξk left Þ, > > > > for x ¼ ξk left > > > > > for ξk left < x < ξk right < derivative of ð8:16Þ, (   ) f ξ k ð xÞ ¼ Y 1  Fwspeedi1 ðvfi1 Þ  Fwspeedi1 ðvci1 Þ  > > >   > > Fwspeedi2 ðvfi2 Þ  Fwspeedi2 ðvri2 Þ > ð8:17Þ > i12B, i22A > >  > > > >  δ x  ξk right , > > > > > for x ¼ ξk right > > > : 0, for x < ξk left or x > ξk right 8 0, for x  ξk left > < ð13Þ, for ξk left < x < ξk right Fξ k ð x Þ ¼ > : 1, for x  ξ k right

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

8.2.1.4

313

Spatial Correlations Between Wind Power Generation Sources

The complementarity of variable wind generation sources at different places effectively can smooth the total wind power fluctuation in a power system. Thus the effect of power variation from multiple wind power generation sources on power system small-signal stability will be different if the wind complementarity is considered. The spatial correlations between different wind power sources are an important factor of wind complementarity. Hence the probabilistic small-signal stability of a power system is affected by the spatial correlations of wind generation sources. The correlation between two wind power sources is closely related to their geographical distance [18]. Two locations over 1200 km have a correlation coefficient close to 0, whilst very close locations (less than 100 km) have a correlation coefficient close to 1. This can be used to approximately estimate their correlation coefficient ρij and thus to establish the correlation coefficient matrix [ρij]m  m for m grid-connected wind power sources [18]. If there are sufficient wind speed data available, the correlations between pairs of wind speed can be calculated to be   Cov vi ; v j ρij ¼ ð8:18Þ σvi σv j where vi and vj are wind speed random variables corresponding to two wind power source locations, Cov(vi, vj) represents the covariance of the speed vi and vj, and σvi and σv j are the standard deviations of the speeds. For each wind speed random variable, vi and vj, a wind speed sample series can be generated correspondingly. Each wind speed sample series should satisfy the Weibull distribution and contain the spatial correlation. There are several methods available to generate the eligible wind speed sample series, such as the normal transformation method in [19, 20] and the Copulas in [21]. With the wind speed sample series available, the wind power sample series with correlations (i.e., ½Pwi NSample 1) can be generated by employing the power-wind speed curve of each wind power sources described in [22]. The wind power variation sample series ½Pwi NSample 1 can be obtained to be ½ΔPwi NSample 1 ¼ ½Pwi NSample 1  ½Pw0i NSample 1 , i ¼ 1, 2, . . . m

ð8:19Þ

where Nsample is the size of each wind power sample series and ½Pw0i NSample 1 is a vector with all the samples equal to the deterministic value Pw0i. When the correlation of the different wind power sources is considered, (8.11) is modified to calculate the nth order cross cumulant of Δξk to be

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

314

2 γn

Δξk

¼

m X m X i1 ¼1 i2 ¼1



3

      m 6 X ∂λk ∂λk ∂λk 6 γn Re   Re 6Re 4 ∂Pwi1 ∂Pwi2 ∂Pwin in ¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

7 7 5

ΔPwi1 i2 in 7

n

ð8:20Þ where γnΔPwi i in denotes the nth order cross cumulants of the multiple wind power 12 variations. For independent wind power sources, i1 ¼ i2 ¼    ¼ in ¼ i. The cumulant γnΔPwi i in is equal to γnΔPwi and (8.11) is replaced by (8.20). 12 Normally the higher order cross cumulants of Δξk have less impact on the accuracy of the CDF and PDF curve of ξk. Hence it is only needed to calculate the first several order cross cumulants of Δξk by using (8.20) to include the correlation. The cumulants of the rest with higher orders can still be calculated by (8.11) so as to reduce the computation cost. The 1st order cross cumulant of wind power variation is still γ1ΔPwi . Their 2nd and 3rd order cross cumulants can be easily calculated from the following equations [1, 2]. h  i γ2 ΔPwi i ¼ β2 ΔPwi i ¼ E ΔPwi1  μΔPwi ΔPwi2  μΔPwi 12

γ3

12

1

2

¼ β3 ΔPwi i i 123 h   i ¼ E ΔPwi1  μΔPwi ΔPwi2  μΔPwi ΔPwi3  μΔPwi

ΔPwi1 i2 i3

1

2

ð8:21Þ

3

where βnΔPwi i in is the nth order cross central moment and μΔPwin is the mean of ΔPwin . 12 The expected values in (8.21) can be calculated directly when each wind power variation sample series is obtained. Hence by using (8.20) and (8.21), first three order cross cumulants of Δξk are obtained. When the spatial correlations between wind power generation sources are considered, (8.17) needs to be modified as follows. Firstly, a vector of the approximate values of ξk is calculated to be    m

X ∂λk ½ξk NSample 1 ¼ ½ξk0 NSample 1 þ Re ½ΔPwi NSample 1 ð8:22Þ ∂Pwi i¼1 where ½ξk0 NSample 1 is a vector with all the samples equal to the deterministic value ξk0. Secondly the maximums and minimums of the vector ½ξk NSample 1 are determined. It is noted that there are possibilities that multiple maximums and multiple minimums exist in some cases. All the wind power data sets corresponding to the maximums and minimums of ½ξk NSample 1 are recorded (each wind power data set [Pw1, Pw2,   Pwm]1  m is consist of m wind power data which are respectively from the same row of each ½Pwi NSample 1 as the maximum or minimum in ½ξk NSample 1 ). Thirdly, the deterministic eigenvalue analysis is carried out to compute ξk by using

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

315

the recorded wind power data sets as output power of m wind power sources. ξk_left and ξk_right are the smallest and largest ξk in the deterministic eigenvalue analysis respectively. The probabilistic density of ξk_left and ξk_right are calculated by Nξk left δðx  ξk left Þ, for x ¼ ξk left Nsample Nξk right  δ x  ξk right , for x ¼ ξk right Nsample

ð8:23Þ

Where Nξk_left and Nξk_right are the repeated numbers of ξk_left and ξk_right in the deterministic eigenvalue analysis respectively. Hence, the PDF and CDF of ξk with modified left and right ends are obtained. The form of PDF and CDF of ξk is similar to (8.17), but the probabilistic density values at left and right ends are changed as given by (8.23).

8.2.1.5

Procedure of Cumulant Theory Based Analytical Method

The procedure of probabilistic analysis of power system small-signal stability introduced above can be summarized as follows: 1. If the grid-connected onshore wind power sources are spatially dependent, the wind power variation sample series should be generated from the correlation coefficient matrix by using (8.19). If not, go to step 2) directly. 2. Determine the stochastic distributions of grid-connected wind power sources defined by (8.2) and (8.3). 3. Compute the nth order moment by using (8.6) and then the nth cumulant by (8.7) of each wind power variation. 4. Compute the nth order cumulant of the variation of the real part of the critical eigenvalue, Δξk, by using (8.11) or (8.20) (if considering correlations) and then its nth order central moment by (8.12). 5. Compute the CDF and PDF of the standardized Δξk by using (8.13) and (8.14). 6. Convert the CDF and PDF of Δξk into the CDF and PDF of ξk by using (8.15) and (8.16). 7. Modify the CDF and PDF of ξk by using (8.17) or (8.17) and (8.23) (if considering correlations). 8. Determine the probability of the real part of the critical eigenvalue to be negative according to (8.1). 9. Step 2–7 can also be achieved by applying the non-analytical method of Monte Carlo simulation.

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

316

8.2.2

Example 8.1

Figure 8.2 shows the configuration of an example 16-machine 5-area power system with 3 grid-connected onshore wind power sources at node 69, 70 and 71. The network data, system load condition, synchronous generator model and parameters are given in Appendix 8.1. For the purpose of better demonstration, no PSSs are installed in the power system. A 70 MW doubly fed induction generator (DFIG) model is used for all three wind farms [23] and the DFIGs’ parameters in p. u. are. H ¼ 1.7s, D ¼ 0.0, xs ¼ 2.9, xr ¼ 2.9, xm ¼ 2.6, rs ¼ 0.0, rr ¼ 0.0013, Pw0 ¼ 0.3333. The controller model of the DFIG rotor-side converter used in the example system is shown by Fig. 8.3, where KP ¼ 30, KQ ¼ 30. Parameters of the wind speed distribution are vc ¼ 4m/s, vr ¼ 10m/s, vf ¼ 22m/s, Pr ¼ 1.0p. u., μ ¼ 6m/s, σ ¼ 2.5. From the deterministic modal analysis, the 29th eigenvalue is identified to be the critical eigenvalue, i.e., λ29 ¼  0.0106  j3.3004. Hence deterministically the system is stable. Denote the wind farm at node 69, 70 and 71 to be the first, second and third source of wind generation respectively. The sensitivity computation of the critical eigenvalue with respect to three sources of wind generation, Pw1, Pw2 and Pw3, is obtained to be

NYPS

WG 14

8 69

A3

66

41

1

40

2

38

70

30

32 62

46

63

10

A4

36

35

64

45

50

37

12

65

A2

21

14

5

34

51

24

15

4

11

49

61

9

9

33

6 7

54

22

12 19

11

2

10

23

20 6

13 8

58

57

56

55 4

59 7

5

3

39

52 44

A1 43

71 16

29

16

3

WG

68

28 27

17 31

15

A5

26

18

1

42 67

25

53

47

48

60

13

NETS

WG

Fig. 8.2 Line diagram of example 16-machine 5-area power system integrated with onshore wind power generation

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

– s X rr –

317

X V X m2 I rd + s m s – Rr I rq X ss X ss

I rq

Ps0

Qs0

S

Xss Vs

+

– X ss X m Vs

I rqref

–1 X m Vs

I rdref

+

+

2

S

KP

S

KQ



+

+

S

S

+

V rq

Vrd

+

I rd s X rr –

X m2 I rq – Rr I rd X ss

Fig. 8.3 Controller model of DFIG rotor-side converter Table 8.1 Moments and cumulants of wind power variations Moments of wind power variations α1ΔPw1, 2, 3 ¼ 0.0319

Cumulants of wind power variations γ1ΔPw1, 2, 3 ¼ 0.0319

α2ΔPw1, 2, 3 ¼ 0.1080

γ2ΔPw1, 2, 3 ¼ 0.1070

α3ΔPw1, 2, 3 ¼ 0.0281

γ3ΔPw1, 2, 3 ¼ 0.0178

α4ΔPw1, 2, 3 ¼ 0.0262

γ4ΔPw1, 2, 3 ¼ 0.0111

γ5ΔPw1, 2, 3 ¼ 0.0126

γ5ΔPw1, 2, 3 ¼ 0.0104

Table 8.2 Cumulants and central moments of real part of critical eigenvalue Cumulants of critical eigenvalue Mean ¼ 8.10  104 Variance ¼ 2.32  105 γ3_Δξ29 ¼ 3.35  108 γ4_Δξ29 ¼ 1.82  1010 γ5_Δξ29 ¼ 1.51  1012

Central moments of critical eigenvalue β1_Δξ29 ¼ 0.0 β2_Δξ29 ¼ 2.32  105 β3_Δξ29 ¼ 3.35  108 β4_Δξ29 ¼ 1.44  109 β5_Δξ29 ¼ 6.26  1012

∂λ29 ∂λi29 ¼ 0:0096  j0:0489, ¼ 0:0083  j0:0466, ∂Pw1 ∂Pw2 ∂λ29 ¼ 0:0075  j0:0394 ∂Pw3 Table 8.1 gives the computational results of the first five orders of moments and cumulants of three wind power variations computed by using (8.6) and (8.7). Table 8.2 gives the computational results of the first five orders of cumulants and the central moments of the real part of the critical eigenvalue obtained from (8.11) and (8.12). Table 8.3 gives the computational results of the first six coefficients of the

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

318

Table 8.3 Coefficients of Gram-Charlier expansion

g1 0

g0 1

g2 0

g3 0.2989

100

80

0.0117δ(x+0.0185)

80

60

60

40

40 0.0003δ(x-0.0094)

20 0 -0.02

-0.015

-0.01

-0.005

0

0.005

20

Monte Carlo sample number

Analytical Method Monte Carlo Simu

120

probabilistic density

g5 0.5794

λ Real Part PDF

140

100

g4 0.3382

0 0.01

λ real part Fig. 8.4 PDF of real part of critical eigenvalue

Gram-Charlier expansion according to (8.14). The PDF curve of the real part of the critical eigenvalue is constructed by use of (8.17) finally, which is shown by in Fig. 8.4. The PDF obtained by the Monte Carlo simulation with 5000 samples is also presented by Fig. 8.4, which confirms the result by using the introduced method of probabilistic analysis. A comparison of computation time between the Monte Carlo simulation and the analytical method has been carried out. Based on the same computational resource (Dell OptiPlex 745, Intel Core 2 CPUs 2.66GHz, 3GB RAM), the time of the Monte Carlo simulation with 5000 iterations is 15236.48 s, while only 38.56 s for the analytical method with first five-order Gram-Charlier expansion is needed. The analytical method is about 395 times faster than the Monte Carlo simulation. According to the PDF of Fig. 8.4 it can be obtained that ð0 Pðξ29 < 0Þ ¼ f ξ29 ðxÞdx ¼ 0:9710 1

Hence when the stochastic variation of wind generation is considered, the critical eigenvalue of the example power system has a probability of 97.10% to remain in the left half-plane. Although the system is considered to be stable by the deterministic analysis, as the critical eigenvalue is λ29 ¼  0.0106  j3.3004, it still has a probability of 2.90% to be unstable due to the uncertainty of wind generation.

8.2 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

319

To demonstrate the case with the correlations between three wind power sources being considered, the correlation coefficient matrix [ρij]3  3 is constructed according to the assumed geographical distance between the three wind farms as 2 3 1 0:8 0 h i 6 7 ρij ¼ 4 0:8 1 0 5 ð8:24Þ 33

0

0

1

(8.24) in fact suggests that the first two wind power sources are strongly correlated, while the third is independent with the other two due to the long distance. By employing the procedure presented in Sect. 8.2.1.4, the 2nd and 3rd order cross cumulants of Δξ29 are computed by using (8.21) to be γ2

Δξ29

¼ 3:63  105 , γ3

Δξ29

¼ 9:02  108

The probabilistic density of ξ29_left and ξ29_right is calculated by using (8.23) to be Nξ29 left δðx  ξ29 left Þ ¼ 0:0396δðx þ 0:0185Þ Nsample Nξ29 right  δ x  ξ29 right ¼ 0:0028δðx  0:0094Þ Nsample Finally, the PDF curve of the real part of the critical eigenvalue is obtained as shown by Fig. 8.5. The result of Monte Carlo simulation (with 5000 samples) in λ Real Part PDF Analytical Method Monte Carlo Simu

probabilistic density

240

200

160

200 160

120

120 80 80 0.0028δ(x-0.0094) 40

40 0 -0.02

-0.015

-0.01

-0.005

λ real part Fig. 8.5 PDF of real part of critical eigenvalue

0

0.005

0 0.01

Monte Carlo sample number

0.0396δ(x+0.0185)

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

320

Fig. 8.5 has verified the analytical method introduced when the correlations of wind power sources are considered. According to the PDF of Fig. 8.5 it can be obtained that ð0 Pðξ29 < 0Þ ¼ f ξ29 ðxÞdx ¼ 0:9334 1

This result indicates that the consideration of correlation has changed the probability of system small-signal stability.

8.3

Probabilistic Analysis of Small-Signal Stability of a Power System Affected by MTDC–Connected Offshore Wind Power Generation

VSC-HVDC transmission is an attractive way to connect offshore wind power generation due to its flexibility in topology and no need for reactive compensation. It is envisaged that an increasing number of offshore wind farms are deployed and connected to AC grids through VSC based subsea MTDC networks. Grid integration of offshore wind power generation through MTDC networks increases the complexity of system stability analysis, as dynamic of MTDC networks may affect the stability of the whole AC/DC power system. In this section, the probabilistic small-signal stability of a hybrid AC/DC power system where MTDC networks connect the variable offshore wind power to the main AC grid is carefully examined.

8.3.1

Cumulant Theory Based Analytical Method of Probabilistic Small-Signal Stability Analysis of a Hybrid AC/DC Power System

As voltage stability is always a concern of HVDC networks, the cumulant theory based analytical method is used for the probabilistic analysis of both small-signal angular and small-signal voltage stability of an AC/DC power system with offshore wind farms supplying power through a MTDC network in this section.

8.3.1.1

Indices for Small-Signal Angular and Voltage Stability of a Hybrid AC/DC Power System

In the cumulant theory based analytical method for probabilistic analysis, the key step is to identify the stability index (i.e., critical eigenvalue) and its sensitivity with respect to the stochastic variables in the system.

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

321

In the small-signal angular stability presented in the previous sections, the following linearized dynamic model of power system is established at an operating point when wind power output is taken under mean wind speed Δx_ ¼ AΔx

ð8:25Þ

where Δx is the system state variable and A the system state space matrix. The critical eigenvalue (λA_cr ¼ ξA_cr þ jωA_cr) of A is identified and then its sensitivity to the variation of wind power output is computed to link the wind generation with the power system small-signal angular stability. Hence the analytical method can perform the probabilistic analysis of power system small-signal angular stability in one-step computation from the well-known Weibull distribution of wind speed. Small-signal voltage stability of a power system can be assessed based on eigenvalue decomposition of reduced Jacobian matrix at a steady-state system operating point [24–26]. The linearized steady-state voltage equations can be written as " # " #" # JPθ JPU ΔP Δθ ¼ ð8:26Þ JQθ JQU ΔQ ΔU where ΔP, ΔQ, Δθ and ΔU are vectors of variation of bus active, reactive power injection, bus voltage angle and magnitude respectively. When the focus of study is DC network, all AC buses except the converter AC buses can be eliminated and thus (8.26) becomes " # " #" # JPθ c JPU c ΔPc Δθc ¼ ð8:27Þ JQθ c JQU c ΔQc ΔUc where the subscript c denotes the vectors or matrices regarding the converter buses. Set ΔPc ¼ 0 to reduce (8.27), thus giving ΔQc ¼ JR ΔUc

ð8:28Þ

where JR is the reduced Jacobian matrix of the converter buses and JR ¼ JQUc  JQθc J1 Pθc JPUc . According to [25, 26], the DC network will be voltage-stable if all the eigenvalues of JR are positive. Therefore, the minimal eigenvalue of JR, λJR min , can be used as an index of AC/DC power system small-signal voltage stability. With the assumption of ΔPc ¼ 0 in (8.28), it can be seen that λJR min indicates the margin of small-signal voltage stability for each specific operating point (or load level) of system. The sensitivity of λJR min with respect to m wind power sources on the given steady-state operating point can be computed conveniently in a numerical way similar to (8.10). On this basis, the probability of small-signal angular and voltage stability of the AC/DC power system with m grid-connected offshore wind farms can be assessed by the following equations

322

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

PðξA PðλJR

cr

< 0Þ ¼

min

ð0

> 0Þ ¼

f ξA cr ðxÞdx

1 ð1 0

f λJR

min

ð8:29Þ

ðxÞdx

The assessment of both probabilistic small-signal angular and voltage stability can be derived from the complete differential-algebraic model. However, as the two types of stability correspond to different matrices of the complete model (i.e., system state space matrix and reduced Jacobian matrix), the probability values calculated in (8.29) are not directly correlated, which can be also easily explained from their essential physical nature. Moreover, these two types of stability may have different shapes of probability distribution due to different sensitivities and their changing rates. Nevertheless, the increase of wind speed variation range, numbers of wind power sources and spatial correlation of wind farms in the system may deteriorate both stability issues. Therefore, the procedure to examine the probabilistic smallsignal angular and voltage stability of a hybrid AC/DC power system as affected by the stochastic fluctuations of grid-connected offshore wind power generation is illustrated by Fig. 8.6.

8.3.1.2

Multi-Point Linearization Technique

Probabilistic analysis of small-signal stability presented in Fig. 8.6 is based on the linearization of power system non-linear model at a single operating point. Accuracy of analytical results based on the single-point linearization is certainly lower than that of Monte Carlo simulation especially for a complex hybrid AC/DC power system with a strong nonlinearity. Work in [27] indicates that if different points of linearization can be used, the probabilistic information around these points can be evaluated with enhanced accuracy. Hence, if multiple steady-state operating points under different wind speed conditions are considered, the analytical method can be improved to better handle power system non-linearity to assess the effect of gridconnected wind power generation on power system probabilistic small-signal stability. Procedure of this improved analysis with multi-point linearization can be explained as follows. Take the ith offshore wind power generation source (wind farm) in the AC/DC power system as an example. The active power output Pwi from the wind farm can vary from 0 to Pri (the rated wind power). If Nlin-point linearization is applied (i.e., Nlin steady-state operating points are considered), the interval of variation of Pwi, [0, Pri], firstly is divided equally into 2Nlin sub-intervals. The endpoints of all the sub-intervals are recorded as a vector, ½Pwi0 ; Pwi1 ;   Pwi2Nlin 1 ; Pwi2Nlin ð2Nlin þ1Þ1 . If the sensitivity of ξA_cr (or λJR min) with respect to the ith wind output power is positive (negative), the endpoints are recorded in a normal order (i.e., Pwi_0 ¼ 0, Pwi2Nlin ¼ Pri). Conversely, the endpoints are recorded in a reverse order (i.e., Pwi_0 ¼ Pri, Pwi2Nlin ¼ 0). Assume that there are m wind farms in the power system. Thus in total 2Nlin þ 1 initial

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

A Given System Steadystate Operating Point

323

Given Wind Power Distributions

Linearizing Power System Model If Correlated

Deterministic Small-signal Angular and Voltage Stability Analysis

Critical Eigenvalue (and Its Sensitivity with Respect to Wind Output Power) Angular and Voltage Stability Analysis

Yes

Generating Wind Power Variation Sample Series

Computing the Moments and Cumulants of Wind Power Variations

Computing the Cumulants and Central Moments of Critical Eigenvalue

Computing Coefficients of Gram-Charlier Series

Generating Distribution of Standardized Eigenvalue

Probabilistic Distribution of Critical Eigenvalue

Conventional Cumulant Theory Based Analytical Method Probability of Power System Small-signal Angular and Voltage Stability

Fig. 8.6 Procedure of probabilistic small-signal angular and voltage stability analysis of a hybrid AC/DC power system

324

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . . Generating N lin Steady-state Operating Points

Given Wind Power Distributions

Setting n=1 Conventional Cumulant Theory Based Analytical Method for Probabilistic Small-signal Angular and Voltage Stability Analysis

n=n+1

If n>Nlin

No

Yes

Combining N lin piecewise PDFs of Stability Indices

Probability of Power System Small-signal Angular and Voltage Stability

Fig. 8.7 Procedure of probabilistic stability analysis based on multi-point linearization technique

operating points will be obtained (i.e., [Pw1_j, Pw2_j, . . .Pwm  1_j, Pwm_j]m  1, j ¼ 0, 1, . . .2Nlin). Secondly, the deterministic eigenvalue analysis is carried out under the jth initial operating point to compute ξA_cr_j and λJR min when j is even (Nlin þ 1 times). The other Nlin initial operating points (when j is odd) are the selected Nlin steady-state operating points. Thirdly, the method introduced by Fig. 8.6 is employed under the selected Nlin steady-state operating points to derive the PDF for each selected point. Finally, Nlin PDFs obtained are properly combined at each point of ξA_cr_j and λJR min (when j is even) to give the final PDFs of multi-point linearization of ξA_cr and λJR min . The computational procedure can be illustrated by Fig. 8.7. In most cases, the final PDF of a stability index obtained by using this multi-point linearization technique does not have an area under the curve exactly equal to unity. Since the selected steady-state operating points are evenly located, the linear solutions are located more closely when the sensitivity of stability index with respect to wind output power becomes smaller, which sequentially makes the curve comparatively more precise. Therefore, the area under the curve can be calculated from the side with a smaller sensitivity. When the area reaches unity, the remaining part of the PDF can be set to zero [27].

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

WG1

WG2

WG3

69

70

71

~ =

~ =

DC1

~ =

DC2

325

DC3

NYPS 14 66 41

~ =

DC6

A3

~ =

40

8

DC4

DC5

47

48

~ =

1

60

25

53

28

2

31 15

32 62

46

30

5

34

36

6 7

54

22

12

64

45

8

2

10

58

19

11

20

13

35 51

24 21

14

4

33 49

15

9

11

10

16

3

63

61 9

17

38

A4

29

27

18

1

42 67

26

56

55 4

12

57

23 59

6 7

5

3 50

A5

37

39

52

A1 68

44

16

43

65 13

A2

NETS

Fig. 8.8 Line diagram of example 16-machine 5-area hybrid AC/DC power system integrated with offshore wind power generation

8.3.2

Example 8.2

Figure 8.8 shows the configuration of an example 16-machine 5-area AC/DC power system with three grid-connected offshore wind farms at node 69, 70 and 71. A six-terminal DC network connects the offshore wind farms to the onshore AC grid. The AC network data, system load condition, synchronous generator model and parameters are given in Appendix 8.1. Parameters of the VSC-HVDC network in p.u. are R12 ¼ 0.0009, R16 ¼ 0.0006, R23 ¼ 0.0012, R25 ¼ 0.0006, R26 ¼ 0.0011, R34 ¼ 0.0006, L12 ¼ 0.0033, L16 ¼ 0.0022, L23 ¼ 0.0044, L25 ¼ 0.0022, L26 ¼ 0.0040, L34 ¼ 0.0022. A 70 MW doubly fed induction generator (DFIG) model is used for all three wind farms and the DFIGs’ parameters in p. u. are H ¼ 1.7s, D ¼ 0.0, xs ¼ 0.29, xr ¼ 0.29, xm ¼ 2.6, rs ¼ 0.0, rr ¼ 0.0013, Pw0 ¼ 0.3333. The controller model of the DFIG rotor-side converter used in the example system is shown by Fig. 8.3, where KP ¼ 30, KQ ¼ 30. Parameters of the wind speed distribution are vc ¼ 3m/s, vr ¼ 12m/s, vf ¼ 22m/s, Pr ¼ 1.0p. u., μ ¼ 6m/s, σ ¼ 2.5. From the deterministic small-signal angular and voltage stability analysis, the 31st eigenvalue of the system state space matrix A is identified to be the critical eigenvalue, i.e., λA_cr ¼  0.0113  j2.0164, and the minimal eigenvalue λJR min

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

326

of the reduced Jacobian matrix JR is 0.2821. Hence deterministically the system is stable. The wind farms at node 69, 70 and 71 are denoted to be the 1st, 2nd and 3rd source of wind generation respectively. The sensitivity computation of two stability indices with respect to three sources of wind generation is ∂ξA cr ∂ξ ∂ξ ¼ 0:0092, A cr ¼ 0:0081, A cr ¼ 0:0078, ∂Pw1 ∂Pw2 ∂Pw3 ∂λJR min ∂λJR min ∂λJR min ¼ 0:2539, ¼ 0:2215, ¼ 0:2186 ∂Pw1 ∂Pw2 ∂Pw3 13-point linearization technique is applied by employing the procedure presented in Sect. 8.3.1.2. Hence 13 steady-state operating points are determined under 13 different output power conditions of wind generation. The values of stability indices (x-axis) and their sensitivities with respect to the first wind power output (y-axis) are calculated at each steady-state operating point by deterministic analysis and plotted in Fig. 8.9 as an example. It can be seen from Fig. 8.9 that the sensitivity of stability indices has a noticeable change with the increase of wind power penetration level. This explains that the conventional method based on one-point linearization may not provide accurate result. The final PDFs of ξA_cr and λJR min are computed by use of the analytical method illustrated in Fig. 8.7 with the 13-point linearization and displayed in Figs. 8.10 and 8.11 respectively. For the purpose of comparison, results obtained by using the

dξA_cr /dP w 1

0.015 1

0.01

2

3

4

5

6

7

8

9

10 11 12

13

0.005 0

-20

-15

-10

-5

ξA_cr

0 x 10

-3

dλJr /dP w 1

-0.1 -0.2 -0.3 -0.4 -0.5

13 -0.2

12

11 -0.1

10 0

9

8

0.1 λJr

7

6

0.2

4

5

0.3

3

2

0.4

1

0.5

Fig. 8.9 Stability indices and their sensitivities with respect to 1st wind output power at each steady-state operating point (from 1 to 13)

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . . ξ

A_cr

327

PDF

160

probabilistic density

140

80

120 60

100 80

40

0.0015δ(x+0.0198)

60 40

20

2e-6δ(x-0.0022)

Monte Carlo sample number

100 Proposed Analytical Method Conventional Analytical Method Monte Carlo Simulation

20 0 -0.025

-0.02

-0.015

-0.01 ξA_cr

-0.005

0 0.005

0

Fig. 8.10 PDF of ξA_cr obtained by analytical methods and Monte Carlo simulation λJ _min PDF R

120

Proposed Analytical Method Conventional Analytical Method Monte Carlo Simulation

100

probabilistic density

5 80

4 0.0015δ(x-0.4880)

60

3

2

40

1 2e-6δ(x+0.2497)

20

0 -0.3

-0.2

-0.1

0

0.1 0.2 λJ _min R

0.3

0.4

0.5

0 0.6

Fig. 8.11 PDF of λJR min obtained by analytical methods and Monte Carlo simulation

Monte Carlo sample number

6

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

328

Table 8.4 Probability stability assessed by use of three different methods Probability P(ξA_cr < 0) PðλJR min > 0Þ

Monte Carlo simulation 0.9936 0.9676

Conventional analytical method 0.9881 0.9762

Proposed analytical method 0.9940 0.9678

conventional cumulant theory based analytical method and Monte Carlo simulation are also presented. From the PDFs in Figs. 8.10 and 8.11, the probability of smallsignal angular and voltage stability of the example system is calculated and the results are given in Table 8.4. From Figs. 8.10, 8.11 and Table 8.4, it can be seen that the multi-point analytical method proposed in Sect. 8.3 gives more accurate assessment of probabilistic small-signal angular and voltage stability. Table 8.4 also indicates that although the AC/DC system is considered to be stable by the deterministic analysis of small-signal angular and voltage stability, there still exists a probability of 0.64% to be unstable in terms of angular stability and 3.24% in terms of voltage stability, when the random variations of wind generation are considered.

8.3.3

Variance Indices Analysis of Probabilistic Small-Signal Stability of a Hybrid AC/DC Power System

Traditional point-to-point HVDC connection is a mature technology to receive and supply offshore wind power. On the other hand, the multi-terminal VSC-HVDC (MTDC) technology has been rapidly developed recently, which are considered as a more favorable way for the transmission and share of large-scale wind power among different regions, such as European supergrid scheme in the North Sea [28, 29]. There are broadly two control strategies for the VSC-HVDC, namely, the constant DC voltage control scheme (or so called master-slave control) and the DC voltage droop control scheme. Traditional constant DC voltage control has been extended to the MTDC from point-to-point HVDC network. The DC voltage droop control as a novel control developed for the MTDC has more advantages than the former in the aspects of reliability and facilitation of converter outage. Thus three main factors of offshore wind power generation that may influence the probabilistic small-signal stability are the wind complementarity, connection technology and network control strategy. A variances indices analysis as a derivation from cumulant theory will be introduced in this section to study the probabilistic small-signal stability as affected by these three factors. According to [1, 2], if the relationship between a random variable ρ and m other random variables ηj, j ¼ 1, 2, . . .m is linear, that is ρ ¼ a1η1 þ a2η2 þ    þ amηm, their nth order cumulants satisfy the following equation

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

γðρnÞ ¼

m X m X



j1 ¼1 j2 ¼1

m X jn

2

329

3

4ai1 ai2   ain γðnÞ 5 |fflfflfflfflfflffl{zfflfflfflfflfflffl} ηj1 jn ¼1

ð8:30Þ

n

where γðρnÞ denotes the nth order cumulant of ρ and γðηnj Þj denotes the nth order cross 1

n

cumulant among different ηj, j ¼ 1, 2, . . .m. For a particular case of (8.30) when the m random variables ηj are independent, (i.e.,γðηnj Þj 6¼ 0only when j1 ¼ j2 ¼    ¼ jn ¼ j), 1

n

(8.30) becomes γðρnÞ ¼ a1n γðηn1Þ þ a2n γðηn2Þ þ    þ amn γðηnmÞ

ð8:31Þ

where γðηnjÞ denotes the nth order cumulant of ηj. In this section, (8.30) and (8.31) are employed as the foundation of the variance indices analysis.

8.3.3.1

Variance Index for Complementarity of Combined Wind Power

The complementarity level of combined wind power sources connected to a power system can be evaluated by many ways including the variance (or standard deviation) of the combined wind power, which is a commonly-used index to measure the fluctuation of wind power [30]. If there are m grid-connected wind farms in the power system, the combined wind power is Pwc ¼ Pw1 þ Pw2 þ    þ Pwm. Hence according to (8.30), the 2nd order cumulant of Pwc can be expressed as ð2Þ

γPwc ¼

m X m X j1 ¼1 j2 ¼1

ð2Þ

γPwj

1 j2

ð8:32Þ

ð2Þ

As the variance of Pwc, σ2Pwc , is equal to γPwc and the covariance of Pwj1 and Pwj2 ,   ð2Þ cov Pwj1 ; Pwj2 , is equal to γPwj j , (8.32) can be written as 12

σ2Pwc ¼

m X m X

  cov Pwj1 ; Pwj2 ¼ σ2Pw1 þ σ2Pw2 þ    þ σ2Pwm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} j1 ¼1 j2 ¼1 m

þ 2ρPw1 Pw2 σPw1 σPw2 þ 2ρPw1 Pw3 σPw1 σPw3 þ    þ 2ρPwm1 Pwm σPwm1 σPwm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð8:33Þ

mðm1Þ=2

where ρPwj Pwj denotes the correlation coefficient between Pwj1 and Pwj2 and σPwj 1 2 denotes the standard deviation of Pwj. Normally, wind output power of different wind farms is positively correlated, i.e. ρPwj Pwj 2 ½0; 1, if there is no special control 1 2 of wind power. Therefore, the variance of combined wind power sources assessed by (8.33) can be used as an index of their complementarity level. The smaller value of variance suggests the higher level of complementarity. It can be seen from (8.33) that the

330

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

complementarity of combined wind power sources relies on the standard deviation of each wind power and their spatial correlations.

8.3.3.2

Variance Index for Probabilistic Small-Signal Stability

The probability of small-signal stability of a power system is closely related to the mean and variance of the PDF of the critical eigenvalue real part. The mean of the PDF is mainly decided by the mean wind speed. The variance of probabilistic distribution of the critical eigenvalue real part can be assessed as follows. As discussed for (8.9), the relation between the real part of a critical eigenvalue and a wind power source can be established as Δξc ¼

m  X    Re ∂λc =∂Pwj ΔPwj

ð8:34Þ

j¼1

where ξc denotes the real part of a critical eigenvalue λc, Re() represents the real part of the sensitivity of λc with respect to each wind power Pwj and Δ is the difference between the variable wind power Pwj and its deterministic value. According to (8.30), the variance of Δξc can be derived as     m X m

X ∂λc ∂λc ð2Þ ð2Þ σ2Δξc ¼ γΔξc ¼ Re Re γΔPwj j ð8:35Þ 12 ∂Pwj1 ∂Pwj2 j ¼1 j ¼1 1

2

Since σ2ξc ¼ σ2Δξc and σPwj ¼ σΔPwj , h  i2 h  i2 ∂λc ∂λc þ    þ Re σ σ σ2ξc ¼ Re ∂P P P w1 wm ∂Pwm w1     ∂λc ∂λc þ 2Re σP σP þ    Re ρ ∂Pw1 ∂Pw2 Pw1 Pw2 w1 w2     ∂λc ∂λc þ 2Re σP σP Re ρ ∂Pwm1 ∂Pwm Pwm1 Pwm wm1 wm

ð8:36Þ

It can be seen from (8.36) that the variance of ξc PDF, σ2ξc , relies on the standard deviation of Pwj, the correlation coefficients between different wind power sources and the sensitivities of λc with respect to each wind power source. Normally, if the system under mean wind speed is stable (i.e., the mean of ξc is negative), the smaller σ2ξc leads to a higher probability of small-signal stability as the variance could affect the shape of the PDF predominantly and thus the probability considerably. Since the standard deviation and correlation are related to the complementarity of wind power and the sensitivity of the real part of the critical eigenvalue is affected by HVDC connection topology and control strategy, σ2ξc has roughly established a relationship between the complementarity, connection technology and control strategy of offshore wind power generation and the probabilistic small-signal stability of a power system.

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

8.3.4

331

Example 8.3

Variance indices analysis will be demonstrated in this section by using the example of 16-machine 5-area power system of Fig. 8.2. Offshore wind farms is connected at node 23, 24 and 29, which is named to be the first, second and third power injection point of offshore wind power generation respectively. The AC network data, system load condition, synchronous generator model and parameters are given in Appendix 8.1. Two types of the VSC-HVDC network topologies for the integration of offshore wind farms in Fig. 8.12 are discussed. The parameters of the wind generators, powerwind speed curve and wind speed distributions are (in p.u.) H ¼ 1:7s, D ¼ 0:0, xs ¼ 0:29, xr ¼ 0:29, xm ¼ 2:6, rs ¼ 0:0, rr ¼ 0:0013, Pw0 ¼ 0:3333

KP ¼ 30, KQ ¼ 30: vc ¼ 4m=s, vr ¼ 10m=s, v f ¼ 22m=s, Pr ¼ 1:0p:u:, μ ¼ 6m=s, σ ¼ 2:5m=sðCase AÞ, σ ¼ 2:0m=sðCase B; C; D; E; FÞ 2 3 1 0:9 0:3 6 7 ρ ¼ 4 0:9 1 0:5 5ðCase AÞ, 0:3

0:5

1 6 ρ ¼ 4 0:5

0:5

2

0

1

1 3 0 7 0 5ðCase B; C; D; E; FÞ

0

1

a

b

WF1

69

~ =

WF3

WF2

DC1

70

~ =

DC2

WF1

71

~ =

DC3 Offshore

69

~ =

WF3

WF2

DC1

71

70

~ =

DC2

~ =

DC3

Offshore

MTDC network

= ~ 1

DC4

= ~ 2

DC5

= ~ 3

DC6 Onshore

= ~ 1

DC4

= ~ 2

DC5

= ~

DC6 Onshore

3

Fig. 8.12 Two typical HVDC connection topologies: (a) point-to-point HVDC transmission lines; (b) multi-terminal HVDC network

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

332

Parameters of the VSC-HVDC network are (in p.u.) R14 ¼ R25 ¼ R36 ¼ 0:0006, X14 ¼ X25 ¼ X36 ¼ 0:0022ðCase AÞ R17 ¼ R37 ¼ R47 ¼ R67 ¼ 0:0007, R27 ¼ R57 ¼ 0:0003ðCase B; C; DÞ X17 ¼ X37 ¼ X47 ¼ X67 ¼ 0:0025, X27 ¼ X57 ¼ 0:0011ðCase B; C; DÞ where the central star node of star topology (ST) MTDC network is denoted as the DC7 bus. The VSC-converter model in [31] is used for all the converters. The loss of the converters is neglected. The parameters in p.u. are Rc ¼ 0.0001, Xc ¼ 0.1643. The DC voltage droop constants for Case E and F are kdroop ¼ 0.04(Case E), 0.004(Case F).

8.3.4.1

Case A (Offshore Wind Generation of Low Complementarity Level Connected by Point-to-Point HVDC)

In Case A, three point-to-point HVDC transmission lines shown in Fig. 8.12a are used to connect three offshore wind farms respectively. From the deterministic small-signal stability analysis, the 29th eigenvalue of the system state space matrix is identified to be the critical eigenvalue, i.e., λc ¼ ξc  jωc ¼  0.0043  j3.3742. Hence deterministically the system is stable. The offshore wind farms at node 69, 70 and 71 (see Fig. 8.12a) are denoted to be the first, second and third source of wind generation respectively. The sensitivity computation of critical eigenvalue real part with respect to three sources of wind generation is ∂ξc ∂ξc ∂ξc ¼ 0:0030, ¼ 0:0028, ¼ 0:0068 ∂Pw1 ∂Pw2 ∂Pw3 Low complementarity level of offshore wind generation is taken into account in this case. Three wind speed sample series that satisfy the Weibull distribution and spatial correlation given above are generated by normal transformation method [19, 20] or Copulas method [21]. Then the corresponding wind power sample series can be obtained by employing the power-wind speed curve. Thus, the standard deviation of each wind power and their correlation coefficients can be calculated to be σPw1 ¼ σPw2 ¼ σPw3 ¼ 0:3271, ρPw1 Pw2 ¼ 0:8827, ρPw1 Pw3 ¼ 0:2750, ρPw2 Pw3 ¼ 0:4643 The variance indices of complementarity level and probabilistic distribution are computed by use of (8.33) and (8.36) to be

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

333

σ2Pwc ¼ σ2Pw1 þ σ2Pw2 þ σ2Pw3 þ 2ρPw1 Pw2 σPw1 σPw2 þ 2ρPw1 Pw3 σPw1 σPw3 þ 2ρPw2 Pw3 σPw2 σPw3 ¼ 6:6825  101 h i2 h i2 h i2 ∂ξc ∂ξc ∂ξc σ σ σ σ2ξc ¼ ∂P þ þ P P P w1 w2 w3 ∂Pw2 ∂Pw3 w1 þ2

∂ξc ∂ξc ∂ξc ∂ξc ρ σP σP þ 2 ρ σP σP ∂Pw1 ∂Pw2 Pw1 Pw2 w1 w2 ∂Pw1 ∂Pw3 Pw1 Pw3 w1 w3

þ2

∂ξc ∂ξc ρ σP σP ¼ 1:1492  105 ∂Pw2 ∂Pw3 Pw2 Pw3 w2 w3

Monte Carlo simulation (with 5000 iterations) is carried out and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.13. A comparison of computation time between Monte Carlo simulation and variance index computation has been carried out. Based on the same computational resource (Dell OptiPlex 745, Intel Core 2 CPUs 2.66 GHz, 3 GB RAM), the time of Monte Carlo simulation with 5000 iterations is 28954.39 s, while only 14.32 s for the variance calculation, which is more than 2000 times faster than Monte Carlo simulation. According to the PDF in Fig. 8.13, it can be obtained that σ2ξc ¼ 1:1310  105 Ð0 Pðξc < 0Þ ¼ 1 f ξc ðxÞdx ¼ 0:8484

ξc PDF 450

Monte Carlo sample number

400 350 300 250 200 150 100 50 0

-8

-6

-4

-2

0

ξc

Fig. 8.13 PDF of ξc obtained by Monte Carlo simulation (Case A)

2

4 x 10-3

334

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

The result has verified the variance computation by using (8.36). It is indicated that although the system is considered to be stable by the deterministic analysis of small-signal stability, the system has a high probability of 15.16% to be unstable due to the low complementarity level of combined wind power, when the uncertainty of wind generation is considered.

8.3.4.2

Case B (Offshore Wind Generation of High Complementarity Level Connected by Point-to-Point HVDC)

In this case, high complementarity level of offshore wind generation is considered and the load condition of Case A is used. Hence, the results of deterministic smallsignal stability analysis and sensitivity computation are exactly the same as Case A. By employing the same procedure as in Case A, the standard deviation of three wind power and their correlation coefficients are calculated to be σPw1 ¼ σPw2 ¼ σPw3 ¼ 0:2860, ρPw1 Pw2 ¼ 0:4847, ρPw1 Pw3 ¼ 0:0005, ρPw2 Pw3 ¼ 0:0011 By use of (8.33) and (8.36), the variance indices of complementarity level and probabilistic distribution of ξc are computed to be σ2Pwc ¼ 3:2489  101 , σ2ξc ¼ 5:8724  106 which are smaller than that obtained in Case A. This is brought about by the decrease of variation of each wind power and their comparatively weak correlations. Monte Carlo simulation (with 5000 iterations) is carried out to confirm the variances calculated above and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.14. According to the PDF in Fig. 8.14, it can be obtained that σ2ξc ¼ 5:8222  106 Ð0 Pðξc < 0Þ ¼ 1 f ξc ðxÞdx ¼ 0:9454 The result indicates that in the case of high complementarity condition, the probability of ξc remaining in the left half-plane increases to 94.54% due to variance reduction when the stochastic variation of wind generation is considered. Hence, it is successfully demonstrated by Case B that the effectiveness of complementarity of wind generation to enhance probabilistic small-signal stability of power systems even when the offshore wind farms are separately connected to different buses of power system by point-to-point HVDC topology. In other words, impact of complementarity does not rely on the physical combination of wind power by MTDC network. Moreover, it can be seen from (8.36) that high complementarity level may also lead to the reduction of probability of small-signal stability if the sensitivities of ξc have different signs.

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

335

ξc PDF

80

Monte Carlo sample number

70 60 50 40 30 20 10 0

-8

-6

-4

-2

0

ξc

2

4 x 10

-3

Fig. 8.14 PDF of ξc obtained by Monte Carlo simulation (Case B)

8.3.4.3

Case C (Offshore Wind Generation of High Complementarity Level Connected by MTDC with Constant DC Voltage Control)

The MTDC network in Fig. 8.12b is applied to connect the three offshore wind farms and traditional constant DC voltage control scheme is employed. MTDC network can be achieved by various topologies such as general ring topology (GRT), star topology (ST), wind farm ring topology (WFRT) and substation ring topology (SSRT) [32]. In this case, the ST MTDC network configuration is selected. The results of deterministic small-signal stability analysis and the standard deviation of each wind power and their correlation coefficients are the same as in Case B since the same load condition and wind speed distribution are applied. However, due to the change of the HVDC connection, the sensitivity of ξc with respect to three sources of wind generation changes to be ∂ξc ∂ξc ∂ξc ¼ 0:0028 ∂Pw1 ∂Pw2 ∂Pw3 The variance index of probabilistic distribution of ξc are computed by using (8.36) to be σ2ξc ¼ 2:4999  106 Compared with Case B, σ2ξc decreases which is caused by the decrease of sensitivities when σ2Pwc ¼ 3:2489  101 . Monte Carlo simulation (with 5000

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

336

ξc PDF

70

Monte Carlo sample number

60 50 40 30 20 10 0

-7

-6

-5

-4

-3 ξc

-2

-1

0

1 x 10-3

Fig. 8.15 PDF of ξc obtained by Monte Carlo simulation (Case C)

iterations) is carried out and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.15. From Fig. 8.15, it can be obtained that σ2ξc ¼ 2:4773  106 Ð0 Pðξc < 0Þ ¼ 1 f ξc ðxÞdx ¼ 0:9944 It confirms the computational results of σ2ξc previously by using (8.36). The result also shows that in the case of the MTDC connection, a very high probability, nearly 100%, to be stable in terms of small-signal stability is achieved owing to the reduction of σ2ξc . Additionally, it can be concluded that in the case of the MTDC network, high complementarity level can certainly enhance the probabilistic smallsignal stability of a power system due to the identical sensitivity.

8.3.4.4

Case D (Offshore Wind Generation Connected by MTDC with Constant DC Voltage Control and DC Slack Bus at Different Location)

In Case D, the results of deterministic small-signal stability analysis and the standard deviation of each wind power and their correlation coefficients are the same as in Case B and C. Three offshore wind farms are still connected by the ST MTDC network with the constant DC voltage control. However, the location of DC voltage

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

337

ξc PDF

70

Monte Carlo sample number

60 50 40 30 20 10 0

-0.01

-0.005

0

0.005

0.01

ξc

Fig. 8.16 PDF of ξc obtained by Monte Carlo simulation (Case D)

control of the MTDC has been changed from DC5 to DC6. Hence, the sensitivity of ξc with respect to three sources of wind generation becomes ∂ξc ∂ξc ∂ξc ¼ 0:0068 ∂Pw1 ∂Pw2 ∂Pw3 The variance index of probabilistic distribution of ξc are computed to be σ2ξc ¼ 1:5114  105 , which increases due to the increase of sensitivities. Monte Carlo simulation (with 5000 iterations) is carried out and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.16. From Fig. 8.16, it can be obtained that σ2ξc ¼ 1:5481  105 ð0 Pð ξ c < 0 Þ ¼ f ξc ðxÞdx ¼ 0:8402 1

It confirms the computational result of σ2ξc previously and also indicates that the probability of ξc remaining negative drops considerably to 84.02% even the complementarity level of combined wind power is high. Therefore, in order to improve the probability of stability, the DC voltage control should be located at the AC bus with comparatively small sensitivity of ξc with respect to wind power, so that the stochastic variation of wind power can have less impact on the probabilistic smallsignal stability.

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

338

8.3.4.5

Case E (Offshore Wind Generation Connected by MTDC with DC Voltage Droop Control)

In Case E, three offshore wind farms are connected by the ST MTDC network with DC voltage droop control and other preconditions stay the same as in Case B, C and D. there are two types of DC voltage droop control schemes as the linear droop relation can be established either between DC bus voltage and injected DC power or between DC bus voltage and injected DC current [33]. When the injected DC power related voltage droop control is used, the sensitivity of ξc with respect to three sources of wind generation becomes ∂ξc ∂ξc ∂ξc ¼ 0:0042 ∂Pw1 ∂Pw2 ∂Pw3 It is reasonable that the sensitivities above are almost identical due to the symmetry of the DC network and the value of the sensitivities (0.0042) is between 0.0028 and 0.0068 as the change of wind power is distributed on each grid-side converter. The variance index of probabilistic distribution of ξc are computed to be σ2ξc ¼ 5:7591  106 , which is very close to that in Case B because same complementarity level of wind generation is assumed and the change of combination of the sensitivity related to each grid-side converter is quite moderate, which is a combination of the sensitivity related to each grid-side converter. Monte Carlo simulation (with 5000 iterations) is carried out and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.17. ξc PDF 70

Monte Carlo sample number

60 50 40 30 20 10 0 -10

-8

-6

-4

-2

0

ξc

Fig. 8.17 PDF of ξc obtained by Monte Carlo simulation (Case E)

2

4 x 10-3

8.3 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

339

From the PDF in Fig. 8.17, it can be obtained that σ2ξc ¼ 5:6946  106 ð0 Pð ξ c < 0 Þ ¼ f ξc ðxÞdx ¼ 0:9491 1

The probability of ξc remaining negative is only a little bit higher than the probability in Case B. Hence the impact of MTDC connection on the probabilistic small-signal stability is not predominant when the DC voltage droop control scheme is applied due to the moderate sensitivity. The sensitivity and variance index of ξc are computed and Monte Carlo simulation is carried out also for the injected DC current related voltage droop control scheme as follows. ∂ξc ∂ξc ∂ξc ¼ 0:0042 ∂Pw1 ∂Pw2 ∂Pw3 σ2ξc ¼ 5:7653  106 σ2ξc ¼ 5:7007  106 ðMonte CarloÞ ð0 f ξc ðxÞdx ¼ 0:9490 Pð ξ c < 0 Þ ¼ 1

It can be seen that calculating results of the two types of droop control schemes are quite similar.

8.3.4.6

Case F (Offshore Wind Generation Connected by MTDC with DC Voltage Droop Control and Using Different DC Voltage Droop Constant)

Compared with Case E, a smaller DC voltage droop constant is selected in Case F for both types of the droop control schemes. The sensitivity of ξc with respect to three sources of wind generation is computed to be ∂ξc ∂ξc ∂ξc 0:0044 ∂Pw1 ∂Pw2 ∂Pw3 The sensitivity increases slightly since the injected DC power of grid-side converter becomes more sensitive to the DC voltage change brought about by the variation of wind power. The variance index of probabilistic distribution of ξc are computed to be σ2ξc ¼ 6:1923  106 . Monte Carlo simulation (with 5000 iterations) is carried out and the PDF of ξc by Monte Carlo simulation is displayed in Fig. 8.18.

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

340

ξc PDF 70

Monte Carlo sample number

60 50 40 30 20 10 0 -10

-5

0 ξc

5 x 10-3

Fig. 8.18 PDF of ξc obtained by Monte Carlo simulation (Case F)

From the PDF in Fig. 8.18, it can be obtained that σ2ξc ¼ 6:1214  106 ð0 Pð ξ c < 0 Þ ¼ f ξc ðxÞdx ¼ 0:9388 1

It can be seen that the decrease of the DC voltage droop constant has weakened the probabilistic small-signal stability compared to Case E owing to the rise of the sensitivity of ξc with respect to wind power. That is to say, probabilistic stability could benefit from the MTDC connection if DC voltage droop constants are increased. The injected DC current related voltage droop control scheme is also examined with the following results. ∂ξc ∂ξc ∂ξc 0:0047 ∂Pw1 ∂Pw2 ∂Pw3 σ2ξc ¼ 7:2149  106 σ2ξc ¼ 7:1293  106 ðMonte CarloÞ ð0 f ξc ðxÞdx ¼ 0:9218 Pð ξ c < 0 Þ ¼ 1

Appendix 8.1: Data of Examples 8.1, 8.2 and 8.3

341

It can be observed that the critical eigenvalue ξc is more sensitive to the variation of wind generation in this case mainly because one unit change in the injected current will cause two units change in the injected power for a bipole DC grid.

Appendix 8.1: Data of Examples 8.1, 8.2 and 8.3 Example 16-Machine 68-Bus New York and New England Power System [34] (Tables 8.5, 8.6 and 8.7) All the synchronous generators employ sixth-order detailed model with damping D ¼ 0.0. The structure of first-order excitation system model is shown by Fig. 8.19. The parameters of excitation system model are KA ¼ 7.4, TA ¼ 0.1s, efdmax ¼ 10.0, efdmin ¼  10.0 (Example 8.1). KA ¼ 2.85, TA ¼ 0.1s, efdmax ¼ 10.0, efdmin ¼  10.0 (Example 8.2). KA ¼ 3.95, TA ¼ 0.1s, efdmax ¼ 10.0, efdmin ¼  10.0 (Example 8.3).

Table 8.5 Bus data Bus no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Voltage – – – – – – – – – – – – – – – – – – –

Angle – – – – – – – – – – – – – – – – – – –

Active power consumption 2.5270 0.0000 3.2200 5.0000 0.0000 0.0000 2.3400 5.2200 1.0400 0.0000 0.0000 0.0900 0.0000 0.0000 3.2000 3.2900 0.0000 1.5800 0.0000

Reactive power consumption 1.1856 0.0000 0.0200 1.8400 0.0000 0.0000 0.8400 1.7700 1.2500 0.0000 0.0000 0.8800 0.0000 0.0000 1.5300 0.3200 0.0000 0.3000 0.0000

Active power generation – – – – – – – – – – – – – – – – – – –

Reactive power generation – – – – – – – – – – – – – – – – – – – (continued)

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

342 Table 8.5 (continued) Bus no. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Voltage – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 1.0450 0.9800 0.9830 0.9970 1.0110 1.0500 1.0630 1.0300

Angle – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Active power consumption 6.8000 2.7400 0.0000 2.4800 3.0900 2.2400 1.3900 2.8100 2.0600 2.8400 0.0000 0.0000 0.0000 1.1200 0.0000 0.0000 1.0200 60.0000 0.0000 2.6700 0.6563 10.0000 11.5000 0.0000 2.6755 2.0800 1.5070 2.0312 2.4120 1.6400 1.0000 3.3700 24.7000 – – – – – – – –

Reactive power consumption 1.0300 1.1500 0.0000 0.8500 0.9200 0.4700 0.1700 0.7600 0.2800 0.2700 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1946 3.0000 0.0000 0.1260 0.2353 2.5000 2.5000 0.0000 0.0484 0.2100 0.2850 0.3259 0.0220 0.2900 1.4700 1.2200 1.2300 – – – – – – – –

Active power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 2.5000 5.4500 6.5000 6.3200 5.0520 7.0000 5.6000 5.4000

Reactive power generation – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – (continued)

Appendix 8.1: Data of Examples 8.1, 8.2 and 8.3

343

Table 8.5 (continued) Bus no. 61 62 63 64 65 66 67 68

Voltage 1.0250 1.0100 1.0000 1.0156 1.0110 1.0000 1.0000 1.0000

Angle – – – – 0.0000 – – –

Active power consumption – – – – – – – –

Reactive power consumption – – – – – – – –

Active power generation 8.0000 5.0000 10.0000 13.5000 – 17.8500 10.0000 40.0000

Reactive power generation – – – – – – – –

Table 8.6 Line data Branch no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

From bus 1 1 2 2 2 3 3 4 4 5 5 6 6 6 7 8 9 10 10 10 12 12 13 14 15 16 16 16

To bus 2 30 3 25 53 4 18 5 14 6 8 7 11 54 8 9 30 11 13 55 11 13 14 15 16 17 19 21

R 0.0035 0.0008 0.0013 0.0070 0.0000 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0.0000 0.0004 0.0023 0.0019 0.0004 0.0004 0.0000 0.0016 0.0016 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008

X 0.0411 0.0074 0.0151 0.0086 0.0181 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.0250 0.0046 0.0363 0.0183 0.0043 0.0043 0.0200 0.0435 0.0435 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135

B 0.6987 0.4800 0.2572 0.1460 0.0000 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.1130 0.1389 0.0000 0.0780 0.3804 0.2900 0.0729 0.0729 0.0000 0.0000 0.0000 0.1723 0.3660 0.1710 0.1342 0.3040 0.2548

Transformer ratio 1 1 1 1 1.025 1 1 1 1 1 1 1 1 1.07 1 1 1 1 1 1.07 1.06 1.06 1 1 1 1 1 1 (continued)

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

344 Table 8.6 (continued) Branch no. 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

From bus 16 17 17 19 19 20 21 22 22 23 23 25 25 26 26 26 28 29 9 9 9 36 34 35 33 32 30 30 1 31 33 38 46 1 47 47 48 35 37 43 44 39

To bus 24 18 27 20 56 57 22 23 58 24 59 26 60 27 28 29 29 61 30 36 36 37 36 34 34 33 31 32 31 38 38 46 49 47 48 48 40 45 43 44 45 44

R 0.0003 0.0007 0.0013 0.0007 0.0007 0.0009 0.0008 0.0006 0.0000 0.0022 0.0005 0.0032 0.0006 0.0014 0.0043 0.0057 0.0014 0.0008 0.0019 0.0022 0.0022 0.0005 0.0033 0.0001 0.0011 0.0008 0.0013 0.0024 0.0016 0.0011 0.0036 0.0022 0.0018 0.0013 0.0025 0.0025 0.0020 0.0007 0.0005 0.0001 0.0025 0.0000

X 0.0059 0.0082 0.0173 0.0138 0.0142 0.0180 0.0140 0.0096 0.0143 0.0350 0.0272 0.0323 0.0232 0.0147 0.0474 0.0625 0.0151 0.0156 0.0183 0.0196 0.0196 0.0045 0.0111 0.0074 0.0157 0.0099 0.0187 0.0288 0.0163 0.0147 0.0444 0.0284 0.0274 0.0188 0.0268 0.0268 0.0220 0.0175 0.0276 0.0011 0.0730 0.0411

B 0.0680 0.1319 0.3216 0.0000 0.0000 0.0000 0.2565 0.1846 0.0000 0.3610 0.0000 0.5310 0.0000 0.2396 0.7802 1.0290 0.2490 0.0000 0.2900 0.3400 0.3400 0.3200 1.4500 0.0000 0.2020 0.1680 0.3330 0.4880 0.2500 0.2470 0.6930 0.4300 0.2700 1.3100 0.4000 0.4000 1.2800 1.3900 0.0000 0.0000 0.0000 0.0000

Transformer ratio 1 1 1 1.06 1.07 1.009 1 1 1.025 1 1 1 1.025 1 1 1 1 1.025 1 1 1 1 1 0.946 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (continued)

Appendix 8.1: Data of Examples 8.1, 8.2 and 8.3

345

Table 8.6 (continued) Branch no. 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

From bus 39 45 50 50 49 52 42 41 31 32 36 37 41 42 52 1

To bus 45 51 52 51 52 42 41 40 62 63 64 65 66 67 68 27

R 0.0000 0.0004 0.0012 0.0009 0.0076 0.0040 0.0040 0.0060 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0320

X 0.0839 0.0105 0.0288 0.0221 0.1141 0.0600 0.0600 0.0840 0.0260 0.0130 0.0075 0.0033 0.0015 0.0015 0.0030 0.3200

B 0.0000 0.7200 2.0600 1.6200 1.1600 2.2500 2.2500 3.1500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4100

Transformer ratio 1 1 1 1 1 1 1 1 1.04 1.04 1.04 1.04 1 1 1 1

Table 8.7 Machine data Generator no. G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 Generator No. G1 G2

Bus name B53 B54 B55 B56 B57 B58 B59 B60 B61 B62 B63 B64 B65 B66 B67 B68 Td0 ðsÞ

Generator base (MVA) 100 100 100 100 100 100 100 100 100 100 100 100 200 100 100 200 xq

xq

xq

Tq0 ðsÞ

Tq0 ðsÞ

Td0 ðsÞ 10.20 6.56 5.70 5.69 5.40 7.30 5.66 6.70 4.79 9.37 4.10 7.40 5.90 4.10 4.10 7.80 M

0.05 0.05

0.0690 0.2820

0.0280 0.0600

0.0250 0.0500

1.50 1.50

0.035 0.035

42.0 30.2

00

xa 0.0125 0.0350 0.0304 0.0295 0.0270 0.0224 0.0322 0.0280 0.0298 0.0199 0.0103 0.0220 0.0030 0.0017 0.0017 0.0041 0

xd 0.1000 0.2950 0.2495 0.2620 0.3300 0.2540 0.2950 0.2900 0.2106 0.1690 0.1280 0.1010 0.0296 0.0180 0.0180 0.0356 00

0

xd 0.0310 0.0697 0.0531 0.0436 0.0660 0.0500 0.0490 0.0570 0.0570 0.0457 0.0180 0.0310 0.0055 0.00285 0.00285 0.0071 0

00

xd 0.0250 0.0500 0.0450 0.0350 0.0500 0.0400 0.0400 0.0450 0.0450 0.0400 0.0120 0.0250 0.0040 0.0023 0.0023 0.0055 00

0

(continued)

8 Probabilistic Analysis of Small-Signal Stability of a Power System. . .

346 Table 8.7 (continued) Generator no. Generator No. G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16

Bus name Td0 ðsÞ

Generator base (MVA) xq

xq

xq

Tq0 ðsÞ

Tq0 ðsÞ

Td0 ðsÞ M

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.2370 0.2580 0.3100 0.2410 0.2920 0.2800 0.2050 0.1150 0.1230 0.0950 0.0286 0.0173 0.0173 0.0334

0.0500 0.0400 0.0600 0.0450 0.0450 0.0500 0.0500 0.0450 0.0150 0.0280 0.0050 0.0025 0.0025 0.0060

0.0450 0.0350 0.0500 0.0400 0.0400 0.0450 0.0450 0.0400 0.0120 0.0250 0.0040 0.0023 0.0023 0.0055

1.50 1.50 0.44 0.40 1.50 0.41 1.96 1.50 1.50 1.50 1.50 1.50 1.50 1.50

0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035

35.8 28.6 26.0 34.8 26.4 24.3 34.5 31.0 28.2 92.3 248.0 300.0 300.0 225.0

00

xa

0

xd

0

xd

00

0

e fd 0

Vtref

+ Vt



Σ

KA 1 + TAs

Δe fd +

+ Σ

00

xd 00

0

e fd max

e fd

e fd min Fig. 8.19 The first-order excitation system model of synchronous generator

References 1. Kendall M (1987) Kendall’s advanced theory statistics. Oxford University Press, New York, NY 2. Cramer H (1946) Numerical methods of statistics. Princeton University Press, Princeton, NJ 3. Ma J, Dong ZY, Zhang P (2006) Eigenvalue sensitivity analysis for dynamic power system. In: International conference on power system technology, Chongqing, pp 22–26 4. Dong ZY, Pang CK, Zhang P (2005) Power system sensitivity analysis for probabilistic small signal stability assessment in a deregulated environment. Int J Control Automat Syst 3:355–362 5. Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, New York 6. Xu Z, Dong ZY, Zhang P (2005) Probabilistic small signal analysis using Monte Carlo simulation. In: IEEE PES SM Paper 2: San Francisco, pp 1658–1664 7. Xu Z, Ali M, Dong ZY, Li X (2006) A novel grid computing approach for probabilistic small signal analysis. In: IEEE PES SM Paper, Montreal, Que. 8. Burchett RC, Heydt GT (1978) Probabilistic methods for power system dynamic stability studies. IEEE Trans Power Appar Syst PAS-97:695–702

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9. Burchett RC, Heydt GT (1978) A generalized method for stochastic analysis of the dynamic stability of electric power system. IEEE PES SM Paper, A78528 10. Brucoli M, Torelli F, Trovato M (1981) Probabilistic approach for power system dynamic stability studies. IEE Proc 128(5):295–301 11. Wang KW, Tse CT, Tsang KM (1997) Algorithm for power system dynamic stability studies taking account the variation of load power. In: 4th International conference on advances in power system control operation and management 2(2), Hong Kong, pp 445–450 12. Pang CK, Dong ZY, Zhang P, Yin X (2005) Probabilistic analysis of power system Small signal stability region. In: International conference on control and automation 1(1), Budapest, pp 503–509 13. Wang KW, Chung CY, Tse CT, Tsang KM (2000) Improved probabilistic method for power system dynamic stability studies. IEE Proc Gener Transm Distrib 147(1):37–43 14. Yi HQ, Hou YH, Cheng SJ, Zhou H, Chen GG (2007) Power system probabilistic small signal stability analysis using two point estimation method. In: 402–407, UPEC, Brighton 15. Xu XL, Lin T, Zhang XM (2009) Probabilistic analysis of small signal stability of microgrid using point estimate method. In: International conference on SUPERGEN, Nanjing, pp 1–6 16. Li MY, Ma J, Dong ZY (2009) Uncertainty analysis of load models in small signal stability. In: International conference on SUPERGEN, Nanjing, pp 1–6 17. Bu SQ (2012) Probabilistic small-signal stability analysis and improved transient stability control strategy of grid-connected doubly fed induction generators in large-scale power systems. PhD thesis. 18. Freris L, Infield D (2008) Renewable energy in power systems. John Wiley and Sons 19. Usaola J (2010) Probabilistic load flow with correlated wind power injections. Electr Power Syst Res 80(5):528–536 20. Morales JM, Baringo L, Conejo AJ, Minguez R (2010) Probabilistic power flow with correlated wind sources. IET Gener Transm Distrib 4(5):641–651 21. Papaefthymiou G, Kurowicka D (2009) Using copulas for modeling stochastic dependence in power system uncertainty analysis. IEEE Trans Power Syst 24(1):40–49 22. Abouzahr I, Ramakumar R (1991) An approach to assess the performance of utility-interactive wind electric conversion systems. IEEE Trans Energy Conver 6:627–638 23. Akagi H, Sato H (2002) Control and performance of a doubly-fed induction machine intended for a flywheel energy storage system. IEEE Trans Power Electron 17:109–116 24. Gao B, Morison GK, Kundur P (1992) Voltage stability evaluation using modal analysis. IEEE Trans Power Syst 7(4):1529–1542 25. Aik D, Andersson G (1997) Voltage stability analysis of multi-infeed HVDC systems. IEEE Trans Power Deliver 12(3):1309–1318 26. Aik D, Andersson G (1998) Use of participation factors in modal voltage stability analysis of multi-infeed HVDC systems. IEEE Trans Power Deliver 13(1):203–211 27. Allan R, Silva A (1981) Probabilistic load flow using multilinearizations. IEE Proc 128(5) 28. Blau J (2010) Europe plans a north sea grid. IEEE Spectr 47(3):12–13 29. Rudion K, Orths A, Eriksen PB, Styczynski ZA (2010) Toward a benchmark test system for the offshore grid in the north sea. In: IEEE power and energy society general meeting, Providence, RI, pp 1–8. 30. Simonsen T, Stevens B (2004) Regional wind energy analysis for the contral United States. In: Proceedings of the global wind power conference, pp 1–16 31. Beerten J, Cole S, Belmans R (2012) Generalized steady-state VSC MTDC model for sequential AC/DC power flow algorithms. IEEE Trans Power Syst 27(2):821–829 32. Gomis-Bellmunt O, Liang J, Ekanayake J, King R, Jenkins N (2010) Topologies of multiterminal HVDC-VSC transmission for large offshore wind farms. Electr Power Syst Res 81:271–281 33. Beerten J, Belmans R (2012) Modeling and control of multi-terminal VSC HVDC systems. Energy Procedia 24:123–130 34. Rogers G (2000) Power system oscillations. Kluwer, Norwell, MA

Index

A AC network data, system load condition, 331 AC/DC power system, 321, 322 Aggregate model, 69–70 Analytical and numerical comparison analysis, 86 Analytical method, 306, 308 Automatic voltage regulators (AVRs), 10, 104

B Base case comparison study, 80–84 Bus Data, 89, 139, 341

C Closed-loop and open-loop EOM, 112 Closed-loop and open-loop state matrix, 125 Closed-loop converter oscillation mode, 166, 171 Closed-loop interconnected model, 98, 100, 105, 149 DFIG, 61–62 PMSG, 59–60 Closed-loop interconnected system, 112, 150–152 Closed-loop power system, 113 Closed-loop state space model, 8 Closed-loop system, 14, 15 Combined wind power, 329–330 Conventional modal analysis, 98 Converter control-based generators (CCBGs), 5 Converter oscillation modes, 153–156, 158–159 Copulas method, 332

Cumulant theory, 306, 308–315 Cumulative Distribution Function (CDF), 304

D Damping contributions, 75 Damping torque analysis (DTA), 7, 72, 80, 98, 100, 150, 243, 253–254 computation’ methods, 64 DFIG, 73 FSIG and DFIG possess, 63, 64 generic implementation framework, 70 grid-connected wind power induction generators, 63–65 WPIGs, 64, 69 Damping torque contribution, 106, 107 DC capacitor, 216 DC voltage control, 337 DC voltage droop control, 338 Decomposed modal analysis, 97–103, 105, 108 DFIG, 98 DFIG oscillation mode (DOM), 178–182, 188, 237 DFIG rotor-side converter, 317 Diagonal matrix/block diagonal matrix, 57 Distribution function, 312 Doubly-fed induction generators (DFIGs), 316 active and reactive power injected, 54 basic data, 143 bus data, 145 comparison analysis, 79 computational results, 110 constant rotor voltage, 77–79 damping coefficient, 73 damping contributions, 79

© Springer International Publishing AG, part of Springer Nature 2018 W. Du et al., Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, https://doi.org/10.1007/978-3-319-94168-4

349

350 Doubly-fed induction generators (DFIGs) (cont.) damping effectiveness, 83 dynamic components, 84 dynamic interactions, 103, 117, 118, 122, 128 eigenvalue, 81 electrical function, 40 EOM, 118 fourteenth-order model, 105 FSIG dynamics, 83 FSIG featured, 80 FSIG, 63, 64 gains of GSC, 143 generator-side dynamics, 77 grid connection study, 64 grid connection, 97 GSC and associated control system, 49–53 induction generator and two-mass shaft rotational system, 41–45 internal dynamics, 81 line data, 144 linearized model diagram, 76, 77 machine data, 145 offset rotor voltage, 77 PQ control, 80 q-axis and d-axis current control, 121 rotor circuit, 79 rotor motion equations, 121 RSC and associated control system, 45–49 RSC control system model, 74 RSC controller parameters, 80 RSC dynamics, 75–77 RSC vector control system, 119 RSC, 73 side linearized model diagram, 78 simplified transfer function model, 118–121 state equations, 53 steady-state value, 78 terminal voltage control vs. reactive power control, 122–124 transfer functions, 82 wind power generation, 40 WPIG reference, 75 Duel second-order generalized integrator phase locked loop PLL (DSOGI PLL), 222, 223, 235–236 Dynamic interactions, 105, 147, 163, 172–199

E Eigenvalue analysis, 324 Eigenvalue variation, 82

Index Electromechanical low-frequency oscillation, 147 Electromechanical oscillation modes (EOMs), 2, 82, 98, 202 DFIG, 236–239 DSOGI PLL, 223, 235–236 GSC vector control system, 231 modal resonance, 226, 228 NEPS, 222, 225 OLMR, 229 open-loop AC power system, 252 PLL gains, 227–232 power system operating conditions, 224–226 reactive power output, 230 relative rotor position, 230 SCR and wind power penetration, 232–235 VSC-based system, 224 Example power system, 104–105, 157 Excitation system model, 88, 341

F Filter reactance, 215 First-order excitation system model, 94 First-Order Second Moment Method (FOSMM), 307 Fixed-speed induction generator (FSIG), 63

G General ring topology (GRT), 335 Gram-Charlier expansion, 311, 317, 318 Grid connection, 148 Grid side converter (GSC), 13, 33–38, 49–53, 203 Grid-connected wind power, 312, 313 Grid-connected wind power induction generators, 63–64, 70–72 Grid-side converter (GSC), 73, 148 active power control, 155 active power control outer loop, 156 active power control path, 155 configuration, 154, 158 converter control system, 154 dynamic function, 181 OLMR, 159 parameters, 158–160 PMSG1, 167 reactive control path, 155, 156 reactive power control, 155, 167 GSC control system, 154

Index H Hooke-Jeeves direct searching algorithm, 134 HVDC networks, 320

I Impedance model, 55 Index of dynamic interactions (IDI) advantages, 117 analytical expression, 116 computational burden, 117 derivation, 113–116 EOM, 114, 116 estimation error, 115 grid-connected DFIG, 111–113 mathematical derivation, 115 modal sensitivity, 114 Induction generator parameters, 88 Input impedance matrix model, 13 Inter-area oscillation modes, 107

J Jacobian matrix, 326

L Line Data, 91, 140, 343 Linearized dynamic equation, 154 Linearized state-space model, 97 Linearized voltage equation, 153 Load flow change, 108–109

M Machine Data, 93, 142, 345 Machine side converter (MSC), 30–33 Modal resonance, 255 Modal sensitivity, 114, 116 Mode-to-parameter sensitivity, 261, 263 Monte Carlo simulation, 306, 307, 318, 319, 322, 333–335, 338–340 MTDC network, 334, 336 Multi-machine power system, 55, 68, 97, 98, 100, 173, 174 Multiple-input and multiple-output (MIMO), 18 Multi-point linearization technique, 322–324 Multi-terminal DC (MTDC) AC power system, 243–247 active/reactive power control, 289–291 CIGRE five-terminal DC network, 266–269, 283–285 closed-loop model, 251–252

351 constant power injections, 243 converter control system, 249 DC voltage control, 286–288 DC voltage droop control, 263–265 DTA, 253–254 dynamic interactions, 254–256 eight-terminal DC network, 277–283 input-output signal vectors, 247 Laplace operator, 247 linearized active and reactive power output, 249 master-slave control, 257–260 NETS, 256–257 network equation, 247 OLMR and near strong modal resonance, 261–263 open-loop modal analysis, 292–294 parameter tuning, 297–298 PI controllers, 248 small-signal instability risk, 295–296 small-signal stability, 243, 244 transfer function matrix model, 251 VSC active power control, 268–271 DC voltage control/voltage droop control, 272–273 OLMR, 273–277 wind power output, 247 wind power transmission, 243 Multi-terminal VSC-HVDC (MTDC) technology, 328 Multivariable closed-loop interconnected model, 173–174

N New England Power System (NEPS), 124, 125, 128, 143, 157, 166, 182, 194, 197, 222, 223, 240, 341 New England test system (NETS), 256–257 Non-linear simulation, 259 Normal distribution, 306 Normal transformation method, 332

O Offshore wind farms, 331 Offshore wind generation, 332–334 Offshore wind power generation, 328 Open-loop and closed-loop converter oscillation mode, 169 Open-loop converter oscillation mode, 151 Open-loop DFIG oscillation modes, 181

352 Open-loop eigenvalues, 185 Open-loop modal residue, 293 Open-loop modal resonance (OLMR) characteristic equation, 176 condition, 151, 206–207 consequence, 148 converter oscillation mode, 166 DFIG and the power system, 195 DFIG1 and DFIG2, 197, 199 DFIGs, 176, 196 dynamic interactions, 151, 161, 190 EOM, 162, 166, 195 EOMs, 166, 171 equations, 178 estimation, 167 GSC, 148 GSC active power control, 159 impact of, 207–209 multiple DFIGs, 196–199 non-linear simulation, 199 open-loop DFIG oscillation mode, 173 open-loop power system, 176 PLL current control inner loops of GSC, 213–214 DC voltage control outer loop of GSC, 211–212 PMSG1, 164–167 PMSG1 and power system, 163 PMSG2, 169 PMSG2 and power system, 168, 170 PMSG3, 171 PMSG3 and power system, 171 PMSGs, 148, 156, 159, 172 power control outer loop, 159 power system, 152 RSC active power control system, 186–190 RSC reactive power control system, 193–195 Open-loop state matrix, 197 Open-loop system, 15, 84

P Participation factors (PFs), 7, 212, 226, 227, 258, 264 Permanent magnet synchronous generators (PMSGs) active power exchange, 165 active power output, 163 configuration, 27, 28, 153 converter oscillation modes, 147, 159 DFIG, 27

Index dynamic interactions, 148–156, 160–162, 167–170 dynamics, 168 EOM, 150, 171 grid-connection, 148–151, 166, 170–171 GSC and associated control system, 33–38 GSC, 240 line data, 240 MSC, 240 MSC and associated control system, 30–33 multi-machine power system, 148 New England power system, 157 OLMR, 163, 164, 166, 167, 170 open-loop converter oscillation mode, 148, 151, 168 participation, 164 power exchange, 161 power output, 39 power system, 149, 150, 155, 156, 160, 162, 169, 171 reactive power output, 169 SG, 27–30 SRF PLL, 240 state-space and transfer function models, 149 state-space model, 40 total participation, 161 transfer function model, 155 wind power generation, 147, 148 Phase locked loop (PLL) AC power system, 202 bandwidth, dynamic component, 209 closed-loop control system, 201 closed-loop model, 204–206 current vector control algorithm, 201 DC voltage control system, 209 dynamic interactions, 201, 202 EOMs (see Electromechanical oscillation modes (EOMs)) external power system, 203 external power system, PMSG infinite AC busbar, 210 non-linear simulation, 219, 222 OLMR (see Open-loop modal resonance (OLMR)) SRF, 210 weak network connection, 214–219 wind farm connection, 219–221 function, 203–204 grid-connected wind farm, 209 multi-machine power system, 203 OLMR, 206–209

Index open-loop and closed-loop oscillation modes, 202 PMSG, 202 PMSG’s terminal voltage, 204 ROPS, 202 zero phase-tracking error, 202 Phillips-Heffron model, 65–69, 87 PLL oscillation mode (POM), 202, 207 Point estimate method, 307 Point of common coupling (PCC), 8, 34, 203 Power system stabilizers (PSSs), 10, 104 Power-wind speed curve, 331 Practical large-scale power system, 117 Probabilistic analysis AC/DC power system, 323 AC/DC system, 328 analytical method, 306 assumption of linearity, 311 categories, 304 CDF and PDF curve, 314 combined wind power, 337 cumulant theory, 306 distribution function, 310–312 grid-connected onshore wind power generation, 307 HVDC connection topology, 330 index, 304 methodologies, 304 numerical method, 305 one-point linearization, 326 PDF and CDF, 308, 315 procedure, 315, 319 semi-invariant, 310 small-signal stability, 305, 322 spatial correlations, 311 variance index, 330, 335 wind power data, 314 Probabilistic Collocation Method (PCM), 307 Probabilistic density function (PDF), 304 Probabilistic small-signal stability, 304, 313 Probability theory, 310 Pulse width modulation (PWM), 31, 35, 154

R Reactive power control outer loop, 167 Reactive power control/terminal voltage control, 109–110 Reactive power/voltage control, 109 Redrawn model, 100 Relative rotor positions, 169

353 Rest of power system (ROPS), 149 Rest of the power system (ROPS), 98, 202, 205–207, 209, 224, 234 ROPS subsystem, 99 Rotor motion equations, 121 Rotor-side converter (RSC), 45–49 algebraic model, 76 configuration diagram, 74 control parameters, 88 DFIG model, 75–77 generator parameters, 80, 84 induction generator, 75 offset rotor voltage, 80 P-Control algebraic model, 77 PI controllers, 77 state variables, 73 transfer functions, 75 RSC active power control system, 197 RSC basic control, 75 RSC controller parameters, 75 RSC Q-Control algebraic model, 77 RSC vector control system, 120

S Short circuit ratio (SCR), 13, 214, 232–235 Single machine infinite bus (SMIB) system linearized model, 87 WPIG, 86–88 Single-input and single-output (SISO), 18 16-Machine 5-area AC/DC power system, 325 16-Machine 68-bus NYPS-NETS test system, 81 16-Machine 68-bus power system, 104 Small-signal angular stability, 321 Small-signal stability analysis, 336 Small-signal voltage stability, 321 Spatial correlations, 313 State-space and transfer function models, 149 State-space models, 56–58, 98, 99, 174 Steady-state operating points, 324, 326 Step-by-step model transformation analysis, 79 Substation ring topology (SSRT), 335 Sub-synchronous oscillation (SSO), 17, 296 Sub-synchronous resonance (SSR), 295 Synchronous generators (SGs), 64, 88, 94, 101, 102, 151, 161, 222 Synchronous reference frame (SRF), 210 Synchronous reference frame PLL (SRF PLL), 210, 219, 223, 224, 227–232, 235, 237

354

Index

T Taylor series expansion, 153 Terminal voltage control, 110 Terminal voltage control vs. reactive power control, 137–138 Time domain simulation, 83 Transfer function matrix, 100 Transfer function matrix model, 180 Transfer function model, 99, 112 Transitional generator model, 79 Transmission network, 56 Two Point Estimate (TPE) method, 307

renewable power generation, 11 small-signal stability configuration, 13 damping torque analysis, 16 dynamic interactions, PLL, 14–15 frequency-domain analysis, 16 VSC-based HVDC line, 17–19 VSC-based MTDC network, 19–22 weak grid connection and PLL, 12–14 VSC-converter model, 332 VSC-HVDC network, 325, 332 VSC-HVDC transmission, 320

V Variable speed wind generators (VSWGs), 110, 147 affecting factors, 7–8 DFIG, 2 EOMs, 2 grid connection, 2 LEPOs, 1 load flow, 2 PLL, 10–11 power system, 2–7, 9 reactive power/terminal voltage control, 10 SGs, 2–4, 8–10 single-machine infinite-bus power system, 6 wind power generation, 3 Variance index, 330 Vector control system, 119 Voltage source converter (VSC) conventional stability theory, 12 DC/AC power system, 12 dynamic interactions, 147 grid connection, 12 HVDC lines and MTDC, 11

W Weibull (or normal skew) distribution, 308 Weibull distribution, 307, 313 Western Electricity Coordinating Council (WECC) power system, 147 Wind farm ring topology (WFRT), 335 Wind generation, 338 Wind power generation, 147, 157, 310, 322 Wind power induction generators (WPIGs), 64, 69–70 algebraic interface equations, 68 closed-form solution, 72 damping mechanisms, 64, 72 dynamics, 72 G5-G15 power angle curve, 83 linearized model diagram, 76 multi-machine power system, 64 open-loop system, 84 parameters, 80 types, 72 Wind power penetration, 126 Wind power transmission, 256–257 Wind power variations, 309, 317

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