Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation (Power Electronics and Power Systems) [1st ed. 2024] 3031353420, 9783031353420

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Power Electronics and Power Systems

Wenjuan Du Haifeng Wang

Analysis of Power System Sub/ Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation

Power Electronics and Power Systems Series Editors Joe H. Chow

, Rensselaer Polytechnic Institute, Troy, NY, USA

Alex M. Stankovic, Tufts University, Medford, MA, USA David J. Hill, University of Hong Kong, Pok Fu Lam, Hong Kong

The Power Electronics and Power Systems book series encompasses power electronics, electric power restructuring, and holistic coverage of power systems. The series comprises advanced textbooks, state-of-the-art titles, research monographs, professional books, and reference works related to the areas of electric power transmission and distribution, energy markets and regulation, electronic devices, electric machines and drives, computational techniques, and power converters and inverters. The series features leading international scholars and researchers within authored books and edited compilations. All titles are peer reviewed prior to publication to ensure the highest quality content. To inquire about contributing to the series, please contact: Dr. Joe Chow Administrative Dean of the College of Engineering and Professor of Electrical, Computer and Systems Engineering Rensselaer Polytechnic Institute Jonsson Engineering Center, Office 7012 110 8th Street Troy, NY USA Tel: 518-276-6374 [email protected]

Wenjuan Du • Haifeng Wang

Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation

Wenjuan Du School of Electrical Engineering Sichuan University Chengdu, Sichuan, China

Haifeng Wang School of Electrical Engineering Sichuan University Chengdu, Sichuan, China

ISSN 2196-3185 ISSN 2196-3193 (electronic) Power Electronics and Power Systems ISBN 978-3-031-35342-0 ISBN 978-3-031-35343-7 (eBook) https://doi.org/10.1007/978-3-031-35343-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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

In recent years, incidents of sub/super-synchronous oscillation induced by the gridconnected wind power generation were reported in practical power systems. Great effort has been spent by many researchers and engineers in order to understand why the sub/super-synchronous oscillation may occur and to find a method to examine the risk of sub/super-synchronous oscillation. Analysis of power system sub/supersynchronous oscillation caused by grid-connected wind power generation has been an actively pursued issue. Over a decade ago, we started the investigation on the small-signal stability of the power system affected by the grid-connected wind power generation based on the power system linearized model. We published a book, Small-signal Stability Analysis of Power Systems Integrated With Variable Speed Wind Generators, in 2018. In the book, we introduce our study about the impact of grid-connected variable speed wind generators on the small-signal angular stability of a power system. In addition, we report our findings on the open-loop modal resonance in the book and promised to share the results of further investigation in the future. After four years, we are now in the position to present a systematical introduction about the open-loop modal resonance for the grid-connected wind power generation to cause sub/supersynchronous oscillation in this current book. Another important result we present in this book is the analytical derivation of small-signal stability limit of a grid-connected single wind turbine generator system. The derivation analytically reveals how three key factors, i.e., the parameter setting of a converter control system for the grid connection, the loading condition, and weak connection of the wind turbine generator, may affect the system small-signal stability. It indicates that it is possible to analytically examine the risk of sub/supersynchronous oscillation induced by the grid-connected wind power generation in order to understand the stability mechanism. We would like to thank the contributions from our research students to the book. They are Dr. Chen Chen, Dr. Qiang Fu, Dr. Yang Wang, Dr. Wenkai Dong, Dr. Yijun Wang, Mr. Jianyong Lv and Mr. Wei Xie. We thank the financial support from the Natural Science Foundation of China (Project 52077144), Project of v

vi

Preface

Engineering Special Team of Sichuan University on New Energy Power Systems (2020SCUNG103), Project of Youth Innovative Research Team of Science and Technology on New Energy Power System (Grant 22CXTD0066), Sichuan Province, China, and Fundamental Research Funds for the Central Universities and Grant YJ201654, China. Chengdu, China February 2023

Wenjuan Du Haifeng Wang

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Power System Sub-synchronous/Super-synchronous Oscillation (SSO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Purpose, Scope, and Organization of the Book . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small-Signal Stability of a Single Grid-Connected PMSG System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Full-Order State-Space Model of a PMSG . . . . . . . . . . . . . . . . . . 2.1.1 The Permanent Magnet Synchronous Generator . . . . . . . . . 2.1.2 The Machine Side Converter and Associated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Grid Side Converter and Associated Control System of the PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The Phase Locked Loop for the PMSG . . . . . . . . . . . . . . . 2.1.5 Model of the PMSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Block Diagram Model of the PMSG . . . . . . . . . . . . . . . . . 2.2 Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System Dominated by Dynamics of the PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Full-Order State-Space Model of a Single Grid-Connected PMSG System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Simplified Reduced-Order Model of the Single Grid-Connected PMSG System Dominated by Dynamics of the PLL . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Small-Signal Stability Limit of the Single Grid-Connected PMSG System Dominated by the Dynamics of the PLL . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 10 14 15 16 16 18 19 21 24 25

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2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Simplified Reduced-Order Model of a PMSG in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.3 Small-Signal Stability of the Single Grid-Connected PMSG System in the Fast Current Timescale . . . . . . . . . . . 48 2.4 Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.1 Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . 51 2.4.2 Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Slow DC Voltage Timescale . . . . . . . . . . . . . . . 55 2.4.3 Small-Signal Stability of the Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.4 Small-Signal Stability of the Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Slow DC Voltage Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5 An Example of Single Grid-Connected PMSG System . . . . . . . . . 73 2.5.1 Stability of the Example Grid-Connected PMSG System in the Fast Current Timescale . . . . . . . . . . . . . . . . . . . . . . 73 2.5.2 Stability of the Example Grid-Connected PMSG System in the Slow DC Voltage Timescale when the Dynamics of the PLL Are in the Fast Current Timescale . . . . . . . . . . 86 2.5.3 Stability of the Example Grid-Connected PMSG System in the Slow DC Voltage Timescale when the Dynamics of the PLL Are in the Slow DC Voltage Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Contents

3

Small-Signal Stability of a Single Grid-Connected DFIG System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Full-Order State-Space Model of a DFIG . . . . . . . . . . . . . . . . . . . 3.1.1 The Induction Generator and Shaft System . . . . . . . . . . . . 3.1.2 The Rotor Side Converter and Associated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Grid Side Converter and Associated Control System of the DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 The Phase Locked Loop for the DFIG . . . . . . . . . . . . . . . . 3.1.5 Model of the DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Block Diagram Model of a Grid-Connected DFIG . . . . . . . . . . . . 3.2.1 Rotor Motion Dynamics in the Electromechanical Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dynamics of Outer Loop Control Loops of the GSC and the RSC in the DC Voltage Timescale . . . . . . . . . . . . 3.2.3 Dynamics of Inner Loop Control Loops of the GSC and Output Filter in the Current Timescale . . . . . . . . . . . . 3.2.4 Dynamics of Inner Loop Control Loops of the RSC and Rotor Winding in the Current Timescale . . . . . . . . . . . 3.2.5 The PLL and the Transmission Line . . . . . . . . . . . . . . . . . 3.3 Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG System in the DC Voltage Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Simplified Model in the Timescale of the DC Voltage Control When Dynamics of the PLL Are in the Fast Timescale of Electric Current Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Small-Signal Stability Limit in the Timescale of the DC Voltage Control When Dynamics of the PLL Are in the Fast Timescale of Electric Current Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Stability Limit in the Slow DC Voltage Timescale When the Dynamics of the PLL Are Also in the DC Voltage Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 An Example Single Grid-Connected DFIG System . . . . . . . . . . . . 3.4.1 Dynamics of the PLL Are in the Slow Timescale of DC Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Dynamics of the PLL Are in the Fast Timescale of Electric Current Control . . . . . . . . . . . . . . . . . . . . . . . . Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 114 114 117 120 123 124 127 128 128 130 130 133

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Contents

Analysis of Sub-synchronous Oscillations in a Sending-End Power System With Series-Compensated Transmission Line . . . . . . . . . . . . 4.1 Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Torque Interaction and the Induction Generator Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Open-Loop Modal Resonance Analysis Based on a Single-Input Single-Output Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Peak of the Closed-Loop SSO Modal Repulsion . . . . . . . . 4.1.4 Demonstration Using the IEEE First Benchmark Power System . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Equivalence of the Open-Loop Modal Resonance Analysis to the Complex Torque Coefficient Analysis in Detecting the Risk of Growing SSR . . . . . . . . . . . . . . . 4.2 The Induction Generator Effect Analysis of Sub-Synchronous Oscillations in a Grid-Connected DFIG System With Series-Compensated Transmission Line . . . . . . . . . . . . . . . . . . . . 4.2.1 The Induction Generator Effect and the Sub-synchronous Control Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Connection of the Induction Generator Effect With the Open-Loop Modal Resonance . . . . . . . . . . . . . . . 4.2.3 An Example Grid-Connected DFIG System With Series-Compensated Transmission Line . . . . . . . . . . . . . . . 4.3 The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations in a Grid-Connected DFIG System With Series-Compensated Transmission Line . . . . . . . . . . . . . . . . . . . . 4.3.1 Closed-Loop Interconnected Model of a Power System With a Grid-Connected DFIG . . . . . . . . . . . . . . . . 4.3.2 Open-Loop Modal Resonance in an Example Grid-Connected DFIG Wind Farm With Series-Compensated Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations in a Grid-Connected PMSG System With Series-Compensated Transmission Line . . . . . . . . . . . . . . . . . . . . 4.4.1 Closed-Loop Interconnected Model of a Power System With a Grid-Connected PMSG . . . . . . . . . . . . . . . 4.4.2 Sub-synchronous Oscillation in an Example Grid-Connected PMSG System With Series Compensation . . . . . . . . . . . . . Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in a Power System with Grid-Connected Wind Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Open-Loop Modal Resonance in a Power System Without the Series Compensated Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Closed-Loop Interconnected Model and the Open-Loop Modal Resonance . . . . . . . . . . . . . . . . 5.1.2 Modified IEEE First Benchmark Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Open-Loop SSO Modal Resonance Between Two DFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Torsional SSOs Caused by DFIG A . . . . . . . . . . . . . . . . . 5.2 Open-Loop Modal Resonance Caused by a Grid-Connected Wind Farm with Multiple PMSGs . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Two-Inputs Two-Outputs Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Open-Loop Modal Resonance Analysis Based on the TITO Closed-Loop Interconnected Model . . . . . . . . 5.2.3 Method of Open-Loop Modal Resonance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 An Example Power System with a Grid-Connected PMSG Wind Farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 An Example Power System with Two Grid-Connected Wind Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Open-Loop Modal Resonance in a Power System Integrated with Multiple Wind Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Multi-input Multi-output Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Open-Loop Modal Resonance Based on MIMO Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . 5.3.3 Method of Open-Loop Modal Resonance Analysis Based on the MIMO Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Study Cases of Open-Loop Modal Resonance Between the Wind Farms and the Torsional Dynamics of Synchronous Generators . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Study Cases of Open-Loop Modal Resonance Affected by the Number of Variable Speed Wind Generators in the Grid-Connected Wind Farms . . . . . . . . . . . . . . . . . . Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Amplifying Effect of Weak Grid Connection on the Open-Loop Modal Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Amplifying Effect of Weak Grid Connection for a Grid-Connected PMSG to Induce Torsional Sub-synchronous Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Open-Loop Modal Resonance Analysis of a Grid-Connected PMSG System Based on a TITO Closed-Loop Interconnected Model . . . . . . . . . 6.1.2 Magnifying Effect of Weak Grid Connection When the Resonant Open-Loop SSO Mode of PMSG Subsystem Is Dominated by Dynamics of the PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Magnifying Effect of Weak Grid Connection When the Resonant Open-Loop SSO Mode of PMSG Subsystem Is Dominated by Dynamics of the DC Voltage Control of the GSC . . . . . . . . . . . . . . . 6.1.4 Demonstration of Study Cases: Oscillation Modes Associated with the PLL . . . . . . . . . . . . . . . . . . . . 6.1.5 Demonstration of Study Cases: Oscillation Mode Associated with Dynamics of DC Voltage Control of the GSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Amplifying Effect of Weak Grid Connection for a Grid-Connected DFIG to Induce Torsional Sub-synchronous Oscillations . . . . . . . . 6.2.1 Open-Loop Modal Resonance Analysis of a Grid-Connected DFIG System Based on a TITO Closed-Loop Interconnected Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The SSO Mode of DFIG Subsystem Dominated by the PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 The SSO Mode of DFIG Subsystem Dominated by the DC Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The SSO Mode of DFIG Subsystem Dominated by Both the PLL and the DC Voltage Control . . . . . . . . . . 6.2.5 Study Cases: The SSO Mode Dominated by the PLL for the DFIG . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Study Cases: The SSO Mode Dominated by the DC Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Chapter 1

Introduction

Abbreviations AVR CTA DFIG DTA HVDC IGE IMA LEPO MA MIMO PMSG PSS ROPS S3O SISO SG SSCI SSO SSR TI WTG

Automatic voltage regulator Complex torque analysis Doubly-fed induction generators Damping torque analysis High-voltage DC Induction generator effect Impedance model based analysis Low-frequency power oscillation Modal analysis multi-input multi-output Permanent magnet synchronous generator Power system stabilizer remainder of the power system source of the SSO single-input single-output Synchronous generator Sub-synchronous control interaction Sub-synchronous oscillation or super-synchronous oscillation Sub-synchronous resonance Torque interaction wind turbine generator

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Du, H. Wang, Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-031-35343-7_1

1

2

1.1

1

Introduction

Introduction

Power system sub-synchronous oscillation (SSO) is a complex and long-standing problem. The problem evolves from the conventional sub-synchronous resonance (SSR) to the recently appeared risk of sub-synchronous/super-synchronous oscillation brought about by the grid-connected wind power generation. In this chapter, the problem and study of the SSO is briefly reviewed firstly in the following subsection. In the second subsection of the chapter, the work presented in and the organization of the book is introduced.

1.1.1

Power System Sub-synchronous/Super-synchronous Oscillation (SSO)

1.1.1.1

Evolution of the SSO Problem

The problem of the SSO in a power system with series-compensated transmission line was found and studied as early as in 1937 [1]. A growing oscillation of line current was observed under certain conditions in a power system where a sendingend synchronous generator (SG) was connected to the grid through a transmission line installed with a series capacitor. As the frequency of the growing oscillation was below the nominal frequency of 50 Hz (or 60 Hz), it was referred to as the subsynchronous oscillation (SSO). At that time, it was considered that the problem was a kind of self-excited oscillation due to the electrical resonance. When the electrical resonance happened at the sub-synchronous frequency, the sending-end SG behaved like an induction generator with a negative effective resistance. Subsequently, the sending-end SG and the series-compensated line together formed an RLC circuit with the negative resistance. This explained why the growing SSO occurred. The explanation normally is referred to as the induction generator effect (IGE). Incidents of the SSO involved with the torsional dynamics of a sending-end SG in the series-compensated transmission system were firstly reported in the Mohave power plant between 1970 and 1971. From that time on, it has been realized that the series-compensated transmission line may possibly interact with the mechanical dynamics of the torsional system of sending-end SG and lead to torsional SSO. The torsional SSO caused by the dynamic interactions between the torsional system of the sending-end SG and the series-compensated transmission line is attributed to the effect of torque interaction effect (TI). The SSO due to the IGE or the effect of TI is named as the sub-synchronous resonance (SSR) [2]. In the next decade after the Mohave incidents were reported, the torsional SSO caused by the power system stabilizer (PSS) [3, 4], the control system of highvoltage DC transmission (HVDC) [5], or the control system of governor [6] were observed in a power system without the series compensation. The SSO was due to the unfavorable dynamic interaction between the torsional system of the sending-end

1.1

Introduction

3

SG and the PSS, HVDC, or the governor. Hence, the SSO is classified as device dependent SSO in order to differentiate with the SSR which is related with the seriescompensated transmission line [7]. Recently, the SSOs caused by the grid-connected wind power generation were found in many real power systems. There are two well-known such incidents. The first one is the SSO of 20–30 Hz that occurred in the series-compensated 345 kV transmission line with a sending-end wind farm in Texas, USA, in 2009 [8]. The second incident is the SSO of 6–9 Hz in the 500 kV double circuit seriescompensated transmission lines to connect the sending-end wind farms to the main grid in Guyuan, North China, during 2012–2013 [9]. In both incidents, the wind farms were comprised of doubly-fed induction generators (DFIGs). A study on the SSO caused by a grid-connected DFIG connected to the grid through a seriescompensated line has indicated that the SSO is closely related with the converter control system of the DFIG. Hence, this kind of SSO is named as the sub-synchronous control interaction (SSCI) in the literature. A study on the SSCI indicates that at the sub-synchronous frequency, the effective resistance of the grid-connected DFIG is negative. Subsequently, the DFIG together with the series-compensated line forms an RLC circuit with the negative resistance. Hence, the SSCI is the result of the IGE, which explains why the SSCI occurs. The common feature of the SSR, the device dependent SSO, and the SSCI is that either the torsional dynamics of the SG, the series compensation, or both of them are involved in the SSO. It seems that the TI and the IGE can satisfactorily explain the mechanism of those SSO problems. In addition, the SSO occurs in a transmission system with a sending-end power generation unit connected to the main grid, which can be represented by a SG or a DFIG. Hence, a single-generator model can be used for studying the oscillation mechanism. During 2015, several incidents of the SSOs, which were caused by the gridconnected wind power generation, occurred in the Harmi power network in western China. At least in one incident, the torsional system of a SG in the power network was excited. Moreover, oscillations with the frequency over the nominal frequency (50 Hz) were detected. This oscillation is thus named as the super-synchronous oscillation (the abbreviation is also SSO) [10]. Harmi incidents have attracted wide interests and great concern from the research community and industry, as many factors involved in the SSO incidents were new and can be summarized as follows: 1. Wind farms consisting of permanent magnet synchronous generators (PMSGs) in the Harmi power network, which are likely involved in the SSOs. Before the Harmi incident, it had been considered that the permanent magnet synchronous generators (PMSGs) may be immune from the problem of the SSO. 2. No series-compensated lines in the Harmi power network. It seemed that the HVDC did not take part in the SSOs. The torsional dynamics of the SG may or may not be excited by the grid-connected wind power generation. 3. The frequency of the SSO may be below or above the nominal frequency to exhibit as the sub-synchronous or the super-synchronous oscillation.

4

1

Introduction

1937 SG

self excitation

Torsional system of SG 1970-1971 SSR the TI

the IGE 1980s series compensation

DFIG

device dependent SSO

control system of various device

SSCI 2015

2009, 2012-2013

Harmi SSO

unknown

key factors involved SSO mechanism SSO problem

DFIG, PMSG

Torsional system of the SG

Fig. 1.1 Evolution of the SSO problem

4. Harmi power network is complicated and cannot be simply represented as a single-generator system. Figure 1.1 summarizes the evolution of the SSO problem as being briefly reviewed above. Obviously, from Fig. 1.1, it can be seen that three common factors are involved in various SSO problems: the torsional system of the SG, the series compensation line, control systems of various device, and wind turbine generators (WTGs). How those three factors affect and cause various kinds of SSO problems are the key to find the answers to the following questions: 1. What is the SSO mechanism in the Harmi incidents and how can such SSO be examined? 2. Is there any connection among the SSR, the device dependent SSO, the SSCI, and the Harmi SSO? This book introduces the work to look for the key to find the answers to the questions above.

1.1.1.2

Main Methods to Examine the SSO

The SSO is examined usually by using the linearized model of a power system. The main methods to examine the SSO can be classified into two categories: the frequency-domain methods and the time-domain methods. Frequency-domain

1.1

Introduction

5

Fig. 1.2 Interconnected model for applying the CTA

–

1 km ( s )

Δδ

torsional subsystem

ΔTe

ke ( s ) electrical subsystem

methods include the method of complex torque analysis (CTA), the damping torque analysis (DTA), and impedance model-based analysis (IMA). The time-domain analysis is mainly the modal analysis (MA). • Complex Torque Analysis (CTA) The CTA was proposed to examine the SSR based on an interconnected linearized model of a power system with a sending-end SG being connected to the grid through a series-compensated transmission line [11, 12]. The interconnected model is comprised of two subsystems: the torsional subsystem and the electrical subsystem as shown in Fig. 1.2. The torsional subsystem consists of the torsional system of the sending-end SG. The electrical subsystem is the remainder of the power system (ROPS) including the series-compensated transmission line. In Fig. 1.2, Δδ is the deviation of rotor angle of the SG and ΔT is the electrical torque. Let ΔT e = - k m ðsÞΔδ

ð1:1Þ

ΔT e = ke ðsÞΔδ

ð1:2Þ

where - km1ðsÞ and ke(s) are the transfer functions of the torsional and electrical subsystem, respectively. In the frequency domain (s = jω), let km ðjωÞ = K m ðωÞ þ jωDm ðωÞ ke ðjωÞ = K e ðωÞ þ jωDe ðωÞ

ð1:3Þ

where Km(ω) and Dm(ω) are the mechanical spring coefficient and damping coefficient; Ke(ω) and De(ω) are the electrical spring coefficient and damping coefficient, respectively. Resonant frequency, ωi, is found from the solution of following equation: K e ðωi Þ þ K m ðωi Þ = 0,

i = 0, 1, . . .

ð1:4Þ

6

1

Fig. 1.3 Block diagram model for applying the DTA

k

Introduction

Δy

s 2 + as + b Δu

S3O subsystem

R (s) ROPS subsystem

For each found resonant frequency, if De ðωi Þ þ Dm ðωi Þ < 0,

i = 0, 1, . . .

ð1:5Þ

the SSR occurs. Otherwise, there should be no risk of the SSR. For the device dependent SSO, which is the torsional SSO caused by control system of an electrical device, such as the PSS, the HVDC, etc., the CTA can be applied to detect the SSO risk. For the torsional SSO excited by the grid-connected wind power generation, the CTA can also be used to assess the SSO risk. • Damping Torque Analysis (DTA) The DTA is a method to examine the risk of low-frequency power oscillation (LEPO) and for the design of the PSS to suppress the LEPO in a power system. It is applied by calculating the damping torque contribution from a device, such as the automatic voltage regulator (AVR) and the PSS, to determine the impact of the device on the damping of the LEPO [13]. It can also be applied to study the SSO risk brought about by the grid-connected wind power generation [14]. Figure 1.3 shows the block diagram model for applying the DTA. The secondorder subsystem is the source of the SSO (S3O), which is known to be responsible for the SSO in a power system. The ROPS subsystem is the remainder of the power system. Δy = RðsÞΔu

ð1:6Þ

where R(s) is the transfer function of the ROPS subsystem. Let ωS = 2πfS, where fS is the frequency of the SSO, and RðjωS Þ = Re½RðjωS Þ] þ jIm½RðjωS Þ]

ð1:7Þ

where Re[] and Im[] denote the real part and imaginary part of a complex number respectively. The DTA considers that if Im[R( jωS)] < 0, there is a risk of the SSO associated with the source of the SSO. The DTA has been applied to examine the SSR, the SSO associated with the phase lock loop (PLL), etc. Usually, application of the DTA requires that the source of the SSO can be described as a second-order system.

1.1

Introduction

7

Fig. 1.4 Interconnected model for applying the IMA

ZS (s)

ΔV

ΔI

S3 O subsystem

ZR (s)

ROPS subsystem

• Impedance Model-Based Analysis (IMA) The IMA is applied on the basis of an interconnected model displayed in Fig. 1.4, where ZS(s) is the transfer function of the subsystem of the source of the SSO (S3O), which may bring about the SSO risk, and ZR(s) is the transfer function of the subsystem of the remainder of the power system (ROPS). In Fig. 1.4, ΔI and ΔV can be the current and voltage at the terminal where the S3O subsystem and the ROPS subsystem are interconnected. From Fig. 1.4, it can have ΔI = Z S ðsÞΔV

ð1:8Þ

ΔV = Z R ðsÞΔI

ð1:9Þ

Thus, both Z S1ðsÞ and ZR(s) can be considered as impedance. This is why the method is called the impedance model-based analysis (IMA). The characteristic equation of the system displayed in Fig. 1.4 is 1 þ Z R ðsÞZ S ðsÞ = 0

ð1:10Þ

By applying the Nyquist criterion to (1.10), the stability of the system can be examined. It is worth to point out that the Nyquist criterion is only applicable to the singleinput single-output (SISO) interconnected model displayed in Fig. 1.4. In the case of a multi-input multi-output (MIMO) interconnected model, the general Nyquist criterion must be used, which needs to conduct the modal computation. • Modal Analysis (MA) Let the state-space model of a MIMO linear system be dX = AX þ Bu dt y = CX þ Du

ð1:11Þ

8

1

Introduction

where X 2 RN × 1 is the state variable vector, u 2 RM × 1 is the input vector, and y 2 RL × 1 is the output vector, respectively; A 2 RN × N is the state matrix, B 2 RN × M, C 2 RN × L, and D 2 RL × M are the control matrix, output matrix, and the direct forward matrix, respectively. The transfer function matrix of the system is GðsÞ = CðsI - AÞ - 1 B þ D

ð1:12Þ

where I 2 RN × N is a unit matrix. The characteristic equation of the system is jA - λIj = 0

ð1:13Þ

The solutions of the characteristic equation, λi; i = 1, 2, ⋯N, are called the eigenvalues of the state matrix, A 2 RN × N. For the ith eigenvalue, if a non-zero vector vi satisfies the following equations Avi = λi vi

ð1:14Þ

wi T A = λi wi T

vi 2 RN × 1 and wi 2 RN × 1 are called the right and left eigenvector of matrix A associated with λi, respectively. If there is one or more eigenvalues of A on the right-hand half of the complex plan, the linear system is unstable. If all the eigenvalues of A are on the left-hand side of the complex plane, the system is stable. Hence by calculating the eigenvalues of A, system stability can be assessed. If a pair of eigenvalues of A are conjugate complex number, i.e., λi, i + 1 = ξi ± jωi, the pair of conjugate eigenvalues are called the oscillation mode of the system. The oscillation frequency fi (Hz) and damping ζ i associated with λi, i + 1 = ξi ± jωi are respectively fi =

ωi , ζi = 2π

ξi ξ i þ ωi 2 2

ð1:15Þ

In this book, the oscillation mode is simply denoted as λi = ξi ± jωi. For the ith oscillation mode, λi = ξi ± jωi, denote vki as the element of vi 2 RN × 1 in the kth row and wki as the element of wk 2 RN × 1 in the ith row. The participation factor (PF) of the kth state variable in the ith oscillation mode is pki =

jvki wki j M i=1

ð1:16Þ

jvki wki j

where || refers to the magnitude of a complex number. The PF measures how much the kth state variable is associated with the ith oscillation mode.

1.1

Introduction

9

Let w1 T V = ½v1 v2 . . . vN ], WT =

w2 T ⋮ wN T

= V-1

ð1:17Þ

It can be seen that vki and wki are respectively the kth row and ith column elements of matrix V and WT. A new state variable vector Z is introduced and equivalent state variable transformation is taken to be X = VZ. From (1.11), it can have dZ = ΛZ þ WT Bu dt y = CVZ þ Du

ð1:18Þ

where Λ = V-1AV = diag [λi] and diag[] refers to a diagonal matrix. The state-space model of (1.18) is called the modal decomposed model of the system which is dynamically equivalent to the original system depicted by (1.11). The stability of the system does not change after the state variable transformation. From (1.18), M

dzi b k uk = λi zi , þ wi T Bu = λi zi , þ wi T dt k=1 c1 T y1 y=

y2 ⋮

N

=C

vi zi þ Du =

c2 T

i=1

N i=1

vi z i ð1:19Þ

N i=1

vi z i

þ Du

⋮

yL cL T

N i=1

v i zi

where zi is the ith state variable in Z,

B = ½ b1

b2

⋯ bM ], u =

u1 u2 ⋮ uM

c1 T ,C=

c2 T ⋮ cL T

10

u1 u2

u2

c1T v 2

c1T v N

w 2 T b1

c 2 T v1

w 2T b 2

1 z2 s – λ2

c2T v 2

...

+

u2

w NTb2

+

......

......

...... w N T b1

+

c2T v N

w 2T b M

u1

c N T v1

+

1 zN s – λN

cNT v2

...

...

uM

1 z1 s – λ1

w1T b M

...

uM

+

...

u1

w1T b 2

Introduction

c1T v1

w1T b1

...

uM

1

+

+

cNT v N

w NTbM

u

1D

+

y

Fig. 1.5 Block diagram of modal decomposition representation of state-space model of the system

From (1.19), the modal decomposed state-space model of the system can be shown by the block diagram displayed in Fig. 1.5. From (1.18) or Fig. 1.5, the residue matrix of the system is obtained to be R = WT BCV

1.2

ð1:20Þ

Purpose, Scope, and Organization of the Book

This book examines the sub/super-synchronous oscillation caused by grid-connected wind power generation in a power system. The modal analysis (MA) is the main method used for the examination. Hence, the model adopted is the state-space representation of the power system with the grid-connected wind power generation. In order to help the understanding of complex dynamic connections of various components in the power system, a block diagram model is used at several places. The MA has been applied to examine the small-signal stability of a power system for over half a century. It is the standard function in all the major commercial software tools for power system stability analysis. For long time, the MA has been

1.2

Purpose, Scope, and Organization of the Book

11

a method used normally in order to know what the small-signal stability of the power system is. If any instability risk is detected, the MA can be applied to identify the responsible trouble makers by computing the participation factors (PFs), etc. Hence, the MA is a computation-based method and can tell straightforward what the status of system stability is, given the system state-space model. However, to pursue why and how the instability risk may occur, the MA often cannot provide the answer. The reason is that no matter how many computational results can be obtained by applying the MA, it is difficult to draw any generally applicable principle about the stability mechanism of the power system. In power system stability analysis, it is important to understand the stability mechanism, not only for educating and training but also to guide the planning, operation, and control of power system in practice. For conventional power systems, system stability mechanism has been revealed mainly from the standpoint of physical understanding. A good example is the small-signal angular stability of a conventional power system. Lack of sufficient damping torque on the rotor motion of synchronous generators may lead to growing low-frequency power oscillation in the power system. The installation of a power system stabilizer (PSS) supplies extra positive damping torque to damp the low-frequency power oscillation. Of course, application of the MA can detect the risk of low-frequency power oscillation and evaluate the performance of the PSS. However, it is not easy to explain why and how the low-frequency power oscillation happens by applying the MA [13]. Therefore, the main drawback of the MA is the lack of capability for revealing the stability mechanism. Of course, the other well-known drawback of the MA is the numerical problem caused by high-dimensional state matrix when the scale of the power system is very large. In a power system with grid-connected wind power generation, why and how the sub/super-synchronous oscillation may occur? It has been a puzzling question, though great effort has been spent by many researchers and power system engineers to seek the answer in the recent years. A widely known explanation about why and how the grid-connected wind power generation may bring about the risk of instability in the range of sub/super-synchronous frequency is the “negative resistance” exhibited by the grid-connected wind power generation. The explanation is similar to the induction generator effect (IGE) and easy to be understood in a send-end power system with series-compensated transmission line. However, for the risk of sub/super-synchronous oscillation caused by the grid-connected wind power generation in a complex power network without the series-compensated transmission line, such as the Harmi power system, the instability mechanism of “negative resistance” still needs more comprehensive investigation. In fact, if a linear system is of a poorly/negative oscillation mode, the system dynamics are dominated by the poorly/negative oscillation mode. Subsequently, the system is equivalent to a second-order linear system to have the same dominant oscillation mode. Physically, the second-order system is equivalent to an RLC circuit. If the system is unstable, the resistance of the RLC circuit is negative. Based on this understanding, the principle of “negative resistance” seems applicable to any linear system. However, it is a challenging job to identify the exact source of

12

1 Introduction

negative resistance and calculate the value of the resistance when growing sub/ super-synchronous oscillation occurs in a complex power system with the gridconnected wind power generation. The challenge comes from two aspects. First, dynamic interconnection of a grid-connected wind power generation with the remainder of the complex power system is normally multi-variable in an AC system. It is needed to reduce the multi-variable dynamic interconnection into a singlevariable interconnection in order to obtain the equivalent impedance of the gridconnected wind power generation. Second, the dynamic description of the gridconnected wind power generation is of high order. It is needed to reduce the order to calculate the effective resistance in order to assess the system stability in the frequency domain by using the Nyquist criterion. The purpose of examination introduced in the book is twofolds. First, the examination is in order to explain why and how the sub/super-synchronous oscillation may occur in the power system as being caused by the grid-connected wind power generation. This is to reveal the instability mechanism of the power system with grid-connected wind power generation associated with the sub/super-synchronous oscillation. Second, it shows how the MA can be applied to detect the risk and source of sub/super-synchronous oscillation brought about by the grid-connected wind power generation based on the revelation of instability mechanism. The examination in the book starts from the simple power system where there is a single grid-connected wind turbine generator (WTG), either a PMSG or a DFIG. Small-signal stability limit, i.e., the sufficient and necessary stability conditions, of the single grid-connected WTG system is analytically derived. Violation of the stability limit means risk of growing sub/super-synchronous oscillation in the single grid-connected WTG system. Derived stability limit unambiguously indicates that the heavy loading condition and weak grid connection of the WTG increase the risk of sub/super-synchronous oscillation. In addition, it clearly shows how the setting of converter control parameters of the WTG affect the small-signal stability of the single grid-connected WTG system together with two other factors, i.e., condition of grid connection and loading of the WTG. Examination of the small-signal stability of a single grid-connected WTG system is introduced in Chaps. 2 and 3 of the book. How the derived stability limit can be used to guide the setting of converter control parameters of the WTG to avoid the risk of sub/super-synchronous oscillation is also discussed in those two chapters. In order to examine why and how the grid-connected wind power generation may bring about the risk of sub/super-synchronous oscillation in a power system by applying the MA, a new concept and theory of open-loop modal resonance is introduced in the book. The proposal and development of the concept of openloop modal resonance and method of open-loop modal resonance analysis are an attempt to explore the function of the MA to study the instability mechanism of the power system. The theory of open-loop modal resonance is introduced in Chap. 4 for a simple power system, where either a single sending-end synchronous generator, or a single sending-end DFIG, or a single sending-end PMSG is connected to the main

1.2

Purpose, Scope, and Organization of the Book

13

grid via a series-compensated transmission line. Under the condition of open-loop modal resonance, strong dynamic interactions between the send-end generator and the series compensation line may lead to growing sub-synchronous oscillation. In this chapter, the open-loop modal resonance analysis is carried out on the basis of a closed-loop interconnected model with single input and single output of the simple power system with the series compensation line. In Chap. 5, the theory of open-loop modal resonance analysis is extended to more complex cases where multiple WTGs or multiple wind farms are connected to a power system. In those cases, the closed-loop interconnected model of the power system is of multiple inputs and multiple outputs. Theoretical proof of the open-loop modal resonance based on the closed-loop multi-input multi-output (MIMO) interconnected model is provided in the chapter. Various study cases of open-loop modal resonance between the WTGs and wind farms are presented. In Chap. 6, one particular aspect of the open-loop modal resonance is discussed. This is the amplifying effect of weak grid connection of the WTG on the open-loop modal resonance to excite the torsional SSO of a synchronous generator. Grid connection of the wind power generation in the Harmi power system is weak. In the Harmi incidents of sub/super-synchronous oscillation, it was reported that the torsional dynamics of nearby synchronous generator were excited. Hence, the examination conducted in the chapter attempts to explain why and how the openloop modal resonance may more likely happen to cause the sub/super-synchronous oscillation in a power system with weakly grid-connected wind power generation. From the introduction presented above about the purpose and scope, the organization of the book can be summarized by Fig. 1.6.

Analysis of sub/super-synchronous oscillations, chapter1, Introduction

Single grid-connected WTG system

chapter 2 chapter 3 Single grid- Single gridconnected connected PMSG DFIG system system

Fig. 1.6 Organization of the book

open-loop chapter 4 model Sending-end resonance power system with series compensated transmission line

Power system with multiple grid-connected WTGs or wind farms chapter 5 Open-loop modal resonance based on MIMO model

chapter 6 Amplifying effect of weak grid connection

14

1

Introduction

References 1. Rustebakke H M, Concordia C. Self-Excited Oscillations in a Transmission System Using Series Capacitors[J]. IEEE Transactions on Power Apparatus and Systems, 1970, PAS-89(7): 1504–1512. 2. IEEE-Subsynchronous-Sesonance-Working. Proposed Terms and Definitions for Subsynchronous Oscillations[J]. IEEE Transactions on Power Apparatus and Systems, 1980, PAS-99(2): 506–511. 3. Lawson R A, Swann D A, Wright G F. Minimization of Power System Stabilizer Torsional Interaction on Large Steam Turbine-Generators[J]. IEEE Transactions on Power Apparatus and Systems, 1978, PAS-97(1): 183–190. 4. Watson W, Coultes M E. Static Exciter Stabilizing Signals on Large Generators - Mechanical Problems[J]. IEEE Transactions on Power Apparatus and Systems, 1973, PAS-92(1): 204–211. 5. Bahrman M, Larsen E V, Piwko R J, et al. Experience with HVDC - Turbine-Generator Torsional Interaction at Square Butte[J]. IEEE Transactions on Power Apparatus and Systems, 1980, PAS-99(3): 966–975. 6. Lee D C, Beaulieu R E, Rogers G J. Effects of Governor Characteristics on Turbo-Generator Shaft Torsionals[J]. IEEE Transactions on Power Apparatus and Systems, 1985, PAS-104(6): 1254–1261. 7. IEEE-Subsynchronous-Sesonance-Working. Terms, Definitions and Symbols for Subsynchronous Oscillations[J]. IEEE Transactions on Power Apparatus and Systems, 1985, PAS-104(6): 1326–1334. 8. Gross L C. Sub-synchronous grid conditions:New event, new problem, and new solutions [C]. Proc. Western Protective Relay Conf., 2010: 1–5. 9. Wang L, Xie X, Jiang Q, et al. Investigation of SSR in practical DFIG-based wind farms connected to a series-compensated power system[J]. IEEE Transactions on Power Systems, 2015, 30(5): 2772–2779. 10. H. Liu, T. Bi, and X. Chang,“ Impact of subsynchronous and supersynchronous frequency components on synchrophasor measurements”, Journal of Modern Power Systems and Clean Energy, vol. 4, no. 3, pp. 362–369, July 2016. 11. Canay I M. A novel approach to the torsional interaction and electrical damping of the synchronous machine Part I: Theory[J]. IEEE Transactions on Power Apparatus and Systems, 1982 (10): 3630–3638. 12. Canay I M. A novel approach to the torsional interaction and electrical damping of the synchronous machine part II: Application to an arbitrary network[J]. IEEE Transactions on Power Apparatus and Systems, 1982 (10): 3639–3647. 13. H F Wang , W Du, Analysis and damping control of power system low-frequency oscillations, Springer US, 2016 14. W Du, H F Wang and S Q Bu, Small-signal stability analysis of power system integrated with variable speed wind generators, Springer US, 2018

Chapter 2

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

Abbreviations GSC MSG PCC PLL PMSG PWM SCR SG WTG

Grid side converter Machine side converter Point of common coupling Phase locked loop Permanent magnet synchronous generator Pulse width modulation Short circuit ratio Synchronous generator Wind turbine generator

Weak grid connection, high loading condition, and improper setting of converter control systems of a grid-connected wind turbine generator (WTG) may bring about the risk of sub/super-synchronous oscillation. This chapter analytically examines those three key factors to explain why and how they may unfavorably affect the damping of sub/super-synchronous oscillation in a single grid-connected PMSG system. The examination is carried out by deriving the necessary and sufficient stability conditions of the single grid-connected PMSG system to reveal the connection of those three key factors and their impact on the system stability. First, the full-order state-space model of a PMSG is introduced. Second, a simplified reduced-order model of a single grid-connected PMSG system, which is dominated by dynamics of the phase locked loop (PLL), is established. Based on the established model, the stability limit of the single grid-connected PMSG system dominated by dynamics of the phase locked loop (PLL) is derived. Third, simplified reduced-order models of the single grid-connected PMSG system in the fast current

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Du, H. Wang, Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-031-35343-7_2

15

16

2

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

timescale and the slow DC voltage timescale are derived, respectively. Based on the derived models, the necessary and sufficient stability conditions of the single gridconnected PMSG system are obtained. Violation of any stability condition implies the occurrence of growing sub/supersynchronous oscillations in the single grid-connected PMSG system. Finally in the chapter, examples of single grid-connected PMSG system are presented. Results of numerical computation and simulation demonstrate that growing sub/super-synchronous oscillations occur in the example single grid-connected single PMSG systems when the stability conditions are violated.

2.1

Full-Order State-Space Model of a PMSG

Figure 2.1 shows the configuration of a PMSG, which is comprised of four main parts: (1) The permanent magnet synchronous generator (SG); (2) The machine side converter (MSG) and associated control system; (3) The grid side converter (GSC) and associated control system; (4) The phase locked loop (PLL). In this section, linearized model of each of those four parts is briefly introduced to lead to the establishment of full-order state-space model of the PMSG. More detailed derivations of the linearized model of the PMSG can be found in [1]. In Fig. 2.1, PCC refers to the point of common coupling.

2.1.1

The Permanent Magnet Synchronous Generator

After Park’s transformation, the voltage equations of stator windings of the permanent magnet SG in Fig. 2.1 are expressed in the d–q coordinate of the permanent magnet SG as:

V psd +jV psq

MSC

N

Pps

Ppc

Cp

V pdc

S

V pcd +jV pcq

GSC

X pf Pp +jQ p

I psd +jI psq V psdref Permanent magnet SG

I pcd + jI pcq PCC

V psqref

Control system

Control system

ω prref

V pq

V pd

I psdref

V pdcref

V pd +jV pq (V ∠θ )

θp

PLL

Q pref PLL

MSC and associated control system

Fig. 2.1 Configuration of a PMSG [1]

GSC and associated control system

θ

2.1

Full-Order State-Space Model of a PMSG

d ψ = - ω0 Rps I psd - ω0 V psd þ ω0 ωpr ψ psq dt psd d ψ = - ω0 Rps I psq - ω0 V psq - ω0 ωpr ψ psd dt psq

17

ð2:1Þ

where Vpsd and Vpsq are the d and q component of voltage, ψ psd and ψ psq 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; Rps is the resistance of stator windings. Note that the direction of d axis of d–q coordinate of permanent magnet SG is that of the permanent magnet rotor, i.e., ψ psd. In addition, subscript d and q are used to indicate the d and q component of a variable, respectively in the d–q coordinate. This notation will be used throughout the book. Flux linkage equations of the permanent magnet SG are ψ psd = X pd I psd - ψ pm ψ psq = X pq I psq

ð2:2Þ

where ψ pm is the constant flux of permanent magnet, Xpd and Xpq are the d and q axis reactance of stator windings, respectively. Motion equations of the rotor of the permanent magnet SG are: J pr

dωpr = T pm - T pe dt

ð2:3Þ

where Jpr is the constant of inertia of the rotor; Tpm and Tpe are the mechanical torque input to/and the electromagnetic power output from the permanent magnet SG, respectively; and T pe = ψ psq I psd - ψ psd I psq

ð2:4Þ

Active power output from the permanent magnet SG is Pps = V psq I psq þ V psd I psd

ð2:5Þ

By linearizing (2.1)–(2.5) with ΔTpm = 0 and ψ pm = 0, it can have d ΔXp1 = Ap1 ΔXp1 þ bp1 ΔV psd þ bp2 V psq dt

ð2:6Þ

ΔPps = cp1 ΔXp1 þ d p1 ΔV psd þ dp2 ΔV psq T

where Xp1 = [Δψ psd Δψ psq Δωpr], prefix Δ refers to a small increment of a variable or variable vector and this notation will be used throughout the book.

18

2.1.2

2

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

The Machine Side Converter and Associated Control System

Configuration of vector control system of the MSC is shown by Fig. 2.2. It controls the d and q axis output current from the stator windings of the permanent magnet SG, Ipsd and Ipsq, so as to regulate the output of the active power from the permanent magnet SG. Hence, the speed control outer loop in Fig. 2.2 controls the angular speed of the rotor of the permanent magnet SG, ωpr. In Fig. 2.2, d x = K pi1 ωpr - ωprref dt p1 d x = K pi2 I psqref - I psq dt p2

ð2:7Þ

d x = K pi3 I psdref - I psd dt p3 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 (2.2), I psqref = K pp1 ωpr - ωprref þ xp1 V psqref = - K pp2 I psqref - I psq - xp2 - ωpr ψ pm þ ωpr X pd I psd

ð2:8Þ

V psdref = - K pp3 I psdref - I psd - xp3 - ωpr X pq I psq 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 current control inner loop

Rotor speed control outer loop

K pp1

ω prref – + ω pr

K pi1 s

+ + x p1

K pp 2

I psqref+ – I psq

K pi 2 s K pp 3

I psdref + – I psd

K pi 3 s

ω pr ψ pm + – +V psqref MSC – + xp2 ω X I pr pd psd + – V psdref MSC + + x p3 ω pr X pq I psq

d-axis current control inner loop

Fig. 2.2 Configuration of control system of the MSC of the PMSG [1]

2.1

Full-Order State-Space Model of a PMSG

19

q axis references of the terminal voltage of the permanent magnet SG. Ignoring the transient of the pulse width modulation (PWM) of the MSC, it can have V psdref ≈ V psd

ð2:9Þ

V psqref ≈ V psq

By linearizing (2.2) and (2.7)–(2.9) with Δωprref = 0 and ΔIpsdref = 0, it can have d ΔXp2 = Ap2 ΔXp2 þ Bp1 ΔXp1 dt ΔV psd = cp2 T ΔXp2 þ cp3 T ΔXp1

ð2:10Þ

ΔV psq = cp4 T ΔXp2 þ cp5 T ΔXp1 where ΔXp2 = ½ Δxp1

2.1.3

Δxp2

Δxp3 ]T .

The Grid Side Converter and Associated Control System of the PMSG

From Fig. 2.1, the line voltage equations across the filter reactance, Xpf, can be written in the d–q coordinate of the GSC as ω0 V pcd ω0 V pd d þ ω0 I pcq I = dt pcd X pf X pf ω0 V pcq ω0 V pq d - ω0 I pcd I = X pf X pf dt pcq

ð2:11Þ

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 as being indicated in Fig. 2.2. Equation of the voltage across the capacitor is Cp V pdc

dV pdc = Pps - Ppc dt

ð2:12Þ

where Vpdc is the DC voltage across the capacitor; Cp is the capacitance; Ppcis the active power output from the GSC; and Ppc = V pcd I pcd þ V pcq I pcq

ð2:13Þ

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

20

2

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

DC voltage control outer loop

K pp 4

V pdcref – + V pdc

Q pref –

+ Qp

K pi 4 s K pp 6

+

s

K pp 5

I cdref +

+ + xp4

K pi 6

d-axis current control inner loop

– I pcd

I cqref+

+ xp6

– I pcq

Reactive power control outer loop

K pi 5

+

s

+ x p5

K pp 7

+

K pi 7 s

+ xp7

V pd V + + pcdref GSC – X pf I pcq V pq V pcqref + + GSC + X pf I pcd

q-axis current control inner loop

Fig. 2.3 Configuration of control system of the GSC of the PMSG [1]

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, d x = K pi4 dt p4 d x = K pi5 dt p5 d x = K pi6 dt p6 d x = K pi7 dt p7

V pdc - V pdcref I pcdref - I pcd Qp - Qpref

ð2:14Þ

I pcqref - I pcq

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 = V pq I pcd - V pd I pcq

ð2:15Þ

From Fig. 2.3, I pcdref = K pp4 V pdc - V pdcref þ xp4 V pcdref = K pp5 I pcdref - I pcd þ xp5 - X pf I pcq þ V pd I pcqref = K pp6 Qp - Qpref þ xp6 V pcqref = K pp7 I pcqref - I pcq þ xp7 þ X pf I pcd þ V pq where Kpp4, Kpp5, Kpp6, and Kpp7 are the gains of proportional controllers.

ð2:16Þ

2.1

Full-Order State-Space Model of a PMSG

21

Ignoring the transient of the PWM of the GSC, V pcdref = V pcd V pcqref = V pcq

ð2:17Þ

By linearizing (2.11)–(2.17), it can have d ΔXp3 = Ap3 ΔXp3 þ bp3 ΔPps þ bp4 ΔV pd þ bp5 ΔV pq dt

ð2:18Þ

T where ΔXp3 = ½ ΔI pcd ΔI pcq ΔV pdc Δxp4 Δxp5 Δxp6 Δxp7 ] . Since ΔIpcd and ΔIpcq are state variables in (2.18), it can have

2.1.4

ΔI pcd = ½ 1

0

0 0

0

0

0 ]ΔXp3 = cp6 T ΔXp3

ΔI pcd = ½ 0

1

0 0

0

0

0 ]ΔXp3 = cp7 T ΔXp3

ð2:19Þ

The Phase Locked Loop for the PMSG

Normally, the d axis of the d–q coordinate of the GSC is chosen to be the direction of the PCC voltage in the common x–y coordinate. In order to implement the vector control scheme of the GSC as being shown by Fig. 2.3, the phase of the PCC voltage needs to be tracked such that the relative position of the d–q coordinate in respect to the x–y coordinate is determined. This phase tracking task is fulfilled by a PLL. The PLL usually is a control system with the input to be the measurement of the PCC voltage of the PMSG. The output is the estimation or tracked phase of the PCC voltage. There are many types of the PLL, among which the synchronous reference frame (SRF) PLL is the simplest and most commonly used. Figure 2.4 shows the block diagram model of the SRF PLL [1], where θ is the phase of the PCC voltage of the PMSG, θp is the estimation of θ by the PLL, i.e., the tracked phase of the PCC voltage, V0 is the magnitude of PCC voltage at the steady state, Kpp and Kpi are the proportional and integral gains of the PLL, respectively. The low-pass filter in Fig. 2.4 is normally designed to filter high-frequency harmonics above 300 Hz. Hence, when the sub-synchronous/super-synchronous oscillations are examined, it can assume that F(s) = 1.

Fig. 2.4 Block diagram model of an SRF PLL

Low-pass Filter

θ+

–

V0

F (s)

K pp

+

K pi s

+ xp

1 s

θp

22

2

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

Fig. 2.5 The relationship between the d–q and x–y coordinates

y q V ∠θ = Vx + jV y d

θ

θp

x

From Fig. 2.4, dxp = K pi V 0 θ - θp dt dθp = K pp V 0 θ - θp þ xp dt

ð2:20Þ

Figure 2.5 shows the relative positions between the d–q coordinate of the GSC and the common x–y coordinate. It can be seen that the tracked phase of the PCC voltage by the PLL is used to give the direction of d axis in the common x–y coordinate. In Fig. 2.5, subscript x and y are used to indicate the x and y component of a variable, respectively, in the common x–y coordinate. This notation will be used throughout the book. From Fig. 2.5, Ix Iy

=

cos θp

- sin θp

I pcd

sin θp

cos θp

I pcq

I pcd

=

I pcq

cos θp

sin θp

Ix

- sin θp

cos θp

Iy

ð2:21Þ

where Ix + jIy is the output current of the PMSG expressed in the common x-y coordinate. Similarly, Vx

=

Vy V pd V pq

=

cos θp

- sin θp

V pd

sin θp

cos θp

V pq

cos θp

sin θp

Vx

- sin θp

cos θp

Vy

ð2:22Þ

2.1

Full-Order State-Space Model of a PMSG

23

where Vx + jVy is the PCC voltage of the PMSG expressed in the common x–y coordinate. The phase of PCC voltage can be expressed as θ = tan

Vy Vx

ð2:23Þ

Linearization of (2.20) and (2.23) is dΔxp = K pi V 0 Δθ - Δθp dt dΔθp = K pp V 0 Δθ - Δθp þ Δxp dt V y0 V x0 ΔV x Δθ = - 2 V 0 V 20 ΔV y

ð2:24Þ

where the subscript 0 refers to the steady-state value of a variable and this notation will be used throughout the book. From (2.24), it can have d ΔXp4 = Ap4 ΔXp4 þ bp6 ΔVx þ bp7 ΔVy dt

ð2:25Þ

where ΔXp4 = ½ Δxp Ap4 =

Δθp ]T

0

- K pi V 0

1

- K pp V 0

-

K pi V y0 V0

-

K pp V y0 V0

, bp6 =

, bp7 =

K pi V x0 V0 K pp V x0 V0

Linearization of (2.21) and the second equation of (2.22) gives ΔI x ΔI y ΔI pcd ΔI pcq

= =

- sin θp0

- cos θp0

I pcd0

cos θp0

- sin θp0

I pcq0

- sin θp0

cos θp0

I x0

- cos θp0

- sin θp0

I y0

Δθp þ Δθp þ

cos θp0

- sin θp0

ΔI pcd

sin θp0

cos θp0

ΔI pcq

cos θp0

sin θp0

ΔI x

- sin θp0

cos θp0

ΔI y

ð2:26Þ

24

2

ΔV pd ΔV pq

=

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

- sin θp0

cos θp0

V x0

- cos θp0

- sin θp0

V y0

Δθp þ

cos θp0

sin θp0

- sin θp0

cos θp0

×

ΔV x ΔV y

ð2:27Þ

2.1.5

Model of the PMSG

Substituting the second equation of (2.6) in (2.18) and using the last two equations of (2.18), then writing the first equation of (2.6), the first equation of (2.10) and (2.18) together, it can have d ΔXp5 = Ap5 ΔXp5 þ bp8 ΔV pd þ bp9 ΔV pq dt where ΔXp5 = ΔXp1 T

Ap5 =

ΔXp2 T

ΔXp3 T

T

,

Ap1 þ bp1 cp3 T þ bp2 cp5 T Bp1

bp1 cp2 T þ bp2 cp4 T Ap2

d p2 bp3 cp5 þ bp3 cp1 þ dp1 bp3 cp3 T

T

bp8 =

0 0

ð2:28Þ

T

dp1 bp3 cp2 þ dp2 bp3 cp4

, bp9 =

bp4

T

0 0 T

,

Ap3

0 0 bp5

Substituting (2.27) in (2.28), ΔV x d ΔXp5 = Ap5 ΔXp5 þ a1 Δθp þ Bp2 dt ΔV y

ð2:29Þ

where a1 = ½ bp8 Bp2 = ½ bp8

bp9 ] bp9 ]

- sin θp0

cos θp0

V x0

- cos θp0

- sin θp0

V y0

cos θp0

sin θp0

- sin θp0

cos θp0

Writing (2.25) and (2.29) together, following full-order linearized model of the PMSG is obtained

2.1

Full-Order State-Space Model of a PMSG

25

ΔV x d ΔXp = Ap ΔXp þ Bp dt ΔV y

ð2:30Þ

where ΔXp = ΔXp5 T Ap = Bp =

ΔXp4 T

Ap5

Ap6

0

Ap4

Bp2 Bp3

T

,

, Ap6 = ½ 0 a1 ],

, Bp3 = ½ bp6

bp7 ]

According to the definition of ΔXp1 in (2.6), ΔXp2 (2.10), ΔXp3 in (2.18), ΔXp4 in (2.25), ΔXp5 in (2.29) and ΔXp in (2.30), from (2.26) it can have ΔI x

= CTp ΔXp

ΔI y

ð2:31Þ

where CTp = ½ 0 cp8 =

2.1.6

0 0 0

cos θp0 sin θp0

0 0

, cp9 =

cp8

cp9

- sin θp0 cos θp0

0

, cp10 =

0 0 0

0 0 cp10 ]

- sin θp0

- cos θp0

I pcd0

cos θp0

- sin θp0

I pcq0

Block Diagram Model of the PMSG

In order to clearly show the dynamic connections between main parts of the PMSG, a block diagram model of the PMSG is derived as follows. From (2.6) and (2.10), ΔXp1 = sI - Ap1

-1

bp1 ΔV psd þ bp2 V psq

ΔPps = cp1 T ΔXp1 þ dp1 ΔV psd þ dp2 ΔV psq

ð2:32Þ

26

2

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

Fig. 2.6 Dynamic connections between the permanent magnet SG and the MSC

ΔV psd

b p1

c p 3T

c p 2T

( sI – Ap1 )–1

Δ X p1

B p1

( sI – Ap 2 )–1

c p 5T

-1

d p1

c p 4T

ΔV psq

bp2

ΔXp2 = sI - Ap2

Δ X p2

c p1T

Δ Pps

d p2

Bp1 ΔXp1

ΔV psd = cp2 T ΔXp2 þ cp3 T ΔXp1

ð2:33Þ

ΔV psq = cp4 T ΔXp2 þ cp5 T ΔXp1 From (2.32) and (2.33), dynamic connections between the permanent magnet SG and the MSC are displayed in Fig. 2.6. Active power output from the PCC is equal to that of the GSC in Fig. 2.2. Hence, Ppc = V pd I pcd þ V pq I pcq

ð2:34Þ

ΔPpc = V pd0 ΔI pcd þ V pq0 ΔI pcq þ I pcd0 ΔV pd þ I pcq0 ΔV pq

ð2:35Þ

Linearization of (2.34) is

Linearization of (2.11), (2.12), (2.14)–(2.17) is respectively d ω ω ΔI = 0 ΔV pcd - 0 ΔV pd þ ω0 ΔI pcq dt pcd X pf X pf ω ω d ΔI = 0 ΔV pcq - 0 ΔV pq - ω0 ΔI pcd X pf dt pcq X pf Cp V pdc0

dΔV pdc = ΔPps - ΔPpc dt

ΔQp = V pq0 ΔI pcd - V pd0 ΔI pcq þ I pcd0 ΔV pq - I pcq0 ΔV pd

ð2:36Þ

ð2:37Þ ð2:38Þ

2.1

Full-Order State-Space Model of a PMSG

27

d Δx = K pi4 ΔV pdc dt p4 d Δx = K pi6 ΔQp dt p6 d Δx = K pi5 ΔI pcdref - ΔI pcd dt p5

ð2:39Þ

d Δx = K pi7 ΔI pcqref - ΔI pcq dt p7 ΔI pcdref = K pp4 ΔV pdc þ Δxp4 ΔI pcqref = K pp6 ΔQp þ Δxp6 ΔV pcdref = K pp5 ΔI pcdref - ΔI pcd þ Δxp5 - X pf ΔI pcq þ ΔV pd

ð2:40Þ

ΔV pcqref = K pp7 ΔI pcqref - ΔI pcq þ Δxp7 þ X pf ΔI pcd þ ΔV pq ΔV pcdref = ΔV pcd

ð2:41Þ

ΔV pcqref = ΔV pcq

At the steady state, the PCC voltage is on the d axis in Fig. 2.5. Hence, Vpq0 = 0 such that ΔPpc = V pd0 ΔI pcd þ ½ I pcd0 ΔQp = - V pd0 ΔI pcq þ ½ I pcd0

I pcq0 ]

ΔV pd = V pd0 ΔI pcd þ k1 ΔVpdq ΔV pq

- I pcq0 ]

ΔV pd ΔV pq

ð2:42Þ

= - V pd0 ΔI pcq þ k2 ΔVpdq

ð2:43Þ From (2.37), the first and second equation in (2.39) and (2.40), it can have K pp4 s þ K pi4 1 ΔPps - ΔPpc = Gd ðsÞ ΔPps - ΔPpc s Cp V pdc0 s K pp6 s þ K pi6 ΔI pcqref = ΔQp = Gq ðsÞΔQp s ð2:44Þ ΔI pcdref = -

From (2.36), (2.41), the third and fourth equation in (2.39) and (2.40), it can have

28

2

ΔI pcd =

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

ω0 K pp5 s þ ω0 K pi5 ΔI = Gid ðsÞΔI pcdref xpf s2 þ ω0 K pp5 s þ ω0 K pi5 pcdref

ð2:45Þ

ω0 K pp7 s þ ω0 K pi7 ΔI pcq = ΔI = Giq ðsÞΔI pcqref xpf s2 þ ω0 K pp7 s þ ω0 K pi7 pcqref From Fig. 2.3 Δθp =

s2

V 0 F ðsÞ K pp s þ K pi Δθ = PLL ðsÞΔθ þ V 0 F ðsÞK pp s þ V 0 F ðsÞK pi

ð2:46Þ

From (2.24), Δθ = -

V y0 V 20

V x0 V 20

ΔV x ΔV y

= k3 ΔVxy

ð2:47Þ

From (2.26) and (2.27), ΔV pd

ΔVpdq =

=

- sinθp0 cosθp0

V x0

ΔV pq - cosθp0 - sinθp0 =k4 Δθp þK1 ΔVxy

V y0

- sinθp0 - cosθp0

I pcd0

ΔI y cosθp0 - sinθp0 =k5 Δθp þK2 ΔIpdq

I pcq0

ΔIxy =

ΔI x

=

Δθp þ

Δθp þ

sinθp0

ΔV x

- sinθp0 cosθp0

ΔV y

cosθp0

cosθp0 - sinθp0

ΔI pcd

sinθp0 cosθp0

ΔI pcq ð2:48Þ

From (2.42)–(2.48), a block diagram model of the PMSG is obtained and displayed in Fig. 2.7.

k1

k4 Δθ

k3

-

Δ Pps

Δ I pcdref Δ I pcd Gid ( s ) Gd ( s )

V pd 0

Δ V pdq

K1

ΔVxy

Δ Ppc

PLL ( s )

k2 Δθ p

K2 V pd 0

Δ Qp

Fig. 2.7 Block diagram model of the PMSG

Gq ( s )

Giq ( s ) Δ I pcqref k5

Δ I pcq

ΔI xy

2.2

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

2.2

29

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System Dominated by Dynamics of the PLL

2.2.1

Full-Order State-Space Model of a Single Grid-Connected PMSG System

Figure 2.8 shows the configuration of a single grid-connected PMSG system where a PMSG is connected to the external main grid via a transmission line, which is represented as a lumped reactance, XL. The full-order state-space model of the PMSG is given by (2.30) and (2.31) which are re-written below. d ΔXp = Ap ΔXp þ Bp ΔVxy dt ΔIxy = Cp ΔXp

ð2:49Þ

where ΔXp is the vector of all the 15 state variables of the PMSG, ΔIxy = T T ½ ΔI x ΔI y ] and ΔVxy = ½ ΔV x ΔV y ] ; Vx + jVy and Ix + jIy are the voltage at/and output current of the PCC of the PMSG, respectively, expressed in the common x–y coordinate. Linearized line voltage equations of the transmission line are d ΔI = dt x d ΔI = dt y

ω0 ðΔV x - ΔV bx Þ þ ω0 ΔI y XL ω0 ΔV y - ΔV by - ω0 ΔI x XL

ð2:50Þ

Ignoring the dynamics of the external main grid, it can have that ΔVbx + jΔVby = 0. Thus, the above equations can be written in the matrix form as

MSC N

Pps

Ppc

Cp

V pdc

GSC

V pcd +jV pcq Vx +jVy X pf

Vbx +jVby XL

S

Pp +jQ p

V psdref

V psqref

Control system

ω prref

V pq

V pd Control system

I psdref

V pdcref

θp

PLL

Q pref

Fig. 2.8 Configuration of a single grid-connected PMSG system

PCC

θ

I x +jI y

External main grid

30

2

s ω0

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

-1

s ΔIxy = XL ðsÞΔIxy ð2:51Þ ω0 0 -1 X s 1 0 XLs þ XL ΔIxy = = L U þ X L U2 ΔIxy ω0 0 1 ω0 1 1 0

ΔVxy = X L

1

1 0 0 -1 , U2 = . 0 1 1 0 From (2.49) and (2.51), it can have

where U1 =

sΔXp = Ap ΔXp þ Bp ΔVxy X s = Ap ΔXp þ Bp L U1 þ X L U2 ΔIxy ω0 XLs = Ap ΔXp þ Bp U þ X L U2 Cp ΔXp ω0 1

ð2:52Þ

Re-arranging (2.52), it can have X d ΔXp = U1 - L Bp Cp ω0 dt

-1

Ap þ X L Bp U2 Cp ΔXp = AΔXp

ð2:53Þ

The above equation is the full-order state-space model of the single gridconnected PMSG system displayed in Fig. 2.8. The linearized model of the PMSG is presented in the form of block diagram as shown by Figs. 2.6 and 2.7. A combined illustration of Figs. 2.6 and 2.7 is displayed in Fig. 2.9. For the single grid-connected PMSG system displayed in Fig. 2.8, from (2.51) it can have ΔVxy = XL(s)ΔIxy, where XL(s) represents the dynamics of the line connecting the PMSG to the external main grid. Hence from Fig. 2.9, a block diagram model of the single grid-connected PMSG displayed in Fig. 2.8 can be easily derived and illustrated by Fig. 2.10. Figure 2.10 clearly indicates that the single grid-connected PMSG system displayed in Fig. 2.8 is comprised of two subsystems in series connection. Subsystem 1 consists of the permanent magnet SG and the MSC. Subsystem 2 is comprised of the GSC, the PLL and XL(s), which represents the grid connection of the PMSG with the external grid. The single grid-connected PMSG system displayed Fig. 2.8 is stable if and only if both subsystems are stable. In the literature, the examination of small-signal stability has been focused on subsystem 2 for the following reasons: (1) The examination is carried out at a given operating condition and hence the wind speed is fixed such that the active power output from the permanent magnet SG is a constant (ΔPpc = 0); (2) There is no dynamic interaction between subsystem 1 and external grid, which is represented by XL(s). For the above reasons, it is sufficient to examine the stability of subsystem 2 only to study the small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8.

2.2

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

31

ΔV psd

b p1

c p 2T

c p 3T

B p1

( sI – Ap 2 )–1

Δ X p1

( sI – Ap1 ) –1 bp2

c p1T

c p 5T

Δ X p2

c p 4T ΔV psq

d p2

d p1

k1

k4 k3

Δθ

-

Δ I pcd Δ I pcdref Gid ( s ) Gd ( s )

V pd 0

ΔV pdq

K1

ΔVxy

ΔPps

ΔPpc

PLL ( s )

K2 V pd 0

-

k2

Δ Qp

Δθ p

Gq ( s )

Giq ( s ) Δ I pcqref

Δ I pcq

ΔI xy

k5

Fig. 2.9 Block diagram model of a PMSG-combined illustration of Figs. 2.6 and 2.7

Subsystem 1

ΔV psd

b p1

c p 2T

c p 3T

B p1

( sI – Ap 2 )–1

c p 5T

c p 4T

Δ X p1

( sI – Ap1 ) –1 bp 2

c p1T

Δ X p2

ΔV psq

d p2

d p1

Subsystem 2

k1

k4 k3

Δθ

Δ Pps

-

Δ I pcd Δ I pcdref Gid ( s) Gd ( s)

V pd 0

Δ V pdq

K1

ΔVbxy ΔVxy

Δ Ppc

PLL ( s)

k2 Δθ p

K2 V pd 0

Δ Qp

Gq ( s)

Giq ( s) Δ I pcqref

Δ I pcq

ΔI xy

k5

X L ( s)

Fig. 2.10 Block diagram model of the single grid-connected PMSG system displayed in Fig. 2.8

32

2

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

However, even if only the stability of subsystem 2 in Fig. 2.10 is examined, the order of the subsystem may be too high to be handled with analytically. Hence in this section, the examination is carried out to consider a simple case that the PMSG is dominated by the dynamics of the PLL. Afterward, more general cases are examined by taking an important and unique feature of the vector control system of the GSC of the PMSG: the bandwidth of power control outer loop is normally one- tenth of the current control inner loop. Subsequently, dynamics of outer loop are slower than those of inner loop by approximately ten times such that the oscillatory stability of the single grid-connected PMSG system can be examined in the low-frequency band (dominated by the outer loop) and the high-frequency band (dominated by the inner loop) separately. As the outer loop and the inner loop are mainly associated with the dynamics of DC voltage and current, in this book, they are referred to as the DC voltage dynamics and current dynamics, respectively. The small-signal stability of subsystem 2 (i.e., the single grid-connected PMSG system) in the fast current timescale is examined in Sect. 2.3. In Sect. 2.4, the small-signal stability of subsystem 2 in the slow DC voltage timescale is analyzed.

2.2.2

Simplified Reduced-Order Model of the Single Grid-Connected PMSG System Dominated by Dynamics of the PLL

If a PMSG is dominated by the dynamics of the PLL, it is possible that the oscillation mode associated with the PLL is the most poorly damped oscillation mode of the PMSG. This case has been found when the PMSG is weakly connected to the external grid. Hence, in order to gain a better understanding about why the oscillation mode associated with the PLL may intend to become poorly damped when the PMSG is under the condition of weak grid connection, it is important to examine the small-signal oscillatory stability of a single grid-connected PMSG system which is dominated the dynamics of the PLL. According to the control system theory, if a linear dynamic system has a dominant oscillation mode, the dynamics of the system are dominated by the dominant oscillation mode. In this case, the linear dynamic system can be approximately represented by a simplified second-order model that describes the dominant oscillation mode. Thus, the dominant oscillation mode determines the stability of the dynamic system, stability of which can be assessed by using the simplified second-order model [2]. Therefore, in this subsection, a reduced second-order model of the PMSG, which is dominated by the dynamics of the PLL, is derived. The derivation is to retain the dynamics of the PLL and to ignore the dynamics of other parts of the PMSG. From the block diagram model of the PMSG shown by Fig. 2.7, it can be seen that when the dynamics of the PMSG except those of the PLL are ignored, it can have

2.2

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

ΔIpdq =

ΔI pcd =0 ΔI pcq

33

ð2:54Þ

By using (2.54) and also from (2.25) and (2.26), following reduced second-order model of the PMSG dominated by the dynamics of the PLL is obtained d ΔXPLL = APLL ΔXPLL þ BPLL ΔVxy dt ΔIxy = CPLL ΔXPLL

ð2:55Þ

where Δθp ]T

ΔXPLL = ΔXp4 = ½ Δxp APLL =

0

- K pi V 0

1

- K pp V 0

CPLL = ½ 0 k5 =

k5 ] =

-

K pi V y0 V0

K pi V x0 V0

-

K pp V y0 V0

K pp V x0 V0

, BPLL =

0

- I y0

0

I x0

- sin θp0

- cos θp0

I pcd0

cos θp0

- sin θp0

I pcq0

=

- I y0 I x0

From (2.53), following reduced-order state-space model of the grid-connected PMSG system displayed in Fig. 2.8, which is dominated by the dynamics of the PLL, is obtained d ΔXPLL = U1 - XωL0 BPLL CPLL dt = ACPLL ΔXPLL

-1

ðAPLL þ X L BPLL U2 CPLL ÞΔXPLL

ð2:56Þ

Let P0 and Q0 be the steady-state active and reactive power output from the PMSG. It can have P0 = I x0 V x0 þ I y0 V y0 Q0 = I x0 V y0 - I y0 V x0

From (2.55) and (2.57), it can have

ð2:57Þ

34

2 BPLL CPLL = -

BPLL U2 CPLL =

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

K pi V y0 V0

K pi V x0 V0

- I y0

0

K pp V y0 K pp V x0 0 I x0 V0 V0 K pi V y0 K pi V x0 0 -1 V0 V0 -

K pp V y0 V0

K pp V x0 V0

1

0

0

K pi

P0 V0

0

K pp

P0 V0

=

0

- I y0

0

I x0

0

K pi

Q0 V0

0

K pp

Q0 V0

=

ð2:58Þ

From (2.55) and (2.58),

U1 -

XL ω0

BPLL CPLL

-1

1 = 0

XL P K 0 ω0 pi V 0 P X 1 - L K pp 0 ω0 V0 -

1-

1

=

-1

P0 XL K ω0 pp V 0 0

P0 XL K ω0 pp V 0 Q 0 X L K pi 0 - K pi V 0 V0 APLL þ X L BPLL U2 CPLL = Q0 1 X L K pp - K pp V 0 V0 1-

XL P K 0 ω0 pi V 0 1

ð2:59Þ Thus, ACPLL = U1 =

=

XL ω0

1-

1 1-

P0 XL K ω0 pp V 0 1

1-

-1

BPLL CPLL

P0 XL K ω0 pp V 0

APLL þ X L BPLL U2 CPLL

P0 XL K ω0 pp V 0 0

XL P K 0 ω0 pi V 0 1

XL P K 0 ω0 pi V 0 1

0 1

Q0 - K pi V 0 V0 Q X L K pp 0 - K pp V 0 V0 X L K pi

a1 X L K pp

Q0 - K pp V 0 V0 ð2:60Þ

where a1 = 1 -

XL ω0

K pp VP00

X L K pi QV 00 - K pi V 0 þ XωL0 K pi VP00 X L K pp QV 00 - K pp V 0

2.2 Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

2.2.3

35

Small-Signal Stability Limit of the Single Grid-Connected PMSG System Dominated by the Dynamics of the PLL

From (2.60), the characteristic equation of the single grid-connected PMSG system displayed in Fig. 2.8, which is dominated by the dynamics of the PLL, is |λU1 - ACPLL| = 0. It can have jλU1 - ACPLL j = λ 1- 1= 1þ

X P0 P XL - L K pi 0 K ω0 pp V 0 ω0 V0 XL P0 K ω0 pp V 0

XL P0 K ω0 pp V 0

X L K pi

Q XL P0 - X L K pp 0 þ K pp V 0 K ω0 pp V 0 V0

λ 1-

Q0 Q X P - K pi V 0 - L K pi 0 X L K pp 0 - K pp V 0 V0 ω0 V0 V0

2

λ2 þ 1 -

XL P0 K ω0 pp V 0

-

Q XL P K 0 - X L K pp 0 þ K pp V 0 λ ω0 pi V 0 V0

Q XL P K 0 X L K pp 0 - K pp V 0 ω0 pi V 0 V0

- 1= 1-

XL P0 K ω0 pp V 0

XL P0 K ω0 pp V 0

- 1-

X L K pi

Q0 - K pi V V0

2

XL P0 K ω0 pp V 0

λ2 þ 1 X L K pi

XL P0 K ω0 pp V 0

0

Q XL P K 0 X L K pp 0 - K pp V 0 ω0 pi V 0 V0

-

Q XL P K 0 - X L K pp 0 þ K pp V 0 λ ω0 pi V 0 V0

Q0 - K pi V 0 = 0 V0

ð2:61Þ Hence, jλU1 - ACPLL j P X = 1 - L K pp 0 λ2 þ ω0 V0

-

Q XL P K 0 - X L K pp 0 þ K pp V 0 λ ω0 pi V 0 V0

ð2:62Þ

Q þ - X L K pi 0 þ K pi V 0 = 0 V0

According to the Routh–Hurwitz stability criterion, the single grid-connected PMSG system displayed in Fig. 2.8 is stable if and only when

36

2

1-

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

XL P0 K >0 ω0 pp V 0

Q XL P K 0 - X L K pp 0 þ K pp V 0 > 0 ω0 pi V 0 V0

- X L K pi

ð2:63Þ

Q0 þ K pi V 0 > 0 V0

Re-arranging (2.63), it can have K pp V0 > X L P0 ω0 ω0 K pp V 0 2 > X L K pi P0 þ ω0 X L K pp Q0

ð2:64Þ

V 0 2 > X L Q0 Since XLKpiP0 > 0, thus if ω0KppV02 > XLKpiP0 + ω0XLKppQ0, it must have ω0KppV02 > ω0XLKppQ0, i.e., V02 > XLQ0. Hence, the single grid-connected PMSG system displayed in Fig. 2.8 is stable if and only when K pp V0 > X L P0 ω0 ω0 K pp V 0 2 > X L K pi P0 þ ω0 X L K pp Q0

ð2:65Þ

Re-arranging (2.65), it can have K pp V0 > X L P0 ω0 K pi Q V 02 > þ 0 X L P0 ω0 K pp P0

ð2:66Þ

Results of case-by-case study by numerical computation/simulation in the literature have indicated that for the single grid-connected PMSG system displayed in Fig. 2.8, there are three main factors which could affect the small-signal oscillatory stability unfavorably. Those three unfavorable affecting factors are: (1) the condition of weak grid connection; (2) the heavy loading condition; and (3) improper setting of control parameters of the GSC and the PLL. The condition of grid connection of the PMSG is measured by the following short circuit ratio (SCR) [3] SCR =

V 02 X L Pw‐rated

ð2:67Þ

2.2

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

37

where Pw ‐ rated is the rated capacity of the PMSG. In general, if the SCR is smaller than 3, the grid connection of the PMSG is considered to be weak [3]. However, the SCR cannot be less than 1, i.e., SCR =

V 02 ≥1 X L Pw‐rated

ð2:68Þ

When the power base is taken to be the nominal power of the PMSG, the SCR can be represented by the per-unit impedance of the transmission line as SCR ≈

1 ≥1 xL

ð2:69Þ

Hence, the bigger xL is, the smaller the SCR is, and thus the weaker the grid connection of the PMSG is. The derived small-signal stability limit of (2.66) clearly explains why the condition of weak grid connection (big value of xL) or/and heavy loading (big value of P0) means the instability risk of the single grid-connected PMSG system. This reveals analytically the mechanism about why condition of weak grid connection and heavy loading may affect the small-signal stability of the single grid-connected PMSG system unfavorably. In addition, it indicates that if the PI gains of the PLL are set to violate the stability limit given by (2.66), the single grid-connected PMSG system becomes unstable. This can be discussed further in details as follows. Normally, V0 ≈ 1. Thus, from (2.68) it can have V 02 V0 ≥ = SCR ≥ 1 X L P0 X L Pw‐rated

ð2:70Þ

Consider that the PI gains of the PLL are set to have K pp ≤1 ω0 K pi Q þ 0 ≤1 ω0 K pp P0

ð2:71Þ

Subsequently, according to (2.66) and (2.70), if (2.71) is satisfied, the single gridconnected PMSG system displayed in Fig. 2.8 should be stable. Normally, a PMSG operates with a high value of power factor in the range from 0.95 to 2. Hence, in (2.71) the maximum value of QP00 is 0.33. Thus, if the PI gains of K K the PLL are set to ensure ω0 Kpipp ≤ 0:67, it should have ω0 Kpipp þ QP00 ≤ 1. Therefore, to ensure the stability of the single grid-connected PMSG system displayed in Fig. 2.8, which is dominated by the dynamics of the PLL, PI gains can be set to satisfy the following condition:

38

2

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

K pi K pp ≤ 1; ≤ 0:67 ω0 ω0 K pp

ð2:72Þ

Improper setting of the PI gains of the PLL means the violation of (2.72) when the single-grid connected PMSG system displayed in Fig. 2.8 may become unstable.

2.3

2.3.1

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System in the Fast Current Timescale Simplified Reduced-Order Model of a PMSG in the Fast Current Timescale

By linearizing (2.11)–(2.17), linearized model of the GSC is obtained to be d ΔI = dt pcd d ΔI = dt pcq

ω0 ω ΔV pcd - 0 ΔV pd þ ω0 ΔI pcq X pf X pf ω0 ω ΔV pcq - 0 ΔV pq - ω0 ΔI pcd X pf X pf

Cp V pdc0

dΔV pdc = ΔPps - ΔPpc dt

ΔPpc = V pcd0 ΔI pcd þ V pcq0 ΔI pcq þ I pcd0 ΔV pcd0 þ I pcq0 ΔV pcq0

ð2:73Þ

ð2:74Þ ð2:75Þ

d Δx = K pi4 ΔV pdc dt p4 d Δx = K pi5 ΔI pcdref - ΔI pcd dt p5 d Δx = K pi6 ΔQp dt p6

ð2:76Þ

d Δx = K pi7 ΔI pcqref - ΔI pcq dt p7 ΔQp = V pq0 ΔI pcd - V pd0 ΔI pcq þ I pcd0 ΔV pq - I pcq0 ΔV pd

ð2:77Þ

ΔI pcdref = K pp4 ΔV pdc þ Δxp4 ΔV pcdref = K pp5 ΔI pcdref - ΔI pcd þ Δxp5 - X pf ΔI pcq þ ΔV pd ΔI pcqref = K pp6 ΔQp þ Δxp6 ΔV pcqref = K pp7 ΔI pcqref - ΔI pcq þ Δxp7 þ X pf ΔI pcd þ ΔV pq

ð2:78Þ

2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

ΔV pcdref = ΔV pcd ΔV pcqref = ΔV pcq

39

ð2:79Þ

Linearized model of the PLL is (2.23), (2.24), and (2.48), which are re-written as follows dΔxp = K pi V 0 Δθ - Δθp dt dΔθp = K pp V 0 Δθ - Δθp þ Δxp dt Δθ = ΔVpdq =

ΔV pd ΔV pq

= ½ V q0 ΔIxy =

ΔI x ΔI y

= ½ - I y0

V y0 V 20

V x0 V 20

ð2:80Þ

ΔV x

ð2:81Þ

ΔV y

= k4 Δθp þ K1 ΔVxy - V d0 ]T Δθp þ

cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y

ð2:82Þ

= k5 Δθp þ K2 ΔIpdq I x0 ]T Δθp þ

cos θp0

- sin θp0

ΔI pcd

sin θp0

cos θp0

ΔI pcq

When the oscillatory stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is studied, the slow dynamics of the outer loop of the GSC can be ignored. That is to assume that the impact of relatively slow variations of output signals from the outer loop of the GSC can be neglected (see Fig. 2.3) to have ΔI pcdref = 0, ΔI pcqref = 0

ð2:83Þ

Hence, Eqs. (2.74), (2.75), and (2.77), the first and third equation in (2.76), which describe the slow dynamics of the outer loop, are not considered when the simplified model of the GSC in the fast current timescale is derived. By using (2.79) and (2.83), the second and fourth equation in (2.76) and (2.78) become, respectively. d Δx = - K pi5 ΔI pcd dt p5 d Δx = - K pi7 ΔI pcq dt p7

ð2:84Þ

40

2

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

ΔV pcd = - K pp5 ΔI pcd þ Δxp5 - X pf ΔI pcq þ ΔV pd ΔV pcq = - K pp7 ΔI pcq þ Δxp7 þ X pf ΔI pcd þ ΔV pq

ð2:85Þ

Substituting (2.85) in (2.73), ω0 K pp5 ω d ΔI pcd þ 0 Δxp5 ΔI pcd = X pf X pf dt ω0 K pp7 d ω ΔI pcq þ 0 Δxp7 ΔI = X pf X pf dt pcq

ð2:86Þ

Thus, following simplified reduced-order model of the GSC in the fast current timescale is obtained from (2.84) and (2.86) d Δx dt p5 d Δx dt p7 d ΔI dt pcd d ΔI dt pcq

0 0 = ω0 X pf 0

0 0 0

- K pi5 0 ω0 K pp5 X pf

ω0 X pf

0

0 - K pi7

Δxp5 Δxp7

0 -

ω0 K pp7 X pf

ð2:87Þ

ΔI pcd ΔI pcq

Parameters of the PLL can be set in a wide range of values depending on the need of phase tracking speed and performance. Thus, it is possible that the dynamics of the PLL may fall in the fast current timescale, which should be included in the model of the PMSG in the fast current timescale. Substituting (2.81) in (2.80) and also from the second equation of (2.82), it can have dΔθp dt dΔxp dt ΔI x ΔI y

=

0

- K pp V 0

Δθp

1

- 0 K pi V 0

Δxp

= ½ - I y0

I x0 ]T Δθp þ

- K pp þ

V y0 V0

cos θp0

V y0 V0 - sin θp0

sin θp0

cos θp0

- K pi

K pp

V x0 V0

K pi

V x0 V0

ΔV x ΔV y

ð2:88Þ

ΔI pcd ΔI pcq

Arranging (2.87) and (2.88) together, following simplified reduced-order model of the PMSG in the fast current timescale is obtained d ΔXf = Af ΔXf þ Bf ΔVxy dt ΔIxy = Cf ΔXf

ð2:89Þ

2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

41

where ΔXf = ½ Δxp5 Δxp7 ΔI pcd ΔI pcq ΔV x ΔI x ΔVxy = , ΔIxy = ΔV y ΔI y

Af =

Bf =

2.3.2

Δxp ]T

Δθp

0

0

- K pi5

0

0

0

0

0

0

- K pi7

0

0

ω0 X pf

0

0

0

0

0

ω0 X pf

0

0

0

0

0

0

0

- K pp V 0

1

0

0

0

0

- K pi V 0

0

-

ω0 K pp5 X pf

0

0

0

0

0

0

0 - K pp

V y0 V0

- K pi

V y0 V0

-

ω0 K pp7 X pf

0 , Cf = 0 0 V K pp x0 V0 K pi

0

cos θp0

- sin θp0

- I y0

0

sin θp0

cos θp0

I x0

0 0

V x0 V0

Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Fast Current Timescale

In the fast timescale, dynamics of line current should be considered. Thus, the linearized voltage equations of the line connecting the PMSG to the external main grid in Fig. 2.8 are d ΔI = dt x d ΔI = dt y

ω0 ΔV x XL ω0 ΔV y XL

ω0 ΔV bx þ ω0 ΔI y XL ω0 ΔV by - ω0 ΔI x XL

ð2:90Þ

42

2

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

where Vbx + jVby is the voltage at the busbar of the external main grid. In studying the small-signal stability of the PMSG system displayed in Fig. 2.8, voltage at this busbar is considered to be constant such that ΔVbx + jΔVby = 0. Hence from (2.90), ΔVxy =

- ω0

XL s ω0 ω 0

s

ð2:91Þ

ΔIxy

Substituting (2.91) in (2.89), X d ΔXf = Af ΔXf þ Bf L ω0 dt XL = Af ΔXf þ B ω0 f

-1

s

Cf ΔXf 1 s 1 0 0 Cf ΔXf þ X L Bf 0 1 1

-1 0

ð2:92Þ

Cf ΔXf

where

1 Bf

0

0 1

Cf =

0

0

0

0

0

0

0

0 V K pp x0 V0

V y0 V0 V y0 - K pi V0

- K pp

=

K pi

V x0 V0

0 0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0 0

b1

b2

b3

0

0 0

b4

b5

b6

0

0

0

cos θp0

0

0

sin θp0

- sin θp0 - I y0 0 cos θp0

I x0

0

2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

43

V y0 V cos θp0 - K pp x0 sin θp0 V0 V0 V y0 V x0 b2 = K pp sin θp0 þ K pp cos θp0 V0 V0 V y0 V b3 = K pp I þ K pp x0 I x0 V 0 y0 V0 V y0 V b4 = - K pi cos θp0 - K pi x0 sin θp0 V0 V0 V y0 V b5 = K pi sin θp0 þ K pi x0 cos θp0 V0 V0 V y0 V x0 b6 = K pi I þ K pi I V 0 y0 V 0 x0 b1 = - K pp

X L Bf

XL

=

0

-1

1

0

ð2:93Þ

Cf =

0

0

0

0

0

0

0

0 V K pp x0 V0

0

-1

0

0

cos θp0

1

0

0

0

sin θp0

- K pp

V y0 V0

- K pi

V y0 V0

K pi

0 0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0 0

b7

b8

b9

0

0 0

b10

b11

b12

0

V x0 V0

- sin θp0 - I y0 cos θp0

I x0

0 0

44

2

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

b7 = X L K pp

V y0 V x0 cos θp0 þ K pp sin θp0 V0 V0

b8 = X L K pp

V y0 V x0 sin θp0 - K pp cos θp0 V0 V0 V y0 V x0 I þ K pp I V 0 y0 V 0 x0

b9 = X L - K pp b10 = X L K pi

V y0 V x0 cos θp0 þ K pi sin θp0 V0 V0

b11 = X L K pi

V y0 V x0 sin θp0 - K pi cos θp0 V0 V0

b12 = X L - K pi

ð2:94Þ

V y0 V x0 I þ K pi I V 0 y0 V 0 x0

From (2.92), 1 X d ΔXf = I - L Bf ω0 dt 0

-1

0 1

Af ΔXf

Cf

1 0 X Cf I - L Bf ω0 0 1

X þ L ω0

-1

Bf

0

-1

1

0

ð2:95Þ Cf ΔXf

where I-

1 XL Bf ω0 0

-1

0 1

Cf

=

I

0

A

B

-1

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

= 0

0 0

1 X - L b2 ω0

0 X 1 - L b3 ω0

0

0

0 X - L b1 ω0

0

0

0

0

0

0

A=

XL b ω0 4

-

XL b ω0 1 X - L b4 ω0

-

-

XL b ω0 5

-

XL b ω0 6

-1

0 1

X XL b 1 - L b3 ω0 2 ω0 ,B= XL XL b b ω0 5 ω0 6

0 1

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

2.3

I

Since

0

A

=

B

-1

I -B

0

-1

A

B-1

and

1 1 XL X b 1 - L b3 ω0 6 ω0

0 X 1 - L b3 ω0

1 1 XL A= X b 1 - L b3 ω0 6 ω0

0 X 1 - L b3 ω0

B-1 =

-B

-1

1 X 1 - L b3 ω0 XL2 b13 = b b þ ω0 2 6 1 2 X b14 = L2 b6 b2 þ ω0 =

0 0

XL b ω0 1 0 b13

XL b ω0 2 b14

0

XL b ω0 3 X 1 - L b3 ω0 1-

45

XL b ω0 2 XL b ω0 5

XL b ω0 1 XL 0 0 b ω0 4

0 0

ð2:96Þ

XL b ω0 4 XL b ω0 5

Thus, 1 0 X s Cf I - L Bf ω0 0 1 =

I -B

-1

0

-1

A B-1

1 0

0

0

0

0

0 1

0

0

0

0

0 0

1

0

0

0

0 1 X b 1- L 3 ω0

0

0 0 = 0 0 ω0

0 0

0 1 XL XL b1 b2 X b X b 1- L 3 ω0 1 - L 3 ω0 ω0 b13 X b 1- L 3 ω0

b14 X b 1- L 3 ω0

ω0

X L b6 X b 1- L 3 ω0

-1

0

X L b3 ω0 X L b3 1ω0 1-

ð2:97Þ Denote Af in (2.89) as

46

2

0 0 ω0 X pf

Af =

0 ω0 X pf 0 0

0 0

Denote Bf

Bf

- K pi5 0 ω0 K pp5 X pf

0 0

0

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

0 1

0

-1

1

0

0 - K pi7

0 0

0 0

0

0

0

ω0 K pp7 X pf 0 0

0 0 0

0

0

- K pp V 0 - K pi V 0

1 0

0 A22

A11 0

=

ð2:98Þ

-1 Cf in (2.94) as 0

Cf =

0

0

B21

B22

=

0 0

0

0

0

0

0 0 0 0

0 0

0 0

0 0

0 0

0 0 0 0

0 b7

0 b8

0 b9

0 0

0 0

b10

b11

b12

0

ð2:99Þ

From (2.92), (2.97), (2.98), and (2.99), I-

1 XLs Bf ω0 0 =

1

Af =

Cf

-B

-1

AA11

B

-B

0

A11

0

- B - 1A

B-1

0

A22

1 B

A22 -1

Cf

0 A

-1

0

I -1

I

0

A11

1 X I - L Bf ω0 0 =

-1

0

-1

Bf 0 B21

0

-1

1

0

0 B22

=

ð2:100Þ Cf

B

0

0

-1

-1

B21

B

B22

Hence from (2.92) and (2.100), the state-space model of the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale PMSG is obtained to be Af1 d ΔXf = Afc ΔXf = dt Af2

0 ΔXf Af3

ð2:101Þ

2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

47

where Afc =

I - XωL0 Bf

=

A11

1 0 0 1

-1

-1 1 0 0 -1 XL X Cf Cf I - L Bf Bf ω0 ω0 0 1 1 0 0 0 Af1 0 þ = Af2 Af3 B - 1 B21 B - 1 B22

Af þ

Cf 0

- B - 1 AA11 B - 1 A22

Af1 = A11 Af2 = - B - 1 AA11 þ B - 1 B21 Af3 = B - 1 A22 þ B - 1 B22

From (2.95), (2.97), and (2.98),

A11 =

B-1 =

0

0

- K pi5

0

0

0

0

- K pi7

ω0 X pf

0

0

ω0 X pf

1 X 1 - L b3 ω0

-

ω0 K pp5 X pf -

0

1 XL b ω0 6

, A22 =

0

- K pp V 0

1

- K pi V 0

0

ω0 K pp7 X pf

ð2:102Þ

0 b9 , B22 = XL 1b b12 ω0 3

0 0

Hence, from (2.101) and (2.102) 0 0 ω 0 Af1 = A11 = X pf 0 Af3 =

1 0 1 X XL X b 1 - L b3 1 - L b3 ω 0 6 ω0 ω0

0

- K pi5

0

0

0 ω0 K pp5 X pf

- K pi7

0 ω0 X pf

0

0 -

ð2:103Þ

ω0 K pp7 X pf

- K pp V 0 1 b9 0 þ - K pi V 0 0 b12 0

- K pp V 0 þ b9 1 = XL XL X 1 - L b3 ω0 b6 - K pp V 0 þ b9 þ 1 - ω0 b3 ω0

1 - K pi V 0 þ b12

XL b ω0 6 ð2:104Þ

48

2

Subsystems of current control inner loop

Subsystems of PLL and transmission line

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

Δ I pcdref = 0 Δ I pcqref = 0

ΔVxy

k3

Δθ

Gid ( s )

Giq ( s )

PLL ( s )

ΔId

K2

ΔIq

Δθ p

k5

ΔI xy

Fig. 2.11 Simplified reduced-order block diagram model in the fast current timescale

Equation (2.101) is the simplified reduced-order model of the single gridconnected PMSG system displayed in Fig. 2.8 in the fast current timescale. It can be seen that the model is comprised of two dynamically decomposed parts. The first part is the current control inner loop of the GSC, since Af1 is the state matrix of simplified reduced-order model of the GSC in the fast current timescale as it is given in (2.87). The second part consists of the PLL and the dynamics of the transmission line, which is represented by XL. In fact, the decomposition of the dynamic model of the single grid-connected PMSG system displayed in Fig. 2.8 can be more clearly observed from the block diagram model of Fig. 2.7: When the slow DC voltage dynamics are ignored to derive the simplified reduced-order model of (2.101) with ΔIpcdref = 0, ΔIpcqref = 0 (see (2.83)), full-order block diagram model of Fig. 2.7 is simplified to become Fig. 2.11. It is clear that in Fig. 2.11, dynamics of the current control inner loop is decomposed from that of the PLL and transmission line. This explains and confirms (2.101).

2.3.3

Small-Signal Stability of the Single Grid-Connected PMSG System in the Fast Current Timescale

From (2.101), it can be seen that small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is determined by the characteristic equation |λI - Af1| = 0 and |λI - Af3| = 0. From (2.87) and (2.103), it can be seen that Af1 is the state matrix of simplified reduced-order model of the GSC in the fast current timescale, stability of which is determined by the current inner loop. In designing the current inner loop control, stability of the loop is considered to ensure the loop is stable as an independent part. Hence, stability of the single gridconnected PMSG system displayed in Fig. 2.8 in the fast current timescale is determined by the characteristic equation |λI - Af3| = 0. Thus, stability analysis can be carried out by examining |λI - Af3| = 0 as follows.

2.3

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

49

In (2.93) and (2.94), V y0 V P I þ K pp x0 I x0 = K pp 0 V 0 y0 V0 V0 V y0 V P b6 = K pi I þ K pi x0 I x0 = K pi 0 V 0 y0 V0 V0 V y0 Q X V b9 = L - K pp x0 I y0 þ K pp I = K pp X L 0 ω0 V0 V 0 x0 V0 b3 = K pp

b12 =

XL ω0

- K pi

ð2:105Þ

V y0 Q V x0 I þ K pi I = K pi X L 0 V 0 y0 V 0 x0 V0

where P0 = Vy0Iy0 + Vx0Ix0 and Q0 = - Vx0Iy0 + Vx0Ix0 are, respectively, the active and reactive power output from the PMSG. From (2.105), XL X b - K pp V 0 þ b9 þ 1 - L b3 ω0 6 ω0 =

P XL K 0 ω0 pi V 0

- K pp V 0 þ K pp X L

= - K pi V 0 þ K pi X L

Q0 V0

- K pi V 0 þ b12 þ 1-

P0 XL K ω0 pp V 0

- K pi V 0 þ K pi X L

Q0 V0

Q0 V0

ð2:106Þ From (2.104), (2.105), and (2.106),

1 Af3 = 1 - XωL0 K pp VP00

Q0 V0 Q0 - K pi V 0 þ K pi X L V0

- K pp V 0 þ K pp X L

1 ð2:107Þ

XL P K 0 ω0 pi V 0

Thus, |λI - Af3| = 0 becomes

jλI - Af3 j =

1 1-

XL P0 K ω0 pp V 0

λ þ K pp V 0 - K pp X L K pi V 0 - K pi X L

λ þ K pp V 0 - K pp X L =

Q0 V0

Q0 V0

Q0 V0

XL P K 0 ω0 pi V 0 X P 1 - L K pp 0 ω0 V0 λ-

-1 λ-

XL P K 0 ω0 pi V 0

þ K pi V 0 - K pi X L

Q0 V0

=0

ð2:108Þ The above equation is equivalent to

50

2

s þ K pp V 0 - K pp X L

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

Q0 V0

s-

XL P K 0 ω0 pi V 0

þ K pi V 0 - K pi X L

Q0 V0

Q0 X L P K 0 λ V0 ω0 pi V 0 Q Q X P P X - L K pi 0 K pp V 0 þ L K pi 0 K pp X L 0 þ K pi V 0 - K pi X L 0 ω0 V0 ω0 V0 V0 V0 Q P X L 0 = λ2 þ K pp V 0 - K pp X L 0 K λ V0 ω0 pi V 0

λ2 þ K pp V 0 - K pp X L

þK pi 1 -

X L P0 K ω0 V 0 pp

V 0 - XL

ð2:109Þ

Q0 V0

=0 Applying the Routh–Hurwitz stability criterion to (2.109), the necessary and sufficient condition that the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is obtained to be K pp V 0 - K pp X L K pi 1 -

Q0 X L P K 0 >0 V0 ω0 pi V 0

X L P0 K ω0 V 0 pp

V 0 - XL

Q0 >0 V0

ð2:110Þ ð2:111Þ

From (2.110), K pp V 0 - K pp X L

Q0 X L P > K 0 >0 V0 ω0 pi V 0

ð2:112Þ

Hence, it can have V 0 - X L QV 00 > 0. Thus from (2.111), it can have 1-

X L P0 K >0 ω0 V 0 pp

ð2:113Þ

Therefore, the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is stable if and only when K pp V 0 - K pp X L 1-

Q0 X L P K 0 >0 V0 ω0 pi V 0

X L P0 K >0 ω0 V 0 pp

From (2.114), it can have

ð2:114Þ

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

K pp V0 > X L P0 ω0 K pi Q V 02 > þ 0 X L P0 ω0 K pp P0

51

ð2:115Þ

Comparing (2.115) and (2.66), it can be seen that the small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is as same as that when the PMSG is dominated by the dynamics of the PLL. This implies that if the dynamics of the PLL are in the fast current timescale, dynamics of the PMSG in the fast current timescale are dominated by the PLL. Hence, taking the derivation as same as to that from (2.66) to (2.72), following conclusion can be obtained to guide the setting of the PI gains of the PLL: Conclusion 2.1 The PI gains of the PLL are set to satisfy K pp K pi ≤ 1; ≤ 0:67 ω0 ω0 K pp

ð2:116Þ

Consequently, small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the fast current timescale is ensured.

2.4

2.4.1

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Fast Current Timescale

When only the stability of subsystem 2 in Fig. 2.10 is considered, ΔPps = 0, as having been discussed previously. Thus, (2.47) becomes dΔV pdc 1 ΔPpc =dt C p V pdc0

ð2:117Þ

where ΔPpc is given by (2.75). At the steady state, the PCC voltage of the PMSG is on the d axis in Fig. 2.5. Hence, Vpq0 = 0 such that (see (2.42)) ΔPpc = V pd0 ΔI pcd þ ½ I pcd0

I pcq0 ]

ΔV pd ΔV pq

ð2:118Þ

52

2

Small-Signal Stability of a Single Grid-Connected PMSG System Δ Pps

k1

ΔP

-

V pd 0

ΔV pdq

K1

ΔI d

Gd ( s )

K2 V pd 0

k4

ΔVxy

k3

Δθ

k2

ΔQ

Gq ( s )

ΔI q

ΔI xy

k5

Fig. 2.12 Simplified block diagram model of the PMSG when dynamics of the PLL are in the fast current timescale

For the same reason, deviation of reactive power output given by (2.77) becomes (see (2.43)) ΔQp = - V pd0 ΔI pcq þ ½ I pcd0

- I pcq0 ]

ΔV pd ΔV pq

ð2:119Þ

In the slow DC voltage timescale, fast dynamics of the current control inner loop can be ignored. This means that the current control is so fast such that in Fig. 2.3, ΔI pcdref = ΔI pcd , ΔI pcqref = ΔI pcq

ð2:120Þ

Hence, ignorance of the fast dynamics of the current control inner loop is equivalent to have Gid(s) = 1 and Giq(s) = 1 (see (2.45)). When the dynamics of the PLL are in the fast current timescale, dynamics of the PLL can also be ignored to have PLL(s) = 1. Thus, Δθp = Δθ. The block diagram of the PMSG of Fig. 2.7 is simplified to that displayed in Fig. 2.12. By using (2.120), the first and third equation in (2.78) becomes ΔI pcd = K pp4 ΔV pdc þ Δxp4

ð2:121Þ

ΔI pcq = K pp6 ΔQp þ Δxp6 Substituting (2.121) in (2.118) and (2.119), it can have ΔPpc = V pd0 K pp4 ΔV pdc þ V pd0 Δxp4 þ ½ I pcd0 ΔQp = -

V pd0 1 Δx þ ½I 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6 pcd0

I pcq0 ]

ΔV pd ΔV pq

- I pcq0 ]

ΔV pd ΔV pq

ð2:122Þ ð2:123Þ

Substituting the first equation of (2.82) in (2.122) and (2.123), respectively

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

53

ΔPpc = V pd0 K pp4 ΔV pdc þ V pd0 Δxp4 þ½ I pcd0

I pcq0 ] ½ V q0

- V d0 ]T Δθp þ

cos θp0 - sin θp0

= V pd0 K pp4 ΔV pdc þ V pd0 Δxp4 þ Q0 Δθp þ ½ I x0

ΔV x

cos θp0

ΔV y

I y0 ]

ΔV y

- I pcq0 ] sin θp0

ΔV x

- sin θp0 cos θp0 V pd0 I pcd0 V q0 þ I pcq0 V d0 =Δx þ Δθp 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6

ΔV y

- V d0 ]T Δθp þ

½ V q0

þ

1 1 þ V pd0 K pp6

½ I pcd0

cos θp0

- I pcq0 ]

ð2:124Þ

ΔV x

V pd0 1 Δx þ ½I 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6 pcd0

ΔQp = -

sin θp0

ð2:125Þ

cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y

Note that in (2.124), Q0 = Ipcd0Vq0 - Ipcq0Vd0 is the steady-state reactive power output from the PMSG and ½ I x0 I y0 ] is obtained by using (2.20). Substituting (2.124) in (2.117), it can have dΔV pdc 1 =dt C p V pdc0

V pd0 K pp4 ΔV pdc þ V pd0 Δxp4 þ Q0 Δθp þ ½ I x0

I y0 ]

ΔV x ΔV y

ð2:126Þ The first equation of (2.76) is d Δx = K pi4 ΔV pdc dt p4

ð2:127Þ

Substituting (2.125) in the third equation of (2.76), it can have V pd0 I pcd0 V q0 þ I pcq0 V d0 d Δx þ Δθp Δx = K pi6 dt p6 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6 þ

1 1 þ V pd0 K pp6

½ I pcd0

- I pcq0 ]

cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y ð2:128Þ

Since Δθp = Δθ, from (2.47) Δθp = Δθ = -

V y0 V 20

V x0 V 20

ΔV x ΔV y

ð2:129Þ

54

2

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

Substituting (2.129) in (2.126) and (2.128) and then writing (2.126), (2.127), and (2.128) together, following simplified reduced-order state equations of the PMSG in the slow DC voltage timescale with the dynamics of the PLL in the fast current timescale being ignored is obtained: d ΔXs3 = As3 ΔXs3 þ Bs3 ΔVxy dt

ð2:130Þ

where ΔXs3 = ½ ΔV pdc Δxp4 K pp4 V pd0 Cp V pdc0 As3 =

Δxp6 ]T V pd0 Cp V pdc0

K pi4

0

0

0

I x0 C p V pdc0 0 K pi6 I y0 1 þ K pp6 V pd0 -

Bs3 =

-

V pd0 C p V pdc0 0

-

K pi6 V pd0 1 þ K pp6 V pd0

I y0 Cp V pdc0 0 K pi6 I x0 1 þ K pp6 V pd0 -

Substituting (2.125) in the second equation of (2.121), it can have ΔI pcq = Δxp6 þ K pp6 þ

1 1 þ V pd0 K pp6

V pd0 I pcd0 V q0 þ I pcq0 V d0 Δx þ Δθp 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6

½ I pcd0

- I pcq0 ]

cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y

ð2:131Þ Substituting (2.129) in (2.131), it can have ΔI pcq = Δxp6 þ K pp6 þ þ

V pd0 Δx 1 þ V pd0 K pp6 p6

V y0 I pcd0 V q0 þ I pcq0 V d0 - 2 1 þ V pd0 K pp6 V0 1 1 þ V pd0 K pp6

½ I pcd0

V x0 V 20

- I pcq0 ]

ΔV x ΔV y cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y ð2:132Þ

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

55

Substituting (2.132) and the first equation of (2.121) in the second equation of (2.79), it can have ΔIxy = Cs3 ΔXs3 þ Ds3 ΔVxy

ð2:133Þ

where K pp4 cos θp0

cos θp0

K pp4 sin θp0

sin θp0

Cs3 =

Ds3 =

K pp6 I y0 sin θp0 1 þ K pp6 V pd0 -

K pp6 I y0 cos θp0 1 þ K pp6 V pd0

-

sin θp0 1 þ K pp6 V pd0

cos θp0 1 þ K pp6 V pd0 K pp6 I x0 sin θp0 1 þ K pp6 V pd0 K pp6 I x0 cos θp0 1 þ K pp6 V pd0

The simplified reduced-order state-space model of the PMSG in the slow DC voltage timescale with the dynamics of the PLL being in the fast current timescale is given by (2.130) and (2.133), which are written together as follows d ΔXs3 = As3 ΔXs3 þ Bs3 ΔVxy dt

ð2:134Þ

ΔIxy = Cs3 ΔXs3 þ Ds3 ΔVxy

2.4.2

Simplified Reduced-Order Model of a Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Slow DC Voltage Timescale

Parameters of the PLL could be set within a wide range of values. Hence, it is possible that the dynamics of the PLL may fall within the slow DC voltage timescale. In this case, dynamics of the PLL cannot be ignored and need to be included in the model of the PMSG in the slow DC voltage timescale. In this subsection, the model of a single grid-connected PMSG system with the dynamics of the PLL being included is derived as follows. When the dynamics of the PLL are included, Δθp is a state variable, as the model of the PLL is (see (2.88))

56

2

dΔθp dt dΔxp dt ΔI x ΔI y

=

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

0

- K pp V 0

Δθp

1

- 0 K pi V 0

Δxp

= ½ - I y0

I x0 ]T Δθp þ

- K pp þ

V y0 V0

cos θp0

V y0 V0 - sin θp0

sin θp0

cos θp0

- K pi

K pp

V x0 V0

K pi

V x0 V0

ΔV x ΔV y

ΔI pcd ΔI pcq ð2:135Þ

Writing (2.126), (2.127), and (2.128) together with the first set of equations in (2.135), following simplified reduced-order state equations of the PMSG in the slow DC voltage timescale with the dynamics of the PLL being included is obtained d ΔXs5 = As5 ΔXs5 þ Bs5 ΔVxy dt

ð2:136Þ

where T

ΔXs5 = ΔV pdc Δxp4 Δxp6 Δθp Δxp -

V pd0 K pp4 V pd0 0 C p V pdc0 Cp V pdc0

0

0

0

0

0

0

0

0

- K pi V 0

0

0

0

1

- K pp V 0

0

0

0

0 -

I x0 C p V pdc0

-

-

0 V y0 - K pi V0 - K pp -

I pcd0 V q0 - I pcq0 V d0 C p V pdc0

K pi4 As5 =

Bs5 =

-

V y0 V0

K pi6 I pcq0 V q0 þ I pcd0 V d0 1 þ K pp6 V pd0

-

I y0 C p V pdc0

0 V K pi x0 V0 K pp

V x0 V0

K pi6 I y0 K pi6 I x0 1 þ K pp6 V pd0 1 þ K pp6 V pd0

Substituting (2.125) in the second equation of (2.121), it can have

K pi6 V pd0 1 þ K pp6 V pd0

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

ΔI pcq = Δxp6 þ K pp6 þ

1 1 þ V pd0 K pp6

57

V pd0 I pcd0 V q0 þ I pcq0 V d0 Δx þ Δθp 1 þ V pd0 K pp6 p6 1 þ V pd0 K pp6

½ I pcd0

- I pcq0 ]

cos θp0

sin θp0

ΔV x

- sin θp0

cos θp0

ΔV y ð2:137Þ

Substituting (2.137) and the first equation of (2.121) in the second equation of (2.79), it can have ΔIxy = Cs5 ΔXs5 þ Ds5 ΔVxy

ð2:138Þ

where K pp4 cos θp0

cos θp0

0 cxθ

K pp4 sin θp0

sin θp0

0 cyθ

Cs5 = K pp6 sin θp0 - I y0 1 þ K pp6 V pd0 K pp6 cos θp0 cyθ = I x0 1 þ K pp6 V pd0 K pp6 I y0 sin θp0 1 þ K pp6 V pd0 Ds5 = K pp6 I y0 cos θp0 1 þ K pp6 V pd0

-

sin θp0 1 þ K pp6 V pd0

cos θp0 1 þ K pp6 V pd0

cxθ =

-

K pp6 I x0 sin θp0 1 þ K pp6 V pd0

K pp6 I x0 cos θp0 1 þ K pp6 V pd0

The simplified reduced-order state-space model of the PMSG in the slow DC voltage timescale with the dynamics of the PLL being included is given by (2.136) and (2.138), which are written together as follows d ΔXs5 = As5 ΔXs5 þ Bs5 ΔVxy dt ΔIxy = Cs5 ΔXs5 þ Ds5 ΔVxy

2.4.3

ð2:139Þ

Small-Signal Stability of the Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Fast Current Timescale

In the slow DC voltage timescale, dynamics of the transmission line connection in the single grid-connected PMSG system displayed in Fig. 2.8 can be ignored. Hence, the line voltage equations can be written as

58

2

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

ΔVxy = X L

-1 ΔIxy 0

0 1

ð2:140Þ

Substituting (2.140) in (2.134), it can have d ΔXs3 = Asc3 ΔXs3 dt

ð2:141Þ

where Asc3 = As3 þ X L Bs3 -

K pp4 V pd0 C p V pdc0

0 -1

K pi4

0

0

0

0 -1

1 0

1

0 1

0

I - X L Ds3

1 0 V pd0 Cp V pdc0

-

-

1þ =

-1

Cs3 =

0

V pq0 C p V pdc0

þ XL

K pi6 V pd0 1 þ K pp6 V pd0

K pp6 I y0 sin θp0 1 þ K pp6 V pd0

-

I x0 C p V pdc0

K pp6 I x0 sin θp0 1 þ K pp6 V pd0

K pp6 I y0 cos θp0 K pp6 I x0 cos θp0 1 þ K pp6 V pd0 1 þ K pp6 V pd0 sin θp0 cos θp0 1 þ K pp6 V pd0

0

-1

1

0

X L K pp6 I x0 sin θp0 1 þ K pp6 V pd0

X L K pp6 I x0 cos θp0 1 þ K pp6 V pd0

-

Iy0 C p V pdc0

0 0 K pi6 I y0 K pi6 I x0 1 þ K pp6 V pd0 1 þ K pp6 V pd0

-

K pp4 sin θp0 sin θp0 I - X L Ds3

1

0

- XL

K pp4 cos θp0

0 -1

-1

0 -1 1

0

cos θp0 1 þ K pp6 V pd0

-1

= -1

X L K pp6 I y0 sin θp0 1 þ K pp6 V pd0 1-

X L K pp6 I y0 cos θp0 1 þ K pp6 V pd0

1 1 þ K pp6 V pd0 þ X L K pp6 I x0 sin θp0 - I y0 cos θp0 1 þ K pp6 V pd0 - X L K pp6 I y0 cos θp0

- X L K pp6 I y0 sin θp0

X L K pp6 I x0 cos θp0

1 þ K pp6 V pd0 þ X L K pp6 I x0 sinθp0

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

0 -1

59

-1

0 -1

= 1 0 1 1þK pp6 V pd0 þX L K pp6 I x0 sinθp0 -I y0 cosθp0

X L Bs3

1 0

I-X L Ds3

-X L I y0 1þK pp6 V pd0 þX 2L K pp6 I 20 cosθp0 X L I x0 1þK pp6 V pd0 þX 2L Kpp6 I 20 sinθp0 C p V pdc0 Cp V pdc0 0

0

X L K pi6 I x0 0 -1 X L Bs3

X L K pi6 I y0 -1

0 -1

Cs3 =

I-X L Ds3 1 0

1 0

X 2L K pp4 K pp6 P20

X 2L K pp6 P20

0

0

X L P0 Cp V pdc0 1þK pp6 V pd0 V 2pd0 Cp V pdc0 1þK pp6 V pd0 V 2pd0 C p V pdc0 1þK pp6 V pd0 V pd0 0

X L K pp4 K pi6 P0

X L K pi6 P0

1þK pp6 V pd0 V pd0

1þK pp6 V pd0 V pd0

0

Thus, Asc3 = As3 þ X L Bs3

=

0

-1

1

0

I - X L Ds3

0

-1

1

0

-1

Cs3

a11

a12

a13

K pi4 X L K pp4 K pi6 P0 1 þ K pp6 V pd0 V pd0

0 X L K pi6 P0 1 þ K pp6 V pd0 V pd0

0 K pi6 V pd0 1 þ K pp6 V pd0 ð2:142Þ

where a11 =

K pp4 C p V pdc0

X 2L K pp6 P20 - V pd0 1 þ K pp6 V pd0 V 2pd0

a12 =

1 C p V pdc0

X 2L K pp6 P20 - V pd0 1 þ K pp6 V pd0 V 2pd0

a13 =

C p V pdc0

X L P0 1 þ K pp6 V pd0 V pd0

60

2

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

Characteristic equation of the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale, when the dynamics of the PLL are excluded, is |λI - Asc3| = 0. From (2.142), it can have jλI-Asc3 j=

-

λ-a11

-a12

-a13

-K pi4

λ

0

X L K pi6 P0 K pi6 V pd0 X L K pp4 K pi6 P0 λþ 1þK pp6 V pd0 1þK pp6 V pd0 V pd0 1þK pp6 V pd0 V pd0

K pi6 V pd0 a13 X L K pi4 K pi6 P0 1þK pp6 V pd0 1þK pp6 V pd0 V pd0 a13 X L K pp4 K pi6 P0 λ a12 K pi4 K pi6 V pd0 -a12 K pi4 λþ 1þK pp6 V pd0 1þK pp6 V pd0 V pd0

=λðλ-a11 Þ λþ

=λ3 þλ2 -a11 þ

ð2:143Þ

K pi6 V pd0 a11 K pi6 V pd0 þλ -a12 K pi4 1þK pp6 V pd0 1þK pp6 V pd0

-

a13 X L K pp4 K pi6 P0 1þK pp6 V pd0 V pd0

-

a13 X L K pi4 K pi6 P0 a12 K pi4 K pi6 V pd0 =0 1þK pp6 V pd0 1þK pp6 V pd0 V pd0

Since Vpd0 = V0, it can have λ 3 þ a2 λ 2 þ a 1 λ þ a 0 = 0

ð2:144Þ

where 2

a2 =

K pp4 V 0 K pi6 V 0 K pp6 K pp4 þ C p V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

a1 =

K pi4 V 0 þ V 20 K pi6 K pp4 þ K pp6 K pi4 K pi6 K pp4 þ K pp6 K pi4 C p V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

a0 =

K pi6 K pi4 V 20 K pi6 K pi4 C p V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

X L P0 V0

xL P0 V0

X L P0 V0

2

2

Applying the Routh–Hurwitz stability criterion to (2.144), it can be concluded that the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale, when the dynamics of the PLL are excluded, is stable if and only when

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

2.4

2

a2 =

K pp4 V 0 K pi6 V 0 K pp6 K pp4 þ Cp V pdc0 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0

a1 =

K pi4 V 0 þ V 20 K pi6 K pp4 þ K pp6 K pi4 K pi6 K pp4 þ K pp6 K pi4 Cp V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

a0 =

K pi6 K pi4 V 20 K pi6 K pi4 Cp V pdc0 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0

a2 a1 - a0 =

X L P0 V0

X L P0 V0

61

>0 X L P0 V0

>0

2

K pp4 V 0 K pi6 V 0 K pp6 K pp4 þ C p V pdc0 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0

>0 X L P0 V0

2

K pi6 K pp4 þ K pp6 K pi4 K pi4 V 0 þ V 20 K pi6 K pp4 þ K pp6 K pi4 Cp V pdc0 1 þ K pp6 V 0 Cp Vpdc0 1 þ K pp6 V 0 -

2

K pi6 K pi4 V 20 K pi6 K pi4 Cp V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

X L P0 V0

X L P0 V0

2

2

>0

ð2:145Þ Rearranging (2.145), it can have X L P0 V0 X L P0 V0 X L P0 V0 X L P0 V0 X L P0 V0

2

2

-

K pp4 V 0 1 þ K pp6 V 0 þ Cp V pdc0 K pi6 V 0 1

ð2:149Þ

Consequently, small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale is ensured. The derived necessary and sufficient stability conditions of (2.147) clearly explains why the condition of weak grid connection (big value of xL) or/and heavy loading (big value of P0) may bring about the instability risk of the single gridconnected PMSG system in the slow DC voltage timescale. It also unambiguously indicates that if the PI gains of the power control outer loop of the GSC are set improperly (big value of k0 such that k0 > 1), growing sub-synchronous/supersynchronous oscillations may occur in the single grid-connected PMSG system even if dynamics of the PLL are in the fast current timescale and its PI gains are set properly to satisfy Conclusion 2.1.

2.4.4

Small-Signal Stability of the Single Grid-Connected PMSG System in the Slow DC Voltage Timescale when Dynamics of the PLL Are in the Slow DC Voltage Timescale

Substituting (2.140) in (2.139), following simplified reduced-order model of the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale, when the dynamics of the PLL are included, is obtained

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

63

d ΔXs5 = Asc5 ΔXs5 dt

ð2:150Þ

where Asc5 = As5 þ X L Bs5 -

0

-1

1

0

K pp4 V 0 C p V pdc0

-

I - X L Ds5

V0 C p V pdc0

0

-1

1

0

-

0

-1

Cs5

I pcd0 V q0 - I pcq0 V d0 C p V pdc0

0

K pi4

0

0

0

0

0

0

0

- K pi V 0

0

0

0

1

- K pp V 0

0

0

0

0

=

-

I x0 C p V pdc0 0

þ XL

V y0 - K pi V0

-

-

K pi6 I pcq0 V q0 þ I pcd0 V d0 1 þ K pp6 V 0

-

K pi6 V 0 1 þ K pp6 V 0

I y0 C p V pdc0

0 V K pi x0 V0

0

-1

1 0 V y0 V K pp x0 V0 V0 K pi6 I x0 K pi6 I y0 1 þ K pp6 V 0 1 þ K pp6 V 0 K pp6 I y0 sin θp0 K pp6 I x0 sin θp0 1 0 1 þ K pp6 V 0 1 þ K pp6 V 0 ð - XL K pp6 I x0 cos θp0 K pp6 I y0 cos θp0 0 1 1 þ K pp6 V 0 1 þ K pp6 V 0 sin θp0 K pp4 cos θp0 cos θp0 0 cxθ 1 þ K pp6 V 0 - K pp

K pp4 sin θp0

sin θp0

0 cyθ

cos θp0 1 þ K pp6 V 0

0

-1

1

0

-1

64

2

I - X L Ds5

0

-1

1

0

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

-1

=

X L K pp6 I x0 sin θp0 1 þ K pp6 V 0

-

X L Kpp6 Ix0 cosθp0 1 þ Kpp6 V0

=

=

X L Bs5

1þ

0

-1

1

0

X L K pp6 I y0 sin θp0 1 þ Kpp6 V0 1-

-1

X L K pp6 I y0 cos θp0 1 þ K pp6 V 0

1 1 þ K pp6 V 0 þ X L K pp6 I x0 sin θp0 - I y0 cos θp0 1 þ K pp6 V 0 - X L K pp6 I y0 cos θp0

- X L K pp6 I y0 sin θp0

X L K pp6 I x0 cos θp0

1 þ K pp6 V 0 þ X L K pp6 I x0 sin θp0

I - X L Ds5

0

-1

1

0

-1

XL 1 þ K pp6 V 0 þ X L K pp6 I x0 sin θp0 - I y0 cos θp0 I y0 I x0 Cp V pdc0 Cp V pdc0 0

0 V K pi x0 V0

V y0 - K pi V0 - K pp

V y0 V0

K pp

K pi6 I y0 1 þ K pp6 V 0

-

V x0 V0

0

-1

1

0

K pi6 I x0 1 þ K pp6 V 0

1 þ K pp6 V 0 - X L K pp6 I y0 cos θp0

- X L K pp6 I y0 sin θp0

X L K pp6 I x0 cos θp0

1 þ K pp6 V 0 þ X L K pp6 I x0 sin θp0

I y0 C p V pdc0 0 V K pi x0 V0 = V x0 K pp V0 K pi6 I x0 1 þ K pp6 V 0 -

X

I x0 Cp V pdc0 0 V y0 K pi V0 V y0 K pp V0 K pi6 I y0 1 þ K pp6 V 0

1 þ K pp6 V 0 - X L K pp6 I y0 cos θp0

- X L K pp6 I y0 sin θp0

X L K pp6 I x0 cos θp0

1 þ K pp6 V 0 þ X L K pp6 I x0 sin θp0

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

=

- I y0 1 þ K pp6 V 0 þ X L K pp6 I 20 cos θp0 C p V pdc0 0 V K pi 1 þ K pp6 V 0 x0 V0 V x0 K pp 1 þ K pp6 V 0 V0 K pi6 I x0

X L Bs5

0

-1

1

0

I - X L Ds5

0

-1

1

0

65

I x0 1 þ K pp6 V 0 þ X L K pp6 I 20 sin θp0 Cp V pdc0 0 V y0 K pi 1 þ K pp6 V 0 V0 V y0 K pp 1 þ K pp6 V 0 V0 K pi6 I y0 -1

Cs5 =

- I y0 1 þ K pp6 V 0 þ X L K pp6 I 20 cos θp0 C p V pdc0

I x0 1 þ K pp6 V 0 þ X L K pp6 I 20 sin θp0 C p V pdc0

0

0

K pi 1 þ K pp6 V 0

V x0 V0

K pi 1 þ K pp6 V 0

V y0 V0

K pp 1 þ K pp6 V 0

V x0 V0

K pp 1 þ K pp6 V 0

V y0 V0

K pi6 I x0

K pi6 I y0

K pp4 cos θp0

cos θp0

0 cxθ

sin θp0 1 þ K pp6 V 0

K pp4 sin θp0

sin θp0

0 cyθ

cos θp0 1 þ K pp6 V 0

=

Thus,

b11 0

b12 0

0 0

b14 0

b15 0

X L K pp4 K pi X L K pp4 K pp X L K pp4 K pi6 P0 1 þ K pp6 V 0 V 0

X L K pi X L K pp X L K iQk P0 1 þ K pp6 V 0 V 0

0 1

0 0

0 0

0

0

0

b11 =

K pp4 K pp6 ðX L P0 Þ2 K pp6 ðX L P0 Þ2 , b = 12 C p V pdc0 1 þ K pp6 V 0 V 20 C p V pdc0 1 þ K pp6 V 0 V 20

b14 =

X L P20 X L P0 , b15 = 1 þ K pp6 V 0 V 20 C p V pdc0 1 þ K pp6 V 0 V 0

C p V pdc0

66

2

Asc5 = As5 þ X L Bs5 -

0

-1

1

0

K pp4 V 0 C p V pdc0

=

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

-

I - X L Ds5

V0 C p V pdc0

-

0

0

-1

1

0

-1

Cs5

I pcd0 V q0 - I pcq0 V d0 Cp V pdc0

0

K pi4

0

0

0

0

0

0

0

- K pi V 0

0

0

0

1

- K pp V 0

0

0

0

0

-

K pi6 I pcq0 V q0 þ I pcd0 V d0 1 þ K pp6 V 0 0 b14 b15

b11

b12

0

0

0

0

0

X L K pp4 K pi

X L K pi

0

0

0

X L K pp4 K pp

X L K pp

1

0

0

X L K pp4 K pi6 P0 1 þ K pp6 V 0 V 0

X L K iQk P0 1 þ K pp6 V 0 V 0

0

0

0

a11

a12

0 a14

a15

a21 = a31

0 a32

0 0 0 a34

0 0

a41 a51

a42 a52

1 a44 0 a54

0 a55

-

þ

K pi6 V 0 1 þ K pp6 V 0

ð2:151Þ

where K pp4 K pp6 X L P0 Cp V pdc0 1 þ K pp6 V 0 V 0

2

a11 =

K pp6 1 X L P0 Cp V pdc0 1 þ K pp6 V 0 V 0

2

a12 = a14 =

Cp V pdc0

- V0 , - V0

X L P20 X L P0 , a15 = , a21 = K pi4 , 1 þ K pp6 V 0 V 20 C p V pdc0 1 þ K pp6 V 0 V 0

a31 = X L K pp4 K pi , a32 = X L K pi , a34 = - K pi V 0 , a41 = X L K pp4 K pp , a42 = X L K pp , a44 = - K pp V 0 , a51 = a54 = a52 =

X L K pp4 K pi6 P0 X L K pi6 P0 , a52 = 1 þ K pp6 V 0 V 0 1 þ K pp6 V 0 V 0

K pi6 P0 K pi6 V 0 , a =1 þ K pp6 V 0 55 1 þ K pp6 V 0

X L K iQk P0 1 þ K pp6 V pd0 V 0

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

67

Subsequently, following characteristic equation of the system is obtained jsI - Asc5 j = a5 s5 þ a4 s4 þ a3 s3 þ a2 s2 þ a1 s þ a0 = 0

ð2:152Þ

where a5 = 1 a4 = - ða11 þ a44 þ a55 Þ ==-

K pp4 K pp6 Cp V pdc0 1 þ K pp6 V 0

= - K 41 a3

K pp4 K pp6 X L P0 C p V pdc0 1 þ K pp6 V 0 V 0

X L P0 V0

X L P0 V0

2

- V 0 þ K pp V 0 þ

2

þ V 0 K pp þ

K pi6 V 0 1 þ K pp6 V 0

K pp4 K pi6 V 0 þ C p V pdc0 1 þ K pp6 V 0

2

þ K 42

=

a11 a55 þ a44 a55 þ a11 a44 - ða14 a41 þ a15 a51 þ a12 a21 þ a34 Þ

=

-

K pp4 K pi6 V 0 C p V pdc0 1 þ K pp6 V 0

-

K pp4 K pp V 0 K pp6 X L P0 C p V pdc0 1 þ K pp6 V 0 V 0

-

2

K pp6 X L P0 1 þ K pp6 V 0 V 0

X 2L K pp4 K pi6 P20 2

C p V pdc0 1 þ K pp6 V 0 V 20

-

- V0 þ

2

- V0 -

K pi6 K pp V 20 1 þ K pp6 V 0

X 2L K pp4 K pp P20 Cp V pdc0 1 þ K pp6 V 0 V 20

K pi4 K pp6 X L P0 C p V pdc0 1 þ K pp6 V 0 V 0

2

- V0

þ K pi V 0 =-

K pp4 K pp K pp4 K pi6 þ K pi4 K pp6 þ C p V pdc0 C p V pdc0 1 þ K pp6 V 0

þV 20 = - K 31

X L P0 V0

2

þ K pi V 0 þ

K pi6 K pp4 K pp K pp4 K pi6 þ þ C V 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0 p pdc0 X L P0 V0

2

þ K 32

K pi4 C p V pdc0

68

2

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

a2 = ða14 a41 þ a12 a21 þ a34 Þa55 þ ða12 a21 þ a15 a51 Þa44 þ a34 a11 - ða14 a31 þ a14 a42 a21 þ a15 a54 a41 þ a15 a52 a21 þ a11 a44 a55 Þ =-

K pi4 K pp6 X 2L K pp4 K pp P20 X L P0 þ Cp V pdc0 1 þ K pp6 V 0 V 20 C p V pdc0 1 þ K pp6 V 0 V 0

K pi4 K pi6 V 0 K pp6 X L P0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0 V 0 K pp V 0 x -

K pp4 K pi V 0 K pp6 X L P0 Cp V pdc0 1 þ K pp6 V 0 V 0

2

- V0 þ

2

- V0 -

2

- V 0 - K pi V 0 X 2L K pp4 K pi6 P20 2

C p V pdc0 1 þ K pp6 V 0 V 20

X 2L K pp4 K pi P20 C p V pdc0 1 þ K pp6 V 0 V 20

X 2L K pp K pi4 P20 X 2L K pp4 K pi6 K pp P20 X 2L K pi4 K pi6 P20 þ 2 2 C p V pdc0 1 þ K pp6 V 0 V 0 Cp V pdc0 1 þ K pp6 V 0 V 0 C p V pdc0 1 þ K pp6 V 0 2 V 20 -

=-

K pp4 K pp K pi6 V 20 Cp V pdc0 1 þ K pp6 V 0

K pp6 X L P0 1 þ K pp6 V 0 V 0

2

- V0

K pp4 K pp K pi6 V 0 þ K pi4 K pi6 K pp4 K pi þ K pp K pi4 þ Cp V pdc0 Cp V pdc0 1 þ K pp6 V 0

þV 20

= - K 21

X L P0 V0

2

K pi K pi6 K pp4 K pp K pi6 V 0 þ K pi4 K pi6 K pp4 K pi þ K pp K pi4 þ þ C V 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0 p pdc0 X L P0 V0

2

þ K 22

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

69

a1 = a14 a42 a21 a55 þa14 a31 a55 þa12 a21 a34 þa15 a52 a21 a44 þ a15 a51 a34 - ða12 a21 a44 a55 þa14 a32 a21 þa11 a55 a34 þ a15 a54 a31 þa15 a54 a42 a21 Þ = -

-

-

þ

= -

X 2L K pp K pi4 K pi6 P20 2

C p V pdc0 1þK pp6 V 0 V 0

-

X 2L K pp4 K pi K pi6 P20 2

C p V pdc0 1þK pp6 V 0 V 0

K pi4 K pi V 0 K pp6 X L P0 Cp V pdc0 1þK pp6 V 0 V 0 X 2L K pp4 K pi6 K pi P20 2

C p V pdc0 1þK pp6 V 0 V 0

-

2

-V 0 -

X 2L K pi6 K pi4 K pp P20 2

Cp V pdc0 1þK pp6 V 0 V 0

K pi4 K pp K pi6 V 20 C p V pdc0 1þK pp6 V 0

K pp4 K pi6 K pi V 20 X 2L K pi4 K pi P20 C p V pdc0 1þK pp6 V 0 V 20 C p V pdc0 1þK pp6 V 0 X 2L K pi6 K pp4 K pi P20 2

2

K pp6 X L P0 1þK pp6 V 0 V 0 K pp6 X L P0 1þK pp6 V 0 V 0

-V 0 2

-V 0

X 2L K pi6 K pp K pi4 P20

þ

2

Cp V pdc0 1þK pp6 V 0 V 0 C p V pdc0 1þK pp6 V 0 V 0 K pi6 K pp4 K pi V 0 þK pi6 K pp K pi4 V 0 K pi4 K pi þ Cp V pdc0 C p V pdc0 1þK pp6 V 0

þ V 20 = -K 11

X L P0 V0

2

K pi6 K pp4 K pi V 0 þK pi6 K pp K pi4 V 0 K pi4 K pi þ C p V pdc0 C p V pdc0 1þK pp6 V 0 2

X L P0 V0

þK 12

a0 = a14 a32 a21 a55 þ a15 a52 a21 a34 - ða12 a21 a34 a55 þ a15 a54 a32 a21 Þ X 2L K pi K pi4 K pi6 P20 X 2L K pi K pi4 K pi6 P20 K pi4 K pi K pi6 V 0 =2 2 C V p pdc0 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0 V 0 Cp V pdc0 1 þ K pp6 V 0 V 0 K pp6 X L P0 1 þ K pp6 V 0 V 0 =-

2

K pi4 K pi K pi6 V 0 Cp V pdc0 1 þ K pp6 V 0

= - K 01

X L P0 V0

2

þ K 02

- V0 þ X L P0 V0

X 2L K pi K pi4 K pi6 P20 2

Cp V pdc0 1 þ K pp6 V 0 V 0 2

þ

K pi4 K pi K pi6 V 30 Cp V pdc0 1 þ K pp6 V 0

70

2

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

Note that in the above equations, following notations are taken: K 41 =

K pp4 K pp6 C p V pdc0 1 þ K pp6 V 0

K 42 = V 0 K pp þ K 31 =

K pp4 K pi6 þ K pi4 K pp6 K pp4 K pp , þ Cp V pdc0 C p V pdc0 1 þ K pp6 V 0

K 32 = K pi V 0 þ K 21 =

K pi K pi6 K pp4 K pp K pi6 V 0 þ K pi4 K pi6 K pp4 K pi þ K pp K pi4 þ þ Cp V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

K pi6 K pp4 K pi V 0 þ K pi6 K pp K pi4 V 0 K pi4 K pi þ C p V pdc0 C p V pdc0 1 þ K pp6 V 0

K 12 = V 20

K 01 =

K pi4 K pi6 K pp4 K pp K pp4 K pi6 þ þ þ V 20 Cp V pdc0 Cp V pdc0 1 þ K pp6 V 0 C p V pdc0 1 þ K pp6 V 0

K pp4 K pp K pi6 V 0 þ K pi4 K pi6 K pp4 K pi þ K pp K pi4 þ C p V pdc0 Cp V pdc0 1 þ K pp6 V 0

K 22 = V 20 K 11 =

K pp4 K pi6 V 0 þ C p V pdc0 1 þ K pp6 V 0

K pi6 K pp4 K pi V 0 þ K pi6 K pp K pi4 V 0 K pi4 K pi þ Cp V pdc0 Cp V pdc0 1 þ K pp6 V 0

K pi4 K pi K pi6 V 0 K pi4 K pi K pi6 V 30 , K 02 = C p V pdc0 1 þ K pp6 V 0 Cp V pdc0 1 þ K pp6 V 0 ð2:153Þ

By applying the Routh–Hurwitz criterion to (2.152), it can be obtained that the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale with dynamics of the PLL being included is stable if and only when

2.4

Risk of Sub/Super-Synchronous Oscillations in a Single Grid-Connected. . .

a4 =-K 41

X L P0 V0

a3 =-K 31

X L P0 V0

a2 =-K 21

X L P0 V0

a1 =-K 11

X L P0 V0

a0 =-K 01

X L P0 V0

2

þK 42 >0 2

þK 32 >0 2

þK 22 >0 2

þK 12 >0 2

þK 02 >0

b1 =a3 a4 -a5 a2 = -K 31 - -K 21

X L P0 V0

X L P0 V0

- -K 21

X L P0 V0

-K 11

X L P0 V0

-K 41

þK 32

-K 31

X L P0 V0

2

þK 22

-K 21

2

þK 12

X L P0 V0

2

X L P0 V0

-K 41

þK 42

þK 22 >0

d1 =c1 ða1 a4 -a5 a0 Þ-a0 b21 =c1

X L P0 V0

2

2

c1 =a2 b1 -a4 ða1 a4 þa0 a5 Þ=

- -K 01

71

-K 41 -K 11

X L P0 V0

þK 42 - -K 21

X L P0 V0

2

þK 22 - -K 41 þK 42 þ -K 01 2

þK 12

X L P0 V0

X L P0 V0

-K 41

2

X L P0 V0

- -K 01

2

þK 32

X L P0 V0

2

þK 02

2

-K 41

þK 42 2

X L P0 V0

X L P0 V0 X L P0 V0

2

þK 02

2

-K 31

þK 42

2

þK 02 2

þK 42

X L P0 V0

2

þK 32

2

2

þK 22

>0 ð2:154Þ

From (2.152), (2.154) and taking V0 ≈ 1, following necessary and sufficient stability conditions are obtained V0 1 V0 > and >1 X L P0 z0 X L P0 In the above equation, z0 is the solution of the following equation

>0

ð2:155Þ

72

2

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

c5 z 0 5 þ c 4 z 0 4 þ c 3 z 0 3 þ c 2 z 0 2 þ c1 z 0 þ c0 > 0

ð2:156Þ

where c5 = - K 11 K 21 K 31 K 241 þ K 01 K 231 K 241 þ K 21 K 341 c4 = K 01 K 21 K 31 K 41 þ 2K 11 K 21 K 31 K 41 þ K 11 K 21 K 32 K 241 þ K 11 K 21 K 31 K 241 þK 11 K 22 K 31 K 241 - 2K 211 K 341 þ 2K 11 K 01 K 241 - K 11 K 41 K 221 - 2K 01 K 41 K 42 K 231 - 2K 01 K 31 K 32 K 241 - 2K 02 K 231 K 241 - 3K 42 K 211 K 241 c3 = - K 01 K 21 K 32 K 41 - 2K 01 K 21 K 31 K 42 - K 02 K 21 K 31 K 41 - K 01 K 22 K 31 K 41 - K 11 K 21 K 31 K 242 - 2K 11 K 22 K 31 K 41 K 42 - 2K 11 K 21 K 32 K 41 K 42 - K 11 K 22 K 31 K 241 - K 11 K 21 K 32 K 241 - 2K 11 K 2 K 31 K 41 K 42 - K 1 K 22 K 32 K 241 þ K 201 K 41 þ K 211 K 341 þK 11 K 42 K 221 þ K 11 K 41 K 221 - 4K 01 K 11 K 41 K 42 - 2K 02 K 11 K 241 - 2K 01 K 11 K 241 þ4K 01 K 31 K 32 K 41 K 42 þ 2K 02 K 41 K 42 K 231 þ 2K 02 K 31 K 32 K 241 þ K 01 K 231 K 242 þK 01 K 232 K 241 þ 6K 42 K 211 K 241 þ 3K 41 K 211 K 242 þ 2K 11 K 21 K 22 K 41

c2 = 4K 02 K 11 K 41 K 42 þ 4K 01 K 11 K 41 K 42 þ 2K 11 K 22 K 32 K 41 þ K 01 K 22 K 31 K 42 - 4K 02 K 31 K 32 K 41 K 42 - 2K 01 K 02 K 41 - K 11 K 42 K 221 - K 02 K 231 K 242 - K 11 K 41 K 222 þK 01 K 22 K 32 K 41 þ K 11 K 21 K 31 K 242 þ K 11 K 22 K 31 K 242 þ K 11 K 22 K 32 K 241 - K 201 K 42 - 2K 11 K 21 K 22 K 42 - 2K 11 K 21 K 22 K 41 - 2K 01 K 31 K 32 K 242 - 2K 01 K 41 K 42 K 232 þ2K 11 K 22 K 31 K 41 K 42 þ 2K 11 K 21 K 32 K 41 K 42 þ K 42 K 11 K 21 K 32 K 242 - K 211 K 342 - 6K 41 K 211 K 242 - 3K 42 K 211 K 241 - K 02 K 232 K 241 þ 2K 02 K 11 K 241 þ 2K 01 K 11 K 242 þK 02 K 22 K 31 K 41 þ K 02 K 21 K 32 K 41 þ K 02 K 21 K 31 K 42 þ K 01 K 21 K 32 K 42

c1 = - K 01 K 22 K 32 K 42 - K 02 K 21 K 32 K 42 - K 02 K 22 K 32 K 41 - K 02 K 22 K 31 K 42 - K 11 K 21 K 32 K 242 - K 11 K 22 K 32 K 242 - K 11 K 22 K 31 K 242 þ K 11 K 42 K 222 þ K 11 K 41 K 222 þ2K 11 K 21 K 22 K 42 - 4K 02 K 11 K 41 K 42 - 2K 02 K 11 K 242 - 2K 01 K 11 K 242 þ K 201 K 41 þ2K 01 K 02 K 42 þ K 01 K 232 K 242 þ 2K 02 K 41 K 42 K 232 þ 3K 41 K 211 K 242 þ 2K 211 K 342 þ2K 02 K 31 K 32 K 242 - 2K 11 K 22 K 32 K 41 K 42 c0 = K 02 K 22 K 32 K 42 þ K 11 K 22 K 32 K 242 - K 11 K 42 K 222 þ 2K 11 K 02 K 242 - K 02 K 232 K 242 - K 42 K 202 - K 211 K 342 In the above equations, Ki1 and Ki2; i = 0, 1, 2, 3, 4 are given in (2.153). It can be seen Ki1 and Ki2; i = 0, 1, 2, 3, 4 are determined entirely by the gains of PLLs, DC voltage, and reactive power control outer loops of GSCs of the PMSGs as well as Cp and Vpdc0. Hence, z0 can be easily computed from the gains of PLLs, DC voltage,

2.5

An Example of Single Grid-Connected PMSG System

73

and reactive power control outer loops of GSCs of the PMSGs, as well as Cp and Vpdc0 by using (2.156) and (2.153). In addition, by comparing (2.154) with (2.146), following conclusion can be obtained. Conclusion 2.3 When the dynamics of the PLL are in the slow DC voltage timescale, the PI gains of the power control outer loops of the GSC are set to satisfy z0 > 1

ð2:157Þ

Consequently, small-signal stability of the single grid-connected PMSG system displayed in Fig. 2.8 in the slow DC voltage timescale is ensured. The derived necessary and sufficient stability conditions of (2.155) clearly explains why the condition of weak grid connection (big value of xL) or/and heavy loading (big value of P0) may bring about the instability risk of the single gridconnected PMSG system in the slow DC voltage timescale when dynamics of the PLL are also in the slow DC voltage timescale. It unambiguously indicates that the PI gains of the power control outer loop of the GSC and the PLL should be set jointly to satisfy (2.155), ideally with big value of z0 such that z0 > 1. Otherwise, growing sub-synchronous/super-synchronous oscillations may occur in the single gridconnected PMSG system when dynamics of the PLL are in the slow DC voltage timescale.

2.5 2.5.1

An Example of Single Grid-Connected PMSG System Stability of the Example Grid-Connected PMSG System in the Fast Current Timescale

Configuration of an example single grid-connected PMSG system is shown in Fig. 2.8. Per unit parameters of the PMSG are given in Table 2.1. Power factor of the PMSG is 0.98 and V0 = 1. Discussions in Sects. 2.2 and 2.3 conclude that the stability of a single gridconnected PMSG system in the fast current timescale is determined by the dynamics of the PLL (see Conclusion 2.1). When the dynamics of the PLL are in the fast current timescale, the necessary and sufficient stability condition of the system in the fast current timescale is given by (2.66), or alternatively ω0 V 0 X L P0 ω V 2 ω Q < 0 0 - 0 0 X L P0 P0

K pp < K pi K pp

ð2:158Þ

74

2

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

Table 2.1 Parameters of the PMSG Symbols Xdk Xqk Jk ωprref Vdc0 C Kpp4, Kpi4 Kpp5, Kpi5 Kpp6, Kpi6 Kpp7, Kpi7

Direct-axis synchronous reactance Quadrature-axis synchronous reactance Rotor inertia constant Reference of rotor speed control outer loop Steady-state DC link voltage Capacitance of the DC capacitor PI gains of DC voltage control outer loop of the GSC PI gains of active current control inner loop of the GSC PI gains of reactive power control outer loop of the GSC PI gains of reactive current control inner loop of the GSC

Value 0.3 0.15 3 0.8 1 3 5, 10 1.2, 300 0.1, 15 0.8, 180

Table 2.2 Results of modal computation when P0 = 0.36 Full-order modal is used -9.83 ± 198j -523.59 ± 2227.5j - 61.795 ± 3693.3j

Simplified second-order model is used -11.695 ± 243.17j

Simplified reduced-order modal is used -10.749 ± 207.42j

Table 2.3 Results of modal computation when P0 = 0.5652 Full-order modal is used -0.51 ± 202.29j -523.59 ± 2227.5j -61.17 ± 3693.3j

Simplified second-order model is used -1.87 ± 245.1j

Simplified reduced-order modal is used -2.54 ± 213.92j

Following four tests are carried out to demonstrate and validate the stability condition given in (2.158). TEST 1 PI gains of the PLL are fixed to be Kpp = 0.3, Kpi = 200 and XL = 1. The steady-state active power output from the PMSG, P0, varies from P0 = 0.36 to P0 = 0.72. When P0 varies, the oscillation modes of the example single grid-connected PMSG system are computed by using: (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified second-order model of the single gridconnected PMSG system given in (2.56) when the system is dominated by the dynamics of the PLL; (3) the simplified reduced-order model of the single gridconnected PMSG system given in (2.101) in the current control timescale. Results of modal computation are given in Tables 2.2, 2.3, and 2.4.

2.5

An Example of Single Grid-Connected PMSG System

75

Table 2.4 Results of modal computation when P0 = 0.72 Simplified second-order model is used 5.48 ± 248.94j

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Full-order modal is used 6.68 ± 205.94j -46.35 ± 33.989j -523.59 ± 2227.5j -59.622 ± 3696.4j

0.6

0.4 0.2 0

Simplified reduced-order modal is used 5.972 ± 215.61j

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.13 Results of PFs computation when Kpp = 0.3, Kpi = 200, XL = 1 and P0 = 0.36

0.6

0.4 0.2 0

0.6 0.4 0.2

PLL Other parts of the system (a)Full-order modal is used

0

PLL Other parts of the system (b)Simplified reduced-order modal is used

Fig. 2.14 Results of PFs computation when Kpp = 0.3, Kpi = 200, XL = 1 and P0 = 0.5652

From Table 2.2, 2.3, and 2.4 it can be seen that 1. There is a dominant oscillation mode, λPLL, of the example single grid-connected PMSG system, which is highlighted in bold in Tables 2.2, 2.3, and 2.4. The stability of the system is determined by the dominant oscillation mode. By using the full-order model and the simplified reduced-order model in the current timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λPLL are computed. Computational results are presented in Figs. 2.13, 2.14, and 2.15. It can be seen that λPLL is associated with the PLL. Hence, in this test, the example single grid-connected PMSG system is dominated by the dynamics of the PLL. In the other words, the stability of the example single grid-connected PMSG system in the current timescale is determined by the PLL, dynamics of which dominate the system.

2

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

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

76

0.6

0.4 0.2 0

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

Fig. 2.15 Results of PFs when Kpp = 0.3, Kpi = 200, XL = 1 and P0 = 0.72

1300

Stable region

1200

ω0 V0 2 ω0 Q0 X L P0 P0

1100 1000 900

K pi

800

K pp

700 666.67

600

Unstable region

500 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

P0 / p.u Fig. 2.16 Variation of ωX0LVP00 2

ω0 Q0 P0

when Kpp = 0.3, Kpi = 200, XL = 1 and P0 varies

2. For the stability assessment, the simplified second-order model when the example system is dominated by the dynamics of the PLL and the simplified reduced-order model of the example system in the current timescale can provide the same results. This confirms the correctness of both simplified models for the stability assessment. In this test,

in Eq. (2.158) is fixed to be

K pi K pp

= 666:67. When P0 varies,

changes accordingly. Figure 2.16 shows the variation of ωX0LVP00 - ωP0 Q0 0 when P0 changes. Figure 2.17 presents the trajectory of λPLL on the complex plane when P0 varies. From Fig. 2.16 it can be seen that when P0 is greater than 0.566, K pi ω0 Q0 ω0 V 0 2 X L P0 - P0 is greater than K pp = 666:67 such that the stability condition given in (2.158) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.17 confirms that (2.158) indeed can be used to assess the stability of the example single grid-connected PMSG system approximately. ω0 V 0 2 X L P0

ω0 Q0 P0

K pi K pp

2

2.5

An Example of Single Grid-Connected PMSG System

77

250 200

Imaginary part

150 100

P0 was increased

50

P0 = 0.57

0 -50

-100 -150

P0 was increased

-200 -250 -10

-8

-4

-6

-2

0

2

4

Real part Fig. 2.17 Trajectory of λPLL when Kpp = 0.3, Kpi = 200, XL = 1 and P0 varies

Deviation of act ive power output/ p.u

0.06

P0 = 0.36 P0 = 0.72

0

- 0.06

Time (second) 0

0.5

1

1.5

2

2.5

3

Fig. 2.18 Results of non-linear simulation when Kpp = 0.3, Kpi = 200, XL = 1and P0 = 0.36 or P0 = 0.72

Two results of non-linear simulation are presented in Fig. 2.18 when P0 = 0.36 and P0 = 0.72. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 5% for 100 ms. They confirm the correctness of stability assessment made by using (2.158). TEST 2 PI gains of the PLL are fixed to be Kpp = 0.3, Kpi = 200, and P0 = 1. The lumped reactance of the transmission line connecting the PMSG to the grid in the example single grid-connected PMSG system, XL, varies from XL = 0.486 to XL = 0.81.

78

2

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

Table 2.5 Results of modal computation whenKpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.486 Full-order modal is used -4.12 ± 200.60j -523.59 ± 2227.5j -61.795 ± 3693.3j

Simplified second-order model is used -6.42 ± 221.2j

Simplified reduced-order modal is used -3.89 ± 206.89j

Table 2.6 Results of modal computation when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.567 Full-order modal is used -0.51 ± 202.29j -523.59 ± 2227.5j -61.17 ± 3693.3j

Simplified second-order model is used -2.87 ± 221.97j

Simplified reduced-order modal is used -0.45 ± 207.2j

Table 2.7 Results of modal computation when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.81 Full-order modal is used 10.96 ± 208.18j -46.35 ± 33.989j -523.59 ± 2227.5j -59.622 ± 3696.4j

Simplified second-order model is used 5.13 ± 224.78j

Simplified reduced-order modal is used 9.96 ± 207.54j

When XL varies, the oscillation modes of the example single grid-connected PMSG system are computed by using: (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified second-order model of the single gridconnected PMSG system given in (2.56) when the system is dominated by the dynamics of the PLL; (3) the simplified reduced-order model of the single gridconnected PMSG system given in (2.101) in the current control timescale. Results of modal computation are given in Tables 2.5, 2.6, and 2.7. From Tables 2.5, 2.6, and 2.7 it can be seen that 1. There is a dominant oscillation mode, λPLL, of the example single grid-connected PMSG system, which is highlighted in bold in Tables 2.5, 2.6, and 2.7. The stability of the system is determined by the dominant oscillation mode. By using the full-order model and the simplified reduced-order model in the current timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λPLL are computed. Computational results are presented in Figs. 2.19, 2.20, and 2.21. It can be seen that λPLL is associated with the PLL. Hence, in this test, the example single grid-connected PMSG system is dominated by the dynamics of the PLL. In the other words, the stability of the example single grid-connected PMSG system in the current timescale is determined by the PLL, dynamics of which dominate the system.

An Example of Single Grid-Connected PMSG System 1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

2.5

0.6

0.4 0.2 0

79

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.19 Results of PFs computation when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.486

0.6

0.4 0.2 0

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.20 Results of PFs computation when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.567

0.6

0.4 0.2 0

0.6 0.4 0.2

PLL Other parts of the system (a)Full-order modal is used

0

PLL Other parts of the system (b)Simplified reduced-order modal is used

Fig. 2.21 Results of PFs when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.567

2. For the stability assessment, the simplified second-order model when the example system is dominated by the dynamics of the PLL and the simplified reduced-order model of the example system in the current timescale can provide the same results. This confirms the correctness of both simplified models for the stability assessment.

80

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

2 800

Stable region

750

K pi

700

K pp

ω0 V0 2 ω0Q0 XL P0 P0

666.67 650 600 550 500

450

Unstable region

400 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

XL Fig. 2.22 Variation of ωX0LVP00 2

ω0 Q0 P0

when XL varies

250 200 150

X L was increased

Imaginary part

100 50

X L = 0.567

0

-50

-100

X L was increased

-150 -200 -250 -20

-15

-10

0

-5

Real part Fig. 2.23 Trajectory of λPLL when XL varies

In this test,

in Eq. (2.158) is also fixed to be

K pi K pp

= 666:67. When XL varies,

changes accordingly. Figure 2.22 shows the variation of ωX0LVP00 - ωP0 Q0 0 when XL changes. Figure 2.23 presents the trajectory of λPLL on the complex plane when XL varies. From Fig. 2.22 it can be seen that when XL is greater than 0.567, K pi ω0 Q0 ω0 V 0 2 X L P0 - P0 is greater than K pp = 666:67 such that the stability condition given in (2.158) is violated. Consequently, the example single grid-connected PMSG system ω0 V 0 2 X L P0

ω0 Q0 P0

K pi K pp

2

2.5

An Example of Single Grid-Connected PMSG System

81

0.06

Deviation of act ive power output/ p.u

X L = 0.486 X L = 0.81

0

- 0.06

Time (second) 0

0.5

1

1.5

2

2.5

3

Fig. 2.24 Results of non-linear simulation when Kpp = 0.3, Kpi = 200, P0 = 1 and XL = 0.486 or XL = 0.81 Table 2.8 Results of modal computation when XL = 1, P0 = 0.72 and .. Full-order modal is used -6.68 ± 186.2j -57.42 ± 33.989j -523.8 ± 2227.5j

Simplified second-order model is used -11.31 ± 178.64j

Simplified reduced-order modal is used -7.00 ± 198.40j

becomes unstable. Figure 2.23 confirms that (2.158) indeed can be used to assess the stability of the example single grid-connected PMSG system approximately. Two results of non-linear simulation are presented in Fig. 2.24 when XL = 0.486 and XL = 0.81. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. They confirm the correctness of stability assessment made by using (2.158). TEST 3 The steady-state active power output from the PMSG is fixed to be P0 = 0.72. The lumped reactance of the transmission line connecting the PMSG to the grid in the example single grid-connected PMSG system is fixed to be XL = 1. PI gains of the PLL, Kpp and Kpi, vary from Kpp = 0.2, Kpi = 90 to Kpp = 0.16, Kpi = 120. When Kpp and Kpi vary, the oscillation modes of the example single gridconnected PMSG system are computed by using: (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified second-order model of the single grid-connected PMSG system given in (2.56) when the system is dominated by the dynamics of the PLL; (3) the simplified reduced-order model of the single grid-connected PMSG system given in (2.101) in the current control timescale. Results of modal computation are given in Tables 2.8, 2.9, and 2.10.

82

2

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

Table 2.9 Results of modal computation when XL = 1, P0 = 0.72 and Kpp = 0.18, Kpi = 94.475 Full-order modal is used -1.74 ± 187.80j -54.86 ± 33.989j -522.03 ± 2227.5j

Simplified second-order model is used -6.742 ± 180.43j

Simplified reduced-order modal is used -1.81 ± 202.80j

Table 2.10 Results of modal computation whenXL = 1, P0 = 0.72 and Kpp = 0.16, Kpi = 120 Simplified second-order model is used 5.271 ± 189.91j

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Full-order modal is used 12.65 ± 197.85j -51.42 ± 33.989j -523.8 ± 2226.9j -72.8 ± 3696.4j

0.6

0.4 0.2 0

Simplified reduced-order modal is used 10.65 ± 210.09j

0.6 0.4 0.2

PLL Other parts of the system (a)Full-order modal is used

0

PLL Other parts of the system (b)Simplified reduced-order modal is used

Fig. 2.25 Results of PFs computation when XL = 1, P0 = 0.72 and Kpp = 0.2, Kpi = 90

From Tables 2.8, 2.9, and 2.10, it can be seen that 1. There is a dominant oscillation mode, λPLL, of the example single grid-connected PMSG system, which is highlighted in bold in Tables 2.8, 2.9, and 2.10. The stability of the system is determined by the dominant oscillation mode. By using the full-order model and the simplified reduced-order model in the current timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λPLL is computed. Computational results are presented in Figs. 2.25, 2.26, and 2.27. It can be seen that λPLL is associated with the PLL. Hence, in this test, the example single grid-connected PMSG system is dominated by the dynamics of the PLL. In the other words, the stability of the example single grid-connected PMSG system in the current timescale is determined by the PLL, dynamics of which dominate the system. 2. For the stability assessment, the simplified second-order model when the example system is dominated by the dynamics of the PLL and the simplified reduced-order model of the example system in the current timescale can provide the same results. This confirms the correctness of both simplified models for the stability assessment.

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

2.5 An Example of Single Grid-Connected PMSG System

0.6

0.4 0.2 0

83

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.26 Results of PFs computation when XL = 1, P0 = 0.72 and Kpp = 0.18, Kpi = 94.475

0.6

0.4 0.2 0

0.6 0.4 0.2 0

PLL Other parts of the system (a)Full-order modal is used

PLL Other parts of the system (b)Simplified reduced-order modal is used

Fig. 2.27 Results of PFs when XL = 1, P0 = 0.72 and Kpp = 0.16, Kpi = 120

K pi 900 K pp 800 700

Unstable region

600

524.86 500 400 0.21

Stable region

130

0.2 0.19

K pp

0.18

110 100

0.17 0.16

K

Fig. 2.28 Variation of K pppi when Kpp and Kpi vary

90

K pi

120

84

2

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

300 -

Imaginary part

200

K pi

100

K pp

was increased K pi

0

K pp

K pi

-100

K pp

= 527

was increased

-200 -300 -7

-6

-5

-4

-3

-2

-1

0

1

2

Real part K

Fig. 2.29 Trajectory of λPLL when K pppi varies

In this test, ωX0LVP00 - ωP0 Q0 0 in Eq. (2.158) is fixed to be ωX0LVP00 - ωP0 Q0 0 = 524:86. K When Kpp and Kpi vary, K pppi changes accordingly. Figure 2.28 shows the variation of 2

K pi K pp

2

when Kpp and Kpi change. Figure 2.29 presents the trajectory of λPLL on the K

complex plane when Kpp and Kpi vary. From Fig. 2.28 it can be seen that when K pppi is

greater than ωX0LVP00 - ωP0 Q0 0 = 524:86 such that the stability condition given in (2.158) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.29 confirms that (2.158) indeed can be used to assess the stability of the example single grid-connected PMSG system approximately. Two results of non-linear simulation are presented in Fig. 2.30 when K pi K pi K pp = 524:86 K pi = 94:475, K pp = 0:18 and K pp = 750 K pi = 120, K pp = 0:16 . At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. They confirm the correctness of stability assessment made by using (2.158). 2

TEST 4 PI gains of the PLL can be set to ensure the small-signal stability of the example single grid-connected PMSG system in the current timescale according to Conclusion 2.1, i.e., if the PI gains of the PLL are set to satisfy (2.116), K pi K pp ≤ 1; ≤ 0:67 ω0 ω0 K pp

ð2:116Þ

the small-signal stability of the example single grid-connected PMSG system in the current timescale is guaranteed. This is demonstrated and validated in this fourth test.

2.5

An Example of Single Grid-Connected PMSG System

K pi

0.8

Deviation of act ive power output/ p.u

K pp

K pi

= 524.86

K pp

85

= 750

0

- 0.8

Time (second) 0

0.5

1

1.5

2

2.5

3

Fig. 2.30 Results of non-linear simulation when XL = 1 and P0 = 0.72 with K pp = 0:18, K pi = 94:475

K pp ω0

K pi K pp

= 524:86 and K pi = 120, K pp = 0:16

K pi K pp

= 750

First, Kpp and Kpi are set according to (2.116) to have Kpp = 1, Kpi = 200 such that K = 0:00265 ≤ 1; ω0 Kpipp = 0:5305 ≤ 0:67 (Note ω0 = 2πf, f = 60). Hence, stability

condition of (2.116) is satisfied. Second, the lumped reactance of the transmission line connecting the PMSG to the grid in the example single grid-connected PMSG system is fixed to be XL = 1. The steady-state active power output from the PMSG is variable from P0 = 0.5 when SCR = 2 to P0 = 1 when SCR = 1. Figure 2.31 presents the trajectory of λPLL when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.32. It can be seen that the example single grid-connected PMSG system is always stable even when the grid connection of the PMSG is as weak when SCR = 1.0. This confirms the correctness of Conclusion 2.1. Finally, the steady-state active power output from the PMSG is fixed to be P0 = 1. XL varies from XL = 0.5 to XL = 1 such that the SCR changes accordingly from SCR = 2 to SCR = 1. Figure 2.33 presents the trajectory of λPLL when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.34. It can be seen that the example single grid-connected PMSG system is always stable even when the grid connection of the PMSG is under the condition of extremely weak gird connected with the SCR to be as low as SCR = 1.0. Hence, correctness of Conclusion 2.1 is confirmed.

86

2

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

400 300

Imaginary part

200 100

SCR = 2

0

SCR = 1

-100 -200

-300 -400 -80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Real part Fig. 2.31 Trajectory of λPLL when the SCR varies with the change of P0

Deviation of act ive power output/ p.u

0.06

P0 = 0.5 P0 = 1

60

0

- 0.06 0

Time (second) 0.5

1

1.5

2

2.5

3

Fig. 2.32 Results of non-linear simulation with SCR = 2 or SCR = 1 when P0 varies

2.5.2

Stability of the Example Grid-Connected PMSG System in the Slow DC Voltage Timescale when the Dynamics of the PLL Are in the Fast Current Timescale

Discussions in Sect. 2.4 conclude that the sufficient and necessary stability conditions of the single grid-connected PMSG system in the slow DC voltage timescale are given by (2.147) and (2.155) for the two different cases. The first case is that the dynamics of the PLL are in the fast current timescale. In this case, the stability conditions are

2.5

An Example of Single Grid-Connected PMSG System

87

250 200

Imaginary part

150 100

50 0

SCR = 1

SCR = 2

-50

-100 -150 -200 -250 -50

-40

-30

-20

-10

0

10

Real part Fig. 2.33 Trajectory of λPLL when the SCR varies with the change of XL

XL = 1 XL = 2

Deviation of act ive power output/ p.u

0.06

0

- 0.06 0

Time (second) 0.5

1

1.5

2

2.5

3

Fig. 2.34 Results of non-linear simulation with SCR = 1 or SCR2 when XL varies

V0 1 V0 > and >1 X L P0 k 0 X L P0

ð2:147Þ

When the dynamics of the PLL are in the fast current timescale, parameters of the control system of the GSC and the PLL should be set to meet the following equations k0 > 1

ð2:149Þ

Following four tests are carried out to demonstrate and validate the stability condition presented in (2.147) and (2.149) about the stability of the example single grid-connected PMSG system in the slow DC voltage timescale.

88

2

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

Parameters of the example single grid-connected PMSG system are given in Table 2.1. PI gains of the PLL are Kpp = 2, Kpi = 2000 such that the dynamics of the PLL are in the fast current timescale. In the demonstration and validation presented in the previous subsection, it can be calculated by using (2.147) that k0 = 100.39 from the values of PI gains of outer control loops of the GSC given in Table 2.1. Hence, k0 > 1 such that the example single grid-connected PMSG system is always stable in the slow DC voltage timescale. In order to demonstrate and validate the assessment of stability of the example single grid-connected PMSG system in the slow DC voltage timescale, PI gains of outer control loops of the GSC are changed in test 1 and test 2 below from the values given in Table 2.1 to Kpp4 = 1.998, Kpi4 = 80 and Kpp6 = 0.0025, Kpi6 = 50. Subsequently, it can be calculated that k0 = 0.52. Hence, it is possible that the example single grid-connected PMSG system may lose the stability in the slow DC voltage timescale. TEST 1 The steady-state active power output from the PMSG, P0, is variable, from P0 = 0.24 to P0 = 0.56 with XL being fixed to be XL = 1.0. When P0 varies, the oscillation modes of the example single grid-connected PMSG system are computed by using (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified reduced-order model of the single gridconnected PMSG system given in (2.134) in the DC voltage timescale when the dynamics of the PLL are in the fast current timescale. Results of modal computation are given in Tables 2.11, 2.12, and 2.13.

Table 2.11 Results of modal computation when P0 = 0.24 Full-order modal is used -7.845 ± 50.96j -31.389 ± 243.17jj -527.59 ± 2223.1j

Simplified reduced-order modal is used -11.87 ± 52.98j

Table 2.12 Results of modal computation when P0 = 0.5 Full-order modal is used 0.0204 ± 51.883j -26.75 ± 243.17jj -527.59 ± 2223.1j

Simplified reduced-order modal is used -1.457 ± 55.13j

Table 2.13 Results of modal computation when P0 = 0.56 Full-order modal is used 1.599 ± 51.88j -23.14 ± 243.17jj -527.59 ± 2223.1j

Simplified reduced-order modal is used 0.568 ± 55.42j

An Example of Single Grid-Connected PMSG System 1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

2.5

0.6 0.4

0.6 0.4

0.2 0

89

0.2 0

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.35 Results of PFs computation when P0 = 0.24

0.6 0.4

0.6 0.4

0.2 0

0.2 0

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.36 Results of PFs computation when P0 = 0.5

0.6 0.4 0.2 0

0.6 0.4 0.2

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

0

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.37 Results of PFs when P0 = 0.56

From Tables 2.11, 2.12, and 2.13 it can be seen that 1. There is an oscillation mode, λDC, of the example single grid-connected PMSG system, which is highlighted in bold. The stability of the system is determined by the oscillation mode. By using the full-order model and the simplified reducedorder model in the DC voltage timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λDC are computed. Computational results are presented in Figs. 2.35, 2.36, and 2.37. It can be seen that λDC is associated with the DC voltage control outer loop.

90

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

2 4.5

S tta able l e region region n Stable

4

V0 X L P0

3.5 3

1 K0

2.5 2 1.92 1.5

Unstable region 0.25

0.3

0.35

0.4

0.45

0.5

0.55

P0 / p.u Fig. 2.38 Variation of XVL P0 0 when P0 varies 60

Imaginary part

20

-

P0 was increased

40

P0 was increased

P0 =0.5

0 - 20

P0 was increased

- 40 - 60 -40

-35

- 30

- 25

-20 -15 Real part

- 10

-5

0

Fig. 2.39 Trajectory of λDC when P0 varies

2. For the stability assessment, the simplified reduced-order model of the example system in the DC voltage timescale can provide the same results. This confirms the correctness of the simplified reduced-order model for the stability assessment in the DC voltage timescale. In this test, k0 in Eq. (2.147) is fixed to be k0 = 0.52. When P0 varies, XVL P0 0 changes accordingly. Figure 2.38 shows the variation of XVL P0 0 when P0 changes. Figure 2.39 presents the trajectory of λDC on the complex plane when P0 varies. From Fig. 2.37 it

2.5

An Example of Single Grid-Connected PMSG System

91

0.01

0

-0.01 0

P0

Deviation of act ive power output/ p.u

P0 = 0.24 P0 = 0.56

Time (second) 0.5

1

1.5

2

2.5

3

Fig. 2.40 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the fast current timescale) when P0 = 0.24 and P0 = 0.56 Table 2.14 Results of modal computation when XL = 0.76 Full-order modal is used -4.21 ± 47.67j -31.389 ± 207.17j -527.59 ± 2223.1j

Simplified reduced-order modal is used -3.56 ± 47.89j

can be seen that when P0 is greater than 0.5, XVL P0 0 is smaller than k10 = 1:92 such that the stability condition given in (2.147) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.39 confirms the correctness of (2.147) to assess the stability of the example single grid-connected PMSG system approximately. Two results of non-linear simulation are presented in Fig. 2.40 when P0 = 0.24 and P0 = 0.56. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. TEST 2 XL varies from XL = 0.76 to XL = 1.08. P0 is fixed to be P0 = 0.5. When XL varies, the oscillation modes of the example single grid-connected PMSG system are computed by using (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified reduced-order model of the single gridconnected PMSG system given in (2.134) in the DC voltage timescale when the dynamics of the PLL are in the fast current timescale. Results of modal computation are given in Tables 2.14, 2.15, and 2.16.

92

2

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

Table 2.15 Results of modal computation when XL = 1.04 Full-order modal is used -0.0204 ± 51.883j -26.75 ± 243.17jj -527.57 ± 2223j

Simplified reduced-order modal is used -0.07654 ± 51.166j

Table 2.16 Results of modal computation when XL = 1.08 Simplified reduced-order modal is used 0.57363 ± 49.797j

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Full-order modal is used 1.142 ± 49.797j -23.14 ± 243.17jj -527.59 ± 2223.7j

0.6 0.4 0.2 0

0.6 0.4 0.2 0

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

Fig. 2.41 Results of PFs computation when XL = 0.76

0.6 0.4 0.2 0

0.6 0.4 0.2

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

0

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.42 Results of PFs computation when XL = 1.04

From Tables 2.14, 2.15, and 2.16 it can be seen that 1. There is an oscillation mode, λDC, of the example single grid-connected PMSG system, which is highlighted in bold. The stability of the system is determined by the oscillation mode. By using the full-order model and the simplified reducedorder model in the DC voltage timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λDC are computed. Computational results are presented in Figs. 2.41, 2.42, and 2.43. It can be seen that λDC is associated with the DC voltage control outer loop.

An Example of Single Grid-Connected PMSG System 1

1

0.8

0.8

Participation factor

Parti cipa tion fa ctor

2.5

0.6 0.4 0.2 0

93

0.6 0.4 0.2 0

DC voltage control Other parts of the system outer loop. (a)Full-order modal is used

DC voltage control Other parts of the system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.43 Results of PFs when XL = 1.08 2.7

Stable Stab abl e region region n

2.6 2.5

V0 X L P0

2.4 2.3 2.2

1 K0

2.1 2 1.92 1.9 1.8

Unstable region 0.8

0.85

0.9

0.95

XL

1

1.05

Fig. 2.44 Variation of XVL P0 0 when XL varies

2. For the stability assessment, the simplified reduced-order model of the example system in the DC voltage timescale can provide the same results. This confirms the correctness of the simplified reduced-order model for the stability assessment in the DC voltage timescale. When XL varies, XVL P0 0 changes accordingly. Figure 2.44 shows the variation of XVL P0 0 when XL changes. Figure 2.45 presents the trajectory of λDC on the complex plane when XL varies. From Fig. 2.44 it can be seen that when XL is greater than 1.05, XVL P0 0 is smaller than k10 = 1:923 such that the stability condition given in (2.147) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.45 confirms the correctness of (2.147) to assess the stability of the example single grid-connected PMSG system approximately. Two results of non-linear simulation are presented in Fig. 2.46 when XL = 0.76 and XL = 1.08. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms.

94

2

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

60

X L was increased

40

Imaginary part

20

X L was increased

X L = 1.04

0 - 20

X L was increased

- 40 - 60 -40

-35

- 30

- 25

-20

-15

- 10

-5

0

Real part Fig. 2.45 Trajectory of λDC when XL varies 0.05

Deviation of act ive power output/ p.u

X L = 0.76 X L = 1.08

0

-0.05 0

Time (second) 0.5

1

1.5

2

2.5

3

3.5

4

Fig. 2.46 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the fast current timescale) when XL = 0.76 and XL = 1.08

TEST 3 Both P0 and XL are fixed to be P0 = 0.5 and XL = 1. PI gains of the reactive power control outer loops of the GSC are fixed to be Kpp6 = 0.0025, Kpi6 = 50. PI gains of the DC control outer loops of the GSC are varied from Kpp4 = 1.5, Kpi4 = 80 to Kpp4 = 3.5, Kpi4 = 160. When PI gains of the DC control outer loops of the GSC vary, k10 changes accordingly. On the plane of PI gains of the DC control outer loops of the GSC, V0 1 k 0 = X L P0 = 2 is a curve, as being shown in Fig. 2.47. According to (2.147), above the curve

1 k0

=

V0 X L P0

= 2, k10 >

V0 X L P0

= 2 such that the example single grid-connected

2.5

An Example of Single Grid-Connected PMSG System

95

3.5

Proport ional gain

3.2

V 1 > 0 K 0 X L P0

Stable region

2.9

2.6

(10 100 00, 0 0,2 2.6) 6 A (100,2.6)

2.3

V0 =2 X L P0

2

1.7 1.4 80

Unstable region 90

100

110

V 1 < 0 K 0 X L P0 120

130

140

B (150,2) 150

160

Integral gain Fig. 2.47 Stable and unstable region of the example single grid-connected PMSG system in the DC voltage timescale when the PI gains of the DC voltage control outer loop of GSC vary

PMSG system is stable. Below the curve k10 = XVL P0 0 = 2, k10 < XVL P0 0 = 2 such that the example single grid-connected PMSG system is unstable. Hence, stable region and unstable region are determine by the curve k10 = XVL P0 0 = 2, as being shown in Fig. 2.47. Let λDC = ξDC ± jωDC. If the real part of λDC = ξDC ± jωDC is negative, i.e., ξDC < 0, the example single-grid-connected PMSG system is stable in the DV voltage timescale. Otherwise, the system is unstable. Hence, to confirm the correctness of Fig. 2.47 which is drawn according to (2.147), variation of the real part of λDC = ξDC ± jωDC, when the PI gains of the DC control outer loops of the GSC vary, is computed. Computational results are presented in Fig. 2.48, which validates the correctness of Fig. 2.47. Two results of non-linear simulation are presented in Fig. 2.49. One result is obtained when the PI gains of the DC voltage control outer loop of the GSC are taken to be Kpp4 = 100, Kpi4 = 2.6 (point A in Fig. 2.47) such that the system is stable. Another result is obtained when the PI gains of the DC voltage control outer loop of the GSC are taken to be Kpp4 = 150, Kpi4 = 2 (point B in Fig. 2.47) such that the system is unstable stable. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. TEST 4 PI gains of the outer control loops of the GSC are set to ensure the smallsignal stability of the example single grid-connected PMSG system in the DC voltage timescale when the dynamics of the PLL are in the fast current timescale. According to Conclusion 2.2, if the PI gains of the outer control loops of the GSC are

96

2

4

ξ DC

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

Unstable region

2

0 -2 -4 1.7

Stable region 1.9

2.1

110

120

130

80

90

100

Fig. 2.48 Variation of real part of λDC = ξDC ± jωDC when the PI gains of the DC voltage control outer loop of GSC vary

0.02

Deviation of act ive power output/ p.u

0.015

A B

0.01 0.005 0 -0.005 -0.01 -0.015 -0.02

Time (second) 0

0.5

1

1.5

2

2.5

3

Fig. 2.49 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the fast current timescale) when the system is respectively in the stable and unstable region in Fig. 2.48

set to satisfy (2.149) such that k0 > 1, the small-signal stability of the example single grid-connected PMSG system in the DC voltage timescale is guaranteed. This is demonstrated and validated in this fourth test. First, the PI gains of the outer control loops of the GSC are set to be Kpp4 = 3, Kpi4 = 80 and Kpp6 = 0.0025, Kpi6 = 50. From (2.147), it is calculated to have k0 = 140.4 > 1. Hence, stability condition of (2.149) is satisfied.

2.5

An Example of Single Grid-Connected PMSG System

97

60 40

Imaginary part

P0 was increased 20

P0 = 1 SCR = 1

0 -20

P0 was increased

-40 -60 -80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Real part Fig. 2.50 Trajectory of when the SCR varies with the change of P0 when k0 = 140.4 > 1

0.02

P0 = 0.5 P0 = 1

Deviation of act ive power output/ p.u

0.015 0.01 0.005 0

-0.005 -0.01 -0.015 -0.02 0

Time (second) 0.5

1

1.5

2

2.5

3

Fig. 2.51 Results of non-linear simulation when SCR = 1 or SCR = 2 with P0 being varied and k0 = 140.4 > 1

Second, the lumped reactance of the transmission line connecting the PMSG to the grid in the example single grid-connected PMSG system is fixed to be XL = 1. The steady-state active power output from the PMSG is variable from P0 = 0.5 when SCR = 2 to P0 = 1 when SCR = 1. Figure 2.50 presents the trajectory of λDC when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.51. It can be seen that the example single grid-connected PMSG system is always stable in the DC voltage timescale even when the grid connection of the PMSG is as weak when SCR = 1.0. This confirms the correctness of Conclusion 2.2.

98

2

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

60

X L was increased

Imaginary part

40

X L = 0.5 SCR = 2

20

XL = 1 SCR = 1

0 -20

X L was increased

-40 -60 -80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Real part Fig. 2.52 Trajectory of λDC when the SCR varies with the change of XL when k0 = 140.4 > 1

0.01

0

-0.01 0

P0

Deviation of act ive power output/ p.u

X L = 0.5 XL = 1

Time (second) 0.5

1

1.5

2

2.5

3

Fig. 2.53 Results of non-linear simulation when SCR = 2 or SCR = 1 with XL being varied and k0 = 140.4 > 1

Finally, the steady-state active power output from the PMSG is fixed to be P0 = 1. varies from XL = 0.5 to XL = 1 such that the SCR changes accordingly from SCR = 2 to SCR = 1. Figure 2.52 presents the trajectory of λDC when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.53. It can be seen that the example single grid-connected PMSG system is always in the DC voltage timescale stable even when the grid connection of the PMSG is under the condition of extremely weak grid connected with the SCR to be as low as SCR = 1.0. This confirms the correctness of Conclusion 2.2.

2.5

An Example of Single Grid-Connected PMSG System

2.5.3

99

Stability of the Example Grid-Connected PMSG System in the Slow DC Voltage Timescale when the Dynamics of the PLL Are in the Slow DC Voltage Timescale

According to the discussions in Sect. 2.4, the sufficient and necessary stability conditions of the single grid-connected PMSG system in the slow DC voltage timescale, when the dynamics of the PLL are in the slow DC voltage timescale, is (2.155) 1 V0 V0 > and >1 X L P0 z0 X L P0

ð2:155Þ

Parameters of the control system of the GSC and the PLL should be set to meet the following equation to ensure the system stability z0 > 1

ð2:157Þ

Parameters of the example single grid-connected PMSG system are given in Table 2.1. PI gains of the PLL are Kpp = 0.02, Kpi = 10 such that the dynamics of the PLL are in the slow DC voltage timescale. Following four tests are conducted to demonstrate and evaluate the stability condition presented in (2.155) about the stability of the example single gridconnected PMSG system in the slow DC voltage timescale when the dynamics of the PLL are in the slow DC voltage timescale. In order to demonstrate and validate the assessment of stability of the example single grid-connected PMSG system in the slow DC voltage timescale, PI gains of outer control loops of the GSC are changed in test 1 and test 2 below from the values given in Table 2.1 to Kpp4 = 0.2, Kpi4 = 60 and Kpp6 = 0.1, Kpi6 = 15. Subsequently, it can be calculated that z0 = 0.52. According to (2.157), it is possible that the example single grid-connected PMSG system may lose the stability in the slow DC voltage timescale. TEST 1 The steady-state active power output from the PMSG, P0, is variable, from P0 = 0.32 to P0 = 0.54 with XL being fixed to be XL = 1. When P0 varies, the oscillation modes of the example single grid-connected PMSG system are computed by using (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model displayed in Fig. 2.10; (2) the simplified reduced-order model of the single gridconnected PMSG system given in (2.139) in the DC voltage timescale when the dynamics of the PLL are in the fast current timescale. Results of modal computation are given in Tables 2.17, 2.18, and 2.19. From Tables 2.17, 2.18, and 2.19 it can be seen that 1. There are two oscillation modes, λDC and λPLL, of the example single gridconnected PMSG system in the DC voltage timescale, which are highlighted in bold. The stability of the system is determined by one of the oscillation modes. By

100

2

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

Table 2.17 Results of modal computation when P0 = 0.35 Full-order modal is used -4.82 ± 52.14j -46.35 ± 35.199j -378.42 ± 2227.5j

Simplified reduced-order model is used -3.87 ± 48.25j -46.35 ± 35.199j

Table 2.18 Results of modal computation when P0 = 0.526 Full-order modal is used -0.02257 ± 48.797j -44.07 ± 31.559j -378.42 ± 2228j

Simplified reduced-order model is used -0.02257 ± 46.54j -42.81 ± 35.30j

Table 2.19 Results of modal computation when P0 = 0.54 Full-order modal is used 0.53692 ± 47.543j -43.35 ± 31.762j -378.42 ± 2227.8j

Simplified reduced-order model is used 0.53692 ± 47.543j -41.57 ± 35.176j

using the full-order model and the simplified reduced-order model in the DC voltage timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λDC and λPLL are computed. Computational results are presented in Figs. 2.54, 2.55, and 2.56. It can be seen that λDC is associated with the DC voltage control outer loop and λPLL is related with the PLL. 2. For the stability assessment, the simplified reduced-order model given in (2.139) of the example system in the DC voltage timescale can provide the same results. This confirms the correctness of the simplified reduced-order model of (2.139) for the stability assessment in the DC voltage timescale. In this test, z0 in Eq. (2.156) is fixed to be z0 = 0.52. When P0 varies, XVL P0 0 changes accordingly. Figure 2.57 shows the variation of XVL P0 0 when P0 changes. Figure 2.58 presents the trajectory of λDC on the complex plane when P0 varies. From Fig. 2.57 it can be seen that when P0 is greater than 0.5298, XVL P0 0 is smaller than z10 = 1:923 such that the stability condition given in (2.156) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.58 confirms the correctness of (2.156) to assess the stability of the example single grid-connected PMSG system approximately Two results of non-linear simulation are presented in Fig. 2.59 when P0 = 0.35 and P0 = 0.54. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. TEST 2 XL varies from XL = 0.76 to XL = 1.08. P0 is fixed to be P0 = 0.5. When XL varies, the oscillation modes of the example single grid-connected PMSG system are computed by using (1) the full-order state-space model of the single grid-connected PMSG system given in (2.53) or the block diagram model

2.5 An Example of Single Grid-Connected PMSG System λ DC λ PLL

0.8 0.6 0.4

1 Participation factor

Parti cipa tion fa ctor

1

0.2 0

101 λ DC λ PLL

0.8

0.6 0.4 0.2 0

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.54 Results of PFs computation when P0 = 0.35 λ DC λ PLL

0.8 0.6 0.4

1 Participation factor

Parti cipa tion fa ctor

1

0.2 0

λ DC λ PLL

0.8

0.6 0.4 0.2 0

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.55 Results of PFs computation when P0 = 0.526 λ DC λ PLL

0.8 0.6 0.4 0.2 0

1 Participation factor

Parti cipa tion fa ctor

1

λ DC λ PLL

0.8

0.6 0.4 0.2

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

0

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.56 Results of PFs when P0 = 0.54

displayed in Fig. 2.10; (2) the simplified reduced-order model of the single gridconnected PMSG system given in (2.139) in the DC voltage timescale when the dynamics of the PLL are in the fast current timescale. Results of modal computation are given in Tables 2.20, 2.21, and 2.22. From Tables 2.20, 2.21, and 2.22 it can be seen that 1. There are two oscillation modes, λDC and λPLL, of the example single gridconnected PMSG system in the DC voltage timescale, which are highlighted in

102

2

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

3.2

Stable region 3 2.8

V0 X L P0

2.6 2.4

1 Z0

2.2 2 1.923

Unstable region

1.8

0.35

0.4

0.45

0.5

P0 / p.u Fig. 2.57 Variation of timescale)

V0 X L P0

when P0 varies (dynamics of the PLL are in the slow DC voltage

75

Imaginary part

50

P0 was increased P0 was increased

25 0 -25

P0 was increased

-50

P0 was increased -100 -60

-54

-48

-40

-32 -24 Real part

-16

-8

0

50

Fig. 2.58 Trajectory of λDC when P0 varies (dynamics of the PLL are in the slow DC voltage timescale)

bold. The stability of the system is determined by one of the oscillation modes. By using the full-order model and the simplified reduced-order model in the DC voltage timescale of the example single grid-connected PMSG system, the participation factors (PFs) of λDC and λPLL are computed. Computational results are presented in Figs. 2.60, 2.61, and 2.62. It can be seen that λDC is associated with the DC voltage control outer loop and λPLL is related with the PLL.

2.5

An Example of Single Grid-Connected PMSG System

103

0.3

Deviation of act ive power output/ p.u

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

P0 = 0.35 P0 = 0.54 0.5

1

Time (second) 1.5

2

2.5

3

Fig. 2.59 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the slow DC voltage timescale) when P0 = 0.35 and P0 = 0.54 Table 2.20 Results of modal computation when XL = 0.76 Full-order modal is used -4.21 ± 54.812j -49.15 ± 31.199j -527.59 ± 2223.1j

Simplified reduced-order modal is used -3.274 ± 50.62j -47.17 ± 37.42j

Table 2.21 Results of modal computation when XL = 1.04 Full-order modal is used 0.187 ± 52.74j -44.07 ± 31.559j -527.57 ± 2223j

Simplified reduced-order modal is used -0.02257 ± 48.797j -46.81 ± 37.89j

Table 2.22 Results of modal computation when XL = 1.08 Full-order modal is used 1.892 ± 40.684j -43.65 ± 31.17j -527.59 ± 2223.7j

Simplified reduced-order modal is used 0.53693 ± 47.543j -44.89 ± 38.25j

2. For the stability assessment, the simplified reduced-order model given in (2.139) of the example system in the DC voltage timescale can provide the same results. This confirms the correctness of the simplified reduced-order model of (2.139) for the stability assessment in the DC voltage timescale when the dynamics of the PLL are in the slow DC voltage timescale. When XL varies, XVL P0 0 changes accordingly. Figure 2.63 shows the variation of XVL P0 0 when XL changes. Figure 2.64 presents the trajectory of λDC on the complex plane

104

2

Small-Signal Stability of a Single Grid-Connected PMSG System λ DC λ PLL

0.8 0.6 0.4

1 Participation factor

Parti cipa tion fa ctor

1

0.2 0

λ DC λ PLL

0.8

0.6 0.4 0.2 0

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.60 Results of PFs computation when XL = 0.76 (when dynamics of the PLL are in the slow DC voltage timescale)

λ DC λ PLL

0.8 0.6 0.4

1 Participation factor

Parti cipa tion fa ctor

1

0.2 0

λ DC λ PLL

0.8

0.6 0.4 0.2 0

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.61 Results of PFs computation when XL = 1.04 (when dynamics of the PLL are in the slow DC voltage timescale)

λ DC λ PLL

0.8 0.6 0.4 0.2 0

1 Participation factor

Parti cipa tion fa ctor

1

λ DC λ PLL

0.8

0.6 0.4 0.2

Other parts of the DC voltage control system outer loop. (a)Full-order modal is used

0

Other parts of the DC voltage control system outer loop. (b)Simplified reduced-order modal is used

Fig. 2.62 Results of PFs when XL = 1.08 (when dynamics of the PLL are in the slow DC voltage timescale)

2.5

An Example of Single Grid-Connected PMSG System

105

Stabl e region region n Stable

2.6 2.5

V0 X L P0

2.4 2.3

2.2

1 Z0

2.1 2 1.92 1.9

Unstable region

1.8

0.8

0.85

0.9

0.95

1

1.05

XL Fig. 2.63 Variation of XVL P0 0 when XL varies (when dynamics of the PLL are in the slow DC voltage timescale) 75

X L was increased X L was increased

Imaginary part

50 25 0

X L was increased

-25

-50 -100 -60

X L was increased -54

-48

-40

-32

-24

-16

-8

0

50

Real part Fig. 2.64 Trajectory of λDC when XL varies (when dynamics of the PLL are in the slow DC voltage timescale)

when XL varies. From Fig. 2.63 it can be seen that when XL is greater than 1.05, XVL P0 0 is smaller than z10 = 1:923 such that the stability condition given in (2.156) is violated. Consequently, the example single grid-connected PMSG system becomes unstable. Figure 2.64 confirms the correctness of (2.156) to assess the stability of the example single grid-connected PMSG system approximately.

106

2

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

0.05

Deviation of act ive power output/ p.u

X L = 0.76 X L = 1.08

0

-0.05 0

Time (second) 0.5

1

1.5

2

2.5

3

3.5

4

Fig. 2.65 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the slow DC voltage timescale) when XL = 0.76 and XL = 1.08

Two results of non-linear simulation are presented in Fig. 2.65 when XL = 0.76 and XL = 1.08. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms. TEST 3 Both P0 and XL are fixed to be P0 = 0.5 and XL = 1. PI gains of the reactive power control outer loops of the GSC are fixed to be Kpp6 = 0.1, Kpi6 = 15. PI gains of the DC control outer loops of the GSC are varied from Kpp4 = 0.2, Kpi4 = 60 to Kpp4 = 1.4, Kpi4 = 100. When PI gains of the DC control outer loops of the GSC vary, z10 changes accordingly. On the plane of PI gains of the DC control outer loops of the GSC, V0 1 z0 = X L P0 = 2 is a curve, as being shown in Fig. 2.66. According to (2.156), above the curve

1 z0

=

V0 X L P0

= 2,

1 z0

>

V0 X L P0

= 2 such that the example single grid-connected

PMSG system is stable. Below the curve z10 = XVL P0 0 = 2, z10 < XVL P0 0 = 2 such that the example single grid-connected PMSG system is unstable. Hence, stable region and unstable region are determine by the curve z10 = XVL P0 0 = 2, as being shown in Fig. 2.66. Let λDC = ξDC ± jωDC. To confirm the correctness of Fig. 2.66 which is drawn according to (2.156), variation of the real part of λDC = ξDC ± jωDC, when the PI gains of the DC control outer loops of the GSC vary, is computed. Computational results are presented in Fig. 2.67, which validates the correctness of Fig. 2.66. Two results of non-linear simulation are presented in Fig. 2.68. One result is obtained when the PI gains of the DC voltage control outer loop of the GSC are taken to be Kpp4 = 1.2, Kpi4 = 90 (point C in Fig. 2.66) such that the system is stable.

2.5

An Example of Single Grid-Connected PMSG System

107

1.4

Proport ional gain

1.2

V0 1 S ttab ab able abl l e rregion egion on Z > X P Stable 0 L 0

C

1

SCR = 2

0.8

0.6

V0 1 Unstable region Z < X P 0 L 0 0.2 D 0.4

0 60

65

70

75

80

85

90

95

100

Integral gain Fig. 2.66 Stable and unstable region of the example single grid-connected PMSG system in the DC voltage timescale when the PI gains of the DC voltage control outer loop of GSC vary and dynamics of the PLL are in the slow DC voltage timescale

1.5

1

ε DC

Unstable region

0.5

0

-0.5 0.6

Stable region 0.55 0.5

0.45 0.4

70

80

90

100

Fig. 2.67 Variation of real part of λDC = ξDC ± jωDC when the PI gains of the DC voltage control outer loop of GSC vary and dynamics of the PLL are in the slow DC voltage timescale

Another result is obtained when the PI gains of the DC voltage control outer loop of the GSC are taken to be Kpp4 = 0.2, Kpi4 = 60 (point D in Fig. 2.66) such that the system is unstable stable. At 0.5 second of simulation, active power output from the PMSG in the example power system of Fig. 2.1 suddenly increases by 10% for 100 ms.

108

2

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

0.3

Deviation of act ive power output/ p.u

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

C D 0.5

Time (second) 1

1.5

2

2.5

3

Fig. 2.68 Results of non-linear simulation to assess the stability of the example single gridconnected PMSG system in the DC voltage timescale (dynamics of the PLL are in the slow DC voltage timescale) when the system is, respectively, in the stable and unstable region in Fig. 2.67

TEST 4 PI gains of the outer control loops of the GSC are set to ensure the smallsignal stability of the example single grid-connected PMSG system in the DC voltage timescale when the dynamics of the PLL are in the slow DC voltage timescale. According to Conclusion 2.3, if the PI gains of the outer control loops of the GSC are set to satisfy (2.157) such that z0 > 1, the small-signal stability of the example single grid-connected PMSG system in the DC voltage timescale is guaranteed. This is demonstrated and validated in this fourth test. First, the PI gains of the outer control loops of the GSC are set to be Kpp4 = 1.4, Kpi4 = 60 and Kpp6 = 1, Kpi6 = 15. From (2.156), it is calculated to have z0 = 67.81 > 1. Hence, stability condition of (2.155) is satisfied. Second, the lumped reactance of the transmission line connecting the PMSG to the grid in the example single grid-connected PMSG system is fixed to be XL = 1. The steady-state active power output from the PMSG is variable from P0 = 0.5 when SCR = 2 to P0 = 1 when SCR = 1. Figure 2.69 presents the trajectory of λDC and λPLL when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.70. It can be seen that the example single grid-connected PMSG system is always stable in the DC voltage timescale even when the grid connection of the PMSG is as weak when SCR = 1.0. This confirms the correctness of Conclusion 2.3. Finally, the steady-state active power output from the PMSG is fixed to be P0 = 1. XL varies from XL = 0.5 to XL = 1 such that the SCR changes accordingly from SCR = 2 to SCR = 1. Figure 2.71 presents the trajectory of λDC and λPLL when the SCR varies. Confirmation of non-linear simulation is given in Fig. 2.72. It can be seen that the example single grid-connected PMSG system is always in the DC voltage timescale stable even when the grid connection of the PMSG is under the condition of extremely weak grid connected with the SCR to be as low as SCR = 1.0. This confirms the correctness of Conclusion 2.3.

2.5

An Example of Single Grid-Connected PMSG System

109

60 40

λ PLL was increased

Imaginary part

λ DC was increased 20

P0 = 1 SCR = 1

0 -20 -40

-60 -80

λ DC was increased λ PLL was increased -70

-60

-40

-50

-30

-20

-10

0

10

20

Real part Fig. 2.69 Trajectory of λDC and λPLL when the SCR varies with the change of P0 when z0 = 67.81 > 1 (Dynamics of the PLL are in the slow DC voltage timescale)

0.3

Deviation of act ive power output/ p.u

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

P0 = 0.5 P0 = 1 0.5

Time (second) 1

1.5

2

2.5

3

Fig. 2.70 Results of non-linear simulation when SCR = 2 or SCR = 1 with P0 being varied and z0 = 67.81 > 1 (Dynamics of the PLL are in the slow DC voltage timescale)

110

2

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

60

Imaginary part

40

λ PLL was increased λ DC was increased

20

XL = 1 SCR = 1

0

-20 -40

λ DC was increased λ PLL was increased

-60 -80

-70

-60

-50

-40

-30 -20

Real part

-10

0

10

20

Fig. 2.71 Trajectory of λDC and λPLL when the SCR varies with the change of XL when z0 = 67.81 > 1 (Dynamics of the PLL are in the slow DC voltage timescale) 0.3

Deviation of act ive power output/ p.u

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

X L = 0.5 XL = 1 0.5

Time (second) 1

1.5

2

2.5

3

Fig. 2.72 Results of non-linear simulation when SCR = 2 or SCR = 1 with XL being varied and z0 = 67.81 > 1 (Dynamics of the PLL are in the slow DC voltage timescale)

Summary of the Chapter Risk of sub/super-synchronous oscillations in a single grid-connected PMSG is analytically examined in this chapter. The examination leads to the following main conclusions.

Summary of the Chapter

111

1. When the PMSG is dominated by the dynamics of the PLL, the sufficient and necessary stability conditions of the single grid-connected PMSG system are given by (2.66), i.e., K pp V0 > X L P0 ω0 2 K pi Q V0 > þ 0 X L P0 ω0 K pp P0

ð2:66Þ

2. The sufficient and necessary stability conditions of the single grid-connected PMSG system in the fast current timescale are given by (2.115), which are exactly the same as those given by (2.66). This implies that if in the fast current timescale, the sub/super-synchronous oscillation occurs in the single gridconnected PMSG system, the PLL is responsible. 3. The sufficient and necessary stability conditions of the single grid-connected PMSG system in the slow DC voltage timescale are given by (2.147) and (2.155) for the two different cases. The first case is that the dynamics of the PLL are in the fast current timescale. In this case, the stability conditions are V0 1 V0 > and >1 X L P0 k 0 X L P0

ð2:147Þ

In the second case, the dynamics of the PLL are in the slow DC voltage timescale. In this case, the stability conditions are V0 1 V0 > and >1 X L P0 z0 X L P0

ð2:155Þ

Stability conditions derived in the chapter clearly indicate the connections of three key factors which affect the damping of sub/super-synchronous oscillation in the single grid-connected PMSG system. When the condition of grid connection weakens (i.e., XL increases) of the PMSG or/and loading condition of the PMSG (P0 increases) increases, stability conditions are more prone to be violated. Hence, the risk of sub/super-synchronous oscillation in the single grid-connected PMSG system increases. In the derived stability conditions as listed above, Kpp and Kpi are the PI gains of the PLL, k0 and z0 are determined by the setting of parameters of the control system of the GSC and the PLL. In order to eliminate the risk of the sub/supersynchronous oscillation in the single grid-connected PMSG system, parameters of the control system of the GSC and the PLL should be set to meet the derived stability conditions as follows. 1. When the dynamics of the PLL are in the fast current timescale, parameters of the control system of the GSC and the PLL should be set to meet the following equations

112

2

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

K pi K pp ≤ 1; ≤ 0:67 ω0 ω0 K pp

ð2:72Þ

k0 > 1

ð2:149Þ

2. When the dynamics of the PLL are in the slow DC voltage timescale, parameters of the control system of the GSC and the PLL should be set to meet the following equation z0 > 1

ð2:157Þ

References 1. Wenjuan DU, Haifeng Wang and Siqi Bu, Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, Springer, 2018 2. F. Golnaraghi and B. C. Kuo, Automatic control systems/9th edition. US: John Wiley & Sons, 2010. 3. Committee D, “IEEE Guide for Planning DC Links Terminating at AC Locations Having Low Short Circuit Capacities”, IEEE Standards, pp. 1204–1997, Jan 1997.

Chapter 3

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

Abbreviations CMVM PCC RSC SRF PLL WTG

Capacity-weighted mean value method Point of common coupling Rotor side converter Synchronous reference frame PLL Wind turbine generator

Grid connection of a DFIG and the arrangement of its converter control systems for the grid connection are not only different from those of a PMSG, but also more complicated. It has also been found that a grid-connected DFIG system may become unstable under the condition of weak grid connection [1]. Focusing on the impact of the PLL, it was found that parameters setting of the PLL may destabilize a gridconnected DFIG system [2]. A study on the DC voltage stability of the gridconnected DFIG system indicated that the parameters setting of DC voltage control may affect unfavorably the small-signal stability of the grid-connected DFIG system in the timescale of DC voltage control [3, 4]. This chapter analytically derives the small-signal stability limit (i.e., the sufficient and necessary stability conditions) of a single grid-connected DFIG system in the timescale of DC voltage control. The derived stability limit establishes the analytical connections between the impact of a condition of weak grid connection and the heavy loading of the DFIG generation system. It also reveals how those two affecting factors are analytically linked with the parameters setting of converter control systems of the DFIG, including the PLL. Hence, an overall view is presented about how the key factors (strength of grid connection, loading condition, and parameters setting of converter control systems) may cause the growing oscillations when the small-signal stability limit is violated. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Du, H. Wang, Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-031-35343-7_3

113

114

3

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

Firstly in the chapter, the full-order state-space model of a DFIG is introduced. Secondly, a simplified reduced-order model of the single grid-connected DFIG system in the DC voltage timescale is established. Based on the simplified model, the necessary and sufficient stability conditions of the single grid-connected DFIG system are derived. Finally in the chapter, an example single grid-connected DFIG system is presented.

3.1

Full-Order State-Space Model of a DFIG

Figure 3.1 shows the configuration of a DFIG which includes four main parts: (1) the induction generator, (2) the rotor side converter (RSC) and associated control system, (3) the gird side converter (GSC) and associated control system, and (4) the phase locked loop (PLL). In this section, the linearized model of each of those four parts is briefly introduced to lead to the establishment of a full-order statespace model of the DFIG. More detailed derivations of the linearized model of the DFIG can be found in [5].

3.1.1

The Induction Generator and Shaft System

The voltage equations of stator windings of induction generator in Fig. 3.1 are:

Induction generator

I dsd +jI dsq

Highspeed shaft

Pds +jQds

X df

I drd +jI drq Pdc1

RSC

Vdrd + jVdrq

Cd

Vdrqref

Vdrdref RSC and associated control system

Fig. 3.1 Configuration of a DFIG [1]

Vddc

Vdcd + jVdcq

Vdcdref Control system

Control system

Pdsref

GSC

Qdsref

θ PLL

Pdr Induction generator and the shaft system

(V ∠θ )

Pdc +jQdc

Lowspeed shaft

Vdsd +jVdsq

I dcd +jI dcq

Two-mass shaft rotational system

Vddcref

Vdcqref GSC and associated control system

θp PLL

3.1

Full-Order State-Space Model of a DFIG

d ψ = ω0 Rds I dsd þ ω0 V dsd þ ω0 ψ dsq dt dsd d ψ = ω0 Rds I dsq þ ω0 V dsq - ω0 ψ dsd dt dsq

115

ð3:1Þ

where Vdsd and Vdsq are the d and q component of voltage, ψ dsd and ψ dsq 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; Rds is the resistance of stator windings. The voltage equations of rotor windings of induction generator are: d ψ = ω0 Rdr I drd þ ω0 V drd þ ω0 ð1 - ωdr1 Þψ drq dt drd d ψ = ω0 Rdr I drq þ ω0 V drq - ω0 ð1 - ωdr1 Þψ drd dt drq

ð3:2Þ

where Vdrd and Vdrq are the d and q component of voltage, ψ drd and ψ drd 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; Rdr is the resistance of rotor windings. Flux equations of stator and rotor windings of the induction generator are: ψ dsd = - X dss I dsd - X dm I drd ψ dsq = - X dss I dsq - X dm I drq ψ drd = - X drr I drd - X dm I dsd ψ drq = - X drr I drq - X dm I dsq

ð3:3Þ

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: d ω = K dm θdr - T de - Ddm ðωdr1 - ωdr2 Þ - Ddr1 ωdr1 dt dr1 d J dr2 ωdr2 = T dm - K dm θdr - Ddm ðωdr2 - ωdr1 Þ - Ddr2 ωdr2 dt d θ = ω0 ðωdr2 - ωdr1 Þ dt dr J dr1

ð3:4Þ

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 = θdr2 - θdr1 is the relative angular position of low-speed and high-speed shaft

116

3 Small-Signal Stability of a Single Grid-Connected DFIG System

(θdr1 is the angular position of high-speed shaft. θdr2 is the angular position of low-speed shaft); Tde is the electromagnetic output torque of the induction generator, which can be expressed as: T de = X dm I dsd I drq - I dsq I drd

ð3:5Þ

Active power output from the rotor side of the induction generator is: Pdr = V drd I drd þ V drq I drq

ð3:6Þ

By linearizing (3.1)–(3.6) with ΔTdm = 0, it can have: d ΔXd1 = Ad1 ΔXd1 þ Bd1 Δzd1 þ bd1 ΔV dsq þ bd2 ΔV dsd dt ΔPdr = cd1 T ΔXd1 þ cd2 T Δzd1 where ΔXd1 = Δψ dsd Δψ dsq Δψ drd Δψ drq Δωdr1 Δωdr2 Δθdr Δzd1 = ½ ΔVdrd

ΔVdrq ]T ,

T

ð3:7Þ

3.1

Full-Order State-Space Model of a DFIG

Bd1 =

3.1.2

117

0

0

0

ω0

0 ω0

0 0

ω0 0

0 0

0 0

ω0 , bd1 = 0

0 0

0

0

0

0

0

0

0

0

cd2 T = ½ I drd0

I drq0 ]:

, bd2 =

0 0

:

The Rotor Side Converter and Associated Control System

Configuration of the control system of the RSC is shown by Fig. 3.2, 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 of the induction generator, Pds and Qds, respectively. sdr1 = 1 - ωdr1 is slip of high-speed shaft of the induction generator. From Fig. 3.2,

Fig. 3.2 Configuration of vector control system of the RSC of the DFIG

118

3

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

d x = K di1 ðPdsref - Pds Þ dt d1 d x = K di2 I drqref - I drq dt d2 d x = K di3 ðQdsref - Qds Þ dt d3 d x = K di4 ðI drdref - I drd Þ dt d4

ð3:8Þ

where Kdi1, Kdi2, Kdi3, and Kdi4 are the gains of corresponding integral controllers; Pdsref, Qdsref, Idrqref, and Idrdref are control reference of active and reactive power output from the stator side, q-axis and d-axis of rotor current, respectively. I dsqref = K dp1 ðPdsref - Pds Þ þ xd1 I dsdref = K dp3 ðQdsref - Qds Þ þ xd3 X I drqref = - dss I dsqref X dm V dsq X I drdref = - dss I dsdref X dm X dm V drqref = - K dp2 I drqref - I drq - xd2

ð3:9Þ

X 2dm s X I þ dr1 dm V dsq X dss drd X dss V drdref = - K dp4 ðI drdref - I drd Þ - sdr1 X drr -

- xd4 þ sdr1 X drr -

X 2dm I X dss drq

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 component of voltage of rotor windings, respectively. Normally, transient of modulation control can be ignored such that V drdref = V drd V drqref = V drq

ð3:10Þ

Active and reactive power outputs from the stator side of the induction generator are: Pds = V dsq I dsq þ V dsd I dsd Qds = V dsq I dsd - V dsd I dsq

ð3:11Þ

3.1

Full-Order State-Space Model of a DFIG

119

By linearizing (3.8)–(3.11) with ΔPdsref = 0 and ΔQdsref = 0, it can have: d ΔXd2 = Ad2 ΔXd2 þ Bd2 ΔXd1 þ bd3 ΔV dsq þ bd4 ΔV dsd dt Δzd1 = Cd1 ΔXd2 þ Cd2 ΔXd1 þ Cd3 ΔVdsq þ Cd4 ΔVdsd where ΔXd2 = ½ Δxd1

Bd2 =

Δxd2

Δxd3

Δxd4 ]T ,

K di1 V dsd0 X drr X drr X dss - X dm 2

K di1 V dsq0 X drr X dss X drr - X dm 2 K dp1 K di2 X dss X drr V dsq0

K di1 V dsd0 X dm X dm 2 - X drr X dss

K dp1 K di2 X dss X drr V dsd0 X dm X dm 2 - X drr X dss K di3 V dsq0 X drr X drr X dss - X dm 2 K dp3 K di4 X dss X drr V dsq0

þK di2 X dm 2 X dm X dm 2 - X dss X drr K di3 V dsd0 X drr X dm 2 - X dss X drr

- K dp1 K di2 X dss V dsd0 X dm 2 - X drr X dss K di3 V dsq0 X dm X dm 2 - X drr X dss - K dp3 K di4 X dss V dsq0

þK di4 X dm 2 X dm X dm 2 - X drr X dss

- K dp3 K di4 X dss X drr V dsd0 X dm X dm 2 - X dss X drr

- K di4 X dss X dm 2 - X drr X dss

- K di1 I dsq0 K di2 K dp1 X dss I dsq0 X dm , bd4 = bd3 = - K di3 I dsd0 K di4 K dp3 X dss I dsd0 - K di4 X dm Cd1 =

0

0

K dp2 X dss X dm

-

K dp4 X dss X dm

-1 ,

K di1 V dsq0 X dm X dm 2 - X dss X drr - K dp1 K di2 X dss V dsq0 - K di2 X dss X dm 2 - X dss X drr - K di3 V dsd0 X dm X dm 2 - X dss X drr K dp3 K di4 X dss V dsd0 X dm 2 - X dss X drr

- K di1 I dsd0 K di2 K dp1 X dss I dsd0 X dm , K di3 I dsq0 - K di4 K dp3 X dss I dsq0 X dm

ð3:12Þ

120

3

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

Cd2 T

=

K dp3 K dp4 X dss X drr V dsq0 þ K dp4 X dm 2 X dm X drr X dss - X dm 2

K dp2 K dp1 X dss X drr V dsd0 sdr10 X dm X dss X dm X drr X dss - X dm 2

K dp3 K dp4 X dss X drr V dsd0 sdr10 X dm þ X dss X dm X dm 2 - X dss X drr K dp3 K dp4 X dss V dsq0 þ K dp4 X dss X dm 2 - X drr X dss K dp3 K dp4 X dss V dsd0 - sdr10 X dss X drr - X dm 2 X 2 - I drq0 X drr - dm X dss

K dp2 K dp1 X dss X drr V dsq0 þ K dp2 X dm 2 X dm X dss X drr - X dm 2 K dp2 K dp1 X dss V dsd0 þ sdr10 X dm 2 - X drr X dss K dp2 K dp1 X dss V dsq0 þ K dp2 X dss X dm 2 - X dss X drr V dsq0 X dm X 2 I drd0 X drr - dm X dss X dss 0 0

K dp4 - K dp3 K dp4 X dss I dsd0 X dm Cd3 = , Cd4 = sdr10 X dm K dp2 K dp1 X dss I dsq0 X dss X dm

K dp3 K dp4 X dss I dsq0 X dm : - K dp2 X dss K dp1 I dsd0 X dm

ΔXd1 and Δzd1 are defined in (3.7).

3.1.3

The Grid Side Converter and Associated Control System of the DFIG

Figure 3.3 shows the configuration of the vector control system of the GSC. The qaxis and d-axis current control loop control 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: d ωV ωV I = 0 dcd - 0 dsd þ ω0 I dcq dt dcd X df X df ω0 V dcq ω0 V dsq d - ω0 I dcd I = X df X df dt dcq

ð3:13Þ

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.

3.1

Full-Order State-Space Model of a DFIG

q-axis current control inner loop

DC voltage control outer loop

K dp 5

Vddcref Vddc

+ -

+

K di 5 s xd 5

Vdsq

K dp 6

I dcqref +

+ +

121

I dcq

+

I dcd

GSC

X df I dcd Vdsd

+ -

+ Vdcqref +

K di 6 s xd 6

K dp 7

I dcdref +

+

+

+

K di 7 s xd 7

+ Vdcdref -

GSC

X df I dcq

d-axis current control inner loop Fig. 3.3 Configuration of vector control system of the GSC of the DFIG

Power balance equation associated with the charging and discharging of the capacitor is: Cd V ddc

dV ddc = Pdr - Pdc1 dt

ð3:14Þ

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 (3.6), Pdc1 is the active power injected from the DC side to the GSC and can be expressed as: Pdc1 = V dcd I dcd þ V dcq I dcq

ð3:15Þ

d x = K di5 ðV ddc - V ddcref Þ dt d5 d x = K di6 I dcqref - I dcq dt d6 d x = K di7 ðI dcdref - I dcd Þ dt d7

ð3:16Þ

From Fig. 3.3,

where Kdi5, Kdi6, and Kdi7 are the integral gains of PI controllers in Fig. 3.3, Vddcref, Idcqref, and Idcdref are respectively the control reference of DC voltage control, q and d component of output current of the GSC.

122

3

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

I cqref = K dp5 ðV ddcref - V ddc Þ þ xd5 V dcqref = K dp6 I dcqref - I dcq þ xd6 þ V dsq þ X df I dcd V dcdref = K dp7 ðI dcdref - I dcd Þ þ xd7 þ V dsd - X df I dcq

ð3:17Þ

where Kdp5, Kdp6, and Kdp7 are the proportional gains of PI controllers in Fig. 3.3; 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: V dcdref = V dcd

ð3:18Þ

V dcqref = V dcq Linearizing (3.13)–(3.18) with ΔVddcref = 0 and ΔIdcdref = 0, it can have: d ΔXd3 = Ad3 ΔXd3 þ bd5 ΔV dsd þ bd6 ΔV dsq þ bd7 ΔPdr dt where ΔXd3 = ½ ΔI dcd

ΔI dcq

0 0 bd5 =

- I dcd0 0 0 0

ΔV ddc

Δxd5

0 0 , bd6 =

- I dcq0 0 0 0

Δxd6

Δxd7 ]T ,

0 0 1 C V d ddc0 : , bd7 = 0 0 0

ð3:19Þ

3.1

Full-Order State-Space Model of a DFIG

123

Fig. 3.4 The relationship between the d - q and x - y coordinates

q

V ∠θ

y

x θp

θ

d

3.1.4

The Phase Locked Loop for the DFIG

Normally, the PLL for a DFIG takes the direction of stator flux as the direction of d axis of its d–q coordinate. When the resistance is ignored, the flux is behind the terminal voltage of the DFIG by 90°. Hence, the direction of terminal voltage of the DFIG is that of q axis. The PLL is considered to track the phase of the terminal voltage as the q axis of d–q coordinate of the DFIG, as being illustrated by Fig. 3.4. Similar to (2.25), linearized model of the PLL for the DFIG can be written as: d ΔXd4 = Ad4 ΔXd4 þ bd8 ΔV x þ bd9 ΔV y dt

ð3:20Þ

where Vx + jVy = V ∠ θ is the terminal voltage (stator) of the DFIG as indicated in Fig. 3.1, expressed in the common x–y coordinate; ΔXd4 = ½ Δxp

Δθp ]T

0

- K pi V 0

1

- K pp V 0

Ad4 =

, bd8 =

K pi V y0 V0

K pp V y0 V0

, bd9 =

K pi V x0 V0 K pp V x0 V0

From Fig. 3.4, Ix Iy

= sin θp cos θp - cos θp sin θp

V dsd V dsq

= sin θp - cos θp cos θp sin θp

I dcd I dcq Vx

þ

I dsd I dsq

ð3:21Þ

Vy

where Ix + jIy is the output current from the DFIG, expressed in the common x–y coordinate.

124

3

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

Linearization of (3.21) gives: cos θp0 - sin θp0 I I dcd0 þ dsd0 Δθp sin θp0 cos θp0 I dcq0 I dsq0 sin θp0 cos θp0 ΔI dsd ΔI dcd þ þ - cos θp0 sin θp0 ΔI dcq ΔI dsq ΔI dcd ΔI dsd = p1 Δθp þ P1 þ ΔI dcq ΔI dsq sin θp0 - cos θp0 sin θp0 cos θp0 V x0 = Δθp þ - sin θp0 cos θp0 V y0 cos θp0 sin θp0 ΔV x = p2 Δθp þ P2 ΔV y

ΔI x = ΔI y

ΔV dsd ΔV dsq

ΔV x ΔV y

ð3:22Þ

3.1.5

Model of the DFIG

Substituting the second equation of (3.12) in the second equation of (3.7), it can have: ΔPdr = cd1 T ΔXd1 þ cd2 T Cd1 ΔXd2 þ cd2 T Cd2 ΔXd1 þ cd2 T Cd3 ΔVdsq þ cd2 T Cd4 ΔVdsd

ð3:23Þ From (3.19) and (3.23), d ΔXd3 = bd7 cd1 T þ cd2 T Cd2 ΔXd1 þ bd7 cd2 T Cd1 ΔXd2 þ Ad3 ΔXd3 dt þ bd5 þ bd7 cd2 T Cd4 ΔVdsd þ bd6 þ bd7 cd2 T Cd3 ΔVdsq

ð3:24Þ

Substituting the second equation of (3.12) in the first equation of (3.7), it can have: d ΔXd1 = ðAd1 þ Bd1 Cd2 ÞΔXd1 þ Bd1 Cd1 ΔXd2 dt þðbd1 þ Bd1 Cd3 ÞΔVdsq þ ðbd2 þ Bd1 Cd4 ÞΔVdsd

ð3:25Þ

Writing the first equation of (3.12), (3.24), and (3.25) together, it can have:

3.1

Full-Order State-Space Model of a DFIG

d ΔXd1 dt d ΔXd2 dt d ΔXd3 dt

125

Ad1 þ Bd1 Cd2

Bd1 Cd1

0

ΔXd1

Bd2

Ad2

0

ΔXd2

Ad3

ΔXd3

=

bd7 ðcd1 þ cd2 Cd2 Þ T

T

T

bd7 cd2 Cd1

bd2 þ Bd1 Cd4

bd1 þ Bd1 Cd3

bd4

bd3

bd5 þ bd7 cd2 T Cd4

bd6 þ bd7 cd2 T Cd3

þ

ð3:26Þ

ΔVdsd ΔVdsq

Substituting the second equation of (3.22) in (3.26) and using (3.20), it can have: d ΔXd1 dt d ΔXd2 dt d ΔXd3 dt

Ad1 þ Bd1 Cd2

Bd1 Cd1

0

ΔXd1

Bd2

Ad2

0

ΔXd2

Ad3

ΔXd3

=

bd7 ðcd1 T þ cd2 T Cd2 Þ bd7 cd2 T Cd1 bd2 þ Bd1 Cd4

bd1 þ Bd1 Cd3

bd4

bd3

bd5 þ bd7 cd2 T Cd4

bd6 þ bd7 cd2 T Cd3

þ

½ 0 p2 ]ΔXd4 þ P2

ΔV x ΔV y

ð3:27Þ Writing (3.20) and (3.27) together, it can have: ΔV x d ΔXd = Ad ΔXd þ Bd dt ΔV y

ð3:28Þ

where ΔXd = ΔXd1 T ΔXd2 T ΔXd3 T ΔXd4 T Ad1 þ Bd1 Cd2 Bd1 Cd1 Ad =

Bd2

Ad2

bd7 ðcd1 þ cd2 Cd2 Þ bd7 cd2 Cd1 T

T

T

T

, 0

Ad14

0

Ad24

Ad3

Ad34 d4

126

3

Bd =

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

bd11

bd12

bd21

bd22

bd31

bd32

bd8

bd9

bd2 þ Bd1 Cd4

bd1 þ Bd1 Cd3

bd4

bd3

Ad34

bd5 þ bd7 cd2 T Cd4

bd6 þ bd7 cd2 T Cd3

bd11

bd12

bd2 þ Bd1 Cd4

bd1 þ Bd1 Cd3

bd21

bd22

bd4

bd3

bd31

bd32

bd5 þ bd7 cd2 T Cd4

bd6 þ bd7 cd2 T Cd3

Ad14 =

Ad24

=

½0

p2 ]

P2

Linearization of (3.3) is: Δψ dsd = - X dss ΔI dsd - X dm ΔI drd Δψ dsq = - X dss ΔI dsq - X dm ΔI drq : Δψ drd = - X drr ΔI drd - X dm ΔI dsd Δψ drq = - X drr ΔI drq - X dm ΔI dsq

ð3:29Þ

From (3.29), ΔId = M - 1 Δψd

ð3:30Þ

where

M=

ΔId = ½ ΔI dsd

ΔI dsq

- X dss 0

0 - X dss

- X dm 0

0 - X dm

- X dm

0

- X drr

0

0

- X dm

0

- X drr

I drd

I drq ]T , Δψd = ψ dsd

ψ dsq

,

ψ drd

ψ drq

T

:

From (3.7) and (3.30), ΔI dsd ΔI dsq

=

- X dss 0

0 - X dss

- X dm 0

0 - X dm

0 0 0 ΔXd1 = Cd11 ΔXd1 0 0 0

ð3:31Þ

3.2

Block Diagram Model of a Grid-Connected DFIG

127

From (3.19), ΔI dcd 1 = ΔI dcq 0

0 1

0 0

0 0 0 0

0 ΔXd3 = Cd13 ΔXd3 0

ð3:32Þ

From (3.20), (3.22), (3.31), and (3.32),

where Cd14 = ½ 0 p1 ]. The full-order state-space model of the DFIG is given by (3.28) and (3.33).

3.2

Block Diagram Model of a Grid-Connected DFIG

Figure 3.5 shows the configuration of a DFIG being connected to the grid via a transmission line which is represented by the lumped reactance XL. In this section, a block diagram model of the grid-connected DFIG is derived on the basis of its fullorder state-space model introduced in the previous section. The derived block diagram model can more clearly show the dynamic connections between various parts of the DFIG. This can help the understanding about the derivation of a simplified model of the DFIG in the timescale of DC voltage control so as to analytically examine the stability of the grid-connected DFIG system displayed in Fig. 3.5 in the timescale of DC voltage control.

Vbx +jVby

Vdsd +jVdsq (V ∠θ )

Pds +jQds

I drd +jI drq Pdc1

RSC

Vdrd + jVdrq

Cd

Vdrdref

Vdrqref

Pdsref

GSC

Vddc

Vdcd + jVdcq

Vdcdref

Vdcqref Control system

Control system

Qdsref

Fig. 3.5 Configuration of a grid-connected DFIG

I x +jI y

X df

θ PLL

Pdr

XL

Pdc +jQdc

Highspeed shaft

I dcd +jI dcq

Lowspeed shaft

I dsd +jI dsq

Vddcref

θp

External main grid

128

3.2.1

3

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

Rotor Motion Dynamics in the Electromechanical Timescale

For the simplicity of derivation, following single-mass rotational system of the induction generator is used instead of the two-mass rotational system given in (3.4). Jr

d ω = T dm - T de - Dr ωr dt r

ð3:34Þ

where Jr is the constant of inertia of the shaft respectively; Tdm is the mechanical torque input to the shaft; ωr is the angular speed of the shaft; Dr is the damping coefficient. Linearization of (3.34) is: Δωr =

1 ðΔT dm - ΔT de Þ J r s þ Dr

ð3:35Þ

Linearization of (3.5) is: ΔT de = X dm cs3 ΔIdrdq þ X dm cT ΔIsdq

ð3:36Þ

where

3.2.2

Dynamics of Outer Loop Control Loops of the GSC and the RSC in the DC Voltage Timescale

Linearization of (3.14) is: ΔV ddc =

1 ðΔPdc1 - ΔPdr Þ = Gddc ðsÞðΔPdc1 - ΔPdr Þ sCd V ddc0

ð3:37Þ

From Fig. 3.3, linearized dynamics of the outer control loops of the GSC can be written as: ΔI dcqref = K dp5 þ ΔI dcdref = 0

K di5 ΔV ddc = G5 ðsÞΔV ddc s

ð3:38Þ

The active power injected from the DC side to the GSC, Pdc1, is given by (3.11). It can also be expressed as:

3.2

Block Diagram Model of a Grid-Connected DFIG

129

Pdc1 = V dsd I dcd þ V dsq I dcq Denote (3.39) is:

ΔIdcdq = ½ I dcd

I dcq ]T , ΔVsdq = ½ V dsd

V dsq ]T .

ð3:39Þ Linearization

ΔPdc1 = cdc1 ΔVsdq þ cdc2 ΔIdcdq

of

ð3:40Þ

where cdc1 = ½ I dcd0 I dcq0 ], cdc2 = ½ V dsd0 V dsq0 ]. T T Denote ΔIdrdq = ½ I drd I drq ] , ΔVrdq = ½ V drd V drq ] . Linearization of (3.6) is ΔPdr = cdr1 ΔVdrdq þ cdr2 ΔIdrdq

ð3:41Þ

where cdr1 = ½ I drd0 I drq0 ], cdr2 = ½ V drd0 V drq0 ]. From Fig. 3.2, linearized dynamics of the outer control loops of the RSC can be written as: K di1 ΔPds = G1 ðsÞΔPds s K ΔI dsdref = - K dp3 þ di3 ΔQds = G3 ðsÞΔQds s X ΔI drqref = - dss ΔI dsqref X dm X 1 ΔI drdref = - sss ΔI dsdref ΔV dsq Xm X sm

ΔI dsqref = - K dp1 þ

Denote ΔIsdq = ½ I dsd

I dsq ]T , ΔVsdq = ½ V dsd

ΔQds = cs3 ΔVsdq þ cs4 ΔIsdq

where

Dynamics of Inner Loop Control Loops of the GSC and Output Filter in the Current Timescale

Linearization of (3.13) is:

ð3:43Þ

V dsq ]T . Linearization of (3.11) is:

ΔPds = cs1 ΔVsdq þ cs2 ΔIsdq

3.2.3

ð3:42Þ

ð3:44Þ

130

3

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

ω ΔV ω ΔV d ΔI = 0 dcd - 0 dsd þ ω0 ΔI dcq Xdf X df dt dcd ω0 ΔV dcq ω0 ΔV dsq d - ω0 ΔI dcd ΔI = dt dcq X df X df

ð3:45Þ

From Fig. 3.3 and by using (3.18), linearized dynamics of the inner control loops of the GSC can be obtained to be: K di6 ΔI dcqref - ΔI dcq þ ΔV dsq þ X df ΔI dcd s = G6 ðsÞ ΔI dcqref - ΔI dcq þ ΔV dsq þ X df ΔI dcd K ΔV dcd = K dp7 þ di7 ðΔI dcdref - ΔI dcd Þ þ ΔV dsd þ X df ΔI dcq s = G7 ðsÞðΔI dcdref - ΔI dcd Þ þ ΔV dsd þ X df ΔI dcq ΔV dcq = K dp6 þ

ð3:46Þ

From (3.45) and (3.46), dynamics of the current control inner loops of the GSC and the output filter can be expressed as: ω 0 G 6 ð sÞ = Gdcq ðsÞΔI dcqref ΔI sX df þ ω0 G6 ðsÞ dcqref ω 0 G 7 ð sÞ ΔI dcd = = Gdcd ðsÞΔI dcdref ΔI sX df þ ω0 G7 ðsÞ dcdref ΔI dcq =

3.2.4

ð3:47Þ

Dynamics of Inner Loop Control Loops of the RSC and Rotor Winding in the Current Timescale

From the current control inner loop of the RSC displayed in Fig. 3.2 and (3.10), it can have (sdr1 = 1 - ωr): X 2dm X dm I drd þ ð1 - ωr Þ V dsq X dss X dss X2 X dm = G2 ðsÞ I drqref - I drq - ð1 - ωr Þ X drr - dm I drd þ ð1 - ωr Þ V dsq X dss X dss K di4 X2 X dm V drd = K dp4 þ ðI drdref - I drd Þ þ ð1 - ωr Þ X drr - dm I drq þ ð1 - ωr Þ V dsd s X dss X dss X2 X dm = G4 ðsÞðI drdref - I drd Þ þ ð1 - ωr Þ X drr - dm I drq þ ð1 - ωr Þ V dsd X dss X dss V drq =

K dp2 þ

K di2 s

I drqref - I drq - ð1 - ωr Þ X drr -

ð3:48Þ

3.2

Block Diagram Model of a Grid-Connected DFIG

131

Linearization of (3.48) is: ΔVrdq = Z1 ðsÞΔI drdref þ Z2 ðsÞΔI drqref þ Z3 ðsÞΔIdrdq þ Z4 ΔVsdq þ Z5 Δωr ð3:49Þ where I drq ]T , ΔVsdq = ½ V dsd

ΔIdrdq = ½ I drd Z1 ðsÞ =

- G4 ðsÞ 0 sdr10

Z4 =

X dm X dss

0

; Z2 ðsÞ =

0 - G2 ðsÞ

X dm X dss

; Z3 ðsÞ = - X drr -

0 sdr10

V dsq ]T , ΔVrdq = ½ V drd

; Z5 = X drr -

V drq ]T ;

G4 ðsÞ

σ0

- σ0

G2 ðsÞ

;

X 2dm X I - dm V X dss drq0 X dss dsd0

X 2dm X I - dm V X dss drd0 X dss dsq0

By ignoring the transient of magnetic linkage of the stator winding, i.e., = dtd ψ dsq = 0, voltage equations of stator winding of (3.1) can be written as:

d dt ψ dsd

Rds I dsd þ V dsd þ ψ dsq = 0 Rds I dsq þ V dsq - ψ dsd = 0

ð3:50Þ

From the flux equations of stator winding, i.e., the first group of two equations in (3.3) and (3.50), it can have: X dm X dss X R R X I þ 2 dm ds2 I drq - 2 ds 2 V dsd - 2 dss 2 V dsq 2 2 drd X dss þ Rds X dss þ Rds X dss þ Rds X dss þ Rds X dm Rds X dm X dss X dss R I dsq = - 2 I drd - 2 I drq þ 2 V dsd - 2 ds 2 V dsq X dss þ R2ds X dss þ R2ds X dss þ R2ds X dss þ Rds I dsd = -

ð3:51Þ Linearized voltage equations of rotor winding of (3.2) (note ωdr1 = ωr) are: d Δψ drd = ω0 Rdr ΔI drd þ ω0 ΔV drd þ ω0 ð1 - ωr0 ÞΔψ drq - ω0 ψ drq0 Δωr dt d Δψ drq = ω0 Rdr ΔI drq þ ω0 ΔV drq - ω0 ð1 - ωr0 ÞΔψ drd þ ω0 ψ drd0 Δωr dt

ð3:52Þ

Linearized flux equations of rotor winding, which are the second group of two equations in (3.3), are: Δψ drd = - X drr ΔI drd - X dm ΔI dsd Δψ drq = - X drr ΔI drq - X dm ΔI dsq

ð3:53Þ

132

3

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

Linearization of (3.51) is: X dm X dss X R R X ΔI drd þ 2 dm ds2 ΔI drq - 2 ds 2 ΔV dsd - 2 dss 2 ΔV dsq X 2dss þ R2ds X dss þ Rds X dss þ Rds X dss þ Rds X R X X X R ΔI dsq = - 2 dm ds2 ΔI drd - 2 dm dss2 ΔI drq þ 2 dss 2 ΔV dsd - 2 ds 2 ΔV dsq X dss þ Rds X dss þ Rds X dss þ Rds X dss þ Rds ΔI dsd = -

ð3:54Þ From (3.49), (3.52), (3.53) and (3.54), it can have: ΔI drq = Grq ðsÞΔI drqref - Gωrq ðsÞΔωr - Gsd ðsÞΔV dsd ΔI drd = Grd ðsÞΔI drdref - Gsq ðsÞΔV dsq þ Gωrd ðsÞΔωr where Grd ðsÞ =

G 4 ðsÞ sdr10 X 2dm Rds X 2dm X ss 1 X þ Rdr s þ G4 ðsÞ drr ω0 X 2dss þ R2ds X 2dss þ R2ds sdr10 X dm Rds X 2dss þ R2ds

Gsq ðsÞ = G 4 ðsÞ -

sdr10 X 2dm Rds X 2dm X dss 1 X þ Rdr s þ drr ω0 X 2dss þ R2ds X 2dss þ R2ds

Gωrd ðsÞ = G 4 ðsÞ -

1 ω0

X m Rs V sq0 þ X m I rd0 X 2ss þ R2s 2 sdr10 X 2dm Rds X dm X dss X þ Rdr s þ drr X 2dss þ R2ds X 2dss þ R2ds

ð3:55Þ

3.2

Block Diagram Model of a Grid-Connected DFIG

3.2.5

133

The PLL and the Transmission Line

Configuration of a PLL is shown in Fig. 2.4. Hence, Δθp =

s2

K pp s þ K pi V 0 Δθ = Gpll ðsÞΔθ = Gpll ðsÞKθ ΔVxy þ V 0 K pp s þ V 0 K pi

ð3:56Þ

where θ is the phase of the PCC voltage of the DFIG, θp is the estimation of θ by the PLL, i.e., the tracked phase of the PCC voltage, V0 is the magnitude of PCC voltage at the steady state, Kpp and Kpi are the proportional and integral gains of the PLL, respectively, ΔVxy = ½ ΔV x ΔV y ]T , Vx + jVy is the voltage at the PCC of the DFIG, V y0 V x0 expressed in the common x–y coordinate, Kθ = - 2 is obtained by the V 0 V 20 derivation similar to that from (2.22) to (2.24). The relationship between the d–q coordinate of the DFIG and the common x–y coordinate is displayed in Fig. 3.4. From Fig. 3.4, cos θp - sin θp sin θp cos θp cos θp sin θp = - sin θp cos θp

Ix = Iy V dsd V dsq

I dd I dq Vx Vy

ð3:57Þ

where Ix + jIy and Idd + jIqq = Idsd + jIdsq + Idcd + jIdcq are the output current of the DFIG expressed in the common x–y coordinate and d–q coordinate respectively. Thus, ΔI dd = ΔI dsd þ ΔI dcd

ð3:58Þ

ΔI qq = ΔI dsq þ ΔI dcq Linearization of (3.57) is ΔIxy = K1 ΔIdq þ K2 Δθp ΔVsdq = K3 ΔVxy þ K4 Δθp

ð3:59Þ

where T T ΔVsdq = ½ V dsd V dsq ] , ΔI dq = ½ I dd I dq ] , ΔI xy = ½ I x sin θ0 - I y0 cos θ0 K1 = , K2 = - cos θ0 sin θ0 I x0 sin θ0 - cos θ0 V dsq0 K3 = , K4 = cos θ0 sin θ0 - V dsd0

I y ]T

134

3

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

The linearized voltage equation of the transmission line connecting the DFIG and the grid is (see (2.51)) ΔVxy =

XL U ðsÞΔIxy þ ΔVbxy ω0 L

ð3:60Þ

ΔV by ]T , Vbx + jVby is the voltage at the grid busbar, s -1 ω0 expressed in the common x–y coordinate, UL ðsÞ = s . 1 ω0 From (3.36)–(3.59), a block diagram model of the grid-connected DFIG is obtained and displayed in Fig. 3.6. where ΔVbxy = ½ ΔV bx

cs 4

ΔI sdq

cs 2 c s1

ΔPds

– G1 ( s )

ΔVdsd ΔI dsqref

–

X dss ΔI drqref X dm

Power control outer loop of RSC

X dss 2 + Rds2 X dss

Gsd ( s )

–

Grq ( s )

ΔI drq

Gwrq ( s )

Δω r

cs 3

ΔTe –

– G3 ( s )

ΔI dsdref

ΔI drdref X – dss X dm – ΔVdsq

X dm

ΔTm = 0

1 Δωr Z5 J r s + Dr

Z1 ( s )

Z4

Z 2 ( s)

X dm X dss 2 + Rds2 X dss

–

X dm Rds 2 X dss + Rds2

ΔI drd

Grd ( s )

–

–

ΔVdsq Gsq ( s ) Current control inner loop of RSC

Z3 ( s)

cdr 2

ΔVdq

ΔVxy

ΔVdsd Rds 2 X dss + Rds2

X m cs 3

ΔV ΔI dcqref ΔI dcq Gddc ( s ) ddc G5 ( s ) Gdcq ( s ) ΔI dcdq DC voltage control ΔI dcd Gdcd ( s ) outer loop of GSC ΔI dcdref cdc 2 Current control inner loop of GSC

ΔI dq

K1 K4

Kθ

PLL Δθ p G pll ( s )

Δθ

xL U L (s) w0

Electromechanical timescale

ΔI dsd

ΔTe

K3 ΔVbxy

ΔI sdq

X m cT

ΔPdr

–

Rds ΔVdsq 2 X dss + Rds2

ΔVdsq X – 2 dss 2 X dss + Rds

–

cdr1 ΔPdc1

–

X dm X dss 2 X dss + Rds2

ΔVrdq

cdc1

ΔI dsq

X dm Rds 2 + Rds2 X dss

Gwrd ( s)

ΔQds

–

–

ΔVdsd

DC voltage control timescale

Fig. 3.6 Block diagram of a grid-connected DFIG

K2

ΔI xy

Electric current control timescale

ΔI xy

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

3.3

135

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG System in the DC Voltage Timescale

In this section, risk of sub/super-synchronous oscillations in a grid-connected DFIG system displayed in Fig. 3.5 is analytically examined. For the examination, simplified reduced-order model of the grid-connected DFIG system in the DC voltage timescale is derived. Based on the simplified reduced-order model, small-signal stability conditions of the grid-connected DFIG system in the DC voltage timescale are derived for the analysis of risk of sub/super-synchronous oscillations.

3.3.1

The Simplified Model in the Timescale of the DC Voltage Control When Dynamics of the PLL Are in the Fast Timescale of Electric Current Control

In the timescale of DC voltage control, slow electromechanical dynamics of the rotor motion, the fast dynamics of the PLL, the electric current control of inner loops of the GSC, and the RSC can be ignored. Ignorance of the fast dynamics of the PLL implies that the phase tracking by the PLL is sufficiently fast such that the q axis of the d–q coordinate if fixed to be in the same direction of terminal voltage of the DFIG, i.e., ΔVdsd = 0. Hence, Δωr = 0; ΔV dsd = 0 ΔI drd = ΔI drdref ; ΔI drq = ΔI drqref Δθ = Δθp

ΔI dcd = ΔI dcdref = 0 ΔI dcq = ΔI dcqref

ð3:61Þ

In addition, resistance of stator windings is ignored to have Rds = 0. Dynamics of the transmission line connecting the DFIG to the grid are neglected such that U L ð 0Þ =

0

-1

1

0

ð3:62Þ

With Xdss >> Rds, (3.61) and (3.62), the block diagram model of the gridconnected DFIG system displayed in Fig. 3.6 is simplified. The simplified reduced-order block diagram model is shown by Fig. 3.7. The state-space model of the grid-connected DFIG system in the DC voltage timescale equivalent to the block diagram model displayed in Fig. 3.7 is derived as follows.

136

3

Small-Signal Stability of a Single Grid-Connected DFIG System cs 4

ΔI sdq

cs 2 c s1 cs 3

ΔPds ΔQds

– G1 ( s ) – G3 ( s )

ΔI dsqref ΔI dsdref

–

X dss X dm

–

X dss X dm

Power control outer loop of RSC

ΔI drq ΔI drd –

–

X dm X dss

–

X dm X dss

ΔI dsq ΔI dsd –

ΔVdsq

ΔVdsq

X dm

X dss

Z3 ( s)

ΔI sdq

cdr 2

Z4 ΔVrdq

cdr1 ΔPdc1 –

cdc1

ΔPdr

Gddc ( s )

ΔVddc

G5 ( s )

Voltage control outer loop of GSC cdc 2

ΔVdq

ΔI dcq

ΔI dcdq

ΔI dcd = 0

ΔI dq K1

K3

ΔVbxy

ΔVxy

K4 Kθ

Δθ

ΔI xy

K2

ΔI xy

xL U (0) w0 L

Fig. 3.7 Simplified reduced-order block diagram model of the grid-connected DFIG in the timescale of DC voltage control when the dynamics of the PLL are in the current timescale

Linearization of (3.8), (3.9), (3.10), and (3.11) is d Δx = K di1 dt d1 d Δx = K di2 dt d2 d Δx = K di3 dt d3 d Δx = K di4 dt d4

- ΔPds ΔI drqref - ΔIdrq - ΔQds ΔIdrdref - ΔIdrd

ð3:62Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

ΔIdsqref = Kdp1 ð - ΔPds Þ þ Δxd1 ΔIdsdref = Kdp3 ð - ΔQds Þ þ Δxd3 X ΔIdrqref = - dss ΔIdsqref X dm ΔVdsq X ΔIdrdref = - dss ΔIdsdref X dm X dm ΔVdrqref = - Kdp2 ΔIdrqref - ΔIdrq - Δxd2

137

ð3:63Þ

X 2dm s X ΔI drd þ dr1 dm ΔVdsq X dss X dss ΔVdrdref = - Kdp4 ðΔIdrdref - ΔIdrd Þ - sdr1 X drr -

- Δxd4 þ sdr1 X drr -

X 2dm ΔI drq X dss

ΔV drdref = ΔV drd ΔV drqref = ΔV drq

ð3:64Þ

ΔPds = cs1 ΔVsdq þ cs2 ΔIsdq ΔQds = cs3 ΔVsdq þ cs4 ΔIsdq

ð3:65Þ

where ΔVsdq, ΔIsdq and csj, j = 1, 2, 3, 4 are given in (3.44). From

ΔI drd = ΔI drdref ΔI drq = ΔI drqref

given in (3.61), (3.62), and (3.63) can be simplified to be d Δx = - K di1 ΔPds dt d1 d Δx = - K di3 ΔQds dt d3

X dss - K dp1 ΔPds þ Δxd1 X dm ΔV dsq X ΔI drd = - dss - K dp3 ΔQds þ Δxd3 X dm X dm X 2dm sdr1 X dm ΔV drq = - sdr1 X drr ΔVdsq ΔI drd þ X dss X dss

ð3:66Þ

ΔI drq = -

ΔV drd = sdr1 X drr -

X 2dm ΔI drq X dss

Linearization of (3.13), (3.14), (3.15), (3.16), (3.17), and (3.18) is

ð3:67Þ

138

3

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

ω ΔV ω ΔV d ΔI = 0 dcd - 0 dsd þ ω0 ΔI dcq X df X df dt dcd ω0 ΔV dcq ω0 ΔV dsq d - ω0 ΔIdcd ΔI = X df X df dt dcq C d V ddc0

dΔV ddc = ΔPdr - ΔPdc1 dt

ΔPdr = ΔV drd I drd0 þ V drd0 ΔI drd þ ΔV drq I drq0 þ V drq0 ΔI drq ΔPdc1 = ΔV dcd I dcd0 þ V dcd0 ΔI dcd þ ΔV dcq I dcq0 þ V dcq0 ΔI dcq

d Δx = K di5 ΔV ddc - ΔV ddcref dt d5 d Δx = K di6 ΔI dcqref - ΔI dcq dt d6 d Δx = K di7 ΔI dcdref - ΔI dcd dt d7

ð3:68Þ

ð3:69Þ ð3:70Þ

ð3:71Þ

ΔI cqref = K dp5 ðΔV ddcref - ΔV ddc Þ þ Δxd5 ΔV dcqref = K dp6 ΔI dcqref - ΔI dcq þ Δxd6 þ ΔV dsq þ X df ΔI dcd ΔV dcdref = K dp7 ðΔI dcdref - ΔI dcd Þ þ Δxd7 þ ΔV dsd - X df ΔI dcq ΔV drdref = ΔV drd ΔV drqref = ΔV drq

ð3:72Þ

ð3:73Þ

(3.70) can be written as ΔPdr = cdr1 ΔVdrdq þ cdr2 ΔIdrdq ΔPdc1 = cdc1 ΔVsdq þ cdc2 ΔIdcdq

ð3:74Þ

where ΔVdrdq, ΔIdrdq, cdr1, cdr2, cdc1, and cdc2 are given in (3.40) and (3.41). ΔI dcd = ΔI dcdref = 0 From given in (3.61), (3.71) can be simplified to be ΔI dcq = ΔI dcqref d Δx = K di5 ΔV ddc dt d5 ΔI cq = - K dp5 ΔV ddc þ Δxd5

ð3:75Þ

Linearization of (3.1) and first two equations of (3.3) are d Δψ dsd = ω0 ΔV dsd þ ω0 Δψ dsq = 0 dt d Δψ dsq = ω0 ΔV dsq - ω0 Δψ dsd = 0 dt

ð3:76Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

Δψ dsd = - X dss ΔI dsd - X dm ΔI drd

139

ð3:77Þ

Δψ dsq = - X dss ΔI dsq - X dm ΔI drq From (3.76) and (3.77), it can have X dm 1 ΔI ΔV dsq X dss drd X dss X 1 ΔI dsq = - dm ΔI drq þ ΔV dsd X dss X dss ΔI dsd = -

ð3:78Þ

Substituting the first two equations of (3.67) in (3.78), ΔI dsd = - K dp3 ΔQds þ Δxd3 ΔI dsq = - K dp1 ΔPds þ Δxd1 þ

ð3:79Þ

1 ΔV dsd X dss

Substituting (3.68), cs2, and cs4 given in (3.44) into (3.79), it can have K dp3 1 Δx c ΔV 1 þ K dp3 V dsq0 d3 1 þ K dp3 V dsq0 s3 sdq : K dp1 1 1 1 ΔI dsq = Δxd1 þ ΔV dsd cs1 ΔVsdq 1 þ K dp1 V dsq0 1 þ K dp1 V dsq0 X dss 1 þ K dp1 V dsq0 ΔI dsd =

ð3:80Þ Substituting (3.78) in (3.80), ΔIdrdq =

0 Δxd1 þ X dss 1 X dm 1 þ K dp1 V dsq0

K dp3 X dss c X dm 1 þ K dp3 V dsq0 s3 K dp1 X dss c X dm 1 þ K dp1 V dsq0 s1

-

1 X dss X dm 1 þ K dp3 V dsq0 Δxd3 0 0

ΔVsdq þ

1 1 1 X dm X dm 1 þ K dp1 V dsq0

Substituting (3.70) and (3.81) in (3.69)

-

1 X dm 0

ð3:81Þ ΔVsdq

140

3

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

X dss V drq0 þ sdr0 σI drd0 V dsq0 V dsq0 K dp5 dΔV ddc =Δxd1 ΔV ddc Δx dt C d V ddc0 C d V ddc0 d5 C d V ddc0 X dm 1 þ K dp1 V dsq0 -

X dss V drd0 - sdr0 σI drq0 Δxd3 C d V ddc0 X dm 1 þ K dp3 V dsq0

I dcq0 X dm sdr0 I drq0 I X dm sdr0 I drd0 ΔVsdq - dcd0 C d V ddc0 X dss C d V ddc0 C d V ddc0 X dss C d V ddc0 V drd0 - sdr0 σI drq0 V drq0 þ sdr0 σI drd0 V drq0 þ sdr0 σI drd0 þ ΔVsdq C d V ddc0 X dm C d V ddc0 X dm C d V ddc0 X dm 1 þ K dp1 V dsq0 þ

þ

X dss K dp1 V drq0 þ sdr0 σI drd0 I dsd0 C d V ddc0 X dm 1 þ K dp1 V dsq0

þ -

X dss K dp1 V drq0 þ sdr0 σI drd0 I dsq0 ΔVsdq C d V ddc0 X dm 1 þ K dp1 V dsq0

X dss K dp3 V drd0 - sdr0 σI drq0 I dsq0 C d V ddc0 X dm 1 þ K dp3 V dsq0

X dss K dp3 V drd0 - sdr0 σI drq0 I dsd0 ΔVsdq C d V ddc0 X dm 1 þ K dp3 V dsq0

ð3:82Þ By using (3.65) and substituting (3.80) in (3.66), it can have K di1 V dsq0 - K di1 I dsq0 K di1 V dsq0 - K di1 I dsd0 1 dΔxd1 Δx þ ΔVsdq =1 þ K dp1 V dsq0 1 þ K dp1 V dsq0 X dss 1 þ K dp1 V dsq0 dt 1 þ K dp1 V dsq0 d1 K di3 I dsq0 K di3 V dsq0 - K di3 I dsd0 dΔxd3 Δx þ ΔVsdq =1 þ K dp3 V dsq0 1 þ K dp3 V dsq0 dt 1 þ K dp3 V dsq0 d3

ð3:83Þ From (3.75), (3.82), and (3.83), following state-space equation of the DFIG (excluding the PLL) in the timescale of DC voltage is obtained d ΔXd5 = Ad5 ΔXd5 þ Bd5 ΔVsdq dt ΔIdq = Cd5 ΔXd5 þ Dd5 ΔVsdq where ΔXd5 = ½ ΔV ddc Ad5 =

Δxd5

Δxd1

V dsq0 K dp5 C d V ddc0

Δxd3 ]T -

K di5 ΔV ddc

0

0

0

0

0

0

X dss V drd0 - sdr0 σI drq0 C d V ddc0 X dm 1 þ K dp3 V dsq0

ad13

ad14

0

K di1 V dsq0 1 þ K dp1 V dsq0

0 X dss V drq0 þ sdr0 σI drd0 ad13 = C d V ddc0 X dm 1 þ K dp1 V dsq0 ad14 = -

V dsq0 C d V ddc0

0 -

K di3 V dsq0 1 þ K dp3 V dsq0

ð3:84Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

Bd5 =

bd11

bd12

0

0

K di1 V dsq0 - K di1 I dsd0 1 1 þ K dp1 V dsq0 1 þ K dp1 V dsq0 X dss

- K di1 I dsq0 1 þ K dp1 V dsq0

141

K di3 I dsq0 - K di3 I dsd0 1 þ K dp3 V dsq0 1 þ K dp3 V dsq0 V drq0 þ sdr0 σI drd0 V drq0 þ sdr0 σI drd0 bd11 = C d V ddc0 X dm C d V ddc0 X dm 1 þ K dp1 V dsq0 þ

X dss K dp1 V drq0 þ sdr0 σI drd0 I dsd0 C d V ddc0 X dm 1 þ K dp1 V dsq0

þ

X dss K dp3 V drd0 - sdr0 σI drq0 X dm sdr0 I drd0 I - dcd0 I Cd V ddc0 X dss C d V ddc0 C d V ddc0 X dm 1 þ K dp3 V dsq0 dsq0

bd12 = þ

V drd0 - sdr0 σI drq0 X dss K dp1 V drq0 þ sdr0 σI drd0 þ I dsq0 Cd V ddc0 X dm C d V ddc0 X dm 1 þ K dp1 V dsq0 X dss K dp3 V drd0 - sdr0 σI drq0 X dm sdr0 I drq0 I dcq0 þ I Cd V ddc0 X dss C d V ddc0 C d V ddc0 X dm 1 þ K dp3 V dsq0 dsd0 0

0

0

1 1 þ K dp3 V dsq0

K dp5

1

1 1 þ K dp1 V dsq0

0

Cd5 =

K dp3 I dsq0 1 þ K dp3 V dsq0 Dd5 = K di1 V dsq0 - K dp1 I dsd0 1 1 þ K dp1 V dsq0 1 þ K dp1 V dsq0 X dss

K dp3 I dsd0 1 þ K dp3 V dsq0 - K dp1 I dsq0 1 þ K dp1 V dsq0

-

From (2.24) and (3.60), it can have dΔxp = K pi V 0 Δθ - Δθp dt dΔθp = K pp V 0 Δθ - Δθp þ Δxp dt Δθ = -

V y0 V 20

V x0 V 20

ΔV x ΔV y

= Kθ ΔVxy

ð3:85Þ

ð3:86Þ

142

3

ΔVxy =

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

XL 0 ω0 1

-1 X ΔIxy = L UL ð0ÞΔIxy ω0 0

ð3:87Þ

Substituting (3.86) and (3.87) in (3.85), dΔxp X = K pi V 0 Kθ L UL ð0ÞΔIxy - K pi V 0 Δθp dt ω0 dΔθp X = K pp V 0 Kθ L UL ð0ÞΔIxy - K pp V 0 Δθp þ Δxp dt ω0

ð3:88Þ

Substituting (3.59) in (3.88), dΔxp X X = K pi V 0 Kθ L UL ð0ÞK1 ΔIdq þ K pi V 0 Kθ L UL ð0ÞK2 - 1 Δθp dt ω0 ω0 dΔθp X X = K pp V 0 Kθ L UL ð0ÞK1 ΔIdq þ K pp V 0 Kθ L UL ð0ÞK2 - 1 Δθp þ Δxp dt ω0 ω0 ð3:89Þ Substituting Kθ, K1, and UL(0) in (3.89), dΔxp =½0 dt dΔθp =½0 dt

K pi X L ]ΔIdq þ K pi V 0 K pp X L ]ΔIdq þ K pp V 0

X L Q0 - 1 Δθp V 20 X L Q0 - 1 Δθp þ Δxp V 20

ð3:90Þ

From (3.59) and (3.86),

ΔVsdq =

X L Q0 0 V0 Δθp þ X L P0 XL V0

V0 -

- XL ΔIdq 0

ð3:91Þ

Writing (3.90) and (3.91) in matrix form. Following state-space equation about the dynamics of the PLL and the line connecting the DFIG to the grid can be obtained d ΔXpll = Apll ΔXpll þ Bpll ΔIdq dt ΔVsdq = Cpll ΔXpll þ Dpll ΔIdq

where

ð3:92Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

ΔXpll = ½ Δxp K pi V 0

0 Apll =

143

Δθp ]T

X L Q0 -1 V 20

; Bpll =

X L Q0 -1 V 20 X Q 0 V0 - L 0 0 V0 ; Dpll = Cpll = X L P0 XL 0 V0 1 K pp V 0

0

K pi X L

0

K pp X L

- XL 0

Since Δθ = Δθp (see (3.61), from (3.86) and (3.87), it can have ΔVsdq =

XL X ðK þ K4 Kθ Þ I - L UL ð0ÞK2 Kθ ω0 3 ω0

-1

UL ð0ÞK1 ΔIdq = ZL ΔIdq

ð3:93Þ

where K 3 þ K 4 Kθ =

0 cos θ0

I - XωL0 UL ð0ÞK2 Kθ

ZL =

0 sin θ0

-1

1 = X L Q0 1V 20

XL 1 - XVL Q2 0 0

1-

X L I y0 V x0 V 20 X L I y0 V y0 V 20

1þ

X L Q0 V 20

-

X L I x0 V x0 V 20 X L I x0 V y0 1V 20 -

X L P0 V 20

Substituting (3.93) in (3.84) following state-space equation of the grid-connected system displayed in Fig. 3.5 in the timescale of the DC voltage is obtained (the equivalent block diagram model is displayed in Fig. 3.7) d ΔXd5 = A4 ΔXd5 dt where A4 = Ad5 þ Bd5 ZL ðI2 - Dd5 ZL Þ - 1 Cd5

ð3:94Þ

144

3

1þ I2 - Dd5 ZL =

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

K dp3 X L I dsd0 1 þ K dp3 V dsq0

K dp1 X L I dsq0 1 þ K dp1 V dsq0

XL X L P0 K dp3 I dsd0 V 20 1 þ K dp3 V dsq0 1 - X L Q0 V 20 K dp1 I dsq0 X P XL 1- L20 V 0 1 þ K dp1 V dsq0 1 - X L Q0 V 20 -

ðI2 - Dd5 ZL Þ - 1 = K dp1 I dsq0 XL X P 1- L20 X Q 1 þ K V V0 dp1 dsq0 1- L2 0 1 V0 K m1 K dp1 X L I dsq0 1 þ K dp1 V dsq0 K m1 = 1 -

1þ

K dp3 X L I dsd0 1 þ K dp3 V dsq0

X 2L Pds0 P0 K dp1 I dsq0 1 K dp3 X L Qds0 1 þ X L Q0 V 0 1 þ K dp3 V 0 1 þ K dp1 V 0 V 30 1V 20 bd12 1 -

Bd5 ZL =

X L P0 K dp3 I dsd0 XL 2 1þK X Q V V0 dp3 dsq0 1- L2 0 V0

XL 1 - XVL Q2 0 0

X L Q0 V 20

0 - K di1 I dsq0 X Q 1- L2 0 1 þ K dp1 V dsq0 V0 - K di3 I dsq0 X Q 1- L2 0 1 þ K dp3 V dsq0 V0

X L P0 bd12 V 20 0 X L P0 K di1 I dsq0 V 20 1 þ K dp1 V dsq0 -

ð3:95Þ

X L P0 K di3 I dsq0 V 20 1 þ K dp3 V dsq0

1 c11 c12 c13 c14 K m1 c21 c22 c23 c24 K dp3 I dsd0 K dp3 I dsd0 X P XL XL X P c11 = K dp5 L 2 0 ; c12 = L 2 0 V 0 1 þ K dp3 V dsq0 1 - X L Q0 V 0 1 þ K dp3 V dsq0 1 - X L Q0 V 20 V 20 X L P0 K dp3 I dsd0 XL 1 c13 = X Q 1 þ K dp1 V dsq0 V 20 1 þ K dp3 V dsq0 1- L2 0 V0 ðI2 - Dd5 ZL Þ - 1 Cd5 =

c14 =

1 1 þ K dp3 V dsq0

1-

XL X L P0 K dp1 I dsq0 2 1þK X Q V V0 dp1 dsq0 1- L2 0 V0

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

145

K dp3 X L I dsd0 K dp3 X L I dsd0 ; c22 = 1 þ 1 þ K dp3 V dsq0 1 þ K dp3 V dsq0 K dp3 X L I dsd0 1 c23 = 1þ 1 þ K dp3 V dsq0 1 þ K dp1 V dsq0 K dp1 X L I dsq0 1 c24 = 1 þ K dp3 V dsq0 1 þ K dp1 V dsq0

ð3:96Þ

c21 = K dp5 1 þ

From (3.95) and (3.96), a11 A4 = Ad5 þ Bd5 ZL ðI2 - Dd5 ZL Þ - 1 Cd5 =

a12

a13

a14

a32 a42

a33 a43

a34 a44

K di5 ΔV ddc a31 a41

where a11 =

V dsq0 K dp5 X Q 1 1 XL XL X L P0 b c 1- L2 0 bd12 c21 X L Q0 2 K m1 1 - X L Q2 0 d12 11 K C d V ddc0 V V m1 1 2 0 0 V V 0

a12 =

0

V dsq0 X Q 1 1 XL XL X L P0 bd12 c12 1 - L 2 0 bd12 c22 X Q X Q 2 L L 0 0 K m1 1 - 2 K m1 1 - 2 V 0 C d V ddc0 V0 V V 0

a13 =

0

X Q XL XL X L P0 1 1 b c 1- L2 0 bd12 c23 þ ad13 X L Q0 K m1 1 - X L Q2 0 d12 13 K V V 20 m1 1 0 V V2 0

a14 =

0

X Q XL XL X L P0 1 1 b c 1- L2 0 bd12 c24 þ ad14 X L Q0 K m1 1 - X L Q2 0 d12 14 K V m1 1 - V 2 V 20 0 V 0

a31 =

0

- K di1 I dsq0 X Q XL 1 - L 2 0 c11 X L Q0 1 þ K dp1 V dsq0 V0 1V 20 1 XL X L P0 K di1 I dsq0 þ c X L Q0 V 20 1 þ K dp1 V dsq0 21 K m1 12 V0 1 K m1

146

3

a32 =

a33 =

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

- K di1 I dsq0 X Q XL 1 - L 2 0 c12 X L Q0 1 þ K dp1 V dsq0 V0 1V 20 1 XL X L P0 K di1 I dsq0 þ c X L Q0 V 20 1 þ K dp1 V dsq0 22 K m1 1V 20 1 K m1

- K di1 I dsq0 X Q XL 1 - L 2 0 c13 X L Q0 1 þ K dp1 V dsq0 V0 1V 20 K di1 V dsq0 XL X L P0 K di1 I dsq0 1 þ c23 2 X Q 1 þ K dp1 V dsq0 1 þ K dp1 V dsq0 K m1 1 - L 2 0 V0 V0 1 K m1

a34 =

a41 =

a42 =

a43 =

- K di1 I dsq0 X Q XL 1 - L 2 0 c14 X L Q0 1 þ K dp1 V dsq0 V0 1V 20 1 XL X L P0 K di1 I dsq0 þ c X L Q0 V 20 1 þ K dp1 V dsq0 24 K m1 1V 20 1 K m1

- K di3 I dsq0 X Q XL 1 - L 2 0 c11 X L Q0 1 þ K dp3 V dsq0 V0 1V 20 XL X L P0 K di3 I dsq0 1 þ c X L Q0 V 20 1 þ K dp3 V dsq0 21 K m1 12 V0 1 K m1

- K di3 I dsq0 X Q XL 1 - L 2 0 c12 X L Q0 1 þ K dp3 V dsq0 V0 1V 20 XL X L P0 K di3 I dsq0 1 þ c X L Q0 V 20 1 þ K dp3 V dsq0 22 K m1 1V 20 1 K m1

- K di3 I dsq0 X Q XL 1 - L 2 0 c13 X L Q0 1 þ K dp3 V dsq0 V0 1V 20 XL X L P0 K di3 I dsq0 1 þ c X L Q0 V 20 1 þ K dp3 V dsq0 23 K m1 1V 20 1 K m1

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

a44 =

3.3.2

147

- K di3 I dsq0 X Q XL 1 - L 2 0 c14 X L Q0 1 þ K dp3 V dsq0 V0 1V 20 K di3 V dsq0 1 XL X L P0 K di3 I dsq0 þ c X L Q0 V 20 1 þ K dp3 V dsq0 24 1 þ K dp3 V dsq0 K m1 1V 20 1 K m1

Small-Signal Stability Limit in the Timescale of the DC Voltage Control When Dynamics of the PLL Are in the Fast Timescale of Electric Current Control

When dynamics of the PLL are in the fast timescale of electric current control, the small-signal stability of the grid-connected DFIG system of Fig. 3.5 in the timescale of DC voltage control is determined by the following characteristic equation: jsI - A4 j = a4 s4 þ a3 s3 þ a2 s2 þ a1 s1 þ a0 = 0

ð3:97Þ

where a0 = K di1 K di5 V 0 K di3 X 2L P20 - K di1 K di5 V 0 K di3

Rdr X 2dss 1 2 3 X P ω2r0 X 2dm V 20 L 0

- K di5 V 0 K di1 V 0 K di3 V 0 V 20 - X L Q0 - K di3 K di5 V 0 K di1 X L Q0 V 20 - xL Q0 - K di3 K di5 V 0 K di1 Rdr

X 2dss Q0 Q0 V þ 0 X 2 P0 X 2dm V 0 V 0 X dss L

þK di5 V 0 K di1 V 0 K di3 Rdr

V X dss Q0 þ 0 X 2 P0 X 2dm V 0 X dss L

K dp1 K di5 þ K dp5 K di1 V 0 K di3 þ K di1 K di5 1 þ V 0 K dp3 X 2L P20 a1 =

þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 V 0 K di3 Rdr þ K di5 V 0 K di1 1 þ V 0 K dp3 Rdr

X dss Q0 V þ 0 X 2 P0 X 2dm V 0 X dss L

X dss Q0 V þ 0 X 2 P0 X 2dm V 0 X dss L

148

-

3

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

K dp1 K di5 þ K dp5 K di1 V 0 K di3 þ K di1 K di5 1 þ V 0 K dp3

- K dp3 K di5 þ K di3 K dp5 V 0 K di1 Rdr þ K di3 K di5 1 þ V 0 K dp1 Rdr -

Rdr X 2dss 1 2 3 X P ω2r0 X 2dm V 20 L 0

X 2dss Q0 Q0 V þ 0 X 2 P0 X 2dm V 0 V 0 X dss L

X 2dss Q0 Q0 V þ 0 X 2 P0 X 2dm V 0 V 0 X dss L

K dp3 K di5 þ K di3 K dp5 V 0 K di1 þ K di3 K di5 1 þ V 0 K dp1 X L Q0 V 20 - X L Q0

- ð K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 V 0 K di3 þ K di5 V 0 K di1 1 þ V 0 K dp3 Þ V 20 - X L Q0 a2 = K dp1 K dp5 V 0 K di3 þ K dp1 K di5 þ K dp5 K di1 1 þ V 0 K dp3 X 2L P20 - K dp1 K dp5 V 0 K di3 þ K dp1 K di5 þ K dp5 K di1 1 þ V 0 K dp3 - K dp3 K dp5 V 0 K di1 Rdr

Rdr X 2dss 1 2 3 X P ω2r0 X 2dm V 20 L 0

X 2dss Q0 Q0 V þ 0 X 2L P0 2 V V X X dm 0 0 dss

þ K dp3 K di5 þ K di3 K dp5 1 þ V 0 K dp1 Rdr

X 2dss Q0 Q0 V þ 0 X 2 P0 X 2dm V 0 V 0 X dss L

- K dp3 K dp5 V 0 K di1 þ K dp3 K di5 þ K di3 K dp5 1 þ V 0 K dp1 X L Q0 V 20 - X L Q0 þK dp5 1 þ V 0 K dp1 V 0 K di3 Rdr

V X dss Q0 þ 0 X 2 P0 X 2dm V 0 X dss L X dss Q0 V þ 0 X 2 P0 X 2dm V 0 X dss L

þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1

1 þ V 0 K dp3 Rdr

þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1

1 þ V 0 K dp3 V 20 - X L Q0

-

C d V ddc0 X L Q0 V 20 - X L Q0 V0

K di3 V 0 K di1 þ

C d V ddc0 2 2 X P K V K ωr0 V 0 L 0 di1 0 di3

- Cd V ddc0 V 20 - Q0 X L V 0 K di1 V 0 K di3 - K dp5 1 þ V 0 K dp1 V 0 K di3 V 20 - X L Q0 a3 = X 2L P20 þ - Rdr þ Rdr

Rdr X 2dss 1 2 3 X P K dp1 K dp5 1 þ V 0 K dp3 ω2r0 X 2dm V 20 L 0 X 2dss Q0 Q0 V þ 0 X 2 P0 - X L Q0 V 20 - X L Q0 X 2dm V 0 V 0 X dss L

V X dss Q0 þ 0 X 2 P0 - V 0 V 20 - X L Q0 X 2dm V 0 X dss L

K dp3 K dp5 1 þ V 0 K dp1

K dp5 1 þ V 0 K dp1 1 þ V 0 K dp3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

3.3

149

Cd V ddc0 K dp3 V 0 K di1 þ K di3 1 þ V 0 K dp1 X L Q0 V 20 - X L Q0 V0 C V þ d ddc0 X 2L P20 K dp1 V 0 K di3 þ K di1 1 þ V 0 K dp3 ωr0 V 0 - Cd V ddc0 V 20 - Q0 X L 1 þ V 0 K dp1 V 0 K di3 þ V 0 K di1 1 þ V 0 K dp3 -

a4 =

K dp1 1 þ V 0 K dp3 - 1 þ V 0 K dp1

Cd V ddc0 2 2 C V X P - K dp3 1 þ V 0 K dp1 d ddc0 X L Q0 V 20 - X L Q0 ωr0 V 0 L 0 V0 1 þ V 0 K dp3 Cd V ddc0 V 20 - Q0 X L

According to the Routh-Hurwitz criterion, the grid-connected DFIG system is stable in the timescale of DC voltage control if and only if a4 > 0; a3 > 0; a2 > 0; a1 > 0; a0 > 0; b1 = a2 a3 - a1 a4 > 0 c1 = a1 b1 - a0 a23 > 0

ð3:98Þ

Expressing the coefficients ai; i = 0, 1, 2, 3, 4 as the functions of XL, it can have Li Li 2 ai = K Li 0 þ K 1 X L þ K 2 X L ; i = 0, 1, 2, 3, 4

ð3:99Þ

where K L0 0 = - K di5 V 0 K di1 V 0 K di3 V 0

K L0 1 = K di5 V 0 K di1 V 0 K di3 V 0 Q0 - K di3 K di5 V 0 K di1 Q0 2 K L0 2 = K di1 K di5 V 0 K di3 P0 - K di1 K di5 V 0 K di3

- K di3 K di5 V 0 K di1 Rdr

Rdr X 2dss 1 3 P ω2r0 X 2dm V 20 0

X 2dss Q0 Q0 V þ 0 P0 þ K di3 K di5 V 0 K di1 Q20 X 2dm V 0 V 0 X dss

þK di5 V 0 K di1 V 0 K di3 Rdr

V X dss Q0 þ 0 X 2 P0 X 2dm V 0 X dss L

K L1 0 = - K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 V 0 K di3 - K di5 V 0 K di1 1 þ V 0 K dp3 K dp3 K di5 þ K di3 K dp5 V 0 K di1 þ K di3 K di5 1 þ V 0 K dp1 Q0 K L1 1 = þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 V 0 K di3 þ K di5 V 0 K di1 1 þ V 0 K dp3 Q0

150

3

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

K dp1 K di5 þ K dp5 K di1 V 0 K di3 þ K di1 K di5 1 þ V 0 K dp3 P20 þ K L1 2 = þ

K L2 0 =

K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 V 0 K di3 þK di5 V 0 K di1 1 þ V 0 K dp3

Rdr

V X dss Q0 þ 0 P0 X 2dm V 0 X dss Rdr X 2dss 1 3 P ω2r0 X 2dm V 20 0

K dp1 K di5 þ K dp5 K di1 V 0 K di3 þ K di1 K di5 1 þ V 0 K dp3

K dp3 K di5 þ K di3 K dp5 V 0 K di1 þ K di3 K di5 1 þ V 0 K dp1 Rdr

X 2dss Q0 Q0 V þ 0 P0 X 2dm V 0 V 0 X dss

K dp3 K di5 þ K di3 K dp5 V 0 K di1 þ K di3 K di5 1 þ V 0 K dp1 Q20

- K dp5 1 þ V 0 K dp1 V 0 K di3 þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 - C d V ddc0 V 0 K di1 V 0 K di3

1 þ V 0 K dp3 V 0

- K dp3 K dp5 V 0 K di1 þ K dp3 K di5 þ K di3 K dp5 1 þ V 0 K dp1 Q0 V 20 þ K dp5 1 þ V 0 K dp1 V 0 K di3 þ K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 1 þ V 0 K dp3 V 0 Q0 K L2 1 = C V - d ddc0 Q0 V 20 K di3 V 0 K di1 þ C d V ddc0 Q0 V 0 K di1 V 0 K di3 V0 K dp1 K dp5 V 0 K di3 þ K dp1 K di5 þ K dp5 K di1 1 þ V 0 K dp3 P20 - K dp1 K dp5 V 0 K di3 þ K dp1 K di5 þ K dp5 K di1 1 þ V 0 K dp3

Rdr X 2dss 1 3 P ω2r0 X 2dm V 20 0

- K dp3 K dp5 V 0 K di1 þ K dp3 K di5 þ K di3 K dp5 1 þ V 0 K dp1 Rdr K L2 2 =

X 2dss Q0 Q0 V þ 0 P0 X 2dm V 0 V 0 X dss

þ K dp3 K dp5 V 0 K di1 þ K dp3 K di5 þ K di3 K dp5 1 þ V 0 K dp1 Q20 þ

K dp5 1 þ V 0 K dp1 V 0 K di3 þ

K dp5 V 0 K di1 þ K di5 1 þ V 0 K dp1 1 þ V 0 K dp3 C d V ddc0 2 C V þ P K V K þ d ddc0 K di3 V 0 K di1 Q20 ωr0 V 0 0 di1 0 di3 V0

Rdr

V X dss Q0 þ 0 P0 X 2dm V 0 X dss

2 K L3 0 = - C d V ddc0 V 0 1 þ V 0 K dp1 V 0 K di3 þ V 0 K di1 1 þ V 0 K dp3

- V 0 K dp5 1 þ V 0 K dp1 1 þ V 0 K dp3 - Q0 V 20 K dp3 K dp5 1 þ V 0 K dp1 þ V 0 X L Q0 K dp5 1 þ V 0 K dp1 1 þ V 0 K dp3 C V - d ddc0 Q0 V 20 K dp3 V 0 K di1 þ K di3 1 þ V 0 K dp1 L3 V0 K1 = 1 þ V 0 K dp1 V 0 K di3 þ C d V ddc0 Q0 þV 0 K di1 1 þ V 0 K dp3

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

2 K L3 2 = P0 -

þ

Rdr X 2dss 1 3 P K dp1 K dp5 1 þ V 0 K dp3 ω2r0 X 2dm V 20 0 Cd V ddc0 2 P K V K þ K di1 1 þ V 0 K dp3 ωr0 V 0 0 dp1 0 di3

þ - Rdr þRdr þ

151

X 2dss Q0 Q0 V þ 0 P0 þ Q20 K dp3 K dp5 1 þ V 0 K dp1 X 2dm V 0 V 0 X dss

X dss Q0 V þ 0 K dp5 1 þ V 0 K dp1 1 þ V 0 K dp3 2 V X X dm 0 dss

C d V ddc0 2 Q0 K dp3 V 0 K di1 þ K di3 1 þ V 0 K dp1 V0 2 K L4 0 = - 1 þ V 0 K dp1 1 þ V 0 K dp3 C d V ddc0 V 0

K L4 1 = - K dp3 1 þ V 0 K dp1

Cd V ddc0 Q0 V 20 þ 1 þ V 0 K dp1 1 þ V 0 K dp3 C d V ddc0 Q0 V0

K L4 2 = K dp1 1 þ V 0 K dp3

Cd V ddc0 2 C V P þ K dp3 1 þ V 0 K dp1 d ddc0 Q20 ωr0 V 0 0 V0

Within feasible parameter space of converter control systems (including the PLL), it is found that when the small-signal stability of a grid-connected DFIG decreases with the variation of XL or P0, among the Routh-Hurwitz coefficients given in (3.98), i.e., ai; i = 1, 2, 3, 4, b1, and c1, it is c1 that becomes negative firstly to cause the instability. Hence, to assess the stability of the grid-connected DFIG system, only c1 needs to be examined. The grid-connected DFIG system is stable if and only if c1 is positive. By substituting (3.99) in c1 given in (3.98), it can have c1 = f L1 ðX L Þ = K 6 X 6L þ K 5 X 5L þ K 4 X 4L þ K 3 X 3L þ K 2 X 2L þ K 1 X L þ K 0

ð3:100Þ

where L2 L3 L1 L4 L1 L3 L0 L3 K 0 = K L1 0 K0 K0 - K0 K0 K0 - K0 K0 K0

K1 =

L3 L1 L1 L4 L1 L1 L3 L2 L1 L2 L3 K L2 0 K0 K1 - K0 K0 K1 þ K0 K0 K1 þ K0 K0 K1 L4 L1 L1 L4 L1 L3 L3 L0 L3 L0 L3 L3 L0 L3 - K L1 1 K0 K0 - K0 K1 K0 - K0 K0 K1 - K0 K0 K1 - K0 K0 K1 L3 L1 L1 L4 L1 L1 L3 L2 L1 L2 L3 K L2 0 K0 K2 - K0 K0 K2 þ K1 K0 K1 þ K1 K0 K1 L4 L1 L1 L4 L1 L3 L3 L0 L3 L0 L3 - K L1 1 K0 K1 - K0 K1 K1 - K1 K0 K1 - K1 K0 K1 L0 L3 L3 L3 L0 L3 L0 L3 L1 L3 L2 K 2 = - K L3 0 K1 K1 - K0 K0 K2 - K0 K0 K2 þ K0 K0 K2 L2 L3 L1 L4 L1 L1 L4 L1 L1 L4 L1 þ K L1 0 K0 K2 - K2 K0 K0 - K1 K1 K0 - K0 K2 K0 L3 L0 L3 L1 L2 L3 - K0 K0 K2 þ K0 K1 K1

152

3

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

L2 L3 L1 L3 L2 L1 L4 L1 L1 L4 L1 K L1 2 K0 K1 þ K2 K0 K1 - K1 K0 K2 - K0 K1 K2 L3 L2 L1 L4 L1 L3 L3 L0 L3 L0 L3 þ K L1 1 K0 K2 - K2 K0 K1 - K1 K0 K2 - K1 K1 K1 L0 L3 L3 L0 L3 L3 L0 L3 L3 L0 L3 K 3 = - K L3 2 K0 K1 - K0 K2 K1 - K1 K0 K2 - K0 K1 K2 L1 L2 L3 L1 L2 L3 L1 L4 L1 L1 L4 L1 þ K1 K1 K1 þ K1 K0 K2 - K1 K1 K1 - K0 K2 K1 L2 L3 L1 L2 L3 L1 L4 L1 L1 L4 L1 L3 L3 L0 þ K L1 0 K2 K1 þ K0 K1 K2 - K1 K2 K0 - K2 K1 K0 - K2 K0 K1 L3 L2 L1 L2 L3 L1 L4 L1 L1 L2 L3 K L1 2 K0 K2 þ K2 K1 K1 - K1 K1 K2 þ K1 K1 K2 L4 L1 L1 L4 L1 L3 L0 L3 L3 L0 L3 - K L1 2 K1 K1 - K1 K2 K1 - K2 K1 K1 - K1 K2 K1 L0 L3 L3 L0 L3 L3 L0 L3 L1 L2 L3 K 4 = - K L3 2 K0 K2 - K1 K1 K2 - K0 K2 K2 þ K2 K0 K2 L4 L1 L1 L2 L3 L1 L2 L3 L1 L4 L1 - K L1 0 K2 K2 þ K1 K2 K1 þ K0 K2 K2 - K2 K2 K0 L3 L0 L1 L4 L1 - K L3 2 K0 K2 - K2 K0 K2

K5 =

L2 L3 L1 L2 L3 L1 L4 L1 L1 L2 L3 K L1 2 K2 K1 þ K2 K1 K2 - K2 K1 K2 þ K1 K2 K2 L4 L1 L1 L4 L1 L3 L0 L3 L3 L0 L3 L3 L0 L3 - K L1 2 K2 K1 - K1 K2 K2 - K2 K1 K2 - K1 K2 K2 - K2 K2 K1 L2 L3 L1 L4 L1 L3 L0 L3 K 6 = K L1 2 K2 K2 - K2 K2 K2 - K2 K2 K2

c1 = fL1(XL) is a function of XL as being typically shown by Fig. 3.8. It can be seen that when XL < zL1, c1 = fL1(XL) < 0 such that the grid-connected DFIG system becomes unstable. Hence, it can be concluded that if XL < zL1, the grid-connected DFIG system is unstable where zL1 is one of the solutions of following equation which is of the smallest value. c1 = f L1 ðX L Þ = 0

ð3:101Þ

Similarly, the coefficients ai; i = 0, 1, 2, 3, 4 can be expressed as the function of P0 to be ai ai 2 ai 3 ai = K ai 0 þ K 1 P0 þ K 2 P0 þ K 3 P0 ; i = 1, 2, . . . , 4:

Fig. 3.8 c1 = fL1(XL) is a function of XL

c1

c1 = f L1 (X L ) 0

z L1

0

XL

ð3:102Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

153

where K a0 0 = K di5 K di1 K di3 K a0 1 = - K di5 K di1 K di3 X L tan φ - K di5 K di1 K di3 Rdr 2 K a0 2 = - K di5 K di1 K di3 X L þ K di5 K di1 K di3 Rdr

- K di5 K di1 K di3 Rdr K a0 3 = K di5 K di1 K di3

1 2 X þ K di5 K di1 K di3 X L tan φ X 2m L

X dss 2 X tan φ - K di5 K di1 K di3 X 2L tan 2 φ X 2dm L

X dss 2 X tan φ X 2dm L

X2 2 Rdr X 2dss 2 X L þ K di5 K di1 K di3 Rdr dss X tan 2 φ 2 2 ωr0 X dm X 2dm L

K a1 0 = K di5 K di1 1 þ K dp3 þ K dp5 K di1 þ K di5 1 þ K dp1 K di3 1 2 X dss 1 2 K a1 X 1 = - K dp5 K di1 þ K di5 1 þ K dp1 K di3 Rdr 2 X L - K di5 K di1 1 þ K dp3 Rdr 2 X X dm dss L X dm þ K dp3 K di5 þ K di3 K dp5 K di1 X L tan φ - K dp5 K di1 þ K di5 1 þ K dp1 K qi X L tan φ - K di5 K di1 1 þ K dp3 X L tan φ þK di3 K di5 1 þ K dp1 X L tan φ K a1 2

=-

K pp K di5 þ K dp5 K di1 K di3 - K di1 K di5 1 þ K dp3 X 2L X - K dp5 K di1 þ K di5 1 þ K dp1 K di3 Rdr dss tan φX 2L X 2dm X X - K di5 K di1 1 þ K dp3 Rdr dss tan φX 2L þ K di3 K di5 1 þ K dp1 Rdr dss tan φX 2L X 2dm X 2dm X þ K dp3 K di5 þ K di3 K dp5 K di1 Rdr dss tan φX 2L X 2dm - K dp3 K di5 þ K di3 K dp5 K di1 tan 2 φX 2L - K di3 K di5 1 þ K dp1 tan 2 φX 2L

K a1 3 = K pp K di5 þ K dp5 K di1 K di3

Rdr X 2dss 2 R X 2dss 2 X L þ K di1 K di5 1 þ V 0 K dp3 dr X 2 2 ωr0 X dm ω2r0 X 2dm L

þ K dp3 K di5 þ K di3 K dp5 K di1 Rdr þK di3 K di5 1 þ K dp1 Rdr

X 2dss tan 2 φX 2L X 2dm

X 2dss tan 2 φX 2L X 2dm

154

3

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

K a2 0 = K dp5 1 þ K dp1 K di3 þ C d V ddc0 K di1 K di3 þ K dp5 K di1 þ K di5 1 þ K dp1 1 1 þ K dp3 Rdr 2 X 2L K a2 1 = - K dp5 K di1 þ K di5 1 þ K dp1 X dm 1 - K dp5 1 þ K dp1 K di3 Rdr 2 X 2L þ K dp3 K dp5 K di1 tan φX L X dm

1 þ K dp3

- C d V ddc0 K di1 K di3 tan φX L þ K dp3 K di5 þ K di3 K dp5 1 þ K dp1 X L tan φ - K dp5 K di1 þ K di5 1 þ K dp1 1 þ K dp3 X L tan φ - K dp5 1 þ K dp1 K di3 tan φX L þ K di3 K di1 C d V ddc0 tan φX L 2 K a2 2 = - K dp1 K dp5 K di3 þ K dp1 K di5 þ K dp5 K di1 1 þ K dp3 X L C d V ddc0 2 X L K di1 K di3 - K dp3 K dp5 K di1 tan 2 φX 2L ωr0 X - K dp5 1 þ K dp1 K di3 Rdr dss tan φX 2L X 2dm X þ K dp3 K di5 þ K di3 K dp5 1 þ K dp1 Rdr dss tan φX 2L X 2dm X þK dp3 K dp5 K di1 Rdr dss tan φX 2L - K di3 K di1 C d V ddc0 tan 2 φX 2L X 2dm X - K dp5 K di1 þ K di5 1 þ K dp1 1 þ K dp3 Rdr dss tan φX 2L X 2dm

K a2 3 = K dp1 K dp5 K di3

X2 Rdr X 2dss 2 X L þ K dp3 K dp5 K di1 Rdr dss tan 2 φX 2L 2 2 ωr0 X dm X 2dm

þ K dp3 K di5 þ K di3 K dp5 1 þ K dp1 Rdr þ K dp1 K di5 þ K dp5 K di1 1 þ K dp3

X 2dss tan 2 φX 2L X 2dm

Rdr X 2dss 2 X - K dp3 K di5 þ K di3 K dp5 1 þ K dp1 tan 2 φX 2L ω2r0 X 2dm L

K a3 0 = K dp5 1 þ K dp1 1 þ K dp3 þ C d V ddc0 1 þ K dp1 K di3 þ K di1 1 þ K dp3 1 2 K a3 1 = K dp3 K dp5 1 þ K dp1 tan φX L - K dp5 1 þ K dp1 1 þ K dp3 Rdr 2 X L X dm - K dp5 1 þ K dp1 1 þ K dp3 tan φX L þ K dp3 K di1 þ K di3 1 þ K dp1 C d V ddc0 tan φX L - C d V ddc0 2 K a3 2 = - K dp1 K dp5 1 þ K dp3 X L þ K dp3 K dp5 1 þ K dp1 Rdr

1 þ K dp1 K di3 þK di1 1 þ K dp3

tan φX L

X dss tan φX 2L X 2dm

- K dp3 K dp5 1 þ K dp1 tan 2 φX 2L - K dp5 1 þ K dp1 1 þ K dp3 Rdr

X dss tan φX 2L - K dp3 K pi þ K di3 1 þ K dp1 C d V ddc0 tan 2 φX 2L X 2dm

C d V ddc0 K dp1 K di3 þ K di1 1 þ K dp3 X 2L ωr0 X2 Rdr X 2dss 2 K a3 X þ K dp3 K dp5 1 þ K dp1 Rdr dss tan 2 φX 2L 3 = K dp1 K dp5 1 þ K dp3 ω2r0 X 2dm L X 2dm -

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

155

K a4 0 = 1 þ K dp1 1 þ K dp3 C d V ddc0 K a4 1 = K dp3 1 þ K dp1 C d V ddc0 tan φX L - 1 þ K dp1 1 þ K dp3 C d V ddc0 tan φX L Cd V ddc0 2 2 2 K a4 XL 2 = - K dp3 1 þ K dp1 C d V ddc0 tan φX L - K dp1 1 þ K dp3 ωr0 K a4 3 =0 By substituting (3.102) in c1 given in (3.98), it can have p1 8 p1 7 p1 6 p1 5 9 c1 = f p1 ðP0 Þ = K p1 9 P0 þ K 8 P0 þ K 7 P0 þ K 6 P0 þ K 5 P0 p1 3 p1 2 p1 p1 4 þK p1 4 P0 þ K 3 P0 þ K 2 P0 þ K 1 P0 þ K 0

ð3:103Þ

where a2 a3 a1 a0 a3 a3 K p1 9 = K3 K3 K3 - K3 K3 K3 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 K p1 8 = K3 K3 K2 þ K2 K3 K3 þ K3 K2 K3 - K3 K2 K3 a3 a0 a3 a3 a0 a0 a3 a3 - K a3 3 K2 K3 - K2 K3 K3 - K2 K3 K3 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 a0 a3 a3 K p1 7 = K3 K3 K1 þ K2 K3 K2 þ K3 K2 K2 - K3 K2 K2 - K1 K3 K3 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 a3 a3 a0 þK a2 1 K3 K3 þ K2 K2 K3 þ K3 K1 K3 - K3 K1 K3 - K2 K3 K2 a4 a1 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a1 2 K2 K3 - K3 K1 K3 - K2 K2 K3 - K1 K3 K3 - K3 K2 K2 a1 a2 a3 a2 a3 a1 a2 a3 a1 a1 a4 a1 a2 a3 a1 K p1 6 = K0 K3 K3 þ K2 K3 K1 þ K3 K2 K1 - K3 K2 K1 þ K1 K3 K2 a3 a1 a2 a3 a1 a1 a4 a1 a0 a3 a3 a1 a4 a1 þK a2 2 K2 K2 þ K3 K1 K2 - K3 K1 K2 - K0 K3 K3 - K2 K2 K2 a3 a1 a2 a3 a1 a2 a3 a1 a3 a3 a0 a2 a3 a1 þK a2 0 K3 K3 þ K1 K2 K3 þ K2 K1 K3 - K2 K2 K2 þ K3 K0 K3 a4 a1 a1 a4 a1 a1 a4 a1 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a1 3 K0 K3 - K2 K1 K3 - K1 K2 K3 - K0 K3 K3 - K2 K1 K3 - K1 K2 K3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a3 0 K3 K3 - K3 K1 K2 - K1 K3 K2 - K3 K2 K1 - K2 K3 K1 a2 a3 a1 a2 a3 a1 a1 a4 a1 a2 a3 a1 a2 a3 a1 K p1 5 = K2 K3 K0 þ K3 K2 K0 - K3 K2 K0 þ K1 K3 K1 þ K2 K2 K1 a3 a1 a1 a4 a1 a1 a4 a1 a0 a3 a3 a2 a3 a1 þK a2 3 K1 K1 - K3 K1 K1 - K2 K2 K1 - K0 K2 K3 þ K0 K3 K2 a3 a1 a2 a3 a1 a2 a3 a1 a0 a3 a3 a1 a4 a1 þK a2 1 K2 K2 þ K2 K1 K2 þ K3 K0 K2 - K0 K3 K2 - K3 K0 K2 a4 a1 a1 a4 a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 - K a1 2 K1 K2 - K1 K2 K2 þ K0 K2 K3 þ K1 K1 K3 þ K2 K0 K3 - K2 K0 K3 a4 a1 a1 a4 a1 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a1 1 K1 K3 - K0 K2 K3 - K0 K2 K3 - K1 K1 K3 - K0 K2 K3 - K0 K3 K2 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a3 2 K1 K2 - K1 K2 K2 - K0 K3 K2 - K3 K1 K1 - K2 K2 K1 - K1 K3 K1

156

3

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

a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 a2 a3 a1 K p1 4 = K1 K3 K0 þ K2 K2 K0 þ K3 K1 K0 - K3 K1 K0 þ K0 K3 K1 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 a1 a4 a1 þK a2 1 K2 K1 þ K2 K1 K1 þ K3 K0 K1 - K3 K0 K1 - K2 K1 K1 a4 a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 - K a1 1 K2 K1 þ K0 K2 K2 þ K1 K1 K2 þ K2 K0 K2 - K2 K0 K2 a4 a1 a1 a4 a1 a2 a3 a1 a2 a3 a1 a1 a4 a1 - K a1 1 K1 K2 - K0 K2 K2 þ K0 K1 K3 þ K1 K0 K3 - K1 K0 K3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 - K a3 0 K1 K3 - K0 K1 K3 - K0 K2 K2 - K1 K1 K2 - K0 K2 K2 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a0 a3 a3 - K a3 0 K3 K1 - K2 K1 K1 - K1 K2 K1 - K0 K3 K1 - K0 K3 K1 a3 a3 a0 a3 a3 a1 a4 a1 a1 a4 a1 - K a0 0 K2 K2 - K0 K1 K3 - K2 K2 K0 - K0 K1 K3 a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 K p1 3 = K0 K0 K3 þ K0 K1 K2 þ K0 K2 K1 þ K0 K3 K0 - K0 K2 K1 a1 a4 a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 - K a1 0 K1 K2 þ K1 K0 K2 þ K1 K1 K1 þ K1 K2 K0 - K1 K2 K0 a1 a4 a1 a1 a4 a1 a2 a3 a1 a2 a3 a1 a1 a4 - K a1 1 K1 K1 - K1 K0 K2 þ K2 K0 K1 þ K2 K1 K0 - K2 K1 K0 a1 a4 a2 a3 a1 a1 a4 a1 a1 a1 a4 a3 a3 a0 - K a1 2 K0 K1 þ K0 K0 K3 - K0 K0 K3 - K0 K3 K0 - K0 K0 K3 a3 a0 a3 a3 a0 a3 a3 a0 a3 a3 a0 a0 a3 a3 - K a3 0 K1 K2 - K0 K1 K2 - K0 K2 K1 - K1 K1 K1 - K0 K0 K3 a3 a3 a0 a3 a3 a0 a3 a3 a3 a3 a0 - K a0 0 K2 K1 - K0 K1 K2 - K0 K0 K3 - K0 K2 K1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 K a1 0 K0 K2 þ K0 K1 K1 þ K0 K2 K0 - K0 K2 K0 a2 a3 a1 a2 a3 a1 a1 a4 a1 a1 a4 þ K a1 1 K0 K1 þ K1 K1 K0 - K1 K1 K0 - K1 K0 K1 a0 a3 a3 a2 a3 a1 a1 a4 a1 a1 a4 K p1 = - K 0 K 1 K 1 þ K 0 K 0 K 2 - K 0 K 0 K 2 - K a1 2 0 K1 K1 a1 a1 a4 a0 a3 a3 a3 a3 a0 a3 a3 a0 - K0 K0 K2 - K0 K0 K2 - K0 K0 K2 - K0 K1 K1 a3 a0 a0 a3 a3 - K a3 0 K1 K1 - K0 K0 K2

a1 a2 a3 a1 a2 a3 a1 a2 a3 a1 a1 a4 a3 a3 a0 K p1 1 = K0 K0 K1 þ K0 K1 K0 þ K1 K0 K0 - K1 K0 K0 - K0 K0 K1 a3 a3 a0 a3 a3 a1 a1 a4 a1 a1 a4 - K a0 0 K0 K1 - K0 K0 K1 - K0 K1 K0 - K0 K0 K1 a1 a2 a3 a1 a1 a4 a3 a3 a0 K p1 0 = K0 K0 K0 - K0 K0 K0 - K0 K0 K0

c1 = fP1(P0) is a function of P0. Hence, let zP1 be one of the solutions of following equation which is of the smallest value c1 = f P1 ðP0 Þ = 0

ð3:104Þ

It can be concluded that when P0 < zP1, c1 = fP1(P0) < 0 such that the gridconnected DFIG system becomes unstable. From the discussions above, following two conclusions about the small-signal stability limit of the grid-connected DFIG system in the timescale of DC voltage control can be made. Conclusion 3.1 Let zL1 be a solution of the sixth-order equation given by (3.100) and (3.101), which can be re-written as

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

157

5

f L1 ðzÞ = d 0 þ

d i zi þ z6 = 0

ð3:105Þ

i=1

where coefficients di; i = 0, 1, 2, ⋯5 are related only with the parameters of PI gains of the outer control loops of the GSC and RSC in addition to the values of Cd (capacitance of the capacitor), Vddc0, and P0. The grid-connected DFIG system displayed in Fig. 3.5 is stable in the timescale of DC voltage control if and only if XL < zL1 . Conclusion 3.2 Let zP1 be a solution of the ninth-order equation given by (3.103) and (3.104), which can be re-written as 8

f P1 ðzÞ = e0 þ

ei zi þ z9 = 0

ð3:106Þ

i=1

where coefficients ei; i = 0, 1, 2, ⋯8 are related only with the parameters of PI gains of the outer control loops of the GSC and RSC in addition to the values of Cd, Vddc0, and XL. The grid-connected DFIG system displayed in Fig. 3.5 is stable in the timescale of DC voltage control if and only if P0 < zP1 . A few remarks can be made about Conclusions 3.1 and 3.2 as follows. 1. For the grid-connected DFIG system displayed in Fig. 3.5, when the power base is the nominal power of DFIG, the short circuit ratio (SCR) is

Hence, Conclusion 3.1 analytically explains why under the condition of a weaker grid connection (smaller SCR, i.e., bigger XL), the instability risk of the grid-connected DFIG system in the timescale of DC voltage control increases. 2. Conclusion 3.2 clearly indicates that the heavier loading condition of the DFIG implies a higher instability risk. In fact, zP1 defines the maximum loading of the DFIG, P0max, to ensure the small-signal stability of the grid-connected DFIG system, i.e., P0max < zP1. 3. Coefficients of algebraic equations, fL1(z) = 0 and fP1(z) = 0, given in (3.105) and (3.106) are related with the PI gains of the outer control loops of the GSC and RSC. Hence, Conclusions 3.1 and 3.2 present the analytical implication of “improper PI gain setting” of the outer control loops of the GSC and RSC that the solutions of the equations, zP1 and zL1, are of smaller values to indicate the higher instability risk. By properly setting those PI gains to increase zP1 and zL1, small-signal stability of the grid-connected DFIG can be enhanced. This is to be further discussed and demonstrated in the second part of the paper. 4. Conclusions 3.1 and 3.2 reveal the analytical connections between three key factors, which are the condition of grid connection exhibited as XL, the loading (P0), and the parameters setting of the outer control loops of the GSC and RSC, to

158

3

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

affect the small-signal stability of the single grid-connected DFIG system. The connections are depicted analytically by (3.105) and (3.106). If variation of any factor leads to the violation of the small-signal stability limit, growing oscillations in the single grid-connected DFIG system occurs.

3.3.3

Stability Limit in the Slow DC Voltage Timescale When the Dynamics of the PLL Are Also in the DC Voltage Timescale

When dynamics of the PLL are in the timescale of DC voltage control, examination of the stability of the grid-connected DFIG system of Fig. 3.5 should consider the dynamics of the PLL. In this case, Δθ ≠ Δθp. Subsequently, Eq. (6.1) becomes Δωr = 0; ΔV dsd = 0 ΔI drd = ΔI drdref ; ΔI drq = ΔI drqref

ΔI dcd = ΔI dcdref = 0

ð3:108Þ

ΔI dcq = ΔI dcqref

With (3.108), the block diagram model of the grid-connected DFIG system displayed in Fig. 3.6 is simplified. The simplified block diagram model is shown by Fig. 3.9. Combining (4.89) and (4.91), the state-space model of the grid-connected DFIG system in the DC voltage timescale equivalent to the block diagram model displayed in Fig. 3.9 is obtained to be d ΔXd6 = A6 ΔXd6 dt

ð3:109Þ

where ΔXd6 = ΔXd5 T A6 =

a6,11

a6,12

a6,21

a6,22

ΔXpll T

T

;

a6,11 = Ad5 þ Bd5 I3 - Dpll Dd5 a6,12 = Bd5 I3 - Dpll Dd5

-1

-1

Cpll

a6,21 = Bpll Cd5 þ Bpll I3 - Dpll Dd5 a6,22 = Bpll Dd5 I3 - Dpll Dd5

Dpll Cd5

-1

-1

Dpll Cd5

Cpll þ Apll

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . . cs 4

ΔI sdq

cs 2 c s1 cs 3

ΔPds ΔQds

-G1 ( s )

ΔI dsqref ΔI dsdref

-G3 ( s )

159

-

X dss X dm

-

X dss X dm

Power control outer loop of RSC

ΔI drq ΔI drd -

-

X dm X dss

-

X dm X dss

ΔI dsq ΔI dsd -

ΔVdsq

ΔVdsq

X dm

X dss Z3 ( s)

ΔI sdq

+

cdr 2

Z4 ΔVrdq

cdr1 ΔPdc1 -

cdc1

ΔPdr

Gddc ( s )

ΔVddc

G5 ( s )

Voltage control outer loop of GSC cdc 2

ΔVdq

ΔI dcq Δ I dcd = 0

ΔI dcdq

Δ I dq K1

K3

ΔVbxy

K4

ΔVxy

Kθ

Δθ

G pll ( s)

Δθ p

xL U L (0) w0

Δ I xy

K2

Δ I xy

Fig. 3.9 Simplified reduced-order block diagram model of the grid-connected DFIG in the timescale of DC voltage control when the dynamics of the PLL are in the timescale of DC voltage control

Hence, when dynamics of the PLL are in the timescale of DC voltage control, the small-signal stability of the grid-connected DFIG system of Fig. 3.5 in the timescale of DC voltage control is determined by the following characteristic equation: jsI - A6 j = a6 s6 þ a5 s5 þ a4 s4 þ a3 s3 þ a2 s2 þ a1 s þ a0 = 0 where

ð3:110Þ

160

3

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

a0 = K pi K di5 K di1 K di3 a1 = K pi K di5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þ K pi K dp5 K di1 K di3 þK di1 K di3 K di5 K pp - K di5 K di3 K pi X 2L P20 a2 = K pi K di5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 C d V ddc0 K pi þK pi K dp5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þ K di1 K di3 K dp5 K pp X þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5 K pp þ K di1 K di3 K di5 L X dss 1 þ 1 - xL sdr10 K di1 K di3 K di5 - X 2L P20 K di5 K di3 K pp X dss - X 2L P20 K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pi a3 = K pi K dp5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 C d V ddc0 K pp þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 C d V ddc0 K pi þ K dp1 K qi þ K di1 K dp3 þ K di1 þ K di3 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 K di5 K pp þ K di1 K di3 K dp5 þ 1 - X L sdr10

1 K di1 K di3 K dp5 X dss

þ K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K di5 þ 1 - X L sdr10

XL X dss

1 X dss

XL X dss

K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5

X 2L P20 C V K K - X 2L P20 K dp5 þ 1 K dp3 þ 1 K pi ωr0 d ddc0 pi di3 - X 2L P20 K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pp -

a4 = K dp1 þ 1 K dp3 þ 1 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pi þ 1 - X L sdr10

1 X dss

K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K dp5

þ 1 - X L sdr10

1 X dss

K dp1 þ 1 K dp3 þ 1 K di5 - X 2L P20 K dp5 þ 1 K dp3 þ 1 K pp

þ K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K dp5 þ K dp1 þ 1 K dp3 þ 1 K di5

XL X þ K di1 K di3 Cd V ddc0 L X dss X dss

P2 XL - X 2L 0 K di5 þ K di1 K di3 C d V ddc0 X dss ωr0

P20 C V K K þ K pi K dp3 þ 1 ωr0 d ddc0 pp di3 þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 Cd V ddc0 K pp - X 2L

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

a5 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pp þ K dp1 þ 1 K dp3 þ 1 K dp5

161

XL X dss

þ K dp1 K di3 þ K dp1 K qp þ K dp1 þ K di3 C d V ddc0 þ 1 - X L sr0

1 X dss

K dp1 þ 1 K dp3 þ 1 K dp5

þ K dp1 þ 1 K di3 þ K dp3 þ 1 K dp1 Cd V ddc0

XL X dss

X 2L P20 X 2 P2 C d V ddc0 K pp K dp3 þ 1 - L 0 K dp5 ωr0 ωr0 2 2 X P a6 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 - L2 0 C d V ddc0 ωr0 XL þ K dp1 þ 1 K dp3 þ 1 C d V ddc0 X dss -

According to the Routh-Hurwitz criterion, the grid-connected DFIG system is stable in the timescale of DC voltage control if and only if a6 > 0, a5 > 0, a4 > 0, a3 > 0, a2 > 0, a1 > 0, a0 > 0, b1 = a4 a5 - a3 a6 > 0; c1 = a3 b1 - ða2 a5 - a1 a6 Þa5 > 0; d1 = ða2 a5 - a1 a6 Þc1 - a1 b1 - a0 a25 b1 > 0;

ð3:111Þ

e1 = a1 b1 - a0 a25 d 1 - a0 a5 c21 > 0 Expressing the coefficients ai; i = 0, 1, 2, 3, 4, 5, 6 in (3.111) as the functions of XL, it can have Li Li 2 ai = K Li 0 þ K 1 X L þ K 2 X L ; i = 1, 2, 3, 4, 5, 6

ð3:112Þ

where K L0 0 = K pi K di5 K di1 K di3 K L1 0 = K pi K di5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þ K pi K dp5 K di1 K di3 þK di1 K di3 K di5 K pp 2 K L1 2 = - K di5 K di3 K pi P0

K L2 0 = K pi K di5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 C d V ddc0 K pi þK pi K dp5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þK di1 K di3 K dp5 K pp þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5 K pp 1 1 þK di1 K di3 K di5 K L2 - sdr10 K K K 1 = K di1 K di3 K di5 X X dss di1 di3 di5 dss 2 2 K L2 2 = - P0 K di5 K di3 K pp - P0 K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pi

162

3

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

K L3 0 = K pi K dp5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 C d V ddc0 K pp þ K dp1 K qi þ K di1 K dp3 þ K di1 þ K di3 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 K di5 K pp þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5 þ K di1 K di3 K dp5 þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 Cd V ddc0 K pi 1 1 K L3 þ K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K di5 1 = K di1 K di3 K dp5 X X dss dss 1 1 - sdr10 K K þ K di1 K dp3 þ K di1 þ K di3 K di5 - sdr10 K K K X dss dp1 di3 X dss di1 di3 dp5 P20 2 K L3 2 = - ω C d V ddc0 K pi K di3 - P0 K dp5 þ 1 K dp3 þ 1 K pi r0 - P20 K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pp

K dp1 þ 1 K dp3 þ 1 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pi K L4 0 =

þ K di1 K di3 C d V ddc0 þ K dp1 þ 1 K dp3 þ 1 K di5 þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K dp5 þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 C d V ddc0 K pp 1 - sdr10 K K þ K di1 K dp3 þ K di1 þ K di3 K dp5 X dss dp1 di3 þ K di1 K di3 C d V ddc0

K L4 1 =

þ

1 1 - sdr10 K þ 1 K dp3 þ 1 K di5 X dss X dss dp1

K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K dp5

þ K dp1 þ 1 K dp3 þ 1 K di5 K L4 2 =

-

1 X dss

1 X dss

P20 P2 K di5 - 0 C d V ddc0 K pp K di3 þ K pi K dp3 þ 1 ωr0 ωr0

- P20 K dp5 þ 1 K dp3 þ 1 K pp K L5 0 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pp þ K dp1 þ 1 K dp3 þ 1 K dp5ddc0 þ K dp1 K di3 þ K dp1 K qp þ K dp1 þ K di3 C d V ddc0 x 1 L5 K 1 = K dp1 þ 1 K dp3 þ 1 K dp5 L - sr0 K þ 1 K dp3 þ 1 K dp5 X dss X dss dp1 1 þ K dp1 þ 1 K di3 þ K dp3 þ 1 K dp1 Cd V ddc0 X dss P20 P20 K L5 2 = - ω C d V ddc0 K pp K dp3 þ 1 - ω K dp5 r0 r0

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

163

K L6 0 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 K L6 1 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 K L6 2 = -

1 X dss

P20 C d V ddc0 ω2r0

Within feasible parameter space of converter control systems (including the PLL), it is found that when the small-signal stability of a grid-connected DFIG decreases with the variation of XL or P0, among the Routh-Hurwitz coefficients given in (3.111), i.e., ai; i = 1, 2, 3, 4, 5, 6, b1, c1, d1, and e1, it is e1 that becomes negative firstly to cause the instability. Hence, to assess the stability of the gridconnected DFIG system, only e1 needs to be examined. The grid-connected DFIG system is stable if and only if e1 is positive. By substituting (3.112) in e1 given in (3.111), it can have 16

e1 = f L2 ðX L Þ = k=0

K k X kL

ð3:113Þ

where b1 L0 L5 L5 d1 L0 L5 c1 c1 K 0 = K L1 0 K0 - K0 K0 K0 K0 - K0 K0 K0 K0 b1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 1 = K L1 0 K0 - K0 K0 K0 K1 þ K0 K1 K0 þ K1 K0 K0 - K0 K1 K0 K0 L5 L5 d1 L0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 1 K0 K0 K0 - K0 K0 K1 K0 - K0 K0 K0 K1 - K0 K0 K1 K0 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K0 K0 b1 L0 L5 a5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 2 = K L1 0 K0 - K0 K0 K0 K2 þ K0 K1 K1 þ K1 K0 K1 - K0 K1 K0 K1 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 a1 b1 d1 L1 b1 d1 - K L0 1 K0 K0 K1 - K0 K0 K1 K1 þ K0 K2 K0 þ K1 K1 K0 þ K2 K0 K0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 0 K0 K2 K0 - K0 K2 K0 K0 - K1 K0 K1 K0 - K1 K1 K0 K0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 2 K0 K0 K0 - K0 K0 K0 K2 - K0 K0 K2 K0 - K0 K0 K1 K1 L5 L5 d1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K1 K1 K0 - K0 K1 þ K0 K1 K1 K0 - K0 K1 þ K0 K1 K0 K1 b1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 3 = K L1 0 K0 - K0 K0 K0 K3 þ K0 K1 K2 þ K1 K0 K2 - K0 K1 K0 K2 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K0 K0 K2 - K0 K0 K1 K2 þ K0 K2 K1 þ K1 K1 K1 þ K2 K0 K1 - K0 K0 K2 K1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K0 K1 K1 K1 - K0 K2 K0 K1 - K1 K0 K0 K1 - K1 K1 K0 K1 - K2 K0 K0 K1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 0 K3 K0 þ K1 K2 K0 þ K2 K1 K0 - K0 K1 K2 K0 - K0 K2 K1 K0 - K1 K0 K2 K0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 1 K1 K1 K0 - K2 K0 K1 K0 - K2 K1 K0 K0 - K0 K0 K0 K3 - K0 K0 K1 K2 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K0 K2 K1 - K0 K0 K3 K0 - K0 K1 þ K0 K1 K2 K0 - K0 K1 þ K0 K1 K0 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K1 K1 - K0 K2 þ K0 K2 K0 K1 - K0 K2 þ K0 K2 K1 K0 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K0 K0

164

3

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

b1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 4 = K L1 0 K0 - K0 K0 K0 K4 þ K0 K1 K3 þ K1 K0 K3 - K0 K1 K0 K3 L5 a5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K0 K0 K3 - K0 K0 K1 K3 þ K0 K2 K2 þ K1 K1 K2 þ K2 K0 K2 - K0 K0 K2 K2 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 0 K1 K1 K2 - K0 K2 K0 K2 - K1 K0 K1 K2 - K1 K1 K0 K2 - K2 K0 K0 K2 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 0 K3 K1 þ K1 K2 K1 þ K2 K1 K1 - K0 K1 K2 K1 - K0 K2 K1 K1 - K1 K0 K2 K1 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 - K L0 1 K1 K1 K1 - K2 K0 K1 K1 - K2 K1 K0 K1 þ K0 K4 K0 þ K1 K3 K0 þ K2 K2 K0 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K0 K2 K2 K0 - K1 K2 K1 K0 - K1 K1 K2 K0 - K2 K1 K1 K0 - K2 K0 K2 K0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 0 K0 K0 K4 - K0 K0 K1 K3 - K0 K0 K2 K2 - K0 K0 K3 K1 - K0 K0 K4 K0 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K1 K2 - K0 K1 þ K0 K1 K2 K1 - K0 K2 þ K0 K2 K0 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K1 K1 - K1 K2 þ K1 K2 K0 K1 - K1 K2 þ K1 K2 K1 K0 L5 c1 c1 L0 L5 L5 d1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 2 K2 K0 K0 - K2 K2 K0 K0 - K0 K1 þ K0 K1 K0 K3 - K0 K2 þ K0 K2 K2 K0 b1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 5 = K L1 0 K0 - K0 K0 K0 K5 þ K0 K1 K4 þ K1 K0 K4 - K0 K1 K0 K4 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K0 K0 K4 - K0 K0 K1 K4 þ K0 K2 K3 þ K1 K1 K3 þ K2 K0 K3 - K0 K0 K2 K3 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K0 K1 K1 K3 - K0 K2 K0 K3 - K1 K0 K1 K3 - K1 K1 K0 K3 - K2 K0 K0 K3 L5 a5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 1 K0 K2 K2 - K1 K1 K1 K2 - K2 K0 K1 K2 - K2 K1 K0 K2 þ K0 K4 K1 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 0 K2 K2 K1 - K1 K2 K1 K1 - K1 K1 K2 K1 - K2 K1 K1 K1 - K2 K0 K2 K1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K4 K0 þ K2 K3 K0 - K1 K2 K2 K0 - K2 K2 K1 K0 - K2 K1 K2 K0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K0 K2 K3 - K0 K0 K3 K2 - K0 K0 K4 K1 - K0 K1 þ K0 K1 K0 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K1 K3 - K0 K1 þ K0 K1 K2 K2 - K0 K1 þ K0 K1 K3 K1 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K0 K3 - K0 K2 þ K0 K2 K1 K2 - K0 K2 þ K0 K2 K3 K0 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K0 K2 - K1 K2 þ K1 K2 K2 K0 - K1 K2 þ K1 K2 K1 K1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 þK L1 0 K3 K2 þ K1 K2 K2 þ K2 K1 K2 - K0 K1 K2 K2 - K0 K2 K1 K2 þ K1 K3 K1 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K0 K0 K5 - K0 K0 K1 K4 - K0 K0 K5 K0 - K0 K1 þ K0 K1 K4 K0 L0 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L1 b1 d1 L0 L5 L5 d1 - K L5 0 K2 þ K0 K2 K2 K1 - K2 K2 K1 K0 - K2 K2 K0 K1 þ K2 K2 K1 - K2 K2 K0 K1 b1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 K 6 = K L1 0 K0 - K0 K0 K0 K6 þ K0 K1 K5 þ K1 K0 K5 - K0 K1 K0 K5 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K0 K0 K5 - K0 K0 K1 K5 þ K0 K2 K4 þ K1 K1 K4 þ K2 K0 K4 - K0 K0 K2 K4 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K1 K0 K4 - K2 K0 K0 K4 þ K0 K3 K3 þ K1 K2 K3 þ K2 K1 K3 - K0 K1 K2 K3 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 1 K0 K2 K3 - K1 K1 K1 K3 - K2 K0 K1 K3 - K2 K1 K0 K3 þ K0 K4 K2 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K0 K2 K2 K2 - K1 K2 K1 K2 - K1 K1 K2 K2 - K2 K1 K1 K2 - K2 K0 K2 K2 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 þ K L1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K1 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 0 K0 K0 K6 - K0 K0 K1 K5 - K0 K0 K2 K4 - K0 K0 K3 K3 - K0 K0 K4 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L5 c1 c1 K K - K L5 K þ K K K K K þ K K K K K þ K L0 0 1 0 1 0 5 0 1 0 1 1 4 0 1 0 K1 K2 K3 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K3 K2 - K0 K1 þ K0 K1 K4 K1 - K0 K1 þ K0 K1 K5 K0 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K1 K3 - K1 K2 þ K1 K2 K3 K0 - K0 K2 þ K0 K2 K2 K2 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K 0L5 K L0 2 þ K0 K2 K4 K0 - K1 K2 þ K1 K2 K0 K3 - K1 K2 þ K1 K2 K1 K2 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L1 b1 d1 L1 b1 d1 - K L0 2 K2 K0 K2 - K2 K2 K1 K1 - K2 K2 K2 K0 þ K1 K3 K2 þ K2 K2 K2 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 0 K1 K1 K4 - K0 K2 K0 K4 - K1 K0 K1 K4 - K0 K2 K1 K3 - K2 K2 K0 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K0 K4 - K0 K2 þ K0 K2 K3 K1 - K1 K2 þ K1 K2 K2 K1 L5 c1 c1 L1 b1 L0 L5 L5 d1 L0 L5 c1 c1 - K L0 0 K0 K5 K1 þ K2 K4 - K2 K2 K2 K0 - K0 K0 K6 K0

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

165

b1 L0 L5 L5 d1 L1 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 K 7 = K L1 0 K0 - K0 K0 K0 K7 þ K0 K1 þ K1 K0 - K0 K1 K0 - K1 K0 K0 - K0 K0 K1 K6 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 0 K2 K5 þ K1 K1 K5 þ K2 K0 K5 - K0 K0 K2 K5 - K0 K1 K1 K5 - K0 K2 K0 K5 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K1 K0 K5 - K2 K0 K0 K5 þ K0 K3 K4 þ K1 K2 K4 þ K2 K1 K4 - K0 K1 K2 K4 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 - K L0 1 K1 K1 K4 - K2 K0 K1 K4 - K2 K1 K0 K4 þ K0 K4 K3 þ K1 K3 K3 þ K2 K2 K3 L5 L5 d1 L1 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 - K L0 1 K1 K2 K3 þ K1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K2 b1 L0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 þ K L1 2 K4 - K2 K2 K2 K1 - K0 K0 K1 K6 - K0 K0 K2 K5 - K0 K0 K3 K4 - K0 K0 K4 K3 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K0 K6 - K0 K1 þ K0 K1 K1 K5 - K0 K1 þ K0 K1 K2 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K4 K2 - K0 K1 þ K0 K1 K5 K1 - K0 K1 þ K0 K1 K6 K0 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K1 K4 - K0 K2 þ K0 K2 K2 K3 - K0 K2 þ K0 K2 K3 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K5 K0 - K1 K2 þ K1 K2 K0 K4 - K1 K2 þ K1 K2 K1 K3 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K3 K1 - K1 K2 þ K1 K2 K4 K0 - K2 K2 K0 K3 - K2 K2 K1 K2 L5 c1 c1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 2 K2 K3 K0 - K2 K2 K0 K3 - K1 K0 K1 K5 - K0 K2 K1 K4 - K1 K0 K2 K4 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 0 K2 K2 K3 - K1 K2 K1 K3 - K2 K1 K1 K3 - K2 K0 K2 K3 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 0 K0 K5 K2 - K0 K0 K6 K1 - K0 K1 þ K0 K1 K3 K3 - K2 K2 K2 K1 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K0 K5 - K0 K2 þ K0 K2 K4 K1 - K1 K2 þ K1 K2 K2 K2

b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L0 L5 L5 d1 K 8 = K L1 0 K1 þ K1 K0 - K0 K1 K0 - K1 K0 K0 - K0 K0 K1 K7 - K2 K0 K0 K6 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 0 K2 K6 þ K1 K1 K6 þ K2 K0 K6 - K0 K0 K2 K6 - K0 K1 K1 K6 - K1 K1 K0 K6 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K2 K5 þ K2 K1 K5 - K0 K1 K2 K5 - K0 K2 K1 K5 - K1 K0 K2 K5 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 2 K1 K0 K5 þ K0 K4 K4 þ K1 K3 K4 þ K2 K2 K4 - K0 K2 K2 K4 - K1 K2 K1 K4 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 L0 L5 L5 d1 - K L0 2 K1 K1 K4 - K2 K0 K2 K4 - K2 K2 K0 K4 þ K2 K4 - K2 K2 K2 K2 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L0 L5 c1 c1 þ K L1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K3 - K0 K0 K4 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K1 K6 - K0 K1 þ K0 K1 K2 K5 - K0 K1 þ K0 K1 K3 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K5 K2 - K0 K1 þ K0 K1 K6 K1 - K0 K2 þ K0 K2 K0 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K2 K4 - K0 K2 þ K0 K2 K3 K3 - K0 K2 þ K0 K2 K4 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K6 K0 - K1 K2 þ K1 K2 K0 K5 - K1 K2 þ K1 K2 K1 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K3 K2 - K1 K2 þ K1 K2 K4 K1 - K1 K2 þ K1 K2 K5 K0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 2 K2 K3 K1 - K2 K2 K4 K0 - K2 K2 K2 K2 - K1 K0 K1 K6 - K0 K2 K0 K6 b1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 c1 c1 þ K L1 0 K0 - K0 K0 K0 K8 - K2 K0 K1 K5 - K1 K1 K2 K4 - K0 K0 K3 K5 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K0 K6 K2 - K0 K1 þ K0 K1 K4 K3 - K0 K2 þ K0 K2 K1 K5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K5 K1 - K1 K2 þ K1 K2 K2 K3 - K2 K2 K0 K4 L5 c1 c1 L0 L5 c1 c1 L1 b1 d1 L0 L5 L5 d1 L0 L5 c1 c1 - K L0 0 K0 K2 K6 - K0 K0 K5 K3 þ K0 K3 K5 - K1 K1 K1 K5 - K2 K2 K1 K3

166

3

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

b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L0 L5 L5 d1 K 9 = K L1 0 K1 þ K1 K0 - K0 K1 K0 - K1 K0 K0 - K0 K0 K1 K8 - K1 K0 K1 K7 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 0 K2 K7 þ K1 K1 K7 þ K2 K0 K7 - K0 K0 K2 K7 - K0 K1 K1 K7 - K0 K2 K0 K7 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K1 K0 K7 - K2 K0 K0 K7 þ K0 K3 K6 þ K1 K2 K6 þ K2 K1 K6 - K0 K1 K2 K6 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 0 K2 K1 K6 - K1 K1 K1 K6 - K2 K0 K1 K6 - K2 K1 K0 K6 þ K0 K4 K5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L2 K b1 2 K5 - K0 K2 K2 K5 - K1 K1 K2 K5 - K2 K1 K1 K5 - K2 K0 K2 K5 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K4 K4 þ K2 K3 K4 - K1 K2 K2 K4 - K2 K2 K1 K4 - K2 K1 K2 K4 b1 L0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 þ K L1 2 K4 - K2 K2 K2 K3 - K0 K0 K3 K6 - K0 K0 K4 K5 - K0 K0 K5 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K2 K6 - K0 K1 þ K0 K1 K3 K5 - K0 K1 þ K0 K1 K4 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K5 K3 - K0 K1 þ K0 K1 K6 K2 - K0 K2 þ K0 K2 K1 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K2 K5 - K0 K2 þ K0 K2 K3 K4 - K0 K2 þ K0 K2 K4 K3 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K5 K2 - K0 K2 þ K0 K2 K6 K1 - K1 K2 þ K1 K2 K0 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K1 K5 - K1 K2 þ K1 K2 K2 K4 - K1 K2 þ K1 K2 K3 K3 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K4 K2 - K1 K2 þ K1 K2 K5 K1 - K1 K2 þ K1 K2 K6 K0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L0 2 K2 K0 K5 - K2 K2 K1 K4 - K2 K2 K2 K3 - K2 K2 K3 K2 - K2 K2 K4 K1 L5 c1 c1 L0 L5 c1 c1 L0 L5 L5 d1 L1 b1 L0 L5 L5 d1 - K L0 2 K2 K5 K0 - K0 K0 K6 K3 - K1 K0 K2 K6 þ K0 K0 - K0 K0 K0 K9 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K3 K5 - K2 K2 K0 K5 - K1 K2 K1 K5

b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L1 b1 d1 K 10 = K L1 0 K1 þ K1 K0 - K0 K1 K0 - K1 K0 K0 - K0 K0 K1 K9 þ K0 K2 K8 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K1 K8 þ K2 K0 K8 - K0 K0 K2 K8 - K0 K1 K1 K8 - K0 K2 K0 K8 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 - K L0 1 K0 K1 K8 - K1 K1 K0 K8 - K2 K0 K0 K8 þ K0 K3 K7 þ K1 K2 K7 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 2 K1 K7 - K0 K1 K2 K7 - K0 K2 K1 K7 - K1 K1 K1 K7 - K2 K0 K1 K7 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 2 K1 K0 K7 þ K0 K4 K6 þ K1 K3 K6 þ K2 K2 K6 - K0 K2 K2 K6 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 - K L0 1 K2 K1 K6 - K1 K1 K2 K6 - K2 K1 K1 K6 - K2 K0 K2 K6 - K2 K2 K0 K6 b1 L0 L5 L5 d1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 þ K L1 2 K4 - K2 K2 K2 K4 - K0 K0 K4 K6 - K0 K0 K5 K5 - K0 K0 K6 K4 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L0 L5 c1 c1 þ K L1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K5 - K2 K2 K4 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K3 K6 - K0 K1 þ K0 K1 K4 K5 - K0 K1 þ K0 K1 K5 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K6 K3 - K0 K2 þ K0 K2 K2 K6 - K0 K2 þ K0 K2 K3 K5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K4 K4 - K0 K2 þ K0 K2 K5 K3 - K0 K2 þ K0 K2 K6 K2 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K1 K6 - K1 K2 þ K1 K2 K2 K5 - K1 K2 þ K1 K2 K3 K4 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K4 K3 - K1 K2 þ K1 K2 K5 K2 - K1 K2 þ K1 K2 K6 K1 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 L5 d1 - K L0 2 K2 K0 K6 þ K2 K2 K1 K5 - K2 K2 K2 K4 - K2 K2 K3 K3 - K1 K0 K2 K7 L5 c1 c1 L0 L5 c1 c1 L1 b1 L0 L5 L5 d1 - K L0 2 K 2 K 6 K 0 - K 2 K 2 K 5 K 1 þ K 0 K 0 - K 0 K 0 K 0 K 10

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

167

b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L1 b1 d1 K 11 = K L1 0 K 1 þ K 1 K 0 - K 0 K 1 K 0 - K 1 K 0 K 0 - K 0 K 0 K 1 K 10 þ K 0 K 2 K 9 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K1 K9 þ K2 K0 K9 - K0 K1 K1 K9 - K0 K2 K0 K9 - K1 K0 K1 K9 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 - K L0 1 K1 K0 K9 - K2 K0 K0 K9 þ K0 K3 K8 þ K1 K2 K8 þ K2 K1 K8 - K0 K1 K2 K8 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 0 K2 K1 K8 - K1 K0 K2 K8 - K1 K1 K1 K8 - K2 K0 K1 K8 þ K0 K4 K7 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K3 K7 þ K2 K2 K7 - K0 K2 K2 K7 - K1 K2 K1 K7 - K1 K1 K2 K7 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 L0 L5 L5 d1 - K L0 2 K1 K1 K7 - K2 K0 K2 K7 - K2 K2 K0 K7 þ K2 K4 - K2 K2 K2 K5 b1 L1 b1 La0 L5 L5 L0 L5 L5 L0 L5 L5 d1 L0 L5 L5 d1 þ K L1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K6 - K2 K1 K0 K8 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 0 K0 K5 K6 - K0 K0 K6 K5 - K2 K2 K6 K1 - K0 K1 þ K0 K1 K4 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K5 K5 - K0 K2 þ K0 K2 K3 K6 - K0 K2 þ K0 K2 K4 K5 L0 L0 L5 c1 c1 c1 c1 L5 L0 L0 L5 L0 L0 L5 c1 c1 - K L5 - K L5 0 K2 þ K0 K2 K5 K4 K6 K3 - K0 K2 þ K0 K2 1 K2 þ K1 K2 K2 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K3 K5 - K1 K2 þ K1 K2 K4 K4 - K1 K2 þ K1 K2 K5 K3 L0 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K6 K2 - K2 K2 K1 K6 - K2 K2 K2 K5 - K2 K2 K3 K4 L5 c1 c1 L0 L5 L5 d1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 2 K2 K4 K3 - K0 K0 K2 K9 - K2 K2 K5 K2 - K0 K1 þ K0 K1 K6 K4

b1 d1 L1 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 K 12 = K L1 0 K 2 K 10 þ K 1 K 1 K 10 þ K 2 K 0 K 10 - K 0 K 0 K 2 K 10 - K 0 K 1 K 1 K 10 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 0 K 2 K 0 K 10 - K 1 K 0 K 1 K 10 - K 1 K 1 K 0 K 10 - K 2 K 0 K 0 K 10 þ K 0 K 3 K 9 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K2 K9 þ K2 K1 K9 - K0 K1 K2 K9 - K0 K2 K1 K9 - K1 K0 K2 K9 L5 L5 d1 L1 b1 L0 L5 L5 d1 L0 L5 L5 d1 L1 b1 d1 - K L0 1 K1 K1 K9 þ K2 K4 - K2 K2 K2 K6 - K2 K1 K0 K9 þ K0 K4 K8 b1 d1 L1 b1 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 þK L1 1 K3 K8 þ K2 K2 K8 - K0 K2 K2 K8 - K1 K2 K1 K8 - K1 K1 K2 K8 L5 L5 d1 L0 L5 L5 d1 L0 L5 L5 d1 L0 L5 c1 c1 - K L0 2 K1 K1 K8 - K2 K0 K2 K8 - K2 K2 K0 K8 - K0 K0 K6 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K5 K6 - K0 K2 þ K0 K2 K5 K5 - K2 K2 K4 K4 b1 L1 b1 L0 L5 L5 L0 L5 L5 L0 L5 L5 d1 þ K L1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K7 L0 L0 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K1 þ K0 K1 K6 K5 - K2 K2 K5 K3 - K0 K2 þ K0 K2 K4 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K6 K4 - K1 K2 þ K1 K2 K3 K6 L0 L0 L5 c1 c1 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K4 K5 - K2 K2 K2 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K5 K4 - K1 K2 þ K1 K2 K6 K3 L5 c1 c1 L0 L5 L5 d1 L0 L5 c1 c1 - K L0 2 K2 K3 K5 - K2 K0 K1 K9 - K2 K2 K6 K2 b1 d1 a1 b1 d1 a1 b1 d1 a0 a5 a5 d1 a0 a5 a5 d1 a0 a5 a5 d1 K 13 = K a1 0 K 3 K 10 þ K 1 K 2 K 10 þ K 2 K 1 K 10 - K 0 K 1 K 2 K 10 - K 0 K 2 K 1 K 10 - K 1 K 0 K 2 K 10 a5 a5 d1 a0 a5 a5 d1 a0 a5 a5 d1 a1 b1 d1 a1 b1 d1 a1 b1 d1 - K a0 1 K 1 K 1 K 10 - K 2 K 0 K 1 K 10 - K 2 K 1 K 0 K 10 þ K 0 K 4 K 9 þ K 1 K 3 K 9 þ K 2 K 2 K 9 a5 a5 d1 a0 a5 a5 d1 a0 a5 a5 d1 a0 a5 a5 d1 a0 a5 a5 d1 - K a0 0 K2 K2 K9 - K1 K2 K1 K9 - K2 K1 K1 K9 - K2 K0 K2 K9 - K2 K2 K0 K9 b1 a1 b1 a0 a5 a5 a0 a5 a5 a0 a5 a5 d1 L0 L5 c1 c1 þ K a1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K8 - K2 K2 K5 K4 b1 a0 a5 a5 d1 L0 L5 c1 c1 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 þ K a1 2 K4 - K2 K2 K2 K7 - K2 K2 K3 K6 - K2 K2 K4 K5 - K0 K1 þ K0 K1 K6 K6 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 1 K2 þ K1 K2 K4 K6 - K1 K2 þ K1 K2 K5 K5 - K1 K2 þ K1 K2 K6 K4 L5 c1 c1 a0 a5 a5 d1 L5 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L0 2 K2 K6 K3 - K1 K1 K2 K9 - K0 K2 þ K0 K2 K6 K5 - K0 K2 þ K0 K2 K5 K6

168

3

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

b1 d1 a1 b1 d1 a1 b1 d1 a0 a5 a5 d1 a0 a5 a5 d1 K 14 = K a1 0 K 4 K 10 þ K 1 K 3 K 10 þ K 2 K 2 K 10 - K 0 K 2 K 2 K 10 - K 1 K 2 K 1 K 10 a5 a5 d1 a1 b1 a0 a5 a5 d1 a0 a5 a5 d1 - K a0 1 K 1 K 2 K 10 þ K 2 K 4 - K 2 K 2 K 2 K 8 - K 2 K 1 K 1 K 10 b1 a1 b1 a0 a5 a5 a0 a5 a5 a0 a5 a5 d1 þ K a1 1 K4 þ K2 K3 - K1 K2 K2 - K2 K2 K1 - K2 K1 K2 K9 L0 L0 L5 c1 c1 L5 L0 L0 L5 c1 c1 - K L5 0 K2 þ K0 K2 K6 K6 - K1 K2 þ K1 K2 K5 K6 L0 L0 L5 c1 c1 L0 L5 c1 c1 a0 a5 a5 d1 - K L5 1 K 2 þ K 1 K 2 K 6 K 5 - K 2 K 2 K 6 K 4 - K 2 K 0 K 2 K 10 a5 a5 d1 L0 L5 c1 c1 L0 L5 c1 c1 - K a0 2 K 2 K 0 K 10 - K 2 K 2 K 5 K 5 - K 2 K 2 K 4 K 6 b1 a1 b1 a0 a5 a5 a0 a5 a5 a0 a5 a5 d1 L0 L5 c1 c1 K 15 = K a1 1 K 4 þ K 2 K 3 - K 1 K 2 K 2 - K 2 K 2 K 1 - K 2 K 1 K 2 K 10 - K 2 K 2 K 5 K 6 L5 c1 c1 a1 b1 a0 a5 a5 d1 L5 L0 L0 L5 c1 c1 - K L0 2 K2 K6 K5 þ K2 K4 - K2 K2 K2 K9 - K1 K2 þ K1 K2 K6 K6 b1 a0 a5 a5 d1 L0 L5 c1 c1 K 16 = K a1 2 K 4 - K 2 K 2 K 2 K 10 - K 2 K 2 K 6 K 6

Let zL2 be the solution of following equation, 16

f L2 ðzÞ =

K k zk = 0

ð3:114Þ

k=0

Similar to the derivation from (3.100) to (3.104) to lead to Conclusions 3.1 and 3.2, following conclusion can be made. Conclusion 3.3 Let zL2 be a solution of the 16th-order equation given by (3.114), which can be re-written as 15

f L2 ðzÞ = g0 þ

gi zi þ z16 = 0

ð3:115Þ

i=1

where coefficients gi; i = 0, 1, 2, ⋯15 are related only with the parameters of PI gains of the PLL, the outer control loops of the GSC and RSC in addition to the values of Cd, Vddc0, and P0. The grid-connected DFIG system displayed in Fig. 3.5 is stable in the timescale of DC voltage control if and only if XL < zL2 . Expressing coefficient ai; i = 0, 1, 2, 3, 4, 5, 6 in (3.111) as the function of P0, it can have ai ai 2 ai = K ai 0 þ K 1 P0 þ K 2 P0 ; j = 0, 1, . . . , 6

ð3:116Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

169

where K a0 0 = K pi K di5 K di1 K di3 K a1 0 =

K pi K di5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þ K pi K dp5 K di1 K di3 þ K di1 K di3 K di5 K pp

2 K a1 2 = - K di5 K di3 K pi X L

K pi K di5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 Cd V ddc0 K pi þ 1 - xL sdr10

1 K di1 K di3 K di5 X dss

K a2 0 = þ K pi K dp5 K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 þ K di1 K di3 K di5

XL þ K di1 K di3 K dp5 K pp X dss

þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5 K pp 2 2 K a2 2 = - X L K di5 K di3 K pp - X L K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pi

K a3 0 = K pi K dp5 K dp1 þ 1 K dp3 þ 1 þ K di1 K di3 C d V ddc0 K pp þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 Cd V ddc0 K pi þ K dp1 K qi þ K di1 K dp3 þ K di1 þ K di3 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 K di5 K pp þ K di1 K di3 K dp5 þ 1 - X L sdr10

1 K di1 K di3 K dp5 þ X dss

þ 1 - X L sdr10

1 X dss

K a3 2 = -

XL X dss

K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K di5

K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K di5

X 2L C V K K - X 2L K dp5 þ 1 K dp3 þ 1 K pi ωr0 d ddc0 pi di3

- X 2L K dp5 K di3 þ K di5 K dp3 þ K di5 þ K di3 K pp

XL X dss

170

3

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

K dp1 þ 1 K dp3 þ 1 K dp5 K pp þ K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pi þ K di1 K di3 C d V ddc0 þ K di1 K di3 C d V ddc0 þ 1 - X L sdr10

1 X dss

1 K a4 0 = þ 1 - X L sdr10 X dss

XL X dss

K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 K dp5 K dp1 þ 1 K dp3 þ 1 K di5

þ K dp1 K di3 þ K di1 K dp3 þ K di1 þ K di3 Cd V ddc0 K pp þ

K dp1 þ 1 K di3 þ K dp3 þ 1 K di1 K dp5

þ K dp1 þ 1 K dp3 þ 1 K di5

K a4 2 =

XL X dss

- X 2L K dp5 þ 1 K dp3 þ 1 K pp -

XL X dss

X 2L K ωr0 di5

X 2L C V K K þ K pi K dp3 þ 1 ωr0 d ddc0 pp di3

K a5 0 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 K pp þ K dp1 þ 1 K dp3 þ 1 K dp5

XL X dss

þ K dp1 K di3 þ K dp1 K qp þ K dp1 þ K di3 Cd V ddc0 þ 1 - X L sr0

1 X dss

K dp1 þ 1 K dp3 þ 1 K dp5

þ K dp1 þ 1 K di3 þ K dp3 þ 1 K dp1 C d V ddc0 K a5 2 = -

XL X dss

X 2L X2 Cd V ddc0 K pp K dp3 þ 1 - L K dp5 ωr0 ωr0

K a6 0 = K dp1 þ 1 K dp3 þ 1 C d V ddc0 þ K dp1 þ 1 K dp3 þ 1 C d V ddc0 K a6 2 = -

XL X dss

X 2L Cd V ddc0 ω2r0

By substituting (3.116) in e1 given in (3.111), it can have 8

e1 = f P2 ðP0 Þ = k=0

where

2k K p2 2k P0

ð3:117Þ

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . .

171

a4 a5 a1 a1 a3 a6 a5 a5 d1 K p2 0 = K 0 K 0 K 0 - K 0 K 0 K 0 - K 0 K 0 K pi K di5 K di1 K di3 K 0 c1 a5 - K c1 0 K 0 K 0 K pi K di5 K di1 K di3 a4 a5 a1 d1 a1 a3 a6 d1 a5 a5 K d1 2 K 0 K 0 K 0 - K 2 K 0 K 0 K 0 - K 2 K 0 K 0 K pi K di5 K di1 K di3 a4 a5 a1 d1 a3 a6 a1 d1 a5 a4 a1 d1 a1 a6 a3 þ K d1 0 K0 K0 K2 - K0 K0 K0 K2 þ K0 K0 K2 K0 - K0 K0 K0 K2 d1 a1 a3 a6 d1 a5 a5 K p2 2 = - K 0 K 0 K 0 K 2 - K 0 K 0 K 2 K pi K di5 K di1 K di3 a5 a5 c1 c1 a5 - K d1 0 K pi K di5 K di1 K di3 K 0 K 2 - K 0 K 0 K pi K di5 K di1 K di3 K 2 c1 c1 c1 c1 d1 a1 a4 a5 - K a5 0 K pi K di5 K di1 K di3 K 0 K 2 þ K 0 K 2 þ K 0 K 0 K 0 K 2 a4 a5 a1 d1 a1 a3 a6 d1 a5 a5 K d1 4 K 0 K 0 K 0 - K 4 K 0 K 0 K 0 - K 4 K 0 K 0 K pi K di5 K di1 K di3 a4 a5 a1 d1 a3 a6 a1 d1 a5 a4 a1 þ K d1 2 K0 K0 K2 - K2 K0 K0 K2 þ K2 K0 K2 K0 a1 a4 a5 d1 a1 a6 a3 d1 a1 a3 a6 þ K d1 2 K0 K0 K2 - K2 K0 K0 K2 - K2 K0 K0 K2 d1 a5 a5 d1 a5 a4 a1 a1 a4 a5 - K 2 K 0 K 2 K pi K di5 K di1 K di3 þ K 0 K 0 K 2 K 2 þ K d1 0 K2 K0 K2 K p2 = 4 a1 a6 a3 d1 a1 a3 a6 d1 a4 a5 a1 - K d1 0 K2 K0 K2 - K0 K2 K0 K2 þ K0 K2 K2 K0 d1 a3 a6 a1 d1 a5 - K 0 K 2 K 2 K 0 - K 0 K pi K di5 K di1 K di3 K a5 2 K2 c1 c1 c1 c1 - K pi K di5 K di1 K di3 K a5 2 K0 K2 þ K0 K2 a5 d1 a5 c1 c1 c1 c1 c1 c1 - K pi K di5 K di1 K di3 K a5 0 K 2 K 2 - K 0 K pi K di5 K di1 K di3 K 0 K 4 þ K 2 K 2 þ K 0 K 4 a4 a5 a1 a1 a3 a6 a5 a5 d1 d1 a1 a6 a3 K p2 6 = K 0 K 0 K 0 - K 0 K 0 K 0 - K 0 K 0 K pi K di5 K di1 K di3 K 6 - K 4 K 0 K 0 K 2 a5 a1 a3 a6 a1 d1 d1 a5 a5 d1 a4 a5 a1 þ K a4 2 K 2 K 2 - K 2 K 2 K 2 K 0 - K 2 K pi K di5 K di1 K di3 K 2 K 2 þ K 4 K 0 K 0 K 2 a3 a6 a1 d1 a5 a4 a1 d1 a1 a3 a6 d1 a5 a5 - K d1 4 K 0 K 0 K 2 þ K 4 K 0 K 2 K 0 - K 4 K 0 K 0 K 2 - K 4 K 0 K 2 K pi K di5 K di1 K di3 a5 a4 a1 d1 a1 a4 a5 d1 a1 a6 a3 d1 a3 a6 a1 þK d1 2 K0 K2 K2 þ K2 K2 K0 K2 - K2 K2 K0 K2 - K2 K2 K2 K0 a5 c1 c1 c1 c1 c1 c1 d1 a1 a3 a6 - K pi K di5 K di1 K di3 K 2 K 0 K 4 þ K 2 K 2 þ K 0 K 4 - K 2 K 2 K 0 K 2 c1 c1 c1 c1 c1 c1 c1 c1 - K a5 0 K pi K di5 K di1 K di3 K 0 K 6 þ K 4 K 2 þ K 2 K 4 þ K 0 K 6 a5 a5 d1 a4 a5 a1 d1 a1 a4 a5 - K d1 4 K pi K di5 K di1 K di3 K 0 K 2 þ K 2 K 2 K 2 K 0 þ K 4 K 0 K 0 K 2 a4 a5 a1 a1 a3 a6 a5 a5 d1 d1 a1 a6 a3 K p2 8 = K 0 K 0 K 0 - K 0 K 0 K 0 - K 0 K 0 K pi K di5 K di1 K di3 K 8 - K 6 K 0 K 0 K 2 a5 a1 a3 a6 a1 d1 d1 a5 a5 d1 a4 a5 a1 þ K a4 2 K 2 K 2 - K 2 K 2 K 2 K 2 - K 6 K pi K di5 K di1 K di3 K 0 K 2 þ K 6 K 0 K 0 K 2 a3 a6 a1 d1 a5 a4 a1 d1 a1 a3 a6 d1 a5 a5 - K d1 6 K 0 K 0 K 2 þ K 6 K 0 K 2 K 0 - K 6 K 0 K 0 K 2 - K 6 K 0 K 2 K pi K di5 K di1 K di3 d1 a5 a4 a1 d1 a1 a4 a5 d1 a1 a6 a3 d1 a3 a6 a1 þK 4 K 0 K 2 K 2 þ K 4 K 2 K 0 K 2 - K 4 K 2 K 0 K 2 - K 4 K 2 K 2 K 0 a5 a5 d1 a1 a4 a5 d1 a4 a5 a1 - K d1 4 K pi K di5 K di1 K di3 K 2 K 2 þ K 6 K 0 K 0 K 2 þ K 4 K 2 K 2 K 0 c1 c1 c1 c1 c1 c1 c1 c1 - K pi K di5 K di1 K di3 K a5 2 K0 K6 þ K4 K2 þ K2 K4 þ K0 K6 c1 c1 c1 c1 c1 c1 d1 a1 a3 a6 - K a5 0 K pi K di5 K di1 K di3 K 6 K 2 þ K 4 K 4 þ K 2 K 6 - K 4 K 2 K 0 K 2

172

3

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

a5 a1 a1 a3 a6 a5 a5 d1 K a4 0 K 0 K 0 - K 0 K 0 K 0 - K 0 K 0 K pi K di5 K di1 K di3 K 10 a1 a6 a3 a4 a5 a1 a3 a6 a1 d1 - K d1 8 K0 K0 K2 þ K2 K2 K2 - K2 K2 K2 K4 a5 a5 d1 a4 a5 a1 d1 a3 a6 a1 - K d1 8 K pi K di5 K di1 K di3 K 0 K 2 þ K 8 K 0 K 0 K 2 - K 8 K 0 K 0 K 2 a5 a4 a1 d1 a1 a3 a6 d1 a5 a5 þ K d1 8 K 0 K 2 K 0 - K 8 K 0 K 0 K 2 - K 8 K 0 K 2 K pi K di5 K di1 K di3 d1 a5 a4 a1 d1 a1 a4 a5 d1 a1 a6 a3 K p2 10 = þ K 6 K 0 K 2 K 2 þ K 6 K 2 K 0 K 2 - K 6 K 2 K 0 K 2 a3 a6 a1 d1 a1 a3 a6 d1 a5 a5 - K d1 6 K 2 K 2 K 0 - K 6 K 2 K 0 K 2 - K 6 K pi K di5 K di1 K di3 K 2 K 2 a5 c1 c1 c1 c1 c1 c1 - K pi K di5 K di1 K di3 K 2 K 6 K 2 þ K 4 K 4 þ K 2 K 6 c1 c1 c1 c1 d1 a1 a4 a5 - K a5 0 K pi K di5 K di1 K di3 K 6 K 4 þ K 4 K 6 þ K 8 K 0 K 0 K 2 a4 a5 a1 þ K d1 6 K2 K2 K0 a4 a5 a1 d1 a3 a6 a1 d1 a5 a4 a1 d1 a1 a4 a5 K d1 10 K 0 K 0 K 2 - K 10 K 0 K 0 K 2 þ K 10 K 0 K 2 K 0 þ K 10 K 0 K 0 K 2 a1 a6 a3 d1 a1 a3 a6 d1 a5 a5 - K d1 10 K 0 K 0 K 2 - K 10 K 0 K 0 K 2 - K 10 K 0 K 2 K pi K di5 K di1 K di3 a5 a4 a1 d1 a1 a4 a5 d1 a1 a6 a3 þ K d1 8 K0 K2 K2 þ K8 K2 K0 K2 - K8 K2 K0 K2 a1 a3 a6 d1 a4 a5 a1 d1 a3 a6 a1 - K d1 K p2 8 K2 K0 K2 þ K8 K2 K2 K0 - K8 K2 K2 K0 12 = d1 a5 a5 a4 a5 a1 a6 a1 d1 - K 8 K pi K di5 K di1 K di3 K 2 K 2 þ K 2 K 2 K 2 - K a3 2 K2 K2 K6 a5 a5 a5 c1 c1 c1 c1 - K d1 10 K pi K di5 K di1 K di3 K 0 K 2 - K i K vi K pi K qi K 2 K 6 K 4 þ K 4 K 6 c1 c1 - K a5 0 K pi K di5 K di1 K di3 K 6 K 6 a5 a4 a1 d1 a1 a4 a5 d1 a1 a6 a3 d1 a1 a3 a6 K d1 10 K 0 K 2 K 2 þ K 10 K 2 K 0 K 2 - K 10 K 2 K 0 K 2 - K 10 K 2 K 0 K 2 d1 a4 a5 a1 d1 a3 a6 a1 a4 a5 a1 a3 a6 a1 d1 K p2 14 = þ K 10 K 2 K 2 K 0 - K 10 K 2 K 2 K 0 þ K 2 K 2 K 2 - K 2 K 2 K 2 K 8 c1 c1 d1 a5 a5 - K pi K di5 K di1 K di3 K a5 2 K 6 K 6 - K 10 K pi K di5 K di1 K di3 K 2 K 2 a4 a5 a1 a3 a6 a1 d1 K p2 16 = K 2 K 2 K 2 - K 2 K 2 K 2 K 10 a4 a5 a3 a6 K b1 0 = K0 K0 - K0 K0 a5 a4 a4 a5 a6 a3 a3 a6 K b1 2 = K0 K2 þ K0 K2 - K0 K2 - K0 K2 a4 a5 a3 a6 K b1 4 = K2 K2 - K2 K2 a4 a5 a3 a3 a3 a6 a2 a5 a5 a5 a1 a6 K c1 0 = K0 K0 K0 - K0 K0 K0 - K0 K0 K0 þ K0 K0 K0 a5 a3 a3 a6 a3 a5 a4 a3 a3 a4 a5 K a4 0 K0 K2 - K0 K0 K2 þ K0 K2 K0 þ K0 K0 K2 a3 a6 a3 a3 a3 a6 a5 a5 a2 a5 a2 a5 K c1 2 = - K0 K0 K2 - K0 K0 K2 - K0 K0 K2 - K0 K0 K2 a6 a1 a5 a1 a6 a2 a5 a5 a1 a6 a5 þ K a5 0 K0 K2 þ K0 K0 K2 - K0 K0 K2 þ K0 K0 K2 a4 a3 a3 a4 a5 a3 a6 a3 a3 a3 a6 K a5 0 K2 K2 þ K2 K0 K2 - K2 K0 K2 - K2 K0 K2 a4 a5 a3 a3 a3 a6 a5 a2 a5 a5 a2 a5 K c1 4 = þ K2 K2 K0 - K0 K2 K2 - K0 K2 K2 - K2 K0 K2 a6 a1 a5 a1 a6 a2 a5 a5 a5 a1 a6 þ K a5 2 K0 K2 þ K2 K0 K2 - K2 K2 K0 þ K0 K2 K2 a4 a5 a3 a3 a3 a6 a5 a2 a5 a5 a1 a6 K c1 6 = K2 K2 K2 - K2 K2 K2 - K2 K2 K2 þ K2 K2 K2

3.3

Risk of Sub/Super-synchronous Oscillations in a Grid-Connected DFIG. . . a2 a5 c1 c1 a1 a6 b1 a4 a5 a1 b1 a1 a3 a6 K d1 0 = K0 K0 K0 - K0 K0 K0 - K0 K0 K0 K0 þ K0 K0 K0 K0 a5 a5 þK b1 0 K 0 K 0 K pi K di5 K di1 K di3 a5 c1 a1 a6 c1 a5 a2 c1 b1 a1 a3 a6 K a2 0 K0 K2 - K0 K0 K2 þ K0 K2 K0 þ K0 K0 K0 K2 a5 c1 a4 a5 a1 a1 a3 a6 þ K a2 0 K2 K0 - K0 K0 K0 - K0 K0 K0

K d1 2 =

a5 b1 c1 a1 a6 b1 a4 a5 a1 - K a5 0 K 0 K pi K di5 K di1 K di3 ÞK 2 - K 0 K 0 K 2 - K 0 K 0 K 0 K 2 a3 a6 a1 b1 a5 a4 a1 b1 a1 a4 a5 þ K b1 0 K0 K0 K2 - K0 K0 K2 K0 - K0 K0 K0 K2 a1 a6 a3 b1 a5 a5 a6 a1 c1 þ K b1 0 K 0 K 0 K 2 þ K 0 K pi K di5 K di1 K di3 K 0 K 2 - K 0 K 2 K 0 a5 a5 þ K b1 0 K 0 K 2 K pi K di5 K di1 K di3

a5 c1 a1 a6 c1 a2 a5 c1 c1 a1 a6 a5 a2 c1 K a2 0 K0 K4 - K0 K0 K4 þ K2 K2 K0 - K0 K2 K2 þ K0 K2 K2 a5 c1 a6 a1 c1 a1 a6 c1 a4 a5 a1 þ K a2 0 K2 K2 - K0 K2 K2 - K0 K2 K2 - K0 K0 K0 a3 a6 a5 a5 b1 - K a1 0 K 0 K 0 - K 0 K 0 K pi K di5 K di1 K di3 ÞK 4 a5 a5 b1 a1 a3 a6 þ K b1 2 K pi K di5 K di1 K di3 K 0 K 2 þ K 0 K 2 K 0 K 2 b1 a1 a4 a5 b1 a5 a4 a1 b1 a1 a4 a5 K d1 4 = - K2 K0 K0 K2 - K0 K0 K2 K2 - K0 K2 K0 K2 a1 a6 a3 b1 a3 a6 a1 b1 a5 a5 þ K b1 0 K 2 K 0 K 2 þ K 0 K 2 K 2 K 0 þ K 0 K pi K di5 K di1 K di3 K 2 K 2 a4 a5 a1 b1 a3 a6 a1 b1 a5 a4 a1 - K b1 2 K0 K0 K2 þ K2 K0 K0 K2 - K2 K0 K2 K0 a1 a6 a3 b1 a1 a3 a6 b1 a5 a5 þ K b1 2 K 0 K 0 K 2 þ K 2 K 0 K 0 K 2 þ K 2 K 0 K 2 K pi K di5 K di1 K di3 a4 a5 a1 - K b1 0 K2 K2 K0 a5 c1 a1 a6 c1 a5 a2 c1 a2 a5 c1 a1 a6 c1 K a2 0 K0 K6 - K0 K0 K6 þ K0 K2 K4 þ K0 K2 K4 - K0 K2 K4 a5 c1 a1 a6 c1 b1 a4 a5 a1 b1 a4 a5 a1 þ K a2 2 K2 K2 - K2 K2 K2 - K0 K2 K2 K2 - K4 K0 K0 K2 a3 a6 a1 b1 a5 a4 a1 b1 a1 a4 a5 þ K b1 4 K0 K0 K2 - K4 K0 K2 K0 - K4 K0 K0 K2

K d1 6 =

a1 a6 a3 b1 a1 a3 a6 b1 a5 a5 þ K b1 4 K 0 K 0 K 2 þ K 4 K 0 K 0 K 2 þ K 4 K 0 K 2 K pi K di5 K di1 K di3 a5 a4 a1 b1 a1 a4 a5 b1 a1 a6 a3 - K b1 2 K0 K2 K2 - K2 K2 K0 K2 þ K2 K2 K0 K2 a1 a3 a6 b1 a3 a6 a1 b1 a4 a5 a1 þ K b1 2 K2 K0 K2 þ K0 K2 K2 K2 - K2 K2 K2 K0 a3 a6 a1 b1 a5 a5 þ K b1 2 K 2 K 2 K 0 þ K 2 K pi K di5 K di1 K di3 K 2 K 2 a5 a5 a6 a1 c1 þ K b1 4 K pi K di5 K di1 K di3 K 0 K 2 - K 0 K 2 K 4 a2 c1 a2 a5 c1 a6 a1 c1 a1 a6 c1 a2 a5 c1 K a5 0 K2 K6 þ K0 K2 K6 - K0 K2 K6 - K0 K2 K6 þ K2 K2 K4

K d1 8 =

a6 c1 b1 a4 a5 a1 b1 a3 a6 a1 b1 a5 a4 a1 - K a1 2 K2 K4 - K2 K2 K2 K2 þ K2 K2 K2 K2 - K4 K0 K2 K2 a1 a4 a5 b1 a1 a6 a3 b1 a1 a3 a6 - K b1 4 K2 K0 K2 þ K4 K2 K0 K2 þ K4 K2 K0 K2 a4 a5 a1 b1 a3 a6 a1 b1 a5 a5 - K b1 4 K 2 K 2 K 0 þ K 4 K 2 K 2 K 0 þ K 4 K pi K di5 K di1 K di3 K 2 K 2

K d1 10

a5 c1 a1 a6 c1 a4 a5 a1 a3 a6 a1 b1 = K a2 2 K2 K6 - K2 K2 K6 - K2 K2 K2 - K2 K2 K2 K4

Following conclusion can be obtained.

173

174

3

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

Conclusion 3.4 Let zP2 be a solution of the 16th-order equation e1 = f P2 ðzÞ = 8 k=0

2k K p2 2k z = 0 where the coefficients are given by (3.117). The equation can be

re-written as 15

f P2 ðzÞ = h0 þ

hi zi þ z16 = 0

ð3:118Þ

i=1

where coefficients hi; i = 0, 1, 2, ⋯8 are related only with the parameters of PI gains of the PLL, the outer control loops of the GSC and RSC in addition to the values of Cd, Vddc0, and XL. The grid-connected DFIG system displayed in Fig. 3.5 is stable in the timescale of DC voltage control if and only if P0 < zP2. Following remarks can be made about the conclusions obtained so far. 1. Conclusions 3.3 and 3.4 also analytically indicate and explain why under the condition of weaker grid connection (bigger XL), or/and heavier loading condition (bigger P0), the grid-connected DFIG system is prone to the increased instability risk which is associated with the outer control loops and the PLL. In the case that the dynamics of the PLL are considered, zP2 defines the maximum loading of the DFIG, P0max, to ensure the stability of the single grid-connected DFIG system, i.e., P0max < zP2. 2. Conclusions 3.3 and 3.4 explain analytically that the smaller values of zP2 and zL2 mean the “improper setting” of the PI gains. Proper setting should ensure relatively greater values of zP2 and zL2 to enhance the small-signal stability of the single grid-connected DFIG system. 3. Conclusions 3.3 and 3.4 also reveal the analytical connections between three key factors, i.e., the condition of grid connection, the loading, and the parameters setting of the converter control systems, including the PLL, to jointly affect the small-signal stability of the single grid-connected DFIG system in the timescale of DC voltage control. Violation of any derived small-signal stability limit can be caused by any single factor or jointly by multiple factors to cause growing oscillations 4. Conclusions 3.1, 3.2, 3.3, and 3.4 indicate that when the DFIG is equipped with either a fast PLL (dynamics of the PLL are in the timescale of electric current control) or a slow PLL (dynamics of the PLL are in the timescale of DC voltage control), the grid-connected DFIG system may lose the stability in the timescale of DC voltage control. Instability risk is associated only with the control outer loops of the RSC and GSC when the PLL is fast, but also related with the PLL when its dynamics are in the timescale of DC voltage control.

3.4

3.4

An Example Single Grid-Connected DFIG System

175

An Example Single Grid-Connected DFIG System

In the previous section, small-signal stability limit of a grid-connected DFIG system in the timescale of DC voltage is derived with four analytical conclusions obtained, which are summarized in Table 3.1. In this section, an example single gridconnected DFIG system is presented to demonstrate and evaluate the analysis and conclusions made in the previous section. Configuration of the example single gridconnected DFIG system is shown by Fig. 3.5. Main parameters of the example system are given in Table 3.2.

Table 3.1 Small-signal stability limit of a grid-connected DFIG system in the timescale of DC voltage control Dynamics of the PLL are in the timescale of

Electric current control (fast PLL)

X L < zL1

DC voltage control (slow PLL)

P0 < zP1 X L < zL2 P0 < zP2

Table 3.2 Parameters of the DFIG Symbols Xdss Xdrr Xdm Rds Rdr ωdr1 Vddc0 Cd Kdp5, Kdi5 Kdp1, Kdi1 Kdp3, Kdi3 Kpp, Kpi

Self-inductance of stator windings Self-inductance of rotor windings Mutual-inductance between the stator and rotor windings Resistance of stator windings Resistance of rotor windings Angular speed of high-speed shaft in the two-mass shaft rotational system Steady-state DC link voltage Capacitance of the DC capacitor PI gains of DC voltage control outer loop of the GSC PI gains of active power control outer loop of the RSC PI gains of reactive power control outer loop of the RSC PI gains of the PLL

Value 2.58 2.52 2.4 0.01 0.015 0.8 1 3 0.03, 35 0.1, 20 0.1, 20 25, 5

176

3.4.1

3

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

Dynamics of the PLL Are in the Slow Timescale of DC Voltage Control

PI gains of the PLL, the outer loops of the GSC and RSG (see Figs. 3.2 and 3.3) are fixed to be K pp = 25, K pi = 5, K dp5 = 0:03, K di5 = 35, K dp1 = 0:1, K di1 = 20, K dp3 = 0:1, K di3 = 20

According to Conclusions 3.3 and 3.4, the system may lose the stability in the timescale of DC voltage control. The stability limits are P0 < zP2 when XL is fixed or XL < zL2 when P0 is fixed. Following two tests (Tests 3.1 and 3.2) are conducted to demonstrate and evaluate the conclusions. Test 3.1 XL was fixed to be XL = 0.5. From (3.118), it was calculated that zP2 = 0.80 , which implied that the maximum loading of the DFIG was P0max = zP2 = 0.80 (Conclusion 3.4). In order to evaluate this result, steady-state active power output from the DFIG was varied from P0 = 0.40 to P0 = 0.95. With the variation, the oscillation modes of the example single grid-connected DFIG system were computed by using the full-order block diagram model displayed in Fig. 3.6. One dominant mode, denoted as λDC, was identified which was associated with the PLL, the outer loops of the GSC and RSC. Table 3.3 presents the results of modal computation when P0 varied. It can be seen clearly that when P0 > P0max = zP2 = 0.80, the example single grid-connected DFIG became unstable, confirming the correctness of Conclusion 3.4. Verification of non-linear simulation is presented Fig. 3.10. At 0.5 second of simulation, the active power output from the DFIG increased by 5% for 0.05 second due to the sudden change of the wind speed. Two results of simulation are given in Fig. 3.10: (1) P0 = 0.40 when the example single grid-connected DFIG was stable; (2) P0 = 1.10 when the example single grid-connected DFIG became unstable. Note that the results of non-linear simulation are presented as the deviation of active power. Test 3.2 P0 was fixed to be P0 = 0.80. From (3.115), it was calculated that zL2 = 0.48. According to Conclusion 3.3, the example system was stable if and only if XL < zL2 = 0.48. This explained why the example system became unstable when P0 = 0.80 and XL = 0.50 as being demonstrated in Test 3.1 above. To evaluate the computational result of stability limit that zL2 = 0.48, XL was varied from XL = 0.20 to XL = 0.80. Computational results of the dominant mode, λDC, using the full-order block diagram model displayed in Fig. 3.6, with the Table 3.3 Results of modal computation when P0 varied (Test 3.1) P0 λDC

0.40 -0.57597 ± j63.52

0.75 -0.1097 ± j61.429

1.10 0.46477 ± j58.264

3.4

An Example Single Grid-Connected DFIG System

177

Fig. 3.10 Non-linear simulation of Test 3.1

Table 3.4 Results of modal computation when XL varied (Test 3.2) XL λDC

0.20 -0.93245 ± j62.192

0.45 -0.19396 ± j61.437

0.80 0.68621 ± j57.213

Fig. 3.11 Non-linear simulation of Test 3.2

variation of XL are presented in Table 3.4. It can be seen that when XL > zL3 = 0.48, the example single grid-connected DFIG system became unstable. Correctness of Conclusion 3.3 was confirmed. Results of non-linear simulation are presented Fig. 3.11 when XL = 0.20 and XL = 0.80, confirming the correctness of Conclusion 3.3. The maximum loading of the example single grid-connected DFIG system can be increased by coordinated tuning of PI gains of the PLL, the outer loops of the GSC and RSC with the stability being still maintained and XL being given. This is demonstrated by Test 3.3 as follows. Test 3.3 XL was fixed to be XL = 0.5 as it was in Test 3.1. Denote p as the following vector of parameters. p = ½ K dp1

K di1

K dp3

K di3

K dp5

K di5

K pp

K pi ]T

ð3:119Þ

Coordinated tuning of PI gains is to solve the following optimization problem:

178

3

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

Table 3.5 Results of modal computation when P0 varied after re-tuning of PI gains (Test 3.3) P0 λDC

0.6 -1.3637 ± j65.756

1.10 -0.23007 ± j61.204

1.20 0.94285 ± j57.513

Fig. 3.12 Non-linear simulation of Test 3.3

max: zP2 ; subject to: f P2 ðp, zÞ = 0; p 2 Ρ

ð3:120Þ

where fP2(p, z) = 0 is given in (3.118) and Ρ is the space of feasible parameters (PI gains) of the PLL, the outer loops of the GSC and RSC. An optimal solution of (3.120) was obtained by using direct search algorithm to be K pp = 20, K pi = 5, K dp5 = 0:04, K di5 = 40, K dp1 = 0:4, K di1 = 20, K dp3 = 0:4, K di3 = 20

ð3:121Þ

With the tuned values of PI gains given in (3.121), the stability limit of the example single grid-connected DFIG system was calculated to be zP2 = 1.16 which was considerably increased from P0max = 0.80 in Test 3.1 to P0max = 1.16. Results of evaluation by modal computation are presented in Table 3.5. It can be seen that the coordinated tuning of PI gains according to (3.119) and (3.120) increased the maximum loading and still maintained the stability of the example system. Non-linear simulation, as compared with that of Test 3.1 is presented in Fig. 3.12, confirming that the coordinated parameters tuning increased the maximum loading of the DFIG.

3.4

An Example Single Grid-Connected DFIG System

Fig. 3.13 Trajectory of λDC when P0 varied (Test 3.4)

179

80 70

P0 increases

60 50

λDC whenP0 = 1.10

30

20

-2

3.4.2

λDC whenP0 = 1.20

Imag ax is

40

Real axis

-1

0

1

2

3

4

Dynamics of the PLL Are in the Fast Timescale of Electric Current Control

PI gains of the PLL are fixed to be Kpp = 100, Kpi = 100 such that dynamics of the PLL are in the timescale of electric current control. PI gains of the outer loops of the GSC and RSG (see Figs. 3.2 and 3.3) are fixed to be K dp5 = 0:03, K di5 = 30, K pp = 100, K pi = 100, K dp1 = 0:1, K di1 = 40, K dp3 = 0:1, K di3 = 40

The example single grid-connected DFIG system may lose the stability in the timescale of DC voltage control (Conclusions 3.1 and 3.2) as to be demonstrated and evaluated in the following two tests. Test 3.4 According to Conclusion 3.2, the stability limit of the example single gridconnected DFIG system in the timescale of DC voltage control is P0 < zP1 when XL is fixed. Hence, in the test, XL was fixed to be XL = 0.5. From (3.106), it was calculated that zP1 = 1.17. This implied that when P0 increases over zP1 = 1.17, the example system will lose the stability in the DC voltage timescale. To evaluate the stability assessment made above, P0 was increased from P0 = 0.4 to P0 = 1.30. Oscillation modes of the example grid-connected DFIG system were computed by using the full-order block diagram model displayed in Fig. 3.6. The dominant oscillation modes associated with the outer loops (λDC) of the GSC and the RSC was identified. Trajectory of λDC with the variation of P0 is displayed in Fig. 3.13. It can be seen that λDC became negatively damped when P0 was greater than 1.17, confirming the correctness of stability assessment made above by using Conclusion 3.2. Results of non-linear simulation are presented in Fig. 3.14. It can be seen that when P0 was greater than zP1 = 1.17, an oscillation at frequency of 8.7 Hz occurred in the example system, indicating the loss of stability in the timescale of DC voltage control.

180

3

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

Fig. 3.14 Non-linear simulation of Test 3.4

Fig. 3.15 Trajectory of λDC when XL varied (Test 3.5)

80 70 X L increases

60 50

λDC when X L = 0.50

30

20

λDC when X L = 0.55

Imag ax is

40

-2

Real axis

-1

0

1

2

3

4

Test 3.5 When P0 is fixed, Conclusion 3.1 indicates that the stability limit of the example single grid-connected DFIG system is XL < zL1 in the timescale of DC voltage control. This test was conducted to evaluate the conclusion. In the test, P0 was fixed to be P0 = 1.1. From (3.105), it was calculated that zL1 = 0.53. This implied that when XL increased over zL1 = 0.53, the example single grid-connected DFIG system would lose the stability in the DC voltage timescale. The stability assessment made above according to Conclusion 3.1 was evaluated by varying XL from XL = 0.2 to XL = 0.6. Oscillation modes of the example gridconnected DFIG system were computed by using the full-order block diagram model. The dominant oscillation mode associated with the outer loops (λDC) of the GSC and the RSC was identified. Trajectory of λDC with the variation of XL is displayed in Fig. 3.15. It can be seen that λDC moved to the right half of complex plan when XL increased over 0.53, confirming the correctness of stability assessment made above by using Conclusion 3.1. Figure 3.16 presents the results of non-linear simulation when XL varied. It can be seen that when XL was greater than zL1 = 0.53, the example system lost the stability in the timescale of DC voltage control with a growing oscillation of 8.5 Hz. Test 3.6 When dynamics of the PLL are in the fast timescale of electric current control, coordinated tuning of PI gains of the outer control loops of the GSC and the RSC can enhance the small-signal stability of a single grid-connected DIFG system

3.4

An Example Single Grid-Connected DFIG System

181

Fig. 3.16 Non-linear simulation of Test 3.4 when XLvaried (Test 3.5)

so as to increase the maximum loading of the DFIG. This is demonstrated in this test as follows. Following vector of PI gains of the outer control loops of the GSC and the RSC was defined. p = ½ K dp1

K di1

K dp3

K di3

K dp5

K di5 ]T

ð3:122Þ

In order to achieve the coordinated tuning of PI gains, the direct search algorithm was employed to solve the following optimization problem: max: zP1 ; subject to: f P1 ðp, zÞ = 0; p 2 Ρ

ð3:123Þ

where fP1(p, z) = 0 is given in (3.106) and Ρ is the space of feasible parameters (PI gains) of the outer loops of the GSC and RSC. Following solutions were obtained K dp5 = 0:025, K di5 = 10, K dp1 = 0:1, K di1 = 20, K dp3 = 0:1, K di3 = 20 With the tuned PI gains as given above, it was calculated that zP1 = 1.35. Hence, the maximum loading was increased to P0max = zP1 = 1.35. Results of evaluation by modal computation are presented in Table 3.6. It can be seen that the coordinated tuning of PI gains according to (3.122) and (3.123) increased the stability limit from P0max = 1.17 (Test 3.4) to P0max = 1.35. Non-linear simulation, as compared with that of Test 3.4 is presented in Fig. 3.17, confirming that the coordinated parameters tuning increased the maximum loading of the DFIG.

182

3

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

Table 3.6 Results of modal computation when P0 varied (Test 3.6) P0 λDC

1.20 -0.77625 ± j34.347

1.30 -0.42379 ± j33.973

1.40 0.24941 ± j33.406

Fig. 3.17 Non-linear simulation of Test 3.6

Summary of the Chapter Risk of sub/super-synchronous oscillations in a single grid-connected DFIG is analytically examined in this chapter. The examination leads to the following main conclusions. 1. Under the condition that the line connecting the DFIG to the external AC grid is fixed (i.e., XL is fixed), the small-signal stability limit is described by the maximum loading of the DFIG as to be either zP1 when dynamics of the PLL are in the fast timescale of electric current or zP2 when dynamics of the PLL are in the slow timescale of DC voltage control (Conclusions 3.2 and 3.4). The derived stability limit analytically explains why increased loading of the DFIG means increased risk of growing oscillations brought about by the grid-connected DFIG. The derived stability limit can guide the operation of the grid-connected DFIG to decide the maximum wind condition to avoid oscillatory instability. The guidance can be obtained by simply calculating either zP1 or zP2 directly with XL, the PI gains of the PLL, the outer control loops of the GSC and RSC being given without having to test the system small-signal stability under various wind conditions. 2. Under the condition that the loading condition of the DFIG is fixed, the smallsignal stability limit is depicted by the maximum XL to be either zL1 when dynamics of the PLL are in the fast timescale of electric current or zL2 when dynamics of the PLL are in the slow timescale of DC voltage control (Conclusions 3.1 and 3.3). The derived stability limit provides the analytical proof that under the condition of weak grid connection, the grid-connected DFIG is prone to the loss of stability, because either zL1 or zL2 gives the minimum SCR under

References

183

which the single grid-connected DFIG system is stable. In fact, if the rated power of DFIG is used as the loading condition, the minimum SCR can be calculated directly from zL1 and zL2.

References 1. Jiabing Hu, Bo Wang, Weisheng Wang, Haiyan Tang, Yongning Chi and Qi Hu. “Small Signal Dynamics of DFIG-Based Wind Turbines During Riding Through Symmetrical Faults in Weak AC Grid”, IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 32, NO. 2, JUNE 2017, pp720–730 2. Yipeng Song and Frede Blaabjerg. “Analysis of Middle Frequency Resonance in DFIG System Considering Phase-Locked Loop”. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 33, NO. 1, JANUARY 2018, pp343–356 3. Jiabing Hu, Yunhui Huang, Dong Wang, Hao Yuan, and Xiaoming Yuan. “Modeling of GridConnected DFIG-Based Wind Turbines for DC-Link Voltage Stability Analysis”, IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 6, NO. 4, OCTOBER 2015, pp1325–1335 4. Dong Wang,Liang Liang, Jiabing Hu, Naichao Chang, and Yunhe Hou. “Analysis of Low-Frequency Stability in Grid-Tied DFIGs by Nonminimum Phase Zero Identification”, IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 33, NO. 2, JUNE 2018, pp716–729 5. Wenjuan DU, Haifeng Wang and Siqi Bu, Small-Signal Stability Analysis of Power Systems Integrated with Variable Speed Wind Generators, Springer, 2018

Chapter 4

Analysis of Sub-synchronous Oscillations in a Sending-End Power System With Series-Compensated Transmission Line

Abbreviations CTA EOM GSC IGE MIMO PCC PF ROPS RSC SCL SG SISO SSCI SSR TI

Complex torque coefficients analysis Electromechanical oscillation mode Grid side converter Induction generator effect Multi-input multi-output Point of common coupling Participation factor Remainder of the power system Rotor side converter Series-compensated transmission line Synchronous generator Single-input single-output Sub-synchronous control interaction Sub-synchronous resonance Torque interaction

Sub-synchronous oscillations (SSOs) were firstly observed in a power system where a sending-end synchronous generator (SG) is connected to the main AC grid via a series-compensated line. This problem of the SSOs is referred to as the sub-synchronous resonance (SSR), as it was considered to be related to the resonance of the nature oscillation frequency of the series-compensated transmission line with the inherent frequencies of the shaft system of the SG. More recently, the problem of the SSOs was reported in a power system where a grid-connected wind farm, being consisted of double-fed induction generators (DFIGs), is connected to the main AC network by a series-compensated line. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Du, H. Wang, Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-031-35343-7_4

185

186

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

Using a single DFIG as the aggregated representation of the wind farm, the power system is comprised of a sending-end aggregated DFIG connected to the main AC grid via the series-compensated transmission line. It has been found that the SSOs are highly associated with the converter control system of the DFIG and caused by the unfavorable dynamic interactions between the DFIG and the series-compensated transmission line. Hence, this problem of the SSOs is referred to as the sub-synchronous control interactions (SSCIs) in the literature. This chapter introduces the analysis of the SSOs in a sending-end power system with the series-compensated transmission line. In the next section, the concept and theory of open-loop modal resonance are introduced to examine the SSR. For the introduction and examination, a single-input single-output (SISO) closed-loop interconnected model of the power system is used. The connection of the openloop modal resonance with the conventional method to examine the SSR, i.e., the complex torque coefficient analysis (CTA), is presented. In addition, the connection of open-loop modal resonance with the torque interaction (TI) and the induction generator effect (IGE), which explain why the SSR happens, is studied. In Sect. 4.2, the SSCI in a power system, where a sending-end DFIG is connected to the main grid via a series-compensated transmission line, is analyzed. The analysis is conducted according to the principle of conventional induction generator effect (IGE) and the open-loop modal resonance. In this section, the open-loop modal resonance analysis is based on the single-input and two-output closed-loop interconnected model, which is equivalent to the SISO model for stability analysis. In Sect. 4.3, the SSCI is examined by applying the open-loop modal analysis. For the examination, dynamics of the phase-locked loop (PLL) are ignored. Subsequently, the closed-loop interconnected model of the grid-connected DFIG system, established in the d-q coordinated of the DFIG, is simplified to have two inputs and one output. The simplification is in order to introduce the theory and principle of the open-loop modal resonance gradually from the simple case of SISO closed-loop interconnected model to the general case of multi-input and multi-output (MIMO) closed-loop interconnected model later in the next chapter. In the final section, the SSO in a power system with a sending-end PMSG connected to the main AC grid via a series transmission line is studied by using the theory of open-loop modal resonance.

4.1 4.1.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance The Torque Interaction and the Induction Generator Effect

The sub-synchronous resonance (SSR) is a kind of sub-synchronous oscillation (SSO) which occurs in a conventional series-compensated sending-end power

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

Fig. 4.1 Conventional series-compensated power system with an SG at the sending end

187

series compensation

~ SG

(a)

effective resistance of SG (b) Fig. 4.2 Closed-loop linearized model to examine the TI by using the CTA

u=0 +

+ Δ Te

M (s) Torsional subsystem

Δδ

Electrical subsystem

E (s)

system, which is comprised of a synchronous generator (SG) being connected to the main AC grid via a series-compensated transmission line (SCL), as being illustrated by Fig. 4.1(a). The mechanism about why the SSR may happen has been examined extensively in the literature. Two widely accepted explanations based on the linearized model of the series-compensated sending-end power system displayed in Fig. 4.1(a) about the SSR mechanism are the induction generator effect (IGE) and the torque interaction (TI). The IGE considers the series-compensated power system of Fig. 4.1(a) as an equivalent RLC circuit. Analysis indicates that at the sub-synchronous oscillation frequency, the effective resistance of rotor winding of the SG is negative. Subsequently, the RLC circuit is of negative resistance as being illustrated in Fig. 4.1(b). That is why the SSR happens to cause growing the SSO. The TI examines the dynamic interaction between the torsional system of the SG and the SCL. The examination is conducted by using the complex torque coefficients analysis (CTA) based on a linearized closed-loop model displayed in Fig. 4.2. The closed-loop model is comprised of the electrical subsystem and the torsional subsystem of the SG [1, 2]. The CTA attributes the SSR to the contribution of negative damping torque from the electrical subsystem to the torsional subsystem of the SG. In Fig. 4.2, the open-loop torsional subsystem is the sixth-order torsional dynamics of the SG; the open-loop electrical subsystem is comprised of the voltage equations of SG windings, dynamics of excitation system of the SG and the RLC equations of transmission line; ΔTe is the electrical torque and Δδ is the rotor angle of the SG. Let the state-space representations of open-loop torsional and electrical subsystem respectively be

188

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

d ΔXm = Am ΔXm þ bm ΔT e dt

Torsional subsystem:

ð4:1Þ

Δδ = cm ΔXm T

d ΔXe = Ae ΔXe þ be Δδ dt ΔT e = ce T ΔXe

Electrical subsystem:

ð4:2Þ

where ΔXm and ΔXe are the vectors of all the state variables of open-loop torsional and electrical subsystem, respectively. Transfer function models of open-loop torsional and electrical subsystem respectively are M ðsÞ = cm T ðsI - Am Þ - 1 bm , E ðsÞ = ce T ðsI - Ae Þ - 1 be

ð4:3Þ

From (4.1) and (4.2), the state-space representation of the closed-loop system of Fig. 4.2 can be obtained to be d ΔX = AΔX dt T

where ΔX = ΔXm T ΔXe T , A =

Am

ð4:4Þ

bm cTe

. Ae be cTm The CTA to examine the TI is briefly reviewed in Chap. 1 and introduced as follows: Let M ðsÞ = - km1ðsÞ , EðsÞ = k e ðsÞ. From (4.1) to (4.3), ΔT e = - k m ðsÞΔδ ΔT e = ke ðsÞΔδ

ð4:5Þ

where km(s) and ke(s) is defined respectively as the mechanical and electrical complex torque. Let s = jω and km ðjωÞ = K m ðωÞ þ jωDm ðωÞ ke ðjωÞ = K e ðωÞ þ jωDe ðωÞ

ð4:6Þ

where Km(ω) and Dm(ω) are named as the mechanical spring coefficient and damping coefficient at frequency ω respectively; Ke(ω) and De(ω) are named as the electrical spring coefficient and damping coefficient at frequency ω respectively. The CTA is conducted to identify the risk of the SSR in the following two steps:

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

189

1. Calculate the resonant frequency ωi by solving the following equation: K e ðωi Þ þ K m ðωi Þ = 0,

i = 0, 1, . . .

ð4:7Þ

2. At the resonant frequency ωi, the SSR will occur if De ðωi Þ þ Dm ðωi Þ < 0,

i = 0, 1, . . .

ð4:8Þ

The IGE explains the SSR mechanism without the torsional dynamics of the SG being involved. This is the main difference between the IGE and the TI to explain the SSR mechanism. The IGE considers that the series-compensated power system displayed in Fig. 4.1(a) constitutes a RLC circuit, which includes the effective resistance of rotor windings of the SG. Denote λe,sta ≈ -

R 2L

±j

1 LC =

- ξe ± jωe

as the oscillation mode of the RLC circuit, where C is the capacitance of series compensation, L and R is the total inductance and resistance of the electrical system respectively, excluding the torsional dynamics of the SG. Note that λe, sta is the oscillation mode of the stationary RLC circuit with a frequency of ωe. λe, sta is an oscillation mode of the electrical system excluding the torsional dynamics of the SG. Thus, it is the oscillation mode of open-loop electrical subsystem in Fig. 4.2. Since the closed-loop model of Fig. 4.2 is established in a synchronous rotating reference frame, when the frequency of the oscillation mode in the stationary circuit is observed from the synchronous rotating reference frame, the frequency should be ω0 - ωe. Thus, the open-loop SSO mode of electrical subsystem in Fig. 4.2 is λe ≈ - ξe ± j(ω0 - ωe). Subsequently, when the series compensation level increases, ωe increases such that ω0 - ωe decreases. The openloop SSO mode of electrical subsystem moves down with decreased frequency on the complex plane. The effective resistance of rotor windings of the SG is ωωe -e Rωr 0 , where ωe - ω0 < 0 such that the effective resistance is negative. The increase of ωe causes ωωe -e Rωr 0 to be more negative. Thus, the total resistance of the RLC circuit decreases when the series compensation level increases. Consequently, the open-loop SSO mode of electrical subsystem moves towards right on the complex plane when the series compensation level increases. The growing SSR occurs when the effective resistance is negative and greater than the positive resistance of the transmission line.

4.1.2

Open-Loop Modal Resonance Analysis Based on a Single-Input Single-Output Closed-Loop Interconnected Model

Denote λmi as an SSO mode of open-loop torsional subsystem. λmi is a complex eigenvalue of open-loop state matrix Am in (4.1). Denote λe as an SSO mode of

190

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

open-loop electrical subsystem. λe is a complex eigenvalue of open-loop state matrix Ae in (4.2). Additionally, denote λmi and λe as the SSO modes of closed-loop system of Fig. 4.2 corresponding to λmi and λe. λmi and λe are two complex eigenvalues of closed-loop state matrix A in (4.4). Normally, the mechanical shaft dynamics of the SG are almost dynamically decoupled with the electrical dynamics of the power system. Hence, within the range of the SSO frequency, Δδ in Fig. 4.2 should normally be very small, i.e., Δδ ≈ 0. This is why when the angular stability of the SG is examined, dynamics of the torsional system of the SG are not considered. Thus, the closed-loop system of Fig. 4.2 is open. Based on this recognition, a small positive number, 0 < ε ≪ 1, can be introduced to express the transfer function of the torsional subsystem in Fig. 4.2 as M(s) = εG(s). Therefore, it is expected that the closed-loop SSO modes are approximately equal to the open-loop SSO modes. The differences between the closed-loop and open-loop SSO modes, i.e., Δλe = λe - λe and Δλmi = λmi - λmi , are normally very small. This implies that on the complex plane, λe is close to λe and λmi is also close to λmi. Since λmi and λe are the poles of M(s) = εG(s) and E(s) respectively, the transfer functions of open-loop torsional and electrical subsystems in Fig. 4.2 can be expressed as M ðsÞ = εGðsÞ =

εgðsÞ eðsÞ , E ð sÞ = s - λmi s - λe1

ð4:9Þ

The characteristic equation of the closed-loop of Fig. 4.2 is M(s)E(s) = 1. Since M(s) = εG(s)E(s), the characteristic equation can be written as εGðsÞEðsÞ - 1 = 0

ð4:10Þ

εhðsÞgðsÞ = ðs - λe1 Þðs - λmi Þ

ð4:11Þ

f ðsÞ = ðs - λe1 Þðs - λmi Þ, F ðsÞ = eðsÞgðsÞ

ð4:12Þ

Substitute (4.9) in (4.10)

Let

With (4.12), characteristic equation (4.10) becomes εF ðsÞ = f ðsÞ

ð4:13Þ

λmi = λmi þ Δλmi is a solution of the characteristic equation of (4.13). Hence,

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

191

ð4:14Þ

εF ðλmi þ Δλmi Þ = f ðλmi þ Δλmi Þ Taylor series expansion of (4.14) at λmi is ε½F ðλmi Þ þ F 0 ðλmi ÞΔλmi þ

F 00 ðλmi Þ Δλmi 2 þ ⋯ 2!

f 00 ðλmi Þ = f ðλmi Þ þ f ðλmi ÞΔλmi þ Δλmi 2 þ ⋯ 2!

ð4:15Þ

0

According to the theory of eigenvalue sensitivity, for a distinct eigenvalue, the expansion of eigenvalue small increment can be written as [3] Δλmi = α1 ε þ α2 ε2 þ α3 ε3 þ ⋯

ð4:16Þ

Substitute (4.16) in (4.15), - ε F ðλmi Þ þ F 0 ðλmi Þ α1 ε þ α2 ε2 þ α3 ε3 þ ⋯ F 00 ðλmi Þ 2 α1 ε þ α2 ε2 þ α3 ε3 þ ⋯ þ ⋯] 2! f ðλmi Þ þ f 0 ðλmi Þ α1 ε þ α2 ε2 þ α3 ε3 þ ⋯

þ =

þ

f 00 ðλmi Þ α1 ε þ α2 ε2 þ α3 ε3 þ ⋯ 2!

2

ð4:17Þ

þ⋯

Both sides of (4.17) are the polynomials of ε. By equating the coefficients of the first-order term of ε on the both sides of (4.17), it can have α1 =

F ðλmi Þ f 0 ðλmi Þ

ð4:18Þ

From (4.9), f ð s Þ0

s = λmi

= ðs - λmi Þ þ ðs - λe1 Þjs = λmi = ðλmi - λe1 Þ

ð4:19Þ

From (4.16), (4.18), and (4.19), the first-order approximation of Δλmi in terms of 0 < ε ≪ 1 is obtained to be Δλmi = λmi - λmi ≈ ε

F ðλmi Þ eðλ Þgðλmi Þ = ε mi ðλmi - λe1 Þ ðλmi - λe1 Þ

ð4:20Þ

λe1 = λe1 þ Δλe1 is also a solution of the characteristic equation of (4.13). Hence, taking the derivation similar to that from (4.14) to (4.20), it can have

192

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

Δλe1 = λe1 - λe1 ≈ ε

eðλe1 Þgðλe1 Þ ðλe1 - λmi Þ

ð4:21Þ

Equations (4.20) and (4.21) give the first-order approximation of Δλe = λe - λe and Δλmi = λmi - λmi . They indicate that Δλe = λe - λe and Δλmi = λmi - λmi are the functions of 0 < ε ≪ 1. From (4.20) and (4.21), it can be seen that when λe is close to λmi on the complex plane, Δλe = λe - λe and Δλmi = λmi - λmi may not be small any more. This special case can be further examined as follows. Assume that initially λe and λmi are away from each other on the complex plane. Δλe = λe - λe and Δλmi = λmi - λmi are small as indicated by (4.20) and (4.21). When the level of series compensation, c, varies, λe and λe shall move together on the complex because Δλe = λe - λe is small; whilst λmi should be close to the stationary λmi and does not move considerably because Δλmi = λmi - λmi is also very small. However, when λe moves close to λmi, Δλe = λe - λe and Δλmi = λmi - λmi increase as indicated by (4.20) and (4.21). In addition, when λe and λmi get close to each other such that e(λmi)g(λmi) → e(λe)g(λe), from (4.20) and (4.21) it can be seen that Δλe and Δλmi may be of the opposite signs. This implies that λe and λmi intend to move towards the opposite directions on the complex plane when λe moves close to λmi. On the other hand, when λe moves away from λmi later with the variation of c, from (4.53) and (4.54) it can be seen that λe and λmi should move back to be close to λe and λmi. Hence, the SSO modal repulsion is formed as the result of moving close and apart of two open-loop SSO modes, λe and λmi, on the complex plane. The nearness of λe and λmi is named as the open-loop modal resonance. Illustration of the open-loop modal resonance as introduced above is presented in Fig. 4.3.

4.1.3

Peak of the Closed-Loop SSO Modal Repulsion

When λe moves close to λmi on the complex plane with the variation of the level of series compensation, c, Δλe1 and Δλmi will not increase to the infinity as indicated by (4.20) and (4.21). This is because what Eqs. (4.20) and (4.21) give is the first-order Fig. 4.3 Illustration of the open-loop modal resonance

λˆmi

Imag axis

Open-loop λ e ≈ λmi modal resonance

Closed-loop modal repulsion

λˆe

Closed-loop modal repulsion SSO modes of open-loop subsystems Closed-loop SSO modes

Real axis

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

193

approximation of Δλe1 and Δλmi. When Δλe1 and Δλmi are not small, they cannot be estimated by (4.20) and (4.21). However, Eqs. (4.20) and (4.21) indicate that the peak of increase of Δλe and Δλmi may be reached when λe is closest to λmi. That is the peak of closed-loop SSO modal repulsion. The peak of SSO modal repulsion provides very important information about the risk of the SSR when the open-loop modal resonance happens. For example, if the peak falls in the right half of complex plane, growing SSR occurs. In this subsection, estimation of the peak of closed-loop SSO modal repulsion without calculating the root loci of the closed-loop SSO modes is discussed. Since the peak is when λe is closest to λmi, the peak of closed-loop SSO modal repulsion can be estimated under the condition that λe = λmi as follows. Express the transfer function of the torsional and electrical subsystems in Fig. 4.2 as M ðs Þ =

Rmi þ ðs - λmi Þ

n

Rmj s - λmj

j=1

ð4:22Þ

j≠i m

E ðsÞ =

Rej Re þ ðs - λe Þ j = 1 s - λej

where Re and Rmi are the residues for λe and λmi respectively; λmj, j = 1, 2, . . ., n, j ≠ i and λej, j = 1, 2, . . ., m are the other poles of open-loop torsional and electrical subsystem respectively; Rmj, j = 1, 2, . . ., n and Rej, j = 1, 2, . . ., m are the residues for λmj, j = 1, 2, . . ., n and λej, j = 1, 2, . . ., m separately. According to the theory of eigenvalue sensitivity, for a double eigenvalue (λe1 = λmi), the expansion of eigenvalue small increment can be written as [3] 1

2

3

Δλmi = β1 ε2 þ β2 ε2 þ α3 ε2 þ ⋯

ð4:23Þ

Hence, when λe1 = λmi, from (4.19) it can have f(λmi)′ = 0 and f(λmi)″ = 2. Substitute (4.23) in (4.15), - ε½F ðλmi Þ þ F 0 ðλmi Þ β1 ε2 þ β2 ε2 þ α3 ε2 þ ⋯ 1

þ

2

3

1 2 3 F 00 ðλmi Þ β1 ε2 þ β2 ε2 þ α3 ε2 þ ⋯ 2!

2

þ ⋯]

= f ðλmi Þ þ f 0 ðλmi Þ β1 ε þ β2 ε þ α3 ε þ ⋯ 1 2

þ

2 2

3 2

1 2 3 f 00 ðλmi Þ β 1 ε2 þ β 2 ε2 þ α 3 ε2 þ ⋯ 2!

2

ð4:24Þ

þ⋯

By equating the coefficients of the first-order term of ε on the both sides of (4.24), it can have

194

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

F ðλmi Þ =

f 00 ðλmi Þ 2 β1 2!

ð4:25Þ

F ðλmi Þ

ð4:26Þ

From (4.25) and f(λg)″ = 2, β1 = ± From (4.9) and (4.12), it can have 1 F ðλmi Þ = gðλmi Þeðλmi Þ = ðs - λmi Þðs - λe1 ÞM ðλmi ÞE ðλmi Þ ε

s = λmi

ð4:27Þ

Substituting (4.22) in (4.27), it can have F ðλmi Þ =

1 R R ε mi e1

ð4:28Þ 1

Hence, the first-order approximation of λmi in terms of 0 < ε2 ≪ 1 is obtained from (4.23), (4.26), and (4.28) to be p 1 λmi ≈ λmi þ β1 ε2 = λmi ± Rmi Re1

ð4:29Þ

Taking the derivation similar to that from (4.23) to (4.28) for λe1 , the first-order 1 approximation of λe1 in terms of 0 < ε2 ≪ 1 can be obtained to be p p λe1 ≈ λe1 ± Rmi Re1 = λmi ± Rmi Re1

ð4:30Þ

Rearranging (4.29) and (4.30), it can have p Δλmi = λmi - λmi ≈ ± Rmi Re p Δλe = λe - λe ≈ ± Rmi Re

ð4:31Þ

Following conclusions can be made from (4.31): 1. The maximum damping degradation may occur when λe is closest to λmi on the complex plane. This is when the open-loop modal resonance is strongest. 2. One of closed-loop SSO modes may be in the right half of the complex plane such that growing SSR may occur, if Real part of

p

Re Rmi > jReal part of λe or λmi j,

ð4:32Þ

Physical meaning of the closed-loop SSO modal repulsion when the open-loop modal coupling occurs is as follows: λe is the open-loop pole of electrical subsystem

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

195

in Fig. 4.2. Hence, |E(λe)| = 1. When λmi is close to λe on the complex plane, i.e., λe ≈ λmi, |E(λmi)| should be significant. The closer λmi is to λe, the bigger |E(λmi)| is. Thus, when λe is closest to λmi such that the open-loop modal resonance is the strongest, |E(λmi)| is most significant. Hence, under the condition of the open-loop SSO modal resonance, it is very likely that ΔTe = E(λmi)Δωr may be significant. ΔTe is the physical exhibition of dynamic interactions between the electrical and torsional subsystem. Hence, the physical exhibition of the open-loop SSO modal resonance is the strong dynamic interactions between the torsional dynamics of the SG and the electrical dynamics of the power system around the open-loop torsional SSO frequency.

4.1.4

Demonstration Using the IEEE First Benchmark Power System

In this subsection, IEEE first benchmark power system for the SSR study is used to demonstrate the analysis of open-loop modal resonance introduced in the previous subsection. Details about it can be found in [4]. When the level of series compensation, c, in the IEEE first benchmark power system varied from 0.01 to 1, trajectories of closed-loop SSO modes, λmi , i = 1, 2, 3, 4, 5 and λe , on the complex plane are displayed in Fig. 4.4 as solid curves. From Fig. 4.4, it can be observed that when c varied, λe moved close to λmi , i = 1, 2, 3, 4, 5 consecutively. The closeness between λe and each of λmi , i = 1, 2, 3, 4, 5 caused the separation of λe and λmi , i = 1, 2, 3, 4, 5 on the complex plane as if they repulsed each other. This is the phenomenon of the closed-loop SSO modal repulsion displayed in [5]. The phenomenon was explained in [5] as the results of the near strong modal resonance between λe and λmi , i = 1, 2, 3, 4, 5. Strong modal resonance is a concept established according to the theory of eigenvalue sensitivity [6]. It is the coincidence of two eigenvalues and their associated eigenvectors of a matrix, such as the closed-loop state matrix A in (4.4). It was mathematically proved that when two eigenvalues move close to each other on the complex plane with the variation of a system parameter, their movement turns 90° towards the opposite directions at the point of their coincidence. However, eigenvalue-to-parameter sensitivity increases to infinity when two eigenvalues move close to each other towards the point of their coincidence [6]. Hence, in the IEEE first benchmark power system, two closed-loop SSO modes, λmi and λe , cannot coincide [5]. Alternatively, when they moved close to each other on the complex plane with the variation of c, they separated at a point near their coincidence, which was referred to as the point of near strong modal resonance. Afterwards, they moved towards the opposite directions. Obviously, this was what happened in Fig. 4.4 when λe moved close to and seemed “pushing” λmi , i = 1, 2, 3, 4, 5 away towards the right on the complex plane, causing the damping degradation of λmi , i = 1, 2, 3, 4, 5.

196

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

The trajectory of open-loop SSO mode of the electrical subsystem, λe, with the variation of c was calculated for the IEEE first benchmark power system and is shown by dashed curve in Fig. 4.4. It can be seen that when λe was away from λmi, i = 1, 2, 3, 4, 5, λe was close to λe and moved along with λe. However, when λe moved close to λmi, i = 1, 2, 3, 4, 5, λe and λmi , i = 1, 2, 3, 4, 5 moved towards the opposite directions on the complex plane. When λe moved away from λmi, i = 1, 2, 3, 4, 5 later, Δλe and Δλmi, i = 1, 2, 3, 4, 5 decreased such that λe and λmi moved back to be close to λe and λmi, i = 1, 2, 3, 4, 5 respectively. In Fig. 4.4, positions of λe when it was closest to λmi, i = 1, 2, 3, 4, 5 (indicated by hollow circles) are shown by hollow triangles on the trajectory of λe (dashed curve) for the IEEE first benchmark power system. Actual peaks of the SSO modal repulsion when λe was closest to λmi, i = 1, 2, 3, 4, 5 are indicated by solid circles and triangles. They confirm that when λe was closest to λmi, i = 1, 2, 3, 4, 5, damping degradation of torsional SSO modes was maximum. For each p pair of closest λe and λmi, i = 1, 2, 3, 4, 5 indicated by hollow circle and are given in the triangle, Re Rmi was computed and the computational results p R 4.1. It can be seen that among R second column of Table e mi , i = 1, 2, 3, 4, 5, p p Real part of Re Rm1 is smallest and Real part of Re Rm5 is largest. This explains why the SSO modal repulsion between λe and λm1 was almost invisible and that between λe and λm5 was biggest. p From the computational results of Re Rmi , i = 1, 2, 3, 4, 5 given in the second column of Table 4.1, positions of λe and λmi , i = 1, 2, 3, 4, 5 were estimated by using (4.29) and (4.30) when the open-loop modal resonance was strongest. The estimated

310

λˆ λm1m1

:Open-loop mode of the electrical subsystem :Open-loop mode of the torsional subsystem :Positions of open-loop mode of the electrical subsystem when the modal repulsion occurs

270

λˆe1 λe1

I ma gina ry axis

230

λˆm 2

λm 2

:The peak of modal repulsion of electrical subsystem and torsional subsystem :Estimated position of the peak of modal repulsion

190

λm 3 150

λm 4 λˆm 4 λm 5

110 70 -8

λˆm 3

-6

-4

-2

λˆm 5

0

2

4

6

Real axis 8

Fig. 4.4 Closed-loop SSO modal repulsion and open-loop modal resonance in the IEEE first benchmark power system

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

Table 4.1 Computational results of SSO residues p Re Rmi

i 1 2 3 4 5

p

Re Rmi (Fig. 4.4) 0.0288 - j0.0023 4.7783 - j0.1148 4.6157 - j0.1772 1.5756 + j0.0696 6.0396 + j0.0922

197 p

Re1 Rmi (Fig. 4.5) 0.0288 - j0.0017 4.7791 - j0.0850 4.6169 - j0.1507 1.5743 + j0.0858 6.0340 + j0.1587

positions are indicated by crosses in Fig. 4.4. It can be seen that the estimated positions were very close to the actual positions of closed-loop SSO modes which are indicated by solid triangles and circles in Fig. 4.4, confirming the correctness of (4.29) and (4.30) for the estimation of peak of SSO modal repulsion. In order to demonstrate that the open-loop modal resonance may cause the damping degradation of not only the torsional SSO modes, but also the electrical SSO modes, line resistance in the IEEE first benchmark power system was decreased from 0.02, which was the given value in [4], to 0.009. The positions and trajectories of open-loop and closed-loop SSO modes with variation of c are displayed in Fig. 4.5. It can be seen that this time, λe approached λmi, i = 1, 2, 3, 4, 5 from the right-hand side, instead of the left-hand side in Fig. 4.4, on the complex plane. The damping of closed-loop electrical SSO mode, λe , degraded as the result of the openloop modal resonance. p The computational results of Re Rmi , i = 1, 2, 3, 4, 5 for each point when λe was closest to λmi, i = 1, 2, 3, 4, 5 are presented in the third column of Table 4.1. The positions of λmi and λe were estimated by using (4.29) and (4.30) and are indicated by crosses in Fig. 4.5. It can be seen that the estimated positions are close to the actual positions of the closed-loop SSO modes indicated by filled triangles and circles, confirming the correctness of (4.29) and (4.30). Following-on test was conducted with the torsional damping constants of each of six mass units being increased from (1, 2, 0, 15, 0, 0.12) to (55, 56.5, 54.5, 55, 52, 53.5) such that λmi, i = 1, 2, 3, 4, 5 moved away from the open-loop electrical SSO mode, λe. The positions and trajectories of open-loop and closed-loop SSO modes on the complex plane with the variation of c are shown in Fig. 4.6. It can be seen that since λe was away from λmi, i = 1, 2, 3, 4, 5, little modal repulsion was observed. When the electrical frequency was close to the SSO frequency of the torsional dynamics of the SG, no considerable damping degradation of the electrical SSO mode occurred. This also explains why the damping increase of the torsional dynamics of the SG can improve the damping of the SSR associated with the electrical SSO mode. Increasing the resistance of the transmission line can directly improve the damping of the open-loop electrical SSO mode, such that λe moves away from λmi, i = 1, 2, 3, 4, 5. The peak of SSO modal repulsion should decrease. To demonstrate this aspect, the sensitivity of the peak of closed-loop SSO modal repulsion to the transmission line resistance under the condition that λe ≈ λm5 was computed. The computational results are presented in Fig. 4.7. It can be seen that the sensitivity

198

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

4 310

λˆm1

:Open-loop mode of the electrical subsystem

λm1

:Open-loop mode of the torsional subsystem :Positions of open-loop mode of the electrical subsystem when the modal repulsion occurs

270

:The peak of modal repulsion of electrical subsystem and torsional subsystem :Estimated position of the peak of modal repulsion

Ima gina ry axis

230

λˆm 2

λm 2

190

λˆm 3 150

110 70 -8

λm 3 λˆm 4 λm 4

λˆm 5

-6

λm 5 -4

-2

λe1

λˆe1

0

2

4

6

Real axis 8

Fig. 4.5 Open-loop modal resonance when λe approached λmi, i = 1, 2, 3, 4, 5 from right-hand side

decreased with the increase of the line resistance. This explains why the increase of line resistance can enhance the damping of the SSR. The open-loop modal resonance interprets the mechanism of the TI to cause the SSR, as being demonstrated above by using the IEEE first benchmark power system. The IGE is mainly determined by the effective resistance of rotor windings of the SG, i.e., ωωs -s Rωr 0 , where Rr is the resistance of rotor windings, ω0 is the synchronous angular frequency, and ωs is the sub-synchronous frequency of the electrical subsystem. Over the SSO frequency range, ωs - ω0 < 0 such that the effective resistance is negative. If this results in the totally negative resistance of the RLC circuit, unstable SSR occurs. In the IEEE first benchmark power system, when Rr increases, the IGE increases accordingly such that the open-loop SSO mode of electrical subsystem, λe, moves towards the right-hand side on the complex plane. There are two possible consequences of λe moving right on the complex plane as caused by the IGE. The first possible consequence is that λe moves towards the right on the complex plane without getting close to any torsional SSO mode of the SG which are of light damping in the IEEE first benchmark power system. Thus, no open-loop modal resonance occurs such that the closed-loop and open-loop SSO mode of electrical subsystem are approximately equal, i.e., λe ≈ λe . In this case, the closed-loop model of Fig. 4.2 may only become unstable in the worst case when λe moves into the right half of complex plane. The IGE is completely irrelevant with the open-loop modal resonance. Figure 4.8 presents the modal trajectories when the level of series compensation was c = 0.6 and Rr increased. It shows that the IGE causes λe moving towards the

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

199

Fig. 4.6 The case where little open-loop modal resonance were observed

Fig. 4.7 Sensitivity of the peak of SSO modal repulsion to the line resistance when the line resistance was varied

Sensitivity

110 100 90 80 70

60 50

0.02

Line resistance 0.04 0.03

0.05

right on the complex plane. No open-loop modal resonance caused by the IGE such that the closed-loop and open-loop SSO modes are close to each other on the complex plane. The second possible consequence is that if the IGE causes λe moving into the proximity of a lightly-damped open-loop torsional SSO mode, λmi, i = 1, 2, 3, 4, 5, on the complex plane, the OLMP occurs to cause the damping decrease of λmi , i = 1, 2, 3, 4, 5. In this case, strong TI occurs as caused by the OLMP, leading to under-damped or unstable SSR.

200

4

Fig. 4.8 IGE increased to cause no open-loop model resonance in the IEEE first benchmark power system

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

170 Imag

λm 3

160

λˆm 3

150 140

λˆe

λe λm 4

130 120 -5

-4

-3

-2

-1

λˆm 4 0

1

Real 2

Figure 4.9 shows the results of modal computation of the IEEE first benchmark power system when the level of series compensation was c = 0.51. In Fig. 4.9, hollow circles and crosses are the positions of open-loop and closed-loop SSO modes of the IEEE first benchmark power system respectively. When Rr increased, the increased IGE led to λe moving close to the third torsional SSO mode of the SG, λm3, on the complex plane. Subsequently, closed-loop torsional SSO mode, λm3 , was pushed towards the right into the right half of complex plane owing to the open-loop modal resonance. Hence, Fig. 4.9 is the case that the increase of the IGE resulted in the open-loop modal resonance, leading to growing SSR. This study case demonstrates that though the IGE and the open-loop modal resonance are different, they happened consecutively. Figure 4.10 summarizes how growing SSR in the IEEE first benchmark power system occurs as caused by the IGE and the open-loop modal resonance which clearly shows the connections between the IGE and the open-loop modal resonance in the series-compensated power system with the SG at the sending end.

4.1.5

Equivalence of the Open-Loop Modal Resonance Analysis to the Complex Torque Coefficient Analysis in Detecting the Risk of Growing SSR

Both the open-loop modal resonance analysis and the complex torque coefficient analysis (CTA) can detect the risk of growing SSR as caused by the TI. Detection by the complex torque coefficient analysis is given by (4.7) and (4.8). Risk identification by the open-loop modal resonance analysis is based on (4.32). In this subsection, the equivalence of the open-loop modal resonance analysis to the complex torque coefficient analysis in detecting the risk of growing SSR is examined. Let Re = ae + jbe and Rmi = am + jbm. Equation (4.32) can be written as

4.1

Open-Loop Modal Resonance Analysis of Sub-synchronous Resonance

Fig. 4.9 IGE increased to cause the open-loop modal resonance in the IEEE first benchmark power system

201

170 Imag 160

λe

λˆe

λm 3

λˆm 3

150 Rr increases

140

λm 4 λˆ

130 120 -5

Rotor resistance increases Series compensation level increases

m4

-4

-3

-2

-1

Weak TI SSO stability of open-loop Open-loop SSO electrical mode of electrical subsystem subsystem moves degrades

IGE

towards the right

Open-loop modal resnance Strong TI

Open-loop SSO mode of electrical subsystem moves down on the complex plane

0

Real 2

1

SSO stability of closed-loop system degrades

Fig. 4.10 Connections of affecting factors to cause growing SSR in IEEE first benchmark power system

Real part of ðRe Rmi Þ = ae am - be bm > ðReal part of λmi Þ2 ,

ð4:33Þ

Since normally the imaginary part of an SSO mode is much greater than the real part, am ≪ bm and ae ≪ be. Equation (4.33) can be simplified as - be bm > ðReal part of λmi Þ2 ,

ð4:34Þ

From Fig. 4.2 and Eq. (4.22), -

Rmi 1 = M ðs Þ = þ k m ðsÞ ðs - λmi Þ

n j=1

Rmj s - λmj

j≠i

ð4:35Þ

m

ke ðsÞ = E ðsÞ =

Rej Re þ ðs - λe Þ j = 1 s - λej

Denote λmi = σ m + jωm and λe = σ e + jωe. When ω is around λm ≈ λe (open-loop modal resonance happens),

202

4

-

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

1 Rmi = M ðjωÞ = þ k m ðjωÞ ðjω - λmi Þ

n j=1 j≠i

Rmj Rmi ≈ ðjω - λmi Þ jω - λmj

ð4:36Þ

m Rej Re Re k e ðjωÞ = EðjωÞ = þ ≈ ðjω - λe Þ j = 1 jω - λej ðjω - λe Þ

Subsequently, it can have σ m þ jðω - ωm Þ am þ jbm ae þ jbe k e ðjωÞ = - σ e þ jðω - ωe Þ k m ðjωÞ = -

ð4:37Þ

From (4.6) and (4.37), it can have Dm ðωÞ = De ðωÞ =

- bm σ m - am ð ω - ω m Þ ω am 2 þ bm 2 - be σ e - ae ðω - ωe Þ

ð4:38Þ

ω σ e 2 þ ðω - ω e Þ2

Since ω ≈ ωm ≈ ωe, (ω - ωm) and (ω - ωe) are very small and can be ignored. From (4.38), Dm ðωÞ þ De ðωÞ =

bm σ m σ e þ b e am 2 þ bm 2 - σ e ω am 2 þ bm 2

ð4:39Þ

Hence, Eq. (4.8) can be written as b m σ m σ e þ be am 2 þ bm 2 jReal part of λDFIG or λSCL j

ð4:48Þ

Consider the simplified circuit model of the DFIG system with series compensation shown by Fig. 4.13. In the Laplace domain, slip is expressed as slip = s -sjωr [8]. The open-loop DFIG subsystem is comprised of impedance model of the DFIG, Zs, and the shunt capacitance of the transmission line, Cp. Hence, the transfer function of open-loop DFIG subsystem is

Z DFIG ðsÞ = Z s ==

Ls Lr - L2m . s s - jωr - L LLs-RrL2 1 s r m = sC p Rr C1 ðsÞ þ Lr s - jωr - Lr

ð4:49Þ

where C1 ðsÞ = sC p . Ls Lr - L2m . s s - jωr -

Z s ðsÞ =

Ls Lr - L2m . s s - jωr Lr s - jωr -

Ls Rr Ls Lr - L2m

Ls R r Ls Lr - L2m

Rr Lr

Since Cp is very small, C1(s) ≈ 0 in (4.49). In [10] and [11], modal analysis was applied to study the oscillatory stability of the DFIG system with series compensation displayed in Fig. 4.15. The results of modal analysis indicated that there is an oscillation mode of the DFIG, frequency of which changes with the variation of wind speed. The oscillation mode, however, is affected only slightly by the compensation level of the transmission line. In addition, the frequency of the oscillation mode is complimentary to the frequency of shaft speed of the wind turbine. Thus, the oscillation mode is related to the mechanical dynamics of wind turbine. The computational results of the participation factors in [10] and [11] show that the oscillation mode is also associated with the stator and rotor current. Hence, this particular oscillation mode is related to both mechanical and electrical dynamics of the DFIG and is named as the electromechanical oscillation mode of the DFIG in [10] and [11]. Note that the electromechanical oscillation mode of the DFIG named in [10] and [11], frequency of which is around 10–20 Hz,

208

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

4

l SCL

210 Imag

190

c incr ease d

170 150 130 110

90

①

②

lˆ DFIG

①

210 Imag

A

l DFIG

lˆ SCL

70

170 ②

③

lˆ DFIG

l DFIG lˆ

150 130

DFIG

110 90

lˆ SCL

70

③

50 30 -10 -8 -6

c incr ease d

190

lˆ SCL

-4 -2

0

2

Real 4 6

50 30 -10 -8 -6

a.

l SCL -4 -2

0

2

4

Real 6

b.

Fig. 4.15 Results of open-loop modal resonance analysis. (a) Rr + kprsc ≠ 0, (b) Rr + kprsc = 0

is different to the conventional electromechanical oscillation mode of a synchronous generator (SG), which is of much lower oscillation frequency. From (4.49), it can be seen that ZDFIG(s) is of an oscillation mode approximately to be equal to λDFIG = jωr - RLrr . The frequency of the oscillation mode is roughly determined by the rotor speed of the DFIG, ωr. Hence, λDFIG = jωr - RLrr is the approximate expression of electromechanical oscillation mode (EOM) of the DFIG studied in [10] and [11]. From (4.49), the residue of the open-loop DFIG subsystem for λDFIG can be found to be (C1(s) = 0) RDFIG = Z DFIG . ðs - λEOM Þjs = λEOM = Rr .

L2m R jωr - r Lr L2r

ð4:50Þ

When Rr = 0, there is no IGE in the grid-connected DFIG system with series compensation displayed in Fig. 4.15. Equation (4.50) indicates that in this case, RDFIG = 0 too. According to (4.47), the impact of open-loop modal resonance is zero. This implies that as far as the electromechanical oscillation mode of the DFIG is concerned, the IGE is equivalent to the impact of the open-loop modal resonance. As the simplified model of Fig. 4.13 is for explaining the idea of negative resistance in the IGE analysis, examination above only demonstrates the concept about the connection between the IGE and the open-loop modal resonance. Works in the literature indicated that the negative resistance of the DFIG is highly related with the current control inner loop of the RSC of the DFIG [8]. Hence, the connection between the IGE and the open-loop modal resonance is examined by focusing on the current control inner loop of the RSC as follows. When the RSC is considered, the transfer function of open-loop DFIG subsystem can be derived from Fig. 4.12 to be

4.2

The Induction Generator Effect Analysis of Sub-Synchronous Oscillations. . .

Z DFIG ðsÞ =

Ls Lr - L2m . sðs - s1 Þðs - s2 Þ C 2 ðsÞ þ Lr ðs - jωs Þ s - jωr -

Rr þk prsc Lr

þ k irsc

209

ð4:51Þ

where kprsc and kirsc are the proportional and integral gain of the current inner control loop of the RSC respectively; C 2 ðsÞ = sC p . Ls Lr - L2m . sðs - s1 Þðs - s2 Þ ≈ 0 s1 = s2 = K=

s0 þ jωb þ

ðs0 - jωb Þ2 - 4K , 2

s0 þ jωb -

ðs0 - jωb Þ2 - 4K , 2

Ls kirsc , Ls Lr - L2m

s0 = jωr -

Ls Rr þ k prsc : Ls Lr - L2m

Comparing ZDFIG(s) expressed by (4.49) and (4.51), it can be noted that when the RSC control is considered, not only ZDFIG(s) is of one more oscillation mode in addition to the electromechanical oscillation mode, but also the electromechanical R þk oscillation mode approximately is λDFIG ≈ jωr - r Lr prsc when kirsc is small. This means that the proportional gain of current control inner loop of the RSC together with Rr affects the damping of the electromechanical oscillation mode of the openloop DFIG subsystem. It explains why the IGE is found being affected by the current control inner loop of the RSC. In addition, the residue for λDFIG is RDFIG = Rr þ kprsc .

Rr þ kprsc L2m jωr 2 Lr Lr

ð4:52Þ

Equation (4.52) shows that if Rr + kprsc = 0, RDFIG = 0. According to (4.47), the open-loop modal resonance taken part in by λEOM does not affect the closed-loop stability of the grid-connected DFIG system with series compensation. Study in the literature indicated that in the grid-connected DFIG system with series compensation, when the converter control system is considered, the proportional gain of the current control loop, kprsc, is equivalent a part of effective resistance of the DFIG. Hence, when both Rr = 0 and kprsc = 0, i.e., Rr + kprsc = 0, there is no IGE in the grid-connected DFIG system with series compensation. Hence, analysis above indicates that the connection between the IGE and openloop modal resonance is related with the electromechanical oscillation mode of the DFIG: If the open-loop modal resonance is taken part in by the electromechanical oscillation mode, the IGE and the impact of open-loop modal resonance are the same.

210

4.2.3

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

An Example Grid-Connected DFIG System With Series-Compensated Transmission Line

An example grid-connected DFIG system with series compensation is used to demonstrate and validate the analysis in the previous subsection about the relationship between the IGE and open-loop modal resonance. The structure of the example is shown by Fig. 4.11. The detailed model and parameters of the grid-connected DFIG system with series compensation given in [12] are used. Analysis conducted in the previous subsection is based on the single-phase circuit model of the grid-connected DFIG system with series compensation established in the abc coordinate as being shown in Figs. 4.12 and 4.13. Analytical conclusions obtained should be applicable regardless of coordinate systems being used. Hence, following demonstration and validation are conducted on the basis of the model of example grid-connected DFIG system with series compensation established in the dq coordinate. The SSO mode of open-loop SCL subsystem, λSCL, is the pole of Y SCL ðsÞ = 1=Z SCL ðsÞ. The variation of λSCL with the increase of level of series compensation, c, is calculated and is displayed in Fig. 4.15 as the dashed trajectory. In Fig. 4.15, hollow triangle is the position of electromechanical oscillation mode, λDFIG, of the DFIG and is an open-loop SSO mode of DFIG subsystem. λDFIG and λSCL are the closed-loop SSO modes corresponding to λDFIG and λSCL separately. The variations of λDFIG and λSCL with the increase of c are displayed in Fig. 4.15 as the solid trajectories. Denote λDFIG and λSCL respectively as λDFIG = - ξDFIG ± jωDFIG and λSCL = ξSCL ± jωSCL. The sub-synchronous frequency in (4.42) is ωs = ω0 - ωSCL. When c increases, ωSCL decreases such that ωs increases. ωDFIG is determined by the rotor speed of DFIG, ωr, and changes little with the increase of c. Hence, when ωSCL > ωDFIG, ωs is smaller than ωr but approaches ωr when c increases. According to (4.42), the effective resistance of the DFIG should be more negative when c increases such that SSO stability of the example grid-connected DFIG system with series compensation degrades. This explanation about the SSO stability degradation of the grid-connected DFIG system with series compensation is well demonstrated by Fig. 4.15a: With increase of c, λSCL moves close to λDFIG on the complex plane. Consequently, the damping of λSCL decreases to become more negative as shown in Fig. 4.15a, confirming the correctness of (4.42) and interpretation made by the IGE. However, if c continues to increase from ωSCL > ωDFIG to ωSCL < ωDFIG, effective resistance of the DFIG should change from being negative to positive, because when ωSCL < ωDFIG, ωs is bigger than ωr such that the effective resistance of the DFIG is positive as indicated by (4.42). This implies that damping of λSCL should change signs from being negative to positive with the increase of c around ωSCL = ωDFIG. In addition, when ωSCL < ωDFIG, increase of c results in increase of ωs - ωr such that the effective resistance of DFIG decreases and damping of λSCL

4.2

The Induction Generator Effect Analysis of Sub-Synchronous Oscillations. . .

211

decreases to become less positive. A sudden change of signs of the damping of λSCL with the continuous increase of c cannot happen in the example grid-connected DFIG system with series compensation due to the continuity principle of eigenvalues. This is a defect of the IGE but can be explained by the results of open-loop modal resonance analysis presented in Fig. 4.15a. When c = 0.45, the open-loop modal resonance, i.e., λSCL ≈ λDFIG, happens around point A in Fig. 4.15a where ωSCL ≈ ωDFIG. It is found that λDFIG and λSCL swap positions at point ② on the complex plane as indicated in Fig. 4.15a. Due to the positions swapping, the damping of λSCL changes abruptly from being negative to positive around point A where ωSCL ≈ ωDFIG and the damping of λEOM changes accordingly from being positive to negative. Thus, the continuity principle of eigenvalues is not violated. In order to evaluate the positions swapping shown in Fig. 4.15, the participation factors (PFs) for two closed-loop SSO modes, λDFIG and λSCL , are computed. The summation of PFs of all the state variables of the DFIG subsystem is denoted as DFIG PF. The summation of PFs of all the state variables of the SCL subsystem is denoted as SCL PF. The computational results of PFs indicate that for both λDFIG and λSCL , DFIG PF is greater than SCL PF. The identification of λDFIG and λSCL is made on the following principle: if it is found that SCL PF for a closed-loop SSO mode is larger, the closed-loop SSO mode is identified as λSCL . The other closed-loop SSO mode is λDFIG . The results of PFs and identification of λDFIG and λSCL with variation of c are displayed in Fig. 4.16. In Fig. 4.16, triangles are the SCL PF for the continuous varying closed-loop SSO mode on the left-hand side in Fig. 4.15; circles are the SCL PF for the continuous varying closed-loop SSO mode on the right-hand side in Fig. 4.15. It can be seen that between point ① to ②, the SCL PF for the continuous varying closed-loop SSO mode on the right-hand side in Fig. 4.15a is greater. Hence, it is identified to be λSCL . However, between point ② and ③, the SCL PF for the continuous varying closedloop SSO mode on the left-hand side in Fig. 4.15a is greater. Hence, the closed-loop SSO mode on the left-hand side is identified to be λSCL . This confirms the identification of λSCL as indicated in Fig. 4.15a. Similarly, in Fig. 4.16, triangles and circles are the SCL PF for the continuous varying closed-loop SSO mode on the left-hand and right-hand side in Fig. 4.16 respectively. The identification of λDFIG is made in the similar way that λSCL is identified. The result of identification is indicated in Fig. 4.15a. The results of open-loop modal resonance analysis when Rr + kprsc = 0 are presented in Fig. 4.15b. It can be noted that the proximity of λEOM and λSCL causes no stability decrease of the example grid-connected DFIG system with series compensation. Hence, Fig. 4.16 confirms the analytical conclusion obtained in the previous subsection that when the electromechanical oscillation mode takes part in the open-loop modal resonance, the IGE and the impact of open-loop modal resonance are the same. When there is no IGE, the open-loop modal resonance does not cause the SSCI and vice versa.

212

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

Fig. 4.16 Computational results of the PFs when c increases

① 0.2

②

③

PFs

0.18 0.16 0.04 0.02 0 0.2

0.45 Compensation level c

0.8

The confirmation of non-linear simulation is displayed in Fig. 4.17. Voltage at the terminal of series capacitor dropped by 10% at 0.1 s of simulation and recovered in 0.2 s when c = 0.5. Figure 4.17 indicates that when Rr + kprsc = 0 (there is neither IGE nor impact of OLMP), no SSCI was induced by the DFIG. Whilst, under the condition that Rr = 0, kprsc = 0.001 or Rr = 0.00549, kprsc = 0.001, unstable SSCI occurred because Rr + kprsc ≠ 0. Note that Rr is fixed and not variable. Here it was varied just for the purpose of demonstration and evaluation. To examine the open-loop modal resonance with the variation of load conditions, the wind speed is tuned from 9 to 6 m/s when c = 0.42. The variation of λDFIG with the decrease of wind speed is calculated and is displayed in Fig. 4.18 as hollow triangle. Hollow circle is the position of λSCL. The variations of λDFIG and λSCL with the decrease of wind speed are displayed as the crosses. It can be observed that when Rr + kprsc ≠ 0, the open-loop modal resonance happens at the wind speed of 6 m/s. Consequently, λSCL shifts to the right half of complex plane and unstable SSCI is induced by the DFIG. Whilst, when Rr + kprsc = 0, proximity between λSCL and λDFIG affects neither λSCL nor λDFIG , indicating no impact of open-loop modal resonance. Non-linear simulation with different wind speeds when Rr + kprsc ≠ 0 is displayed in Fig. 4.19. Voltage at the terminal of series capacitor dropped by 10% at 0.1 s of simulation and recovered in 0.2 s. Figure 4.19 indicates that when the wind speed is 6 m/s, unstable SSCI occurred because the open-loop modal resonance happened. In the closed-loop model of example grid-connected DFIG system with the series-compensated transmission line shown by Fig. 4.14, the open-loop modal resonance may happen between the SSO mode of open-loop SCL subsystem and the other SSO mode of open-loop DFIG subsystem, rather than the electromechanical oscillation mode. Two study cases of such open-loop modal resonance are presented in this subsection as follows. The first study case is the open-loop modal resonance between the SCL subsystem and the GSC in the DFIG subsystem. To examine the open-loop modal resonance, the integral gain of current control inner loop of the GSC is tuned between kigsc = 50 and kigsc = 230 when c = 0.4. The trajectories of open-loop and closed-loop SSO mode associated with the current control inner loop of the GSC, λGSC and λGSC , are displayed in Fig. 4.20.

The Induction Generator Effect Analysis of Sub-Synchronous Oscillations. . . Terminal voltage of DFIG/P.u. Active power output of DFIG/p.u.

4.2

1

213

Rr = 0,k prsc = 0 Rr = 0,k prsc = 0.001 Rr = 0.00549,k prsc = 0.001

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

0

0.1

0.2

Time(sec) 0.3 0.4

0

0.1

0.2

0.3

0.5

0.6

0.7

0.5

0.6

0.7

1.3 1.2 1.1 1.0 0.9 0.8

Time(sec)

0.7

0.4

Fig. 4.17 Results of non-linear simulation

150 Imag

lˆ SCL

l SCL

l DFIG

100

0

wind speed decreased

-6

-4

-2

a.

0

2

lˆ SCL l SCL

100

lˆ DFIG 50

150 Imag

Real 4

l DFIG lˆ

wind speed decreased

DFIG

50

0

-6

-4

-2

0

2

Real 4

b.

Fig. 4.18 Results of open-loop modal resonance analysis with the variation of wind speed. (a) Rr + kprsc ≠ 0, (b) Rr + kprsc = 0

Figure 4.20 indicates that in both cases of the IGE (Rr = 0.00549, kprsc = 0.001) and no IGE (Rr = 0, kprsc = 0), there is no impact of open-loop modal resonance between the SSO mode of current control inner loop of the GSC and the SSO mode of SCL subsystem. The open-loop modal resonance is irrelevant with the IGE

4 Terminal volt age of DFIG/ P.u. Acti ve power output of DFIG/p.u.

214

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

0.6

0.4

0.2

0 0 1.05

Time(sec) 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.2

Time(sec) 0.3 0.4

0.5

0.6

0.7

8m/s 6m/s

1

0.95 0

0.1

Fig. 4.19 Results of non-linear simulation under different wind speed

because the open-loop modal resonance does not cause the SSCI regardless of the existence of the IGE. The second study case is the open-loop modal resonance between the SCL subsystem and the RSC in the DFIG subsystem. The results of modal analysis are given in Fig. 4.21 when the integral gain of the current control inner loop of the RSC is tuned between kirsc = 1 and kirsc = 30. In Fig. 4.21, λRSC and λRSC , are the openloop and closed-loop SSO mode associated with the current control inner loop of the RSC respectively. It shows that in both cases of the IGE (Rr = 0.00549, kprsc = 0.01) and no IGE (Rr = 0, kprsc = 0), the open-loop modal resonance happens between λRSC and the SSO mode of SCL subsystem, λSCL, which causes the unstable SSCI in the example grid-connected DFIG system with the series-compensated transmission line. Hence, this is also the case that the open-loop modal resonance is irrelevant with the IGE, as the open-loop modal resonance causes the SSCI regardless of the existence of the IGE. The open-loop modal resonance can also happen between the electromechanical oscillation mode and the SSO mode associated with the converter control systems of the DFIG, instead of the SSO mode of the SCL subsystem. To examine such openloop modal resonance, a division of the example grid-connected DFIG system with the series compensation different to that shown in Fig. 4.14 can be made. By taking the current control inner loop of the GSC and the output filter together as a subsystem and remainder of the example grid-connected DFIG system with the series compensation as the other subsystem, a closed-loop model similar to that

4.2

The Induction Generator Effect Analysis of Sub-Synchronous Oscillations. . .

160 Imag

160 Imag

140

140

ˆ l SCLl SCL

120

kigsc

100

215

lˆSCL l SCL

120

100

increase d

80

80

60

60

lˆGSC

lGSC

40

20 -3.5

-3

-2.5

-2

-1.5

lGSC

40

Rea l 20 -1 -0.5 -1.8 -1.6 -1.4

a.

-1.2

-1

lˆGSC

Rea l -0.8 -0.6

b.

Fig. 4.20 The IGE being irrelevant with the open-loop modal resonance between the current control inner loop of the GSC and the SCL subsystem (study case 1). (a) Rr + kprsc ≠ 0, (b) Rr + kprsc = 0

120 Imag

lˆSCL

lˆRSC

100

100 80

120 Imag

lˆRSC

l SCL

80

kirsc

60

lˆSCL

l SCL

60

increased 40

40 20 -60

l RSC -40

-20

0

a.

20

Rea l 40 60

20 -60

l RSC -40

-20

0

20

40

Real 60

b.

Fig. 4.21 The IGE being irrelevant with the open-loop modal resonance between the current control inner loop of the RSC and the SCL subsystem (study case 2). (a) Rr + kprsc ≠ 0, (b) Rr + kprsc = 0

displayed in Fig. 4.14 is established. The established closed-loop model is comprised of the current control of the GSC subsystem and remainder subsystem. Following two tests are conducted on the basis of such established closed-loop model. In the first test, Rr is varied from Rr = 0 to Rr = 0.00549. Figure 4.22a displays the trajectories of open-loop and closed-loop SSO modes. In Fig. 4.22a, λCC is the SSO mode of the open-loop current control of the GSC subsystem and λEOM is the electromechanical oscillation mode, which is an SSO mode of open-loop remainder subsystem. It can be noted that when Rr increases, λCC and λEOM move towards the

216

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

0

100 Imag

95

lCC

lˆCC

-2

increased

lˆEOM

80

75 -10

increased

Rr

85

a.

-3 -4

0

-5

Rr

-1

l EOM

90

Imag

Real 5

-5 0

1

3

2

4

Real 5

b.

Fig. 4.22 Test results of the open-loop modal resonance between electromechanical oscillation mode and the GSC when Rr varies. (a) Traje ctories of modes, (b) variation of residue

opposite directions in respect to the positions of λCC and λEOM. This is caused by the increasing effect of the open-loop modal resonance between λCC and λEOM. Hence, Fig. 4.22a shows that the IGE is closely related with the impact of open-loop modal resonance. A further confirmation is presented in Fig. 4.22b where variation of residue REOM for λEOM is displayed when Rr increases. Obviously, when Rr = 0, REOM = 0 indicating that there is neither the IGE nor the impact of open-loop modal resonance. With increased Rr, both the IGE and impact of open-loop modal resonance increases leading to the unstable SSCI induced by the DFIG in the example grid-connected DFIG system with the series compensation. In the second test, the integral gain of current control inner loop of the GSC, kigsc, increases such that λCC moves on the complex plane to get close to λEOM. Modal positions and trajectories are displayed in Fig. 4.23. It can be observed that when Rr + kprsc ≠ 0, the open-loop modal resonance happens. Consequently, λCC shifts to the right half of complex plane and unstable SSCI is induced by the DFIG. Whilst, when Rr + kprsc = 0, proximity between λCC and λEOM affects neither λCC nor λEOM , indicating no impact of open-loop modal resonance at all. This shows again that the IGE and impact of OLMP are equivalent.

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

110 Imag

90 lˆEOM

110 Imag

lCC

70 50 30

lˆEOM l EOM

90

l EOM

lˆCC

217

70 50

kigsc increased

lCC

30 -8 -6 -4 -2 0

2

Real 4 6

lˆCC

-8 -6 -4 -2 0

a.

2

Real 4 6

b.

Fig. 4.23 Test results of the open-loop modal resonance between electromechanical oscillation mode and the GSC when kigsc varies. (a) Rr + kprsc ≠ 0, (b) Rr + kprsc = 0

4.3

4.3.1

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations in a Grid-Connected DFIG System With Series-Compensated Transmission Line Closed-Loop Interconnected Model of a Power System With a Grid-Connected DFIG

In Sects. 4.1 and 4.2, the open-loop modal resonance is introduced and applied to examine the SSOs on the basis of a closed-loop interconnected model, which is of single input and single output. This section presents the analysis of open-loop modal resonance to examine the SSOs in a grid-connected DFIG system with series compensated line. The examination is based on a two-inputs single-output closedloop interconnected model as to be introduced as follows. The state-space model of a DFIG expressed in the common x-y coordinate is given by (3.28) and (3.33), which is re-written below d ΔXd = Ad ΔXd þ Bd ΔVxy dt ΔIxy = Cd ΔXd T

T

ð4:53Þ

where ΔVxy = ½ ΔV x ΔV y ] ; ΔIxy = ½ ΔI x ΔI y ] ; Vx + jVy and Ix + jIy separately is the terminal voltage and output current of the DFIG expressed in the common x-y coordinate.

218

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

Fig. 4.24 Configuration of a grid-connected DFIG

Fig. 4.25 Closed-loop interconnected model of a power system with a gridconnected DFIG

G (s) Δ I xy

W (s)

Δ Vxy

ROPS subsystem

DFIG subsystem

Consider a power system integrated with a grid-connected DFIG as shown by Fig. 4.24. At the terminal of the DFIG, a state-space model of the remainder of the power system excluding the DFIG can be derived and generally expressed as d ΔXs = As ΔXs þ Bs ΔIxy dt ΔVxy = Cs ΔXs

ð4:54Þ

Obviously, a closed-loop interconnected model of the power system with the grid-connected DFIG can be established and shown by Fig. 4.25. The closed-loop interconnected model is of two inputs and two outputs. In the closed-loop interconnected model displayed in Fig. 4.25, the variables selected in the interface of two subsystems, i.e., the subsystem of the DFIG (DFIG subsystem) and the subsystem of the remainder of the power system (ROPS subsystem), are the terminal voltage and output current from the DFG expressed in the common x-y coordinate. Alternatively, the output active and reactive power instead of the output current from the DFIG can be selected to derive the closed-loop interconnected model, as being presented as follows. In this section, a relatively simpler closed-loop interconnected model of the power system with the grid-connected DFIG is used to introduce the open-loop modal resonance analysis. The interconnected model is derived based on the assumption that the PLL implemented by the DFIG for the control of grid connection is of the perfect phase tracking performance. Subsequently, the PLL can always track the terminal voltage of the DFIG instantly such that in Fig. 3.4 terminal voltage of the DFIG, Vx + jVy = V ∠ θ, is always on the q axis of d-q coordinate the DFIG. This

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

219

implies that Δθp = Δθ such that dynamics of the PLL do not need to be considered in the dynamic model of the DFIG. In addition, ΔVdsd = 0 and ΔVdsq = ΔV. With dynamics of the PLL being excluded, the state equation of the DFIG is (3.26), which can be re-written below ΔV dsd d ΔXdd = Add ΔXdd þ bdd dt ΔV dsq

ð4:55Þ

T

where ΔXdd = ΔXd1 T ΔXd2 T ΔXd3 T . Active and reactive power output from the DFIG are Pd = Pds þ Pcd = V dsd I dsd þ V dsq I dsq þ V dsd I dcd þ V dsq I dcq Qd = Qds þ Qcd = V dsq I dsd - V dsd I dsq þ V dsq I dcd - V dsd I dcq

ð4:56Þ

where Pcd and Qcd are the active and reactive power output from the GSC to the power system. Linearization of (4.56) can be written as ΔPd ΔQd

= Fdsv ΔVdsv þ Fdsi ΔIdsi þ Fdci ΔIdci

ð4:57Þ

where ΔVdsv = ½ ΔV dsq ΔV dsd ]T , ΔIdsi = ½ ΔI dsq ΔI dsd ]T , ΔIdci = ½ ΔI dcq I dsd0 þ I dcd0 V dsd0 I dsq0 þ I dcq0 V dsq0 , Fdsi = , Fdsv = I dsd0 þ I dcd0 - I dsq0 - I dcq0 - V dsd0 V dsq0 V dsq0 V dsd0 Fdci = : - V dsd0 V dsq0

ΔI dcd ]T ,

By using (3.19) and (3.30), ΔIdsi and ΔIdci in (4.57) can be substituted by state variable vector defined in (4.55) to have ΔPd = cdp T ΔXd þ ddp1 ΔV dsq þ d dp2 ΔV dsd ΔQd = cdq T ΔXd þ ddq1 ΔV dsq þ d dq2 ΔV dsd where

ð4:58Þ

220

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

cdp T =

X drr V dsq0 -X dm V dsq0 X drr V dsd0 -X dm V dsd0 0 V dsd0 V dsq0 0 2 2 2 X dm -X drr X dss X dm -X dss X drr X dm -X drr X dss X dm 2 -X dss X drr

cdq T =

-X dm V dsq0 X drr V dsq0 -X drr V dsd0 X dm V dsd0 0 V dsq0 -V dsd0 0 X dm 2 -X drr X dss X dm 2 -X dss X drr X dm 2 -X drr X dss X dm 2 -X dss X drr

ddp1 =I dsq0 þI dcq0 , d dp2 =I dsd0 þI dcd0 , d dq1 =I dcd0 þI dsd0 , ddq2 = -I dcq0 -I dsq0

The exclusion of dynamics of the PLL is based on the assumption that the PLL is of perfect phase tracking performance, such that ΔVdsd = 0 and ΔVdsq = ΔV. Subsequently, state equation of the DFIG of (4.55) becomes d ΔXdd = Add ΔXdd þ bddq ΔV dt

ð4:59Þ

Equation (4.58) becomes ΔPd = cdp T ΔXdd þ ddp1 ΔV ΔQd = cdq T ΔXdd þ ddq1 ΔV

ð4:60Þ

Writing (4.59) and (4.60) together, following state-space model of the DFIG, derived by (1) ignoring the dynamics of the PLL; (2) selecting the output power instead of output current of the interfacing variables, is obtained d ΔXdd = Add ΔXdd þ bddq ΔV dt ΔPd = cdp T ΔXdd þ ddp1 ΔV ΔQd = cdq T ΔXdd þ d dq1 ΔV

ð4:61Þ

In the power system integrated with the grid-connected DFIG displayed in Fig. 4.24, following state-space model of the remainder of the power system excluding the DFIG can be derived [1] d ΔXsp = Asp ΔXsp þ bsp ΔPd þ bsq ΔQd dt ΔV = csp T ΔXsp þ dsp ΔPd þ dsq ΔQd

ð4:62Þ

From (4.61) and (4.62), a closed-loop interconnected model of the power system of Fig. 4.24 can be established and is shown by Fig. 4.26. As far as the gridconnected DFIG is concerned, the closed-loop interconnected model displayed in Fig. 4.26 is of one input and two outputs. In Fig. 4.26,

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Fig. 4.26 One-input two-outputs closed-loop interconnected model of the power system with a gridconnected DFIG

ROPS subsystem

DFIG subsystem

+

H P (s)

ΔV

H Q (s)

Δ Pd

ΔQd

H P ðsÞ = cdp T ðI - Add Þ - 1 bddq þ ddp1 H Q ðsÞ = cdq T ðI - Add Þ - 1 bddq þ ddq1 GP ðsÞ = Csp I - Asp

-1

GQ ðsÞ = Csp I - Asp

-1

221

GP ( s )

GQ ( s )

ð4:63Þ

bsp þ dsp

ð4:64Þ

bsq þ d sq

From (4.58), ΔV dsd

= - Ddd - 1

ΔV dsq where Ddd =

d dp2

d dp1

ddq2

d dq1

cdp T cdq

T

ΔXd þ Ddd - 1

ΔPd ΔQd

ð4:65Þ

. Substituting (4.65) in (4.55), it can have

ΔPd d ΔXdd = Addd ΔXdd þ bddd dt ΔQd

ð4:66Þ

cdp T , bddd = bdd Ddd - 1 . cdq T The exclusion of dynamics of the PLL gives ΔVdsd = 0 and ΔVdsq = ΔV. Hence, writing (4.65) and (4.66) together, following state-space model of the grid-connected DFIG can be obtained where Addd = Add - bdd Ddd - 1

d ΔXdd = Addd ΔXdd þ bddd1 ΔPd þ bddd2 ΔQd dt ΔV = cdv T ΔXdd þ ddvp ΔPd þ ddvq ΔQd

ð4:67Þ

An alternative state-space model of the remainder of the power system excluding the DFIG in Fig. 4.24 can be written as

222

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

4

Fig. 4.27 Two-inputs one-output closed-loop interconnected model of the power system with a gridconnected DFIG

Fp ( s )

Fq ( s )

ΔPd

Pp ( s )

ΔQd

ROPS subsystem

+

Pq ( s )

ΔV

DFIG subsystem

d ΔXsp = Aspp ΔXsp þ bsv ΔV dt ΔPd = csvp T ΔXsp þ dspv ΔV ΔQd = csvq T ΔXsp þ d sqv ΔV

ð4:68Þ

From (4.67) and (4.68), an alternative closed-loop interconnected model of the power system similar to that displayed Fig. 4.24 is obtained and shown by Fig. 4.27. As far as the grid-connected DFIG is concerned, the closed-loop interconnected model displayed in Fig. 4.27 is of two inputs and one output. In Fig. 4.27, Pp ðsÞ = cdv T ðI - Addd Þ - 1 bddd1 þ d dvp Pq ðsÞ = cdv T ðI - Addd Þ - 1 bddd2 þ d dvq F p ðsÞ = csvp T I - Aspp

-1

bsv þ dspv

F q ðsÞ = csvq I - Aspp

-1

bsv þ dsqv

T

4.3.2

ð4:69Þ

ð4:70Þ

Open-Loop Modal Resonance in an Example Grid-Connected DFIG Wind Farm With Series-Compensated Transmission Line

Figure 4.28 shows the configuration of an example wind farm connected to the external AC grid through a series-compensated transmission line. The wind farm is divided into two areas. Each of the areas consists of 50 DFIGs and is represented by the aggregated dynamic model of a DFIG [12]. The model and parameters of aggregated single DFIG to represent 50 DFIGs given in [12] are adopted. In order to reflect the difference of wind power generation by the DFIGs in two areas of the wind farm, converter control parameters of the aggregated DFIG given in [12] are increased by 10% and used by DFIG A in Fig. 4.28. Converter control parameters of the aggregated DFIG given in [12] are decreased by 10% and adopted by DFIG B in Fig. 4.28. DFIG A operates at the wind speed 7 m/s and DFIG B at 8 m/s At the terminal of two DFIGs in Fig. 4.28, the power system is divided to two subsystems, the DFIG subsystem and the ROPS subsystem. The DFIG subsystem is comprised of DFIG A and DFIG B. The ROPS subsystem consists of the series-

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Fig. 4.28 A seriescompensated power system with a grid-connected DFIG wind farm

223

V

DFIG A

Pd + jQd

RSC

GSC

xL

xC

External AC grid

DFIG B RSC

GSC

compensated transmission line. The state-space model of the DFIG given in (4.67) is derived for DFIG A and DFIG B. Write the state-space model of DFIG A and DFIG B together, state-space model of the DFIG wind farm (i.e., the DFIG subsystem) is established. Subsequently, the closed-loop interconnected model of the example grid-connected DFIG wind farm is derived and shown by Fig. 4.27. Based on the derived model, the risk of the SSO in the example grid-connected wind farm can be identified in the following two steps: (1) Calculate all the open-loop SSO modes of the open-loop subsystems and identify any pair of open-loop SSO modes close to each other on the complex plane. This is the identification of open-loop modal resonance. (2) Assess the instability risk caused by the open-loop modal resonance to cause the closed-loop modal repulsion. The assessment is made by calculating the residues for the identified pair of open-loop SSO modes. The results of risk identification are presented as follows. Firstly, the open-loop SSO modes of the DFIG subsystem are computed from the open-loop state matrix Addd in (4.67). Computational results of four open-loop SSO modes, λdi, i = 1, 2, 3, 4, are listed in the first column of Table 4.2. By calculating the participation factors for each open-loop SSO modes, dynamics associated with each of the SSO modes are identified and identified results are given in the second column of Table 4.2. Shaft stiffness of the DFIG given in [12] is increased to 60 and 50 for DFIG A and DFIG B respectively. Thus, the torsional oscillation frequency is higher than that given in [12]. The electromechanical oscillation modes in Table 4.2 are the oscillation modes related with both the electrical and mechanical dynamics (rotor motion) of the DFIGs. That was why they are referred to as the electromechanical oscillation modes as being explained in Sect. 4.2. Secondly, the open-loop SSO modes of the ROPS subsystem are computed from the open-loop state matrix Aspp in (4.68). By calculating the participation factors, the open-loop SSO mode associated with the series compensation is identified to be λe. By varying the level of series compensation, c, it is found that when c = 0.496, c = 0.566, c = 0.667, and c = 0.832, λe is close to λdi, i = 1, 2, 3, 4 respectively. Computational results of λe being close to λdi, i = 1, 2, 3, 4 are given in the thirrd column of Table 4.2. Hence, it is expected that for the pairs of λe and λdi, i = 1, 2, 3, 4 listed in each row of Table 4.2, the SSO modal repulsion between the closed-loop

224

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

Table 4.2 Computational results of the open-loop SSO modes of the DFIG subsystem and ROPS subsystem Open-loop SSO modes of DFIG subsystem λd1 = - 1.63 + j129 λd2 = - 1.52 + j114 λd3 = - 1.69 + j92.0 λd4 = - 2.38 + j56.6

Dynamics associated with the SSO modes of the DFIG subsystem Torsional SSO mode of DFIG A Torsional SSO mode of DFIG B Electromechanical SSO modes of DFIG B Electromechanical SSO modes of DFIG A

Open-loop SSO mode of ROPS subsystem λe = - 1.89 + j123 λe = - 1.89 + j105 λe = - 1.89 + j82.1 λe = - 1.89 + j47.6

SSO mode, λe and λdi , i = 1, 2, 3, 4, may very likely happen such that either λe or λdi , i = 1, 2, 3, 4 may possibly be poorly or even negatively damped. Thirdly, for λdi, i = 1, 2, 3, 4, right and left eigenvectors of open-loop state matrix Addd in (4.67) are calculated to be rdi, i = 1, 2, 3, 4 and ldi, i = 1, 2, 3, 4 respectively. For λe when c = 0.496, c = 0.566, c = 0.667, and c = 0.832, right and left eigenvectors of open-loop state matrix Aspp in (4.68) are calculated to be rei, i = 1, 2, 3, 4 and lei, i = 1, 2, 3, 4 respectively. The residues for λdi, i = 1, 2, 3, 4 and λe are calculated by using the standard formulas given in [13] as Rdpi = cd T rdi ldi bp , Rdqi = cd T rdi ldi bq Repi = cp T rei lei bv , Reqi = cq T rei lei bv

ð4:71Þ

Based on the closed-loop interconnected model of the example grid-connected DFIG wind farm derived and shown by Fig. 4.27, from the residues given in (4.71), the peaks of closed-loop modal repulsion can be estimated. The equations to estimate the peaks of closed-loop modal repulsion can be derived in the way similar to the derivation from (4.55) to (4.64), which is based on the single-input and single-output closed-loop interconnected model displayed in Fig. 4.2. As the closed-loop interconnected model of Fig. 4.27 is of two inputs and one output, the derivation of equations to estimate the peaks of closed-loop modal repulsion is slightly different to that of (4.31), which is explained as follows. Due to the fast speed of converter control, dynamic variations of power output from the DFIG are normally very small, i.e., ΔPd + jΔQd ≈ 0. Hence, open-loop transfer function of ROPS subsystems in Fig. 4.27 can be written as Fp(s) = εEp(s), Fq(s) = εEq(s), where 1 ≫ ε > 0 is a small positive number. The characteristic equation of the power system with the DFIGs can be obtained from Fig. 4.27 to be εE p ðsÞPp ðsÞ þ εEq ðsÞPq ðsÞ = 1 Denote the transfer functions of open-loop subsystems in Fig. 4.27 as

ð4:72Þ

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Table 4.3 Computational results of Rdpi Repi þ Rdqi Reqi 4.2986 + j0.0159 5.2916 - j0.1036 1.5023 + j5.1330 4.4215 + j4.7270

i 1 2 3 4

F p ðsÞ = εE p ðsÞ =

225

Rdpi Repi þ Rdqi Reqi λdi 4.7976 + j128.66 3.7387 + j114.65 -3.1421 + j86.663 -4.6988 + j53.953

λe -6.0797 + j128.43 -6.7063 + j114.40 -0.6395 + j96.935 -0.1308 + j59.201

εep ðsÞ εeq ðsÞ , F ðsÞ = εEq ðsÞ = , s - λe q s - λe

gp ðsÞ gq ð s Þ , P q ðsÞ = Pp ðsÞ = s - λdi s - λdi

ð4:73Þ

Using (4.73), the characteristic equation of (4.72) can be written as εep ðsÞgp ðsÞ þ εeq ðsÞgq ðsÞ = ðs - λe Þðs - λdi Þ

ð4:74Þ

f ðsÞ = ðs - λe Þðs - λdi Þ, F ðsÞ = ep ðsÞgp ðsÞ þ eq ðsÞgq ðsÞ

ð4:75Þ

Let

The characteristic equation of (4.75) can be expressed by (4.13). Thus, λe can be expressed by the Taylor series expansion similar to (4.15). When λe = λd, small increment of Δλe can be expressed by the expansion similar to (4.23). Finally, taking the derivation similar to that from (4.23) to (4.31), following equation to estimate the peaks of closed-loop modal repulsion, when the open-loop modal resonance happens in the example grid-connected DFIG wind farm, can be derived. Δλdi = λdi - λdi ≈ ± Δλe = λe - λe ≈ ±

Rdpi Repi þ Rdqi Reqi Rdpi Repi þ Rdqi Reqi

ð4:76Þ

The computational results of Rdpi Repi þ Rdqi Reqi are presented in the second column of Table 4.3. From Tables 4.2 and 4.3, it can have Real part of

Rdpi Repi þ Rdqi Reqi > jReal part of λdi j, i = 1, 2

Real part of

Rdpi Repi þ Rdqi Reqi ≈ jReal part of λdi j, i = 3, 4

ð4:77Þ

The first inequalities in (4.77) indicate that when c = 0.496 (i = 1) and c = 0.566 (i = 2), it is very likely that the closed-loop SSO modes of the DFIGs, λdi , i = 1, 2 may be in the right half of complex plane such that growing SSCIs may very possibly occur. The second inequalities in (4.77) indicate that when c = 0.667 (i = 3) and c = 0.832 (i = 4), the closed-loop SSO modes of the

226

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

190

λˆe

170

λe

Ima gina ry axis

150 130

λd 1

110

λd 2 λˆd 3

90 70

λˆd 1 λˆd 2

:Closed-loop mode :Open-loop mode of the electrical subsystem :Open-loop mode of the DFIG subsystem :Positions of open-loop mode of the electrical subsystem when the modal repulsion occurs :The peak of modal repulsion of electrical subsystem and DFIG subsystem :Estimated position of the peak of modal repulsion

λd 3

λˆd 4 λd 4

50 30 10 -8

-6

-4

-2

0 Real axis

2

4

6

Fig. 4.29 Trajectories and positions of closed-loop and open-loop SSO modes

DFIGs, λdi , i = 3, 4 may be poorly damped. Thus, the instability risk caused by the SSCIs between the DFIGs and the series-compensated line is identified by the openloop modal resonance analysis. The open-loop modal resonance analysis conducted above is based on the statespace model of open-loop subsystems to estimate the stability of closed-loop system, i.e., the series-compensated power system with a grid-connected DFIG wind farm displayed in Fig. 4.28. The analysis attributes the instability risk to the SSO modal repulsion which occurs under the condition of open-loop modal resonance. To evaluate and confirm the identification of instability risk by the open-loop modal resonance analysis above, the closed-loop state matrix of the power system with the wind farm is derived from (4.66) and (4.67). The pairs of closed-loop SSO modes, λdi , i = 1, 2, 3, 4 and λe , for each of modal repulsion are computed. Computational results are given in the third and fourth column of Table 4.3, confirming the estimation made above about the consequence of modal repulsion from (4.77) that λdi , i = 1, 2 are negatively damped and λdi , i = 3, 4 are poorly damped. To clearly show the SSO modal repulsion occurred in the power system with the wind farm displayed in Fig. 4.28, Fig. 4.29 presents the trajectories of λdi , i = 1, 2, 3, 4 and λe as solid curves when the level of series compensation, c, varied from 0.01 to 1. In Fig. 4.29, the dashed curve is the trajectory of λe with the variation of c. Hollow circles are the positions of λdi, i = 1, 2, 3, 4 as given in the first column of Table 4.2. Hollow triangles are the positions of λe being closest to λdi, i = 1, 2, 3, 4 when the closed-loop modal repulsion happened as given in the third column of Table 4.2. Solid circles and triangles are the positions of λdi , i = 1, 2, 3, 4 and λe when the SSO modal repulsion occurred as given in the third and fourth column of Table 4.3. Crosses are the

4.3

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Fig. 4.30 Results of non-linear simulation

c = 0.496 c = 0.4

0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2

Line current (p.u.)

Electromagnetic torque of DFIG A (p.u.)

0.2

227

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

0.5

0

1.0

1.5

1.0

1.5

Time(sec) c = 0.496 c = 0.4

0.5

0

Time(sec) MSC Vs PMSG

Is

Ps Pc

GSC Vc Xf

Vdc C

V

Ic

θpll

PLL

XL Vsc Xc Vb

θ2

Fig. 4.31 A grid-connected PMSG system with series-compensated transmission line

estimated positions of λdi , i = 1, 2, 3, 4 and λe obtained by using (4.76) and computational results given in the second column of Table 4.3. Non-linear simulation is conducted to confirm the results of modal computation. Figure 4.30 presents one of the simulation results when the modal repulsion occurs between λd1 and λe with c = 0.496. At 0.1 s of simulation, the voltage at the terminal of series capacitor dropped by 10% and recovered in 0.1 s. Obviously, torsional SSOs of DFIG A occurred which led to the loss of system stability.

228

4.4

4.4.1

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations in a Grid-Connected PMSG System With Series-Compensated Transmission Line Closed-Loop Interconnected Model of a Power System With a Grid-Connected PMSG

Figure 4.31 shows the configuration of a PMSG being connected to the main grid through a series-compensated transmission line. To examine the risk of the SSO caused by the open-loop modal resonance in the grid-connected PMSG system with the series-compensated transmission line of Fig. 4.31, a closed-loop interconnected model of the system can be established. In the previous section, the closed-loop interconnected model for a grid-connected DFIG system with the seriescompensated transmission line is of two subsystems. One subsystem consists of the DFIG. The other subsystem is comprised of the remainder of the power system (ROPS), i.e., the series-compensated transmission line. In fact, when the closed-loop interconnected model for the open-loop modal resonance analysis is established, any particular part of the system can be selected as one subsystem with the remainder of the power system as the other subsystem. In this subsection, this strategy to establish the closed-loop interconnected model is demonstrated for examining the open-loop modal resonance in the grid-connected PMSG system with the series-compensated transmission line displayed in Fig. 4.31. Firstly, the PLL is selected as an open-loop subsystem. The linearization of (2.19) gives the following state-space model of the PLL as the open-loop subsystem dΔxp = K pi V 0 Δθ - Δθp dt dΔθp = K pp V 0 Δθ - Δθp þ Δxp dt

ð4:78Þ

In matrix form, the state-space model of the PLL is d ΔXp4 = App ΔXpp þ bpp Δθ dt Δθp = cpp T ΔXp4 where

ð4:79Þ

4.4

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Δθp ]T

ΔXp4 = ½ Δxp 0 App = 1

229

- K pi V 0 - K pp V 0

, bpp =

K pi V 0

, cpp =

K pp V 0

0 1

From (4.79) it can be seen that the input to and output from the open-loop PLL subsystem respectively are Δθ and Δθp. Secondly, the state-space model of the remainder of the power system of Fig. 4.31 is derived as follows in three steps. The first step is to derive the expression of the PCC voltage in terms of the state variables of the PMSG, the voltage across the series capacitor and the output of the open-loop PLL subsystem, Δθp. ΔI pcdref = K pp4 ΔV pdc þ Δxp4 ΔV pcdref = K pp5 ΔI pcdref - ΔI pcd þ Δxp5 - X pf ΔI pcq þ ΔV pd ΔI pcqref = K pp6 ΔQp þ Δxp6

ð4:80Þ

ΔV pcqref = K pp7 ΔI pcqref - ΔI pcq þ Δxp7 þ X pf ΔI pcd þ ΔV pq Substituting the first and the third equation of (4.80) into the second and the fourth equation of (4.80) respectively and using (2.17), ΔV pcd = K pp5 K pp4 ΔV pdc þ K pp5 Δxp4 - K pp5 ΔI pcd þ Δxp5 - X pf ΔI pcq þ ΔV pd ΔV pcq = K pp7 K pp6 ΔQp þ K pp7 Δxp6 - K pp7 ΔI pcq þ Δxp7 þ X pf ΔI pcd þ ΔV pq ð4:81Þ Linearization of (2.15) is ΔQp = V pq0 ΔI pcd - V pd0 ΔI pcq þ I pcd0 ΔV pq - I pcq0 ΔV pd

ð4:82Þ

Substituting (4.82) in (4.81) ΔV pcd = - K pp5 ΔI pcd - X pf ΔI pcq þK pp5 K pp4 ΔV pdc þK pp5 Δxp4 þΔxp5 þΔV pd

ð4:83Þ

ΔV pcq = ðK pp7 K pp6 V pq0 þX pf ÞΔI pcd - ðK pp7 K pp6 V pd0 þK pp7 ÞΔI pcq þK pp7 Δxp6 þΔxp7 - K pp7 K pp6 I pcq0 ΔV pd þðK pp7 K pp6 I pcd0 þ1ÞΔV pq

Denote ΔXp3 = ½ ΔI pcd Eq. (2.18)). From (4.83),

ΔI pcq

ΔV pdc

Δxp4

Δxp5

Δxp6

Δxp7 ]T

(see

230

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

ΔV pcd ΔV pd = Ep3 ΔXp3 þ Ep4 ΔV pcq ΔV pq

ð4:84Þ

ω0 V pcd ω0 V pd d þ ω0 I pcq I = X pf X pf dt pcd ω0 V pcq ω0 V pq d - ω0 I pcd I = X pf X pf dt pcq

ð4:85Þ

ω0 V pd d ω V - 0 scd þ ω0 I pcq I = dt pcd XL XL ω ω V d 0 pq 0 V scq - ω0 I pcd I pcq = XL XL dt

ð4:86Þ

where

From Fig. 4.31,

From (4.85) and (4.86), it can have V pd V scd V pcd X pf XL = þ X L þ X pf V scq X L þ X pf V pcq V pq

ð4:87Þ

Substituting (4.84) in (4.87), V pcd V scd X pf Ep5 Ep4 = Ep5 Ep3 ΔXp3 þ X þ X V pcq V scq L pf where Ep5 = I -

XL X L þX pf

Ep4

-1

ð4:88Þ

.

From (4.87) and (4.88), V pd X pf X L X pf = Iþ X L þ X pf V pq X L þ X pf

2

Ep5 Ep4

V scd XL þ E E ΔX ð4:89Þ X L þ X pf p5 p3 p3 V scq

4.4

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

231

By using (2.27), the above equation expressed in the common x-y coordinate can be obtained to be ΔV x ΔV y

= Ep6

ΔV scx ΔV scy

þ Ep7 ΔXp3 þ ep1 Δθp

ð4:90Þ

where Ep6 =

Ep7 = ep1 =

-

cos θp0

sin θp0

- sin θp0

cos θp0

cos θp0

sin θp0

- sin θp0

cos θp0

cos θp0

sin θp0

- sin θp0

cos θp0

-1

-1

-1

- sin θp0

cos θp0

- cos θp0

- sin θp0

cos θp0

sin θp0

- sin θp0

cos θp0

X L X pf X pf Iþ X L þ X pf X L þ X pf

2

Ep5 Ep4

2

Ep5 Ep4

cos θp0

sin θp0

- sin θp0

cos θp0

XL Ep5 Ep3 X L þ X pf X L X pf X pf Iþ X L þ X pf X L þ X pf V scx0

-1

V scy0 - sin θp0

cos θp0

- cos θp0

- sin θp0

The above equation is the expression of the PCC voltage in terms of the state variables of the PMSG, the voltage across the series capacitor and the output of the open-loop PLL subsystem, Δθp. The second step is to derive the state-space representation of the PMSG in terms of the voltage across the series capacitor and the output of the open-loop PLL subsystem, Δθp. The state-space equation of the PMSG excluding the PLL is (2.29), which is re-written below ΔV x d ΔXp5 = Ap5 ΔXp5 þ a1 Δθp þ Bp2 dt ΔV y where ΔXp5 = ΔXp1 T ΔXp2 T Substituting (4.90) in (4.91),

ΔXp3 T

T

.

ΔV scx d ΔXp5 = Ap5 þ Ep8 ΔXp5 þ ep2 Δθp þ Ep9 dt ΔV scy where

ð4:91Þ

ð4:92Þ

232

4

Ep8 = ½ 0

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

0 Bp2 Ep7 ], Ep9 = Bp2 Ep6 , ep2 = a1 þ þBp2 ep1

The third step is to derive the state-space representation of the series capacitor. Linearized voltage equation across the series capacitor in Fig. 4.31 is dΔV scx = X C ΔI x þ ΔV scy dt dΔV scy = X C ΔI y - ΔV scx dt Denote ΔXp6 = ½ ΔV scx

ð4:93Þ

ΔV scy ]T . Equation (4.93) can be written as

ΔI x dΔXp6 = Ap6 ΔXp6 þ Ep10 dt ΔI y 0 1 XC , Ep10 = -1 0 0 Substituting (2.26) in (4.94),

where Ap6 =

ð4:94Þ

0 XC

ΔI pcd dΔXp6 þ ep3 Δθp = Ap6 ΔXp6 þ Ep11 dt ΔI pcq

ð4:95Þ

where Ep11 = Ep10 cos θp0 - sin θp0 sin θp0 cos θp0 , ep3 = Ep10

- sin θp0

- cos θp0

I pcd0

cos θp0

- sin θp0

I pcq0

The above equation can be written as dΔXp6 = Ap6 ΔXp6 þ Ep12 ΔXp5 þ ep3 Δθp dt

ð4:96Þ

where Ep12 = ½ 0 0 Ep11 0 ]. Writing (4.92) and (4.96) together, d ΔXp7 = Ap7 ΔXp7 þ bp10 Δθp dt where

ð4:97Þ

4.4

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

Fig. 4.32 Closed-loop interconnected model of the power system displayed in Fig. 4.31

G PLL (s)

+

PLL subsystem

Δθ

233

Δθ p

ROPS subsystem

G ROPS (s)

ΔXp7 = ΔXp5 T Ap7 =

ΔXp6 T

Ap5 þ Ep8 Ep12

Ep9

T

, , bp10 =

ep2

Ap6 ep3

The above is the state-space equation of the remainder of the power system displayed in Fig. 4.31 excluding the PLL, i.e., the open-loop ROPS subsystem. The output equation of the open-loop subsystem is derived from (2.24) and (4.90) to be Δθ = csc T ΔXp7 þ dp Δθp

ð4:98Þ

where Ep13 = -

V y0 V 20

csc T = ½ 0 0

V y0 V x0 V x0 Ep6 , 2 Ep7 , Ep14 = V0 V 20 V 20 V y0 V x0 Ep13 Ep14 ], d p = - 2 e Δθ V 0 V 20 p1 p

From (4.79), (4.97), and (4.98), the closed-loop interconnected model of the power system displayed in Fig. 4.31 is established and shown by Fig. 4.32. The transfer function expression of the open-loop PLL subsystem is obtained from statespace model of (4.79) to be. Δθp = GPLL ðsÞΔθ

ð4:99Þ

where GPLL(s) = cppT(sI - App)-1bpp. The transfer function expression of the open-loop ROPS subsystem is obtained from state-space model of (4.97) and (4.98) to be. Δθ = GROPS ðsÞΔθp where GROPS(s) = cscT(sI - Ap7)-1bp10 + dp.

ð4:100Þ

234

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

180

Imaginary part

160

Closed-loop comp. mode Closed-loop PLL mode Open-loop comp. mode Open-loop PLL mode Predicted position

C

140 120 100

B

λˆC

A

λC

C B

80 ki increases

λp

A

λˆp

60 40 -20

-15

-10 -5 Real part

0

5

Fig. 4.33 Open-loop modal resonance between the PLL and series compensation when Kpi was varied

4.4.2

Sub-synchronous Oscillation in an Example Grid-Connected PMSG System With Series Compensation

The configuration of an example grid-connected PMSG system with series compensation is shown by Fig. 4.31. System parameters are given in Appendix. In this subsection, the sub-synchronous oscillation (SSO) caused by the open-loop modal resonance in the example system is studied by conducting two tests. The study case is the open-loop modal resonance between the PLL and the series compensation. Hence, for the study, the closed-loop interconnected model displayed in Fig. 4.32 was established. The oscillation modes of open-loop PLL subsystem and the ROPS subsystem were calculated. An oscillation mode of the ROPS subsystem was identified to be associated with the series compensation. This oscillation mode is named as the compensation mode, denoted as λc. The oscillation mode of the PLL subsystem is named as the PLL mode, denoted as λp. In the first test, the integral gain of the PLL, Kpi, was varied. With the gain variation, the position of PLL mode moved on the complex plane. Firstly, the trajectory of movement of open-loop PLL mode, λp, with the variation of Kpi is shown by dashed line in Fig. 4.33, where the hollow circle is the position of openloop compensation mode, λc. Secondly, the oscillation modes of the closed-loop interconnected model of the example system were calculated. The trajectories of the closed-loop compensation mode, λc , and the closed-loop PLL mode, λp , with the variation of Kpi are shown by the solid curves in Fig. 4.33. It can be observed that

4.4

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

50

80 PLL Comp.

60 40 20

B

50 ki

C 90

PF(100%)

PF(100%)

100

0 10 A

235

40 PLL Comp.

30 20

10 0 10 A

B

50 ki

C 90

Fig. 4.34 Computational results of the PFs to demonstrate the open-loop modal resonance between the PLL and series compensation when Kpi was varied

when the open-loop PLL mode, λp, moves close to the open-loop compensation mode, λc, the closed-loop compensation mode, λc , and the closed-loop PLL mode, λp , move apart on the complex plane. It is exactly the exhibition of the open-loop modal resonance: open-loop oscillation modes move close to each other, the corresponding closed-loop oscillation modes repulse each other. Subsequently, the closed-loop PLL mode moves into the right half of complex plan. This is when the SSO occurs as caused by the open-loop modal resonance. Thirdly, the peak of open-loop modal resonance was estimated by using (4.31). the estimated positions of the closed-loop compensation mode, λc , and the closedloop PLL mode, λp , are indicated in Fig. 4.33 by crosses. It can be seen that the estimation is close to the actual peak of the open-loop modal resonance (point B is Fig. 4.32). Fourthly, the participation factors (PFs) of the closed-loop compensation mode, λc , and the closed-loop PLL mode, λp , were computed. The computational results with the variation of Kpi are presented in Fig. 4.34. It can be seen that at point A and C in Fig. 4.32 when the open-loop modal resonance did not happen, the closed-loop PLL mode is only associated with the PLL and the closed-loop compensation mode is mainly related with the series compensation. When the open-loop modal resonance happened at point B in Fig. 4.32, both the closed-loop PLL mode and the closed-loop compensation mode are participated by the PLL and the series compensation. This indicates the strong dynamic interactions between the PLL and the series compensation when the open-loop modal resonance happened. Fifthly, non-linear simulation was conducted to demonstrate the open-loop modal resonance. At 0.5 s of the simulation, the mechanical torque input to the PMSG dropped by 20% for 0.1 s. Simulation results are presented in Fig. 4.35. Two simulation results are compared in Fig. 4.35 when there was no open-loop modal resonance (point C in Fig. 4.32) and the open-loop modal resonance happened (point B in Fig. 4.32). In the second test, the PI gains of the PLL were fixed. The level of compensation, c, was varied. With the variation of compensation level, c, the position of compensation mode moved on the complex plane. The trajectory of movement of open-loop compensation mode, λc, with the variation of c is shown by dashed line in Fig. 4.36,

236

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

0.18

Operating point B Operating point C

Active pow er output

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1

0

0.2

0.4

0.6

0.8

1 1.2 Time

1.4

1.6

1.8

2

Fig. 4.35 Simulation results to demonstrate the open-loop modal resonance between the PLL and series compensation when Kpi was varied

180

λˆg

160

A

λg

140 120

B

C

λp

A

Closed-loop comp. mode Closed-loop PLL mode Open-loop comp. mode Open-loop PLL mode Predicted position

λˆp B

100 k increases

80 60 -20

-15

-10 -5 Real part

C

0

5

Fig. 4.36 Open-loop modal resonance between the PLL and series compensation when level of series compensation varied

where the hollow circle is the position of open-loop PLL mode, λp. The oscillation modes of the closed-loop interconnected model of the example system were calculated. The trajectories of the closed-loop compensation mode, λc , and the closed-loop PLL mode, λp , with the variation of c are shown by the solid curves in Fig. 4.36. It

The Open-Loop Modal Resonance Analysis of Sub-synchronous Oscillations. . .

PF(100%)

237

50

100

PLL Comp.

80

PF(100%)

4.4

60 40

20 B 0 0.2 A 0.3 0.4 0.5 C 0.6 Comp. level

40 30 20

PLL 10 Comp. B 0 0.2 A 0.3 0.4 0.5 C 0.6 Comp. level

Fig. 4.37 Computational results of the PFs to demonstrate the open-loop modal resonance between the PLL and series compensation when level of series compensation varied 0.19 Operating point B Operating point C

Active power output/p.u.

0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11

0.1 0

0.4

0.8

1.2

1.6

2

Time/s

Fig. 4.38 Simulation results to demonstrate the open-loop modal resonance between the PLL and series compensation when the level of series compensation varied

can be observed that when the open-loop compensation mode, λc, moves close to the open-loop PLL mode, λp, the closed-loop compensation mode, λc , and the closedloop PLL mode, λp , move apart on the complex plane. Subsequently, the closed-loop PLL mode moves into the right half of complex plane and the SSO occurs as caused by the open-loop modal resonance. The estimated positions of the closed-loop compensation mode, λc , and the closed-loop PLL mode, λp , which were calculated by using (4.31), are indicated in Fig. 4.36 by crosses. Figure 4.37 gives the computational results of the PFs when the level of series compensation varied. It can be seen that when the open-loop modal resonance happened, both the closed-loop PLL mode and the closed-loop compensation mode are participated by the PLL and the series compensation, indicating strong

238

4

Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

dynamic interactions between the PLL and the series compensation. Figure 4.38 presents the results of nonlinear simulation to confirm that the SSO was caused by the open-loop modal resonance.

Summary of the Chapter By dividing a power system into two subsystems, a closed-loop interconnected model of the power system can be established. Open-loop modal resonance is the closeness of a pair of oscillation modes, each of which belongs to one of two subsystems. Hence, open-loop modal resonance is the closeness of two oscillation modes separately from two open-loop subsystems. Two open-loop subsystems may normally be weakly coupled such that interconnection of the subsystems results in no danger of instability of closed-loop interconnected system. However, theoretical analysis in this chapter indicates that under the condition of open-loop modal resonance, dynamic interaction between two weakly coupled subsystems may become strong. On the complex plan, two resonant open-loop oscillation modes shall lead to the repulsion of two corresponding closed-loop oscillation modes. Hence, the strong dynamic interaction caused by the open-loop modal resonance may bring out the instability risk of the closed-loop interconnected system. The risk of sub/super-synchronous oscillations caused by the open-loop modal resonance is examined for three types of the sending-end power system with seriescompensated transmission line. The first type of power system is of a sending-end SG being connected to the main grid via a series-compensated transmission line. The SSO in this type of sending-end power system is the conventional problem of the SSR. The examination in the chapter indicates that the TI can be analyzed satisfactorily according to the theory of open-loop modal resonance introduced in the chapter. The open-loop modal resonance explains why the weakly coupled torsional dynamics of the SG can interact strongly with the electric system to result in the SSR. The second type of power system is comprised of a sending-end DFIG being connected to the main grid through a series-compensated transmission line. The SSO in this type of sending-end power system is the so-called SSCI due to the dynamic interaction between the converter control system and the series compensation. Examination in the chapter indicates that the open-loop modal resonance can reveal the SSCI mechanism from the standpoint of modal condition: when the open-loop oscillation mode of the DFIG is in resonance with the oscillation mode of series compensation, the SSCI may likely result in growing SSO. The third type of the power system consists of a sending-end PMSG being connected to the main grid via a series-compensated transmission line. Examination indicates that the open-loop modal resonance between the converter control system and the series compensation may lead to growing SSO. Introduction on the theory of open-loop modal resonance and examination of instability risk caused by the open-loop modal resonance in three types of power

References

239

system in the chapter demonstrates that the open-loop modal resonance is a generally applicable theory. It examines the SSO risk and mechanism from the system modal condition and dynamic interactions between subsystems. However, it is worthwhile to point out that in this chapter the theory of open-loop modal resonance and application are introduced and demonstrated on the basis of a single-input singleoutput (SISO) closed-loop interconnected model. More general case of open-loop modal resonance based on a multiple-input multiple-output (MIMO) closed-loop interconnected model is to be introduced in the next chapter. In addition, the oscillation mode associated with the series compensation is of the sub-synchronous frequency. Subsequently, the result of open-loop modal resonance with the participation of the oscillation mode of series compensation is the sub-synchronous oscillation. The open-loop modal resonance can also be applied to study the super-synchronous oscillation. The application is demonstrated in the next chapter.

References 1. I. M. Canay, “A novel approach to the torsional interaction and the electrical damping of synchronous machine: Part I – Theory,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 10, pp. 3630–3638, Oct. 1984. 2. I. M. Canay, “A novel approach to the torsional interaction and the electrical damping of synchronous machine: Part II – Application to an arbitrary network,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 10, pp. 3639–3646, Oct. 1984. 3. M. I. Višik and L. A. Ljusternik, “The solution of some perturbation problems in the case of matrices and self-adjoint and non-self-adjoint differential equations,” Russian Mathematical Surveys, vol. 15, no. 3, pp. 1–73, 1960. 4. IEEE subsynchronous resonance task force, “First benchmark model for computer simulation of subsynchronous resonance,” IEEE Trans. Power Syst., vol. 96, no. 5, pp. 1565–1572, Sept. 1977. 5. I. Dobson, “Strong Resonance Effects in Normal Form Analysis and Subsynchronous Resonance,” in Proc. Bulk Power System Dynamics Control, Onomichi, Japan, Aug. 26–31, 2001. 6. A. P. Seyranian, “Sensitivity analysis of multiple eigenvalues”, Mech. Struct. Mach., vol. 21, no. 2, pp. 261–284, 1993. 7. X. Xie, X. Zhang, H. Liu, H. Liu, Y. Li and C. Zhang, “Characteristic Analysis of Subsynchronous Resonance in Practical Wind Farms Connected to Series-Compensated Transmissions,” IEEE Transactions on Energy Conversion, vol. 32, no. 3, pp. 1117–1126, Sept. 2017. 8. Y. Song and F. Blaabjerg, “Overview of DFIG-Based Wind Power System Resonances Under Weak Networks,” IEEE Transactions on Power Electronics, vol. 32, no. 6, pp. 4370–4394, Jun. 2017. 9. L. Fan and Z. Miao, “Nyquist-Stability-Criterion-Based SSR Explanation for Type-3 Wind Generators,” IEEE Transactions on Energy Conversion, vol. 27, no. 3, pp. 807–809, Sept. 2014. 10. L. Fan, C. Zhu, Z. Miao and M. Hu, “Modal Analysis of a DFIG-Based Wind Farm Interfaced With a Series-compensated Network,” IEEE Transactions on Energy Conversion, vol. 26, no. 4, pp. 1010–1020, Dec. 2011.

240

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Analysis of Sub-synchronous Oscillations in a Sending-End Power. . .

11. H. A. Mohammadpour and E. Santi, “Modeling and Control of Gate-Controlled Series Capacitor Interfaced With a DFIG-Based Wind Farm,” IEEE Transactions on Industrial Electronics, vol. 62, no. 2, pp. 1022–1033, Feb. 2015. 12. L. Fan, R. Kavasseri, Z. L. Miao and C. Zhu, “Modeling of DFIG-Based Wind Farms for SSR Analysis,” IEEE Transactions on Power Delivery, vol. 25, no. 4, pp. 2073–2082, Oct. 2010. 13. Brian Porter and Roger Crossley, Modal control: theory and applications, Taylor and Francis, 1974.

Chapter 5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in a Power System with Grid-Connected Wind Farms

Abbreviation SG WTG SSR SSO SISO SSI FBMPS RSC TITO ROPS MIMO

Synchronous generator Wind turbine generator Sub-synchronous resonance Sub-synchronous oscillation Single-input single output Sub-synchronous interaction First benchmark power system Rotor side converter Two-inputs two-outputs Remainder of power system Multi-input multi-output

In the previous chapter, the concept of open-loop modal resonance is introduced to examine the sub-synchronous resonance (SSR) in a power system with a sendingend synchronous generator (SG) being connected to the main grid via a seriescompensated transmission line. Afterwards, the open-loop modal resonance to cause the sub-synchronous oscillation (SSO) in a power system with a sending-end wind turbine generator (WTG), either a DFIG or a PMSG, being connected to the main grid via a series-compensated transmission line is studied. Hence, the theory and application of open-loop modal resonance to cause the SSO introduced in the previous chapter are mainly about the modal resonance between the sending-end generator, either the SG or the WTG, and the series compensation. The theory of open-loop modal resonance is established on the basis of single-input single-output (SISO) closed-loop interconnected model.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Du, H. Wang, Analysis of Power System Sub/Super-Synchronous Oscillations Caused by Grid-Connected Wind Power Generation, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-031-35343-7_5

241

242

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . .

In this chapter, the open-loop modal resonance to cause the sub/super-synchronous oscillation in a power system without the series-compensated transmission line is introduced. In the next section, the open-loop modal resonance between two DFIGs and between a DFIG and the torsional dynamics of a SG is demonstrated in a power system without the series-compensated transmission line. The analysis of open-loop modal resonance is based on a one-input two-output closed-loop interconnected model. The theory of open-loop modal resonance introduced in the previous chapter is applied for the analysis. In Sect. 5.2, the open-loop modal resonance of a grid-connected wind farm with multiple PMSGs is studied. For the study, a two-input two-output (TITO) closedloop interconnected model is established. The theory of open-loop modal resonance based on the TITO closed-loop interconnected model is introduced. In Sect. 5.3, the open-loop modal resonance in a power system with multiple gridconnected wind farms is introduced. The analysis of open-loop modal resonance is based on a multi-input multi-output (MIMO) closed-loop interconnected model. Thus, the theory of open-loop modal resonance is established for this general case and can be applied to any complex power system with multiple grid-connected wind farms.

5.1 5.1.1

Open-Loop Modal Resonance in a Power System Without the Series Compensated Lines The Closed-Loop Interconnected Model and the Open-Loop Modal Resonance

In Chap. 4, the open-loop modal resonance to cause the SSO in a power system with a series-compensated transmission line is introduced. In this section, the SSO caused by the open-loop modal resonance in a power system with grid-connected DFIGs and without any series-compensated lines are examined. The examination is carried out by using the closed-loop interconnected model of the power system shown in Fig. 4.26, which is re-displayed in Fig. 5.1. The state-space model of the open-loop DFIG subsystem and the ROPS subsystem is respectively (see Sect. (4.61) and (4.62)) d ΔXdd = Add ΔXdd þ bddq ΔV dt ΔPd = cdp T ΔXdd þ ddp1 ΔV ΔQd = cdq T ΔXdd þ d dq1 ΔV

ð5:1Þ

5.1

Open-Loop Modal Resonance in a Power System Without the Series. . .

Fig. 5.1 Closed-loop interconnected model of the power system with a gridconnected DFIG

ROPS subsystem

DFIG subsystem

+

H P (s)

ΔV

H Q (s)

243

Δ Pd

ΔQd

GP ( s )

GQ ( s )

d ΔXsp = Asp ΔXsp þ bsp ΔPd þ bsq ΔQd dt

ð5:2Þ

ΔV = csp T ΔXsp þ d sp ΔPd þ d sq ΔQd Obviously, transfer function of the DFIG subsystem and the ROPS subsystem is (see Sect. (4.63) and (4.64)) H P ðsÞ = cdp T ðI - Add Þ - 1 bddq þ ddp1 H Q ðsÞ = cdq T ðI - Add Þ - 1 bddq þ ddq1

GP ðsÞ = Csp I - Asp

-1

GQ ðsÞ = Csp I - Asp

-1

bsp þ d sp bsq þ d sq

ð5:3Þ

ð5:4Þ

From (5.1) and (5.2), the state-space model of the interconnected closed-loop system displayed in Fig. 5.1, i.e., the entire power system with the grid-connected DFIG, can be obtained to be d ΔX = AΔX dt

ð5:5Þ

where ΔX = ΔXdd T ΔXsp T Asp þ A=

T

ddp1 bsp þ ddq1 bsq csp T 1 - d sp d dp1 - d sq d dq1

bddq csp T 1 - d sp d dp1 - d sq d dq1

bsp cdp T þ bsq cdq T þ Add þ

d dp1 bsp þ d dq1 bsq dsp cdp T þ dsq cdq T 1 - d sp d dp1 - d sq d dq1 d sp bddq cdp T þ dsq bddq cdq T 1 - dsp ddp1 - dsq ddq1

Denote λdj, j = 1, 2, . . .m and λgj, j = 1, 2, ⋯n as the open-loop SSO modes of the DFIG subsystem and the ROPS subsystem in Fig. 5.1, respectively. λdj, j = 1, 2, . . .m are the eigenvalues of the open-loop state matrix of the DFIG subsystem, Add

244

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . .

in (5.1), i.e., the poles of HP(s) and HQ(s) given in (5.3). λgj, j = 1, 2, ⋯n are the eigenvalues of the open-loop state matrix of the ROPS subsystem, Asp in (5.2), i.e., the poles of GP(s) and GQ(s) given in (5.4). Denote Rdpj, Rdqj, j = 1, 2, . . .m as the residues corresponding to λdj, j = 1, 2, . . .m. Denote Rgpj, Rgqj, j = 1, 2, . . .n as the residues corresponding to λgj, j = 1, 2, . . .n. It can have GP ðsÞ =

Rgpi þ s - λgi

n j=1

Rgpj þ dsp s - λgj

j≠i

GQ ðsÞ =

Rgqi þ s - λgi

n j=1

Rgqj þ d sq s - λgj

j≠i

Rdpk H P ðsÞ = þ ðs - λdk Þ

m j=1

Rdpj þ d dp1 s - λdj

ð5:6Þ

j≠k

H Q ðsÞ =

Rdqk þ ðs - λdk Þ

m j=1

Rdqj þ d dq1 s - λdj

j≠k

Consider that open-loop modal resonance happens such that λgi ≈ λdk. Denote λgi and λdk as the closed-loop SSO modes corresponding to λgi and λdk respectively. Taking the derivation similar to that from (4.71), (4.72), (4.73), (4.74), (4.75), and (4.76), it can be proved that λgi ≈ λgi ± SRIik , λdk ≈ λdk ± SRIik ≈ λgi ± SRIik

ð5:7Þ

where SRIik = Rgpi Rdpk þ Rgqi Rdqk SRIik = Rgpi Rdpk þ Rgqi Rdqk is named as the SSI (sub-synchronous interaction) residue index and can be used to identify the risk of growing SSOs caused by the open-loop modal resonance. This is because if |Real part of SRIik| is greater than |Real part of λgi ≈ λdk|, it is very likely that there may be a closed-loop SSO mode, either λgi or λdk , located in the right half of the complex plane such that growing SSO occurs in the powers system. In the ROPS subsystem, there may be the synchronous generators (SGs) and other wind turbine generators (WTGs), such as the DFIGs and PMSGs. Hence, λgi could be a torsional SSO mode of an SG in the ROPS subsystem. Subsequently, the open-loop SSO modal resonance between the SG and the DFIG in the DFIG subsystem may lead to growing torsional SSOs. Alternatively, λgi could be an open-loop SSO mode associated with a DFIG in the ROPS subsystem. Thus, the

5.1

Open-Loop Modal Resonance in a Power System Without the Series. . .

245

open-loop SSO modal resonance between two DFIGs may cause poorly and even negatively damped sub/super-synchronous oscillation. In the next three subsections, the cases of the sub/super-synchronous oscillation induced by the DFIGs under the condition of open-loop modal resonance in an example power system without series compensation are presented.

5.1.2

Modified IEEE First Benchmark Power System

Figure 5.2 shows the configuration of the modified IEEE first benchmark power system (FBMPS) for the SSR study [1]. The modifications include the removal of the series compensation and the connection of two wind farms at the sending end of the power system. The models and parameters of the SG, transformer and transmission line recommended in [1] are used. Each of the wind farms consists of 50 DFIGs and is represented by an aggregated DFIG [2]. The models and parameters of the aggregated DFIG given in [2] for the sub-synchronous interaction (SSI) study are adopted. In practice, it is unlikely that there will be two identical wind farms; therefore, the parameters of the converter control systems of the two DFIGs given in [2] are adjusted slightly to obtain a difference of 2% between DFIGA and DFIGB in Fig. 2.5. The SSO induced by DFIGA under the condition of open-loop modal resonance is examined as follows. Firstly, the open-loop model of (5.1) for DFIGA is established. The eigenvalues of the open-loop state matrix of DFIGA, Add in (5.1), are computed. From the results of eigenvalues, one open-loop SSO mode of DFIGA

DFIG A

Z3

Line Current

Z1

ZL

ZT

SG

DFIG B

Equivalent system

Z2

Fig. 5.2 Configuration of modified IEEE first benchmark power system

246

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . .

Table 5.1 SSO modes of the modified IEEE first benchmark power system Open-loop SSO modes λd1 = - 6:9303 þj105:65

Related component of system dynamics Inner current PI controller in the RSC-RPCS of DFIGA

Closed-loop SSO modes λd1 = - 12:671 þj131:91

λg1 = - 0:8955 þj60:782

Torsional dynamics of mass HP, IP, LPA, and GEN

λg1 = - 1:087

λg2 = - 1:5178

Torsional dynamics of mass EXC

λg2 = - 1:535

λg3 = - 0:7776 þj98:559

Torsional dynamics of mass HP and LPA

λg3 = - 0:8125 þj98:633

λg4 = - 0:9045 þj124:54

Torsional dynamics of mass LPB

λg4 = - 0:9253

λg5 = - 6:5845 þj103:9

Inner current PI controller in the RSC-RPCS of DFIGB

λg5 = 0:8964

þj77:978

þj60:898 þj78:004

þj124:65 þj70:02

is identified and selected for studying the open-loop SSO modal resonance of DFIGA with the rest of the modified IEEE FBMPS. The mode is denoted by λd1 and listed in the first column of Table 5.1. Participation factors (PFs) for λd1 are calculated from the open-loop state-space model of (5.1) for DFIG A. From the computational results for the PFs, λd1 is identified to be related to the inner current PI controller in the reactive power control system of the rotor side converter (RSC) of DFIGA. In Table 5.1, the symbols HP, IP, LPA, LPB, GEN, and EXC refer to the high pressure cylinder, intermediate pressure cylinder, first low pressure cylinder, second low pressure cylinder, generator, and exciter, respectively. Secondly, the open-loop state-space model of (5.2) for the rest of the modified IEEE first benchmark power system, including DFIGB, is established. From the computational results for the eigenvalues of the open-loop state matrix, Asp in (5.2), five open-loop SSO modes for the power system are selected, given by λgi, i = 1, 2, 3, 4, 5 in the first column of Table 5.1. By calculating the PFs from the established open-loop state-space model of (5.2), λgi, i = 1, 2, 3, 4 are identified to be the torsional SSO modes of the SG; λg5 is found to be related to the inner current PI controller in the RSC of DFIGB. From the first column of Table 5.1, it can be seen that all the open-loop SSO modes are well damped. Thirdly, the closed-loop interconnected model of (5.5) for the modified IEEE first benchmark power system is established. The closed-loop SSO modes are computed from the closed-loop state matrix A in (5.5); the computational results are presented in the third column of Table 5.1. The PFs are computed for the closed-loop SSO modes to identify their association with the dynamic components and corresponding open-loop SSO modes.

5.1

Open-Loop Modal Resonance in a Power System Without the Series. . .

5.1.3

247

Open-Loop SSO Modal Resonance Between Two DFIGs

From the third column of Table 5.1, it can be seen that the closed-loop SSO mode associated with the inner current PI controller in the RSC of DFIGB, λg5 , is negatively damped, as a result of connecting DFIGA to the modified IEEE first benchmark power system. From the first column of Table 5.1, it can be seen that there is an open-loop SSO modal resonance, λg5 ≈ λd1. From the computational result for the SSI residue index in the second row of Table 5.2, it can be seen that |Real part of SRI51| > |Real part of λg5 ≈ λd1|. Therefore, the negatively damped λg5 is identified as being caused by the open-loop SSO modal resonance, λg5 ≈ λd1. To clearly demonstrate the identified open-loop SSO modal resonance, λg5 ≈ λd1, the positions of the open-loop and closed-loop SSO modes involved in the resonance on the complex plane are displayed in Fig. 5.3. From Fig. 5.3, it can be seen that the closed-loop SSO modes λg5 and λd1 are at the approximately opposite positions with respect to those of λg5 ≈ λd1. In addition, the estimated positions of λg5 and λd1 obtained using (5.7) and SRI51 given in Table 5.2 are indicates by crosses in Fig. 5.3 and can be seen to be close to the actual positions of λg5 and λd1 . Figure 5.3 confirms Table 5.2 Open-loop modal resonance between two DFIGs Open-loop SSO modal resonance λi5 ≈ λd1

SSI residue SRI51 = 7:1358 - j29:384

Parameters of inner current PI controllers of DFIG A and B A : K QP = 0:045, K QI = 16:5 B : K QP = 0:044, K QI = 16:2

300

Imaginary part

250

KQI increases

200

λˆd1

150

λd1 100 50 0 -16

λˆg 5

λg 5 Estimated positions of closed-loop SSO modes -14

-12

-10

-8

-6

Real part

-4

-2

0

2

Fig. 5.3 Open-loop SSO modal resonance between two DFIGs when PwA = 0.5 p. u. , PwB = 0.5 p. u.

248

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . . 150

λˆd1 Imaginary part

130

λd1 λg 5

110

90

70

50 -16

Estimated positions of closed-loop SSO modes -14

-12

-10

-8

-6

λˆg 5 -4

-2

0

2

Real part Fig. 5.4 Open-loop SSO modal resonance between two DFIGs when PwA = 0.1 p. u., PwB = 0.9 p. u.

the effectiveness of the criteria proposed from (5.7) for identifying open-loop SSO modal resonance. The positions of open-loop SSO modes on the complex plane change with the variation of the power system operating conditions, which may affect the occurrence and results of open-loop SSO modal resonance. Hence, the power output from DFIGA and DFIGB are varied. The results of the SSO modal computation are presented in Fig. 5.4. It can be seen that the positions of the open-loop SSO modes change slightly as compared to those in Fig. 5.3. However, open-loop SSO modal resonance still exists and leads to negatively damped SSOs. The open-loop SSO modal resonance shown in Fig. 5.3 occurs due to the resonance between the inner current PI controllers of the RSC of the two DFIGs. The parameters of the two PI controllers, KQP and KQI, are approximately the same as those given in the second row of Table 5.2, such that λg5 ≈ λd1. By changing the parameters of the inner current PI controller of DFIGA, λd1 can move away from λg5 on the complex plane. Thus, the open-loop SSO modal resonance is dismissed and the damping of the negatively damped λg5 can be improved. In Fig. 5.3, the dashed curves are the trajectories of λd1 and λg5 when KQI is changed from KQI = 16.5 to KQI = 60. From the dashed curves in Fig. 5.3, it can be seen that as λd1 moves away from λg5, λg5 moves towards λg5 with improved damping. This confirms once again that the negatively damped λg5 is caused by the open-loop SSO modal resonance, λg5 ≈ λd1. Figure 5.5 shows the computational results for the PFs of the closed-loop SSO modes, λg5 and λd1 . The case of modal resonance in Fig. 5.5 occurs when λg5 ≈ λd1 as indicated in Fig. 5.3 (DFIGA : KQI = 16.5, DFIGB : KQI = 16.2), such that λg5 is negatively damped. The case of no modal resonance occurs when λd1 is far from λg5

Open-Loop Modal Resonance in a Power System Without the Series. . .

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

DFIG A

DFIG B

PFs

PFs

5.1

PFA1

PFA2

PFB1

PFB2

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

DFIG A

PFA1

PFs for λˆ d 1

PFA2

249

DFIG B

PFB1

PFB2

PFs for λˆ g 5

Fig. 5.5 Computational results for the PFs for λd1 and λg5

as indicated in Fig. 5.3 (DFIG A : KQI = 60), such that λg5 is close to λg5 with improved damping. In Fig. 5.5, PFA1 and PFB1 are the sums of the PFs of state variables associated with the inner current PI controller of the RSC of DFIGA and DFIGB, respectively; PFA2 and PFB2 are the sums of the PFs of all other state variables for DFIGA and DFIGB, respectively. From Fig. 5.5, it can be seen that in the case of no modal resonance (depicted by bars with slashed lines), λd1 or λg5 is respectively participated by only DFIGA or DFIGB, indicating weak SSIs between the two DFIGs. In the case of modal resonance (depicted by filled bars), both λd1 and λg5 are considerably participated by both DFIG A and B, indicating strong SSIs between the two DFIGs. Confirmation from a non-linear simulation is presented in Fig. 5.6. At 0.1 s in the simulation, the reactive power output from DFIGB drops by 0.2 p.u. and recovers in 20 ms. From Fig. 5.6, it can be seen that in the case of open-loop modal resonance, there are considerable variations in the reactive power output from both DFIGs. Growing SSOs occur in the modified IEEE first benchmark power system. When the modal resonance is eliminated by tuning KQI, the SSOs are well damped.

5.1.4

Torsional SSOs Caused by DFIG A

Parameter tuning is often exercised in control system design in order to achieve satisfactory control performance. For example, in the previous subsection, it is demonstrated that tuning KQI can dismiss the open-loop SSO modal resonance and thereby improve the damping of λg5 . However, improper parameter tuning may also cause the open-loop SSO modal resonance to degrade the damping of closed-loop SSO modes; this is demonstrated in the current subsection as follows. The integral gain of the inner current PI controller of the RSC of DFIGA, KQI, is tuned within the range from KQI = 3 to KQI = 25. The open-loop SSO mode of

250

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . . 0.3

No modal resonance Modal resonance

Qw - DFIG A (p.u.)

0.2 0.1

0 -0.1 -0.2

Time (second) -0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4

No modal resonance Modal resonance

Qw - DFIG B (p.u.)

0.3 0.2

0.1 0 -0.1 -0.2 -0.3

Time (second) -0.4

0

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

No modal resonance Modal resonance

1.6

Line Current (p.u.)

0.2

1.5

1.4 1.3

1.2

Time (second) 1.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 5.6 Non-linear simulation (modal resonance between two DFIGs)

0.9

1

5.1

Open-Loop Modal Resonance in a Power System Without the Series. . .

251

140

Imaginary part

120

100

Open-loop modal resonance points Actual Positions Estimated Positions Open-loop SSO modes trajectory Closed-loop SSO modes trajectory

④

③

②

80

①

60 -4

-3

-1

-2

0

1

Real part Fig. 5.7 Open-loop SSO modal resonance between the RSC control system of DFIG A and the torsional dynamics of the SG (PwA = 0.5 p. u., PSG = 0.5 p. u.)

DFIGA, λd1, moves on the complex plane. The trajectory of movement of λd1 with the variation of KQI is shown by the dashed curve in Fig. 5.7. In Fig. 5.7, the positions of the open-loop torsional SSO modes of the SG, λgi, i = 1, 2, 3, 4 (see the first column of Table 5.1), are indicates by hollow circles. The solid curves indicate the trajectories of corresponding movement of the closed-loop SSO modes, λd1 and λgi , i = 1, 2, 3, 4. From Fig. 5.7, it can be observed that λd1 is consecutively close to λgi, i = 1, 2, 3, 4, at points ①, ②, ③, and ④ (indicated by hollow circles) as it moves along the trajectory. Therefore, the open-loop SSO modal resonance occurs around these four points, which causes the corresponding movement of the closed-loop torsional SSO modes, λgi , i = 1, 2, 3, 4, towards the right on the complex plane respectively. Moreover, when the open-loop SSO modal resonance occurs around these four points, the closed-loop SSO modes of DFIGA move further towards the left on the complex plane. The positions of the closed-loop SSO modes, λd1 and λgi , i = 1, 2, 3, 4, when the open-loop SSO modal resonance occurs at points ①, ②, ③, and ④ are indicated by filled circles in Fig. 5.7. Obviously, the open-loop SSO modal resonance improves the damping of the SSO mode of DFIG A. At the same time, the openloop SSO modal resonance causes damping degradation of the torsional SSO modes of the SG. In the second column of Table 5.3, the open-loop SSO modes of DFIGA, λd1, at points ①, ②, ③, and ④ in Fig. 5.7 are given. The first column of Table 5.3 gives the value of KQI when λd1 is at points ①, ②, ③, and ④ in Fig. 5.7, respectively. The third column in Table 5.3 presents the computational results for the SSI residue

252

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . .

Table 5.3 Open-loop SSO modal resonance of DFIG A with the SG

λd1 -1.6834 + j60.796 -1.6834 + j78.248 -1.6834 + j98.794 -1.6834 + j124.96

KQI 4.3 8.2 14.3 25.6

0.6

0.5

0.5

Rest of power system

DFIG A

0.3

0.2

0.4

PFs

PFs

0.4

Rest of power system

DFIG A

0.3 0.2

0.1 0

SSI residue index SRI11 = 1.4524 - j1.126 SRI21 = 0.4623 - j0.338 SRI31 = 0.4358 - j0.766 SRI41 = 0.4575 - j1.109

0.1 PFA1

PFA2

PFG1

PFG2

0

PFA1

PFA2

PFG1

PFG2

Fig. 5.8 Computational results for the PFs for λd1 (left) and λg1 (right) when open-loop SSO modal resonance occurs at ①

index at points ①, ②, ③, and ④. Using (5.7), the estimated positions of the closedloop SSO modes, λd1 and λgi , i = 1, 2, 3, 4, when the open-loop SSO modal resonance occurs at points ①, ②, ③, and ④ are calculated and indicated by crosses in Fig. 5.7. They are very close to the actual positions of the closed-loop SSO modes, indicated by solid circles. In addition, from the first column of Table 5.1 and the third column of Table 5.3, it can be seen that for the open-loop SSO modal resonance at point ①, |real part of SRI11| > |real part of λg1|. This explains why the open-loop SSO modal resonance causes negative damping of λg1 . For the open-loop SSO modal resonance at points ②, ③, and ④, |real part of SRIi1| ≈ |real part of λgi|, i = 2, 3, 4. This is why the torsional SSO modes, λgi , i = 2, 3, 4, are poorly damped when the open-loop SSO modal resonance occurs at points ②, ③, and ④. The results described above confirm the correctness of (5.7) and the analytical conclusions obtained from (5.7) regarding open-loop SSO modal resonance. In order to evaluate the SSIs between the torsional dynamics of the SG and DFIGA, the PFs for λd1 and λgi , i = 1, 2, 3, 4, when the open-loop SSO modal resonance occurs at points ①, ②, ③, and ④, are calculated and displayed in Figs. 5.8 and 2.11. In Figs. 5.8, 5.9, 5.10 and 5.11, PFA1 is the sum of the PFs of state variables associated with the inner current PI controller of the RSC of DFIGA; PFA2 is the sum of the PFs of all other state variables of DFIGA; PFG1 is the sum of the PFs of state variables associated with the torsional dynamics of the SG; and PFG2 is the sum of the PFs of all other state variables in the rest of the modified IEEE first benchmark power system. The computational results for the PFs presented in Figs. 5.8, 5.9, 5.10 and 5.11 confirm the SSIs between the torsional dynamics of the SG and the inner current PI controller of the RSC of DFIGA.

Open-Loop Modal Resonance in a Power System Without the Series. . .

0.7

0.7

0.6

0.6

Rest of power system

PFs

0.5

DFIG A

0.4 0.3

Rest of power system

DFIG A

0.4 0.3

0.2

0.2

0.1

0.1

0

253

0.5

PFs

5.1

0

PFA1

PFA2

PFG1

PFG2

PFA1

PFA2

PFG1

PFG2

Fig. 5.9 Computational results for the PFs for λd1 (left) and λg2 (right) when open-loop SSO modal resonance occurs at ②

0.7

Rest of power system

PFs

0.5

DFIG A

0.4 0.3

PFs

0.6

0.2 0.1

0

PFA1

PFA2

PFG1

PFG2

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Rest of power system

DFIG A

PFA1

PFA2

PFG1

PFG2

Fig. 5.10 Computational results for the PFs for λd1 (left) and λg3 (right) when open-loop SSO modal resonance occurs at ③ 0.7

0.7

0.6

0.6

PFs

DFIG A

0.4 0.3

Rest of power system

0.5

PFs

Rest of power system

0.5

DFIG A

0.4 0.3

0.2

0.2

0.1

0.1 0

0

PFA1

PFA2

PFG1

PFG2

PFA1

PFA2

PFG1

PFG2

Fig. 5.11 Computational results for the PFs for λd1 (left) and λg4 (right) when open-loop SSO modal resonance occurs at ④

254

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . . 0.50

No modal resonance Modal resonance

0.49 0.48

0.47 0.46

0.45 0.44

0.43 0.42

0.41 0.40

Time (second) 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6

0.7

0.8

0.9

1

0.015

No modal resonance Modal resonance

0.01 0.005 0 -0.005 -0.01 -0.015

Time (second) 0

0.1

0.2

0.3

0.4

0.5

0.8

0.9

1

Fig. 5.12 Results of non-linear simulation

Confirmation from the non-linear simulation is presented in Fig. 5.12 for two cases: (1) the case of open-loop SSO modal resonance at point ①; (2) the case of no modal resonance. Modal analysis and the non-linear simulation results above demonstrate that the open-loop SSO modal resonance between DFIGA and the torsional dynamics of the SG degrades the damping of the torsional SSO modes of the SG. At the same time, the damping of the closed-loop SSO mode of DFIGA is improved at the points of modal resonance. In order to show that the open-loop SSO modal resonance may also reduce the damping of the SSO mode of DFIG A, the damping coefficients of the torsional dynamics of the SG in the modified IEEE first benchmark power system are increased. Subsequently, the open-loop torsional SSO modes of the SG are of better damping. Their positions on the complex plane are indicated by hollow circles in

5.1

Open-Loop Modal Resonance in a Power System Without the Series. . .

255

140

Imaginary part

120

Open-loop modal resonance points Open-loop SSO modes trajectory Closed-loop SSO modes trajectory

⑧

⑦

100

⑥

80

⑤

60 -5

-4

-3

-2

-1

Real part Fig. 5.13 Open-loop SSO modal resonance between the RSC control system of DFIGA and the shaft dynamics of the SG

Fig. 5.13. Then, the integral gain of the inner current PI controller of the RSC of DFIGA, KQI, is varied from KQI = 3 to KQI = 25. The trajectory of movement of the 2 open-loop SSO mode of DFIGA, λd1, with the change of SCR = X L PVw0- rated is shown by the dashed curve in Fig. 5.13. The trajectories of the corresponding movement of the closed-loop SSO modes, Pw - rated and λgi , i = 1, 2, 3, 4, are displayed in Fig. 5.13 as solid curves. From Fig. 5.13, it can be observed that open-loop SSO modal resonance occurs at points ⑤, ⑥, ⑦, and ⑧. Apparently, the damping of the torsional SSO modes of the SG is improved when open-loop modal resonance occurs. At the same time, open-loop SSO modal resonance degrades the damping of the closed-loop SSO mode of DFIGA, λd1 . In order to demonstrate the open-loop modal SSO resonance as affected by the variation of the power system operating conditions, the active power output from DFIGA and DFIGB is changed from 0.5 p.u. to 0.1 p.u. and 0.9 p.u., respectively. The results of the open-loop modal computation corresponding to the open-loop modal SSO resonance at point ① in Fig. 5.7 are presented in Fig. 5.14. It can be seen that with the variation of the power system operating point, the open-loop SSO mode of DFIGA, λd1, moves away from the open-loop torsional SSO mode, λg1. Consequently, the effect of the open-loop SSO modal resonance is considerably reduced. Compared with the results presented in Fig. 5.7, it can be seen that the damping of the closed-loop torsional SSO mode, λg1 , is improved. The results presented in Figs. 5.4 and 5.14 indicate that if the variations of power system operating points do not change the condition of open-loop SSO modal resonance considerably, damping degradation of the closed-loop SSO mode occurs.

5

Open-Loop Modal Resonance to Cause Sub/Super-synchronous Oscillations in. . .

Fig. 5.14 Open-loop SSO modal resonance between the RSC control system of DFIG A and λg1 at point ① (PwA = 0.1 p. u., PSG = 0.9 p. u.)

62

61

Imaginary part

256

λˆd1 λd1

λg1

60

Unstable Area

λˆg1 59

58 -5

Estimated positions of closed-loop SSO modes

-4

-3

-2

-1

0

1

Real part

However, if the variations change or even eliminate the condition for open-loop SSO modal resonance, damping of the closed-loop SSO mode improves consequently.

5.2

Open-Loop Modal Resonance Caused by a Grid-Connected Wind Farm with Multiple PMSGs

In previous subsection and Sect. 4.3, the open-loop modal resonance is analyzed on the basis of closed-loop interconnected model which is of either two inputs and one output or one input and two-outputs. The closed-loop interconnected model is derived by assuming an ideal PLL is adopted by the grid-connected DFIG. In this section, the open-loop modal resonance caused by a grid-connected wind farm with multiple PMSGs is examined. In this case, the closed-loop interconnected model of the wind farm with multiple PMSGs is of two inputs and two outputs.

5.2.1

Two-Inputs Two-Outputs Closed-Loop Interconnected Model

Figure 5.15 shows the configuration of a grid-connected wind farm, where Ix + jIy is the output current from the wind farm and Vx + jVy is the voltage at the terminal of the wind farm, expressed in the common x - y coordinate of the power system. Take ΔIx + jΔIy as the output variables from and ΔVx + jΔVy as the input variables to the wind farm, following linearized transfer function matrix model of the wind farm can be established

5.2

Open-Loop Modal Resonance Caused by a Grid-Connected Wind Farm. . .

Vx + jVy

Wind farm

...

Fig. 5.15 A power system with a grid-connected wind farm

257

I x + jI y

External power system

ΔI = WðsÞΔV

ð5:8Þ

T T where ΔI = ½ ΔI x ΔI y ] , ΔV = ½ ΔV x ΔV y ] , W(s) is the two-inputs two-outputs (TITO) transfer function matrix of the wind farm. ΔIx + jΔIy are the exhibitions of dynamic interactions between the wind farm and external power system. Normally, the dynamic interactions are weak due to the fast speed of converter control implemented by the variable speed wind generators in the wind farm such that ΔIx + jΔIy ≈ 0. To depict the weak dynamic interactions between the wind farm and the external power system, a small positive number, 0 < ε