ANALYSIS OF SUBSYNCHRONOUS

RESONANCE IN POWER SYSTEMS

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE

Power Electronics and Power Systems Consulting Editors

Thomas A. Lipo and M. A. Pai

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VOLTAGE STABILITY OF ELECTRIC POWER SYSTEMS, Thierry Van Cutsem and Costas Voumas, ISBN: 0-7923-8139-4

AUTOMATIC LEARNING TECHNIQUES IN POWER SYSTEMS, Louis A. Wehenkel, ISBN: 0-7923-8068-1

ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY, M. A. Pai, ISBN: 0-7923-9035-0

ELECTROMAGNETIC MODELLING OF POWER ELECTRONIC CONVERTERS, J. A. Ferreira, ISBN: 0-7923-9034-2

MODERN POWER SYSTEMS CONTROL AND OPERATION, A. S. Debs, ISBN: 0-89838-265-3

RELIABILITY ASSESSMENT OF LARGE ELECTRIC POWER SYSTEMS, R. Billington, R. N. Allan, ISBN: 0-89838-266-1

SPOT PRICING OF ELECTRICITY, F. C. Schweppe, M. C. Caramanis, R. D. Tabors, R. E. Bohn, ISBN: 0-89838-260-2

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ANALYSISOF SUBSYNCHRONOUS

RESONANCEIN POWER SYSTEMS

by

K. R. Padiyar Department of Electrical Engineering

Indian Institute of Science Bangalore 560 012, India

~.

" Springer Science+Business Media, LLC

ISBN 978-1-4613-7577-7 ISBN 978-1-4615-5633-6 (eBook) DOI 10.1007/978-1-4615-5633-6

Library of Congress Cataloging-in-Publication Data

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Copyright © 1999 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1999 Softcover reprint ofthe hardcover Ist edition 1999

AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC .

Printed on acid-free paper.

To my sister, Manorama

Contents

Preface

1. INTRODUCTION 1 1.1 General 1 1.2 Definitions of SSR 4 1.3 Interactions with power system controllers 7 1.4 FACTS Controllers 8 1.5 Methods of Analysis of SSR 12 1.6 Chapter outline 16

2. MODELLING OF TURBINE GENERATOR 17 2.1 Introduction 17 2.2 Synchronous machine model 18 2.3 Park's transformation 22 2.4 Per unit quantities 30 2.5 Operational impedances and equivalent circuits 35 2.6 Modelling of excitation control system 41 2.7 Modelling of turbine generator mechanical system 43 2.8 Modelling of turbine and governor 55 2.9 Modelling and analysis of the mechanical and prime mover system 56 2.10 Synchronous generator modelling for transient simulation 59

3. MODELLING OF THE ELECTRIC NETWORK 63 3.1 Introduction 63 3.2 Transmission lines 64 3.3 Transformation using a - f3 variables 68 3.4 State equations 70 3.5 Interface between the network and generator 74 3.6 Impedance functions 75 3.7 Simulation of electromagnetic transients 78

4. ANALYSIS OF SSR WITH FIXED SERIES COMPENSATION 83

VII

Vlll ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

4.1 Introduction 83 4.2 Analysis of induction generator effect: frequency scanning method 83 4.3 Analysis of torsional interaction(TI) 87 4.4 State equations and eigenvalue analysis 96 4.5 An algorithm for computing torsional modes 108 4.6 Countermeasures for SSR III 4. 7 Torsional oscillations in parallel connected turbine generators 120

5. INTERACTIONS WITH POWER SYSTEM STABILIZER 5.1 Introduction 5.2 Basic concept in the application of PSS 5.3 Design of PSS 5.4 Torsional interaction with PSS 5.5 A case study

6. INTERACTIONS WITH HVDC CONVERTER CONTROL 6.1 Introduction 6.2 HVDC converters and control

121 121 122 126 130 132

137 137 138

6.3 Modelling of HVDC system for study of torsional interactions 147 6.4 Analysis of torsional interactions - A simplified approach 153 6.5 A case study 156 6.6 A simplified damping torque analysis 161 6.7 Control of torsional interaction 167

7. INTERACTIONS WITH SHUNT COMPENSATORS 169 7.1 Introduction 169 7.2 Static Var Compensator 171 7 .3 Torsional Interactions with SVC 186 7.4 Static Condenser(STATCON) 189 7.5 Torsional interactions with STATCON 196 7.6 A simplified analysis of torsional interaction with voltage controller 200

8. INTERACTIONS WITH SERIES COMPENSATORS 8.1 Introduction 8.2 Thyristor Controlled Series Compensator 8.3 Modelling of TCSC for SSR studies 8.4 Mitigation of SSR with TCSC 8.5 Static Synchronous Series Compensator (SSSC) 8.6 Torsional interactions with SSSC

Appendices

A- Data on IEEE Benchmark Models A.1 IEEE First Benchmark Model ( FBM ) A.2 IEEE Second Benchmark Model ( SBM )

B- Calculation of Initial Conditions

205 205 206 216 223 229 234

239

239 239 241 245

Contents IX

c- Abbreviations 249

References and Bibliography 251

Index 261

Foreword

In addition to the power flow at and around the nominal power frequency, all electrical and electromechanical power systems involve a wide range of resonant oscillatory modes which are excited during disturbances and switching events. Most of these oscillations are harmless and die out because of net positive damping. However, under some circumstances, a specific oscillation may have unacceptably high magnitude, rise or sustain for a long period, and result in damage due to insulation, mechanical aging or breakdown, or system instability. Given the natural parameters of lines and equipment, power system oscillations that involve only the passive electrical components, their resonance frequencies are substantially higher than the main power frequency. However, when the oscillations involve both the electrical and rotating mechanical equipment coupled through the magnetic flux, frequencies lower than the power frequency appear. These oscillations that involve mass and inertia of the complete turbine-generator have inter-machine or inter-area electromechanical oscillation frequencies in the range of 0.1 Hz to several Hz.

Sub synchronous oscillations in the range of 10-50Hz result from mechanical oscillations among individual turbine masses and the generator coupled into a long shaft, and these mechanical oscillations, electrically coupled with the electrical system via the generator. Subsynchronous Resonance (SSR) was unheard of until the catastrophic damage to the turbine-generator at Southern California Edison's Mojave Power Plant in 1970.

It has since been recognized that all high speed active controls of a power system such as HVDC, FACTS, excitation control, etc., have a potential of mitigating as well as causing damage or loss of life in large multi-machine generators. Even high speed reclosing after fault clearance has been recognized as having a potential of causing loss of life of turbine-generator shafts

There are simple rules of thumb that convey whether or not such possibilities of SSR exist and, if so, there are available computational tools and expertise. Also, a large number of papers have been published which are available in scattered form. Analytically, SSR is a very complex subject matter, and it is gratifying to see Professor Padiyar bring together complicated analytical and practical material into a monograph. This monograph will be of great value to engineers and post-graduate students who wish to learn about the details and find solutions for SSR problems.

Narain G. Hingorani Los Altos Hills, CA

Preface

Modern power systems are large and complex syst.ems and pose challenges to their secure and economic operation. The regulatory and resource constraints have resulted in power transmission networks operating under stressed condi­tions. The problems of system stability are further complicated by recent trends towards deregulation and restructuring of electric utilities. The system plan­ners are increasingly relying on existing and new solid-state controllers based on high power semiconductors such as thyristors and GTO's. HVDC links and Static Var Compensators based on thyristor controls have contributed to system stability and prevent system collapse. New Flexible AC Transmission System (FACTS) controllers are presently under development and have the po­tential of overcoming many of the control problems. The problem of Subsynchronous Resonance (SSR) was encountered in the sev­enties when fixed series compensation was us~d in long radial lines connecting turbine-generators to load centres. This involV!~s interaction between the elec­trical network and the torsional system of t.he turbine-generator leading to self-excitation. The torsional oscillation modes, generally have frequencies in the range of 10 to 50 Hz. Such torsional interactions were also discovered with Power System Stabilizer (PSS), HVDC cOllvert.er controller and SVC voltage controller. While the SSR problem had discouraged syst.~1lI planners from introducing se­ries compensation, the recent development of Thyristor Controlled Series Com­pensator (TCSC) has demonstrated that the SSR problem can be mitigated. New FACTS controllers based on Voltage Source Converters (VSC) such as Static Compensator (STATCOM) and Static Synchronous Series Compensator (SSSC) for voltage and power flow control ar~ expected to minimize the SSR problem. While there are a large number of papers published on SSR with fixed series compensation, there is hardly any book that gives a comprehensive coverage of the various aspects of the SSR problem. The modelling and analysis of SSR is more complex than the analysis of small signal stability involving low frequency oscillations. The system simulation for SSR studies, cannot be performed using

ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

transient stability type programs. It is generally carried out using EMTP type program and this can be cumbersome. This research monograph is aimed at presenting comprehensive mathematical models and small signal stability analysis of SSR Both damping torque analysis (in the frequency domain) and eigenvalue analysis are discussed. The analysis is backed by a number of illustrative examples. A major feature of this monograph is the coverage of interactions from fixed series compensation to HYDC and FACTS controllers. Apart from present­ing the detailed mathematical analysis, basic concepts of SSR interactions are also explained based on simplified models which capture the phenomenon of interest. This should help those involved with system planning and design to understand the nature and scope of SSR interactions with various controllers. The book is organized into eight chapters. The first chapter explains the back­ground and introduces the topics covered. The second and third chapters cover the modelling of the turbine-generator and the passive electric network. The models are developed from first principles and the application of transforma­tion (Park or Kron) that reduces the system equations to time-invariant form. Apart from comprehensive coverage, the treatment has several new features such as (i) the development of electrical analogue for the rotor system (ii) the derivations of state equations based on circuit topology and (iii)application of immittance functions based on D-Q variables. Chapter four presents a comprehensive analysis of SSR with fixed series com­pensation. The state-space models derived in chapt(~r 2 and 3 are used for the study of SSR based on damping torque and eigenvalue analysis. A novel iter­ative method for computing the eigenvalues corresponding to torsional modes is presented. The countermeasures for SSR are also described. Chapters 5, 6, 7 and 8 are addressed to the study of interactions with (i)PSS (ii) HYDC converter control (iii) shunt FACTS and (iv) series FACTS con­trollers. The torsional interactions with TCSC awl SSSC devices are explained in sufficient detail. Many new results are also presellted. While an attempt has been made to cover the entire gamut of SSR interactions and discuss them in detail, the book does not claim to be complete in covering all aspects of SSR. The emphasis is on the analysis based on linearzied models. Also, the research on FACTS controllers is of recent origin and several new developments are expected to take place in future. However, an attempt has been made to explain the basic concepts of SSR int.eractions with HYDC and FACTS controllers. Acknowledgments Prof. M.A. Pai at University of Illinois, encouraged me to write this mono­graph. He has been a constant source of inspiration. Indian Institute of Technology, Kanpur and Indian Institute of Science, Banga­lore have provided an invigorating academic environment for research. Several of my graduate students have worked on SSR. In part.icular, I wish to thank Drs. A. G. Kothari, M.K. Geetha and A.M. Kulkarni whose work has con­tributed to some of the case studies reported in the book.

PREFACE

I acknowledge the support of Department of Science and Technology, Govern­ment of India for research on FACTS controllers. I am grateful to Dr. Narain Hingorani for readily agreeing to write the foreword. Prabha, Rajesh and Hiren have helped in the preparation of the manuscript. Mr. Alex Greene of Kluwer Academic Publishers has been helpful in ensuring that I complete the manuscript on time. I am indebted to my family, in particular my sist.er Manorama Kamath, for encouragement and support. Finally, I would like to t.hank my wife Usha, for her patience and support.

K R PADIYAR