Mladen Kezunovic · Jinfeng RenSaeed Lot� fard

Design, Modeling and Evaluation of Protective Relays for Power Systems

Design, Modeling and Evaluation of ProtectiveRelays for Power Systems

Mladen Kezunovic • Jinfeng Ren • Saeed Lotfifard

Design, Modelingand Evaluation of ProtectiveRelays for Power Systems

Mladen KezunovicDepartment of ECENTexas A&M UniversityCollege Station

TX, USA

Jinfeng RenAlstom GridRedmond, WA, USA

Saeed LotfifardSchool of Electrical Engineeringand Computer Science

Washington State UniversityPullman, WA, USA

ISBN 978-3-319-20918-0 ISBN 978-3-319-20919-7 (eBook)DOI 10.1007/978-3-319-20919-7

Library of Congress Control Number: 2015946240

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar ordissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material containedherein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media(www.springer.com)

Preface

This book has been written for university students, professionals in the area of

protective relaying, and other interested individuals with minimum engineering

skills to study the material on their own. To achieve this goal, the book has been

written in an unconventional way: It uses a simulation tool called MERIT 2000,

based on widely known MATLAB software, to offer hands-on experience in

understanding and implementing protective relaying designs.

Many books on protective relaying have been published over the years, which

provide an excellent background on power system faults and protective relaying

principles. However, most of these books assume that the reader is interested in

learning about relaying principles and on how protective relay products can be used

to implement protective approaches for various power system apparatus.

This book takes a different approach: It assumes that the reader is interested in

learning how the relays work, what the basic design principles are, and how an

implemented design of a relay may be evaluated. With this goal in mind, the book

tries to make the learning process a design experience wherein the reader starts

using the software engineering tools (MATLAB) from the very beginning as the

basic relay design principles are introduced. The book is not a substitute for a

reference on the fundamentals of relaying but is rather a complementary source on

the topic.

To provide a hands-on experience, the authors have provided MERIT 2000

software developed in MATLAB as a supplement to this book. The software has

been in use since 2000 in laboratory assignments in courses at Texas A&M

University.

College Station, TX, USA Mladen Kezunovic

Redmond, WA, USA Jinfeng Ren

Pullman, WA, USA Saeed Lotfifard

v

Acknowledgments

The effort to create exercises that provide a hands-on experience in this book is the

result of collaboration in specifying, implementing, testing, and evaluating the

MERIT 2000 software. This effort was financially supported at its beginning by

the National Science Foundation Grant ECS-96-19294 awarded to Texas A&M

University in 1996 and in part by EPRI under contract WO 8618-02 awarded to

Texas A&M University and Washington State University in 2002.

Texas A&M faculty, Dr. Mladen Kezunovic, and Dr. Garng Huang, as well as

Dr. Ali Abur who was with Texas A&M University at the time the NSF grant was

awarded, worked on different aspects of the power system analysis and developed

MATLAB-based software to facilitate the experimental part of the respective

analysis issues. Dr. Kezunovic was responsible for the protective relaying area

and development of the MERIT 2000 software used in this book.

Several former graduate students, including the co-authors of the book, as well

as scholars in Dr. Kezunovic’s group, have contributed to the development of the

MERIT 2000 software. Dr. Bogdan Kasztenny is acknowledged for his key role in

this regard while he was a postdoc in Dr. Kezunovic’s lab at Texas A&MUniversity

in 1997–1999.

Final thanks go to the many undergraduate and graduate students who undertook

courses in protective relaying at Texas A&M University since 2000. They all

provided feedback, and some contributed with new developments over the years

making the MERIT 2000 software a very valuable teaching and learning tool. For

those who want to engage in the learning experience, the software will be made

available through a third party as a supplement to our book. The authors will keep

updating the software as new developments are reported.

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basics of Protection Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Modeling and Simulation Methodology and Tools . . . . . . . . . . . . 3

1.3.1 Relay Elements Library . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.2 Signal Source Library and Analysis Tools . . . . . . . . . . . . 3

1.3.3 Relay Models and Power Network Elements Library . . . . 4

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Power System Fault Analysis and Short-Circuit

Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Module 1: Analysis of a System with an Unbalanced

Source Using Symmetrical Components . . . . . . . . . . . . . 13

2.2.2 Module 2: Analysis of a System with Single

Line-Ground Fault Using Symmetrical

Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Short-Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Sequence Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.3 Two-Winding Transformer . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.4 Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.5 Positive Sequence Network Model . . . . . . . . . . . . . . . . . 37

2.4.6 Negative Sequence Network . . . . . . . . . . . . . . . . . . . . . . 39

2.4.7 Sequence Networks in Steady State . . . . . . . . . . . . . . . . 39

2.4.8 Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

ix

2.5 Matrix Method for Short-Circuit Calculation . . . . . . . . . . . . . . . . 42

2.5.1 Matrix Computation Approach . . . . . . . . . . . . . . . . . . . 42

2.5.2 Admittance and Impedance Approaches . . . . . . . . . . . . 44

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Basics of Protective Relaying and Design Principles . . . . . . . . . . . . . 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Overcurrent Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Relaying Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Software Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Impedance Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2 Relaying Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.3 Software Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Differential Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.2 Relaying Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Modeling of Digital Relay and Power System Signals . . . . . . . . . . . . 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Major Elements of a Digital Relay . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Data Acquisition Block . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.2 Phasor Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Library of Modeling Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.1 Bias Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.2 Basic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.3 Data Acquisition Board . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3.4 Directional Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3.5 Differential Equation-Based Impedance

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3.6 Digital Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3.7 Digital Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 112

4.3.8 Orthogonal Components . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.9 Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.10 Triggering Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.3.11 Universal Comparator . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.3.12 Phase Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.3.13 Vector Group Compensator for 2-Winding

Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.3.14 Zone Comparator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

x Contents

4.4 Interfacing Power System and Relay Models . . . . . . . . . . . . . . . . 150

4.4.1 Analytical Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.4.2 Fault Signal Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.4.4 Phasor Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.4.5 Spectrum Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.4.6 Three-Phase Phasor Generator . . . . . . . . . . . . . . . . . . . . 161

4.4.7 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.5 GUI and Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.5.1 Phasor Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5 Design and Implementation of Relay CommunicationSchemes and Trip Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.2 Communication Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.2.2 Working with Software . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6 Design and Implementation of Overcurrent, Pilot,

and Distance Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.2 Line Protection System: Overcurrent Relaying . . . . . . . . . . . . . . . 189

6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.2.3 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.2.4 Minimizing the False Trip in the Directional Relay . . . . . 208

6.3 Line Protection System: Differential Relaying . . . . . . . . . . . . . . . 211

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.3.3 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.4 Line Protection System: Zone Protection . . . . . . . . . . . . . . . . . . . 218

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.4.2 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.5 Line Protection System: Pilot Protection . . . . . . . . . . . . . . . . . . . 229

6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.5.2 Relay Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7 Design and Implementation of Transformer and Busbar

Differential Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

7.2 Transformer Protection Systems . . . . . . . . . . . . . . . . . . . . . . . . . 241

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Contents xi

7.2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 242

7.2.3 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.3 Busbar Protection Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.3.3 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8 Testing of Digital Protective Relays . . . . . . . . . . . . . . . . . . . . . . . . . 263

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

8.2 Modeling and Testing Digital Relays . . . . . . . . . . . . . . . . . . . . . . 264

8.2.1 Modeling and Testing Overcurrent Relay . . . . . . . . . . . . 264

8.2.2 Modeling and Testing Impedance Relay . . . . . . . . . . . . . 275

8.3 Test Using Digital Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

8.3.1 Digital Simulator-Based Relay Test System . . . . . . . . . . 282

8.3.2 System Modeling and Simulation Programs . . . . . . . . . . 290

8.4 Closed-Loop and Open-Loop Analysis . . . . . . . . . . . . . . . . . . . . 296

8.4.1 General Procedures for Performing Tests . . . . . . . . . . . . 296

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

xii Contents

List of Figures

Fig. 1.1 Component representation of a typical protective relay . . . . . . . . . . . . 2

Fig. 1.2 Library of protective relay elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Fig. 1.3 Signal source library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Fig. 1.4 Output phasor display block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Fig. 1.5 Structure of directional overcurrent relay model . . . . . . . . . . . . . . . . . . . . 7

Fig. 1.6 Directional overcurrent relay block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Fig. 1.7 Power network elements library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Fig. 2.1 A SLG fault representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Fig. 2.2 The sequence current phasors for a single-line-to-ground

fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Fig. 2.3 Main window for Module 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Fig. 2.4 Three-phase network with unbalanced source . . . . . . . . . . . . . . . . . . . . . . . 14

Fig. 2.5 Dialog box for the abc and 012-phasor display block . . . . . . . . . . . . . . 14

Fig. 2.6 abc or 012 phasor display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Fig. 2.7 Analysis using symmetrical components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Fig. 2.8 Positive sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Fig. 2.9 Negative sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Fig. 2.10 Zero sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Fig. 2.11 Connected sequence networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Fig. 2.12 Main window for Module 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Fig. 2.13 System with a single line-ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Fig. 2.14 Replacement of fault currents by symmetrical components . . . . . . . 19

Fig. 2.15 Positive-sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Fig. 2.16 Negative-sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Fig. 2.17 Zero-sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Fig. 2.18 Connected symmetrical networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Fig. 2.19 Module interface for short-circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Fig. 2.20 Three-phase model of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Fig. 2.21 Phasor display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Fig. 2.22 Sequence networks for the original network . . . . . . . . . . . . . . . . . . . . . . . . 26

xiii

Fig. 2.23 The data entry for the fault model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 2.24 Simulink model of transposed transmission

line (two rotations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 2.25 Simulink model for unsymmetrical transmission

line (rotation and twist) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 2.26 Models of the parallel transmission lines in the abcand symmetrical components domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fig. 2.27 Model of the parallel transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Fig. 2.28 Dialog box of the transmission line block . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Fig. 2.29 Dialog box for the mutual coupling between the lines . . . . . . . . . . . . . 32

Fig. 2.30 Simplified model of parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Fig. 2.31 Zero sequence equivalent model (general case) . . . . . . . . . . . . . . . . . . . . . 33

Fig. 2.32 Zero sequence equivalent model (identical lines) . . . . . . . . . . . . . . . . . . . 33

Fig. 2.33 Transmission line with wye-connected load . . . . . . . . . . . . . . . . . . . . . . . . . 34

Fig. 2.34 Transmission line with delta-connected load . . . . . . . . . . . . . . . . . . . . . . . . 35

Fig. 2.35 Yg–Δ, Δ–Yg, and Yg–Y transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Fig. 2.36 Positive sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Fig. 2.37 Negative sequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Fig. 2.38 Steady state sequence networks for the synchronous

machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Fig. 2.39 Model in abc domain and symmetrical networks

for the induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Fig. 2.40 Positive and negative sequence equivalent circuits . . . . . . . . . . . . . . . . . 41

Fig. 2.41 Admittance approach to short-circuit studies . . . . . . . . . . . . . . . . . . . . . . . . 43

Fig. 2.42 Impedance approach to short-circuit studies . . . . . . . . . . . . . . . . . . . . . . . . . 43

Fig. 3.1 A sample radial network with overcurrent relays . . . . . . . . . . . . . . . . . . . 46

Fig. 3.2 Illustration of the definite-time overcurrent protection

principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Fig. 3.3 The functional model of a definite-time overcurrent relay . . . . . . . . . 54

Fig. 3.4 Simple transmission network with impedance relays . . . . . . . . . . . . . . . 58

Fig. 3.5 Impedance measured during a fault with significant

fault resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Fig. 3.6 Sample operating characteristic of an impedance relay . . . . . . . . . . . . 60

Fig. 3.7 Illustration of the impedance protection principle . . . . . . . . . . . . . . . . . . 61

Fig. 3.8 The functional model of an impedance relay . . . . . . . . . . . . . . . . . . . . . . . . 65

Fig. 3.9 A simple transmission network with impedance relays . . . . . . . . . . . . 68

Fig. 3.10 The operating characteristic of a biased differential relay . . . . . . . . . 70

Fig. 3.11 The functional model of a single-slope differential relay . . . . . . . . . . 73

Fig. 4.1 Major components of a digital relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Fig. 4.2 The front-end part of a digital relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Fig. 4.3 Illustration of the aliasing frequencies (a)and the application of an anti-aliasing analog filter (b).

The resulting frequency spectrum (c) does not contain

any aliasing frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xiv List of Figures