Power System Analysis Murthy PDF

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Power System Analysis "This page is Intentionally Left Blank" Power System Analysis Prof. P.S.R. Murthy B.Sc. (Engg.) (Hons.) ME., Dr. - ING (Berlin), F.I.E. (India) Life Member - ISTE (Formerly Principal O.U. College of Engineering & Dean, Faculty of Engineering, O.U. Hyderabad) Prin...

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Power System Analysis

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Power System Analysis

Prof. P.S.R. Murthy B.Sc. (Engg.) (Hons.) ME., Dr. - ING (Berlin), F.I.E. (India) Life Member - ISTE (Formerly Principal O.U. College of Engineering & Dean, Faculty of Engineering, O.U. Hyderabad) Principal, Sree Datha Institute of Engineering and Science, Sheriguda, Ibrahim Patnam, Hyderabad (AP).

BSP BS Publications 4-4-309, Giriraj Lane, Sultan Bazar, Hyderabad - 500095 A.P. Phone: 040-23445688

Copyright © 2007, by Publisher All rights reserved

No part of this book or parts thereof may be reproduced, stored in a retrieval system or transmitted in any language or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publishers.

Published by :

SSP BS Publications 4-4-309, Giriraj Lane, Sultan Bazar, Hyderabad - 500 095 A. P. Phone: 040-23445688 e-mail: [email protected] www.bspublications.net

Printed at:

AdithyaArt Printers Hyderabad.

ISBN: 978-81-7800-161-6

Preface Power System analysis is a pre-requisite course for electrical power engineering students. In Chapter I. introductory concepts about a Power system, network models, faults and analysis;the primitive network and stability are presented. Chapter 2 deals with the graph theory that is relevant to various incidence matrices required for network modelling are explained. Chapter 3 explains the various incidence matrices and network matrices. Chapter 4 discusses, step-by-step method of building of network matrices. Chapter 5 deals with power flow studies. Both Gauss-Seidel method and Newton-Raphson methods are explained. In Newton-Raphson method both the Cartesion coordinates method and polar coordinates methods are discussed. In chapter 6 short circuit analysis is explained Per unit quantity and percentage values are defined. Analysis for symmetrical faults is discussed. The utility of reactors for bus bar and generator protection is also explained. Unbalanced fault analysis is presented in chapter 7. Use of symmetrical components and network connections are explained. Chapter 8 deals with the power system stability problem. Steady state stability. transient stability and dynamic stability are discussed. It is earnestly hoped that this book will meet the requirements of students in the subject power system analysis.

- Author

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Acknowledgment My sincere thanks go to Mr. Nikhil Shah, Proprietor of BS Publications for his constant encouragement to me to write and complete this book on 'Power System Analysis'. My thanks also go to Mr. M.Y.L. Narasimha Rao, my well wisher for taking great pains in transferring script material and proof material between my residence and the press with a great smiley face, day and night over the last few months. I thank Mrs. Swarupa for MATLAB assistance.

-Author

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Contents Preface .................................................................................................................. (vii) Acknowledgment................................................................................................... (ix)

1

2

Introduction I. I

The Electrical Power System .................................................................. I

1.2

Network Models ...................................................................................... 3

1.3

Faults and Analysis .................................................................................. 3

1.4

The Primitive Network ............................................................................ 4

1.5

Power System Stability ........................................................................... 5

Graph Theory 2.1

Introduction ............................................................................................. 6

2.2

Definitions ................................................................................................ 6

2.3

Tree and Co-Tree .................................................................................... 8

2.5

Cut-Set .................................................................................................... 9

2.4

Basic Loops ............................................................................................. 9

2.6

Basic Cut-Sets ....................................................................................... 10

vVorked Examples

.. ' .... ..

II

Prohlems .......

IS

Questions

16

Content~

(xii)

3

Incidence Matrices 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

4

18 18 19 20 21 22 24 25 26 28 29 33 51 52

Building of Network Matrices 4.1 4.2 4.3 4.4

5

Element Node Incidence Matrix ............................................................ Bus Incidence Matrix ............................................................................ Branch - Path Incidence Matrix K ....................................................... Basic Cut-Set Incidence Matrix ............................................................ Basic Loop Incidence Matrix ................................................................. Network Performance Equations .......................................................... Network Matrices .................................................................................. Bus Admittance Matrix and Bus Impedance Matrix ............................. Branch Admittance and Branch Impedance Matrices ........................... Loop Impedance and Loop Admittance Matrices ................................. Bus Admittance Matrix by Direct Inspection ........................................ Worked Examples. ...... ... ..... ........ ....... ............ ................................. Problems ........................... : ..................................................... Questions .......... ........................................... '" ..........................

Partial Network ...................................................................................... 53 Addition ofa Branch .............................................................................. 55 Addition ofa Link .................................................................................. 60 Removal or Change in Impedance of Elements with Mutual Impedance ................................................................................. 66 Worked Examples. ........ " .. . 70 Problems .. ............... : ............ . .... 96 Questions. .............. . .............. 97

Power Flow Studies 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Necessity for Power Flow Studies ....................................................... 98 Conditions for Successful Operation of a Power System .................... 99 The Power Flow Equations ................................................................... 99 Classification of Buses ........................................................................ 101 Bus Admittance Formation .................................................................. 102 System Model for Load Flow Studies ................................................ 104 Gauss-Seidel Iterative Method ............................................................ 105 Gauss - Seidel Iterative Method of Load Flow Solution .................... 106 Newton-Raphson Method ................................................................... 109 5.9.1 The Rectangular Coordinates Method .................................. 110 5.9.2 The Polar Coordinates Method ............................................ 112

(xiii)

Contents 5.10

Sparsity of Network Admittance Matrices .......................................... 115

5.11

Triangular Decompostion .................................................................... 116

5.12

Optimal Ordering ................................................................................. 118

5.13

Decoupled Methods ............................................................................. 119

5.14

Fast Decoupled Methods ..................................................................... 120

5.15

Load Flow Solution Using Z Bus ..................................................... 121

5.16

5.15.1

Bus Impedance Formation ................................................... 121

5.15.2

Addition of a Line to the Reference Bus .............................. 122

5.15.3

Addition of a Radial Line and New Bus ............................... 122

5.15.4

Addition of a Loop Closing Two Existing Buses in the System ............................................................. 123

5.15.5

Gauss - Seidel Method Using Z-bus for Load Flow Solution .............................................................. 124

Convergence Characteristics ............................................................... 124

Worked Examples ..... .... . . Problems Questions ................................. .

6

126 .............. 161 . ... ......................... 175

Short Circuit Analysis 6.1

Per Unit Quantities ............................................................................... 176

6.2

Advantages of Per Unit System .......................................................... 178

6.3

Three Phase Short Circuits ................................................................. 178

6.4

Reactance Diagrams ............................................................................ 181

6.5

Percentage Values ................................................................................ 181

6.6

Short Circuit KVA ................................................................................ 182

6.7

Importance of Short Circuit Currents ................................................. 183

6.8

Analysis ofR-L Circuit ........................................................................ 184

6.9

Three Phase Short Circuit on Unloaded Synchronous Generator ...... 185

6.10

Effect of Load Current or Prefault Current ........................................ 185

6.11

Reactors ............................................................................................... 186

6.12

Construction of Reactors .................................................................... 186

6.13

Classification of Reactors .................................................................... 187

Worked Examples. ..... ....... ....... ..... ... .

............ 189

Problems . ................................... .

............. 216

Questions........ ... ... ....... ......... .. .......... .. .............. ................ .. 2 I 6

(xiv)

7

Contents

Unbalanced Fault Analysis 7. I

The Operator "'a'" ................................................................................. 218

7.2

Symmetrical Components of Unsymmetrical Phases ......................... 219

7.3

Power in Sequence Components ........................................................ 221

7.4

Unitary Transformation for Power Invariance ................................... 222

7.5

Sequence Impedances ......................................................................... 224

7.6

Balanced Star Connected Load ........................................................... 224

7.7

Transmission Lines .............................................................................. 226

7.8

Sequence Impedances of Transformer ............................................... 227

7.9

Sequence Reactances of Synchronous Machine ................................ 228

7.10

Sequence Networks of Synchronous Machines ................................. 228 7.10.1

Positive Sequence Network .................................................. 228

7.10.2

Negative Sequence Network ................................................ 229

7.10.3

Zero Sequence Network ....................................................... 230

7.11

Unsymmetrical Faults .......................................................................... 231

7.12

Assumptions for System Representation ............................................ 232

7.13

Unsymmetrical Faults on an Unloaded Generator ............................... 232

7.14

Llne-to-Line Fault ................................................................................ 235

7.15

Double Line to Ground Fault ............................................................... 238

7.16

Single-Line to Ground Fault with Fault Impedance ............................ 241

7.17

Line-to-Line Fault with Fault Impedence ............................................ 242

7.18

Double Line-to-Ground Fault with Fault Impedence .......................... 243

Worked Examples............................. . ................ ............ .. .............. 245 Problems ............... .. .............. ............ ............ ...... ........ .. ........ 257 Questions . ...................................................................................... 257

8

Power System Stability 8.1

Elementary Concepts ........................................................................... 259

8.2 8.3

Illustration of Steady State Stability Concept ..................................... 260 Methods for Improcessing Steady State Stability Limit ..................... 261

8.4

Synchronizing Power Coefficient ....................................................... 262

8.5

Transient Stability ................................................................................ 262

8.6

Stability ofa Single Machine Connected to lnfinite Bus .................... 262

8.7

The Swing Equation ............................................................................ 263

8.8

Equal Area Criterion and Swing Equation ........................................... 267

Contents

(xv) 8.9

Transient Stability Limit ...................................................................... 269

8.10

Frequency of Oscillations .................................................................... 270

8.11

Critical Clearing Time and Critical Clearing Angle .............................. 272

8.12

Fault on a Double-Circuit Line ............................................................ 274

8.13

Transient Stability When Power is Transmitted During the Fault ...... 275

8.14

Fault Clearance and Reclosure in Double-Circuit System .................. 277

8.15

Solution to Swing Equation Step-by-Step Method ............................. 277

8.16

Factors Affecting Transient Stability .................................................. 279

8.17

Dynamic Stability ................................................................................ 280

8.18

Node Elimination Methods ................................................................... 282

Worked Examples ............. ., ........ ....... ...... ... ......................... 285 Problems ..... , ........................ ............................ ................ . ... 303 Questions. ........................ .......... ...... . ...... .................. ....... ....... 304

Objective Questions ................................................................................. 305 Answers to Objective Questions ............................................................ 317 Index ........................................................................................................... 319

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1

INTRODUCTION

Power is an essential pre-requisite for the progress of any country. The modern power system has features unique to it self. It is the largest man made system in existence and is the most complex system. The power demand is more than doubling every decade. Planning, operation and control of interconnected power system poses a variety of challenging problems, the solution of which requires extensive application of mathematical methods from various branches. Thomas Alva Edison was the first to conceive an electric power station and operate it in Newyork in 1882. Since then, power generation originally confined to steam engines expanded using (steam turbines) hydro electric turbines, nuclear reactors and others. The inter connection of the various generating stations to load centers through EHV and UHV transmission lines necessitated analytical methods for analysing various situations that arise in operation and control of the system. Power system analysis is the subject in the branch of electrical power engineering which deals with the determination of voltages at various buses and the currents that flow in the transmission lines operating at different voltage levels.

1.1

The Electrical Power System

The electrical power system is a complex network consisting of generators, loads, transmission lines, transformers, buses, circuit breakers etc. For the analysis of a power system in operation

2

Power System Analysis

a suitable model is needed. This model basically depends upon the type of problem on hand. Accordingly it may be algebraic equations, differential equations, transfer functions etc. The power system is never in steady state as the loads keep changing continuously. However, it is possible to conceive a quasistatic state during which period the loads could be considered constant. This period could be 15 to 30 minutes. In this state power flow equations are non-linear due to the presence of product terms of variables and trigonometric terms. The solution techniques involves numerical (iterative) methods for solving non-linear algebraic equations. Newton-Raphson method is the most commonly used mathematical technique. The analysis of the system for small load variations, wherein speed or frequency and voltage control may be required to maintain the standard values, transfer function and state variable models are better suited to implement proportional, derivative and integral controllers or optimal controllers us...