Modern Power SystemAnalysis

D P KothariProfessor, Centre of Energy Studies

Deputy Director (Admin.)Indian Institute of Technology

Delhi

I J Nagrath

Adjunct Professor, and Former Deputy Director,Birla Institute of Technology and Science

Pilani

Higher EducationBoston Burr Ridge, IL Dubuque, IA New York

San Francisco St. Louis Bangkok Bogota Caracas Kuala LumpurLisbon London Madrid Mexico City Milan Montreal New Delhi

Santiago Seoul Singapore Sydney Taipei Toronto

Contents

Preface v

1. Introduction 1

1.1 A Perspective 11.2 Structure of Power Systems 101.3 Conventional Sources of Electric Energy 131.4 Renewable Energy Sources 251.5 Energy Storage 281.6 Growth of Power Systems in India 291.7 Energy Conservation 311.8 Deregulation 331.9 Distributed and Dispersed Generation 341.10 Environmental/Aspects of Electric Energy Generation 351.11 Power System Engineers and Power System Studies 391.12 Use of Computers and Microprocessors 391.13 Problems Facing Indian Power Industry and its Choices 40

References 43

2. Inductance and Resistance of Transmission Lines 45

2.1 Introduction 452.2 Definition of Inductance 452.3 Flux Linkages of an Isolated

Current-Carrying Conductor 462.4 Inductance of a Single-Phase Two-Wire Line 502.5 Conductor Types 512.6 Flux Linkages of one Conductor in a Group 532.7 Inductance of Composite Conductor Lines 542.8 Inductance of Three-Phase Lines 592.9 Double-Circuit Three-Phase Lines 662.10 Bundled Conductors 682.11 Resistance 702.12 Skin Effect and Proximity Effect 77

Problems 72References 75

3. Capacitance of Transmission Lines 76

3.1 Introduction 763.2 Electric Field of a Long Straight Conductor 76

Contents

3.3 Potential Difference between two Conductorsof a Group of Parallel Conductors 77

3.4 Capacitance of a Two-Wire Line 783.5 Capacitance of a Three-Phase Line

with Equilateral Spacing 803.6 Capacitance of a Three-Phase Line with

Unsymmetrical Spacing 813.7 Effect of Earth on Transmission Line Capacitance 833.8 Method of GMD (Modified) 91

• 3.9 Bundled Conductors 92Problems 93References 94

4. Representation of Power System Components 95

4.1 Introduction 954.2 Single-phase Solution of Balanced

Three-phase Networks 954.3 One-Line Diagram and Impedance or

Reactance Diagram 984.4 Per Unit (PU) System 994.5 Complex Power 1054.6 Synchronous Machine 1084.7 Representation of Loads 121

Problems 125References 127

5. Characteristics and Performance of PowerTransmission Lines 128

5.1 Introduction 1285.2 Short Transmission Line 1295.3 Medium Transmission Line 7375.4 The Long Transmission Line—Rigorous Solution 1395.5 Interpretation of the Long Line Equations 7¥35.6 Ferranti Effect 7505.7 Tuned Power Lines 7575.8 The Equivalent Circuit of a Long Line 7525.9 Power Flow through a Transmission Line 1585.10 Methods of Voltage Control 7 73

Problems 180References 183

6. Load Flow Studies 184

6.1 Introduction 1846.2 Network Model Formulation 755

Contents 1 xi

6.3 Formation of 7BUS by Singular Transformation 1896.4 Load Flow Problem 1966.5 Gauss-Seidel Method 2046.6 Newton-Raphson (NR) Method 2736.7 Decoupled Load Flow Methods 2226.8 Comparison of Load Flow Methods 2286.9 Control of Voltage Profile 230

Problems 236References 239

7. Optimal System Operation 242

7.1 Introduction 2427.2 Optimal Operation of Generators on a Bus Bar 2437.3 Optimal Unit Commitment (UC) 2501.4 Reliability Considerations 2537.5 Optimum Generation Scheduling 2597.6 Optimal Load Flow Solution 2701.1 Optimal Scheduling of Hydrothermal System 276

Problems l284References 286

8. Automatic Generation and Voltage Control 290

8.1 Introduction 2908.2 Load Frequency Control (Single Area Case) 2978.3 Load Frequency Control and

Economic Despatch Control 3058.4 Two-Area Load Frequency Control 3078.5 Optimal (Two-Area) Load Frequency Control 3708.6 Automatic Voltage Control 3758.7 Load Frequency Control with Generation

Rate Constraints (GRCs) 3208.8 Speed Governor Dead-Band and Its Effect on AGC 3278.9 Digital LF Controllers 3228.10 Decentralized Control 323

Problems 324References 325

9. Symmetrical Fault Analysis 327

9.1 Introduction 3279.2 Transient on a Transmission Line 3289.3 Short Circuit of a Synchronous Machine

(On No Load) 3309.4 Short Circuit of a Loaded Synchronous Machine 3399.5 Selection of Circuit Breakers 344

xii I Contents

9.6 Algorithm for Short Circuit Studies 3499.7 ZBUS Formulation 355

Problems 363References 368

10. Symmetrical Components 369

10.1 Introduction 36910.2 Symmetrical Component Transformation 37010.3 Phase Shift in Star-Delta Transformers 37710.4 Sequence Impedances of Transmission Lines 37910.5 Sequence Impedances and Sequence Network

of Power System 35710.6 Sequence Impedances and Networks of

Synchronous Machine 35710.7 Sequence Impedances of Transmission Lines 35510.8 Sequence Impedances and Networks

of Transformers 35610.9 Construction of Sequence Networks of

a Power System 389Problems 393References 396

11. Unsymmetrical Fault Analysis 397

11.1 Introduction 39711.2 Symmetrical Component Analysis of

Unsymmetrical Faults 39811.3 Single Line-To-Ground (LG) Fault 39911.4 Line-To-Line (LL) Fault 40211.5 Double Line-To-Ground (LLG) Fault 40411.6 Open Conductor Faults 41411.7 Bus Impedance Matrix Method For Analysis

of Unsymmetrical Shunt Faults 416Problems 427References 432

12. Power System Stability 433

12.1 Introduction 43312.2 Dynamics of a Synchronous Machine 43512.3 Power Angle Equation 44012.4 Node Elimination Technique 44412.5 Simple Systems 457 '12.6 Steady State Stability 45412.7 Transient Stability 45912.8 Equal Area Criterion 461

Contents I xiii

12.9 Numerical Solution of Swing Equation 48012.10 Multimachine Stability 48712.11 Some Factors Affecting Transient Stability 496

Problems 506References 508

13. Power System Security 510

13.1 Introduction 57013.2 System State Classification 57213.3 Security Analysis 57213.4 Contingency Analysis 57613.5 Sensitivity Factors 52013.6 Power System Voltage Stability 524

References 529

14. An Introduction to State Estimation of Power Systems 531

14.1 Introduction 53114.2 Least Squares Estimation: The Basic

Solution 53214.3 Static State Estimation of Power

Systems 53514.4 Tracking State Estimation of Power Systems 54414.5 Some Computational Considerations 54414.6 External System Equivalencing 54514.7 Treatment of Bad Data 54614.8 Network Observability and Pseudo-Measurements 54914.9 Application of Power System State Estimation 550

Problems 552References 553

15. Compensation in Power Systems 556

15.1 Introduction 55615.2 Loading Capability 55715.3 Load Compensation 55715.4 Line Compensation 55515.5 Series Compensation 55915.6 Shunt Compensators 56215.7 Comparison between STATCOM and SVC 56515.8 Flexible AC Transmission Systems (FACTS) 56615.9 Principle and Operation of Converters 56715.10 Facts Controllers 569

References 574

xiv I Contents

16. Load Forecasting Technique 575

16.1 Introduction 57516.2 Forecasting Methodology 57716.3 Estimation of Average and Trend Terms 57716.4 Estimation of Periodic Components 55716.5 Estimation of ys (k): Time Series Approach 55216.6 Estimation of Stochastic Component:

Kalman Filtering Approach 55316.7 Long-Term Load Predictions Using

Econometric Models 55716.8 Reactive Load Forecast 557

References 589

17. Voltage Stability 591

17.1 Introduction 59117.2 Comparison of Angle and Voltage Stability 59217.3 Reactive Power Flow and Voltage Collapse 59317'.4 Mathematical Formulation of

Voltage Stability Problem 59317.5 Voltage Stability Analysis 59717.6 Prevention of Voltage Collapse 60017.7 State-of-the-Art, Future Trends and Challenges 607

References 603

Appendix A: Introduction to Vector and Matrix Algebra 605

Appendix B: Generalized Circuit Constants 617

Appendix C: Triangular Factorization and Optimal Ordering 623

Appendix D: Elements of Power System Jacobian Matrix 629

Appendix E: Kuhn-Tucker Theorem 632

Appendix F: Real-time Computer Control of Power Systems 634

Appendix G: Introduction to MATLAB and SIMULINK 640

Answers to Problems 679

Index 685