Modern Power System Analysis D P Kothari Professor, 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 Education Boston Burr Ridge, IL Dubuque, IA New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto Contents Preface v 1. Introduction 1 1.1 A Perspective 1 1.2 Structure of Power Systems 10 1.3 Conventional Sources of Electric Energy 13 1.4 Renewable Energy Sources 25 1.5 Energy Storage 28 1.6 Growth of Power Systems in India 29 1.7 Energy Conservation 31 1.8 Deregulation 33 1.9 Distributed and Dispersed Generation 34 1.10 Environmental/Aspects of Electric Energy Generation 35 1.11 Power System Engineers and Power System Studies 39 1.12 Use of Computers and Microprocessors 39 1.13 Problems Facing Indian Power Industry and its Choices 40 References 43 2. Inductance and Resistance of Transmission Lines 45 2.1 Introduction 45 2.2 Definition of Inductance 45 2.3 Flux Linkages of an Isolated Current-Carrying Conductor 46 2.4 Inductance of a Single-Phase Two-Wire Line 50 2.5 Conductor Types 51 2.6 Flux Linkages of one Conductor in a Group 53 2.7 Inductance of Composite Conductor Lines 54 2.8 Inductance of Three-Phase Lines 59 2.9 Double-Circuit Three-Phase Lines 66 2.10 Bundled Conductors 68 2.11 Resistance 70 2.12 Skin Effect and Proximity Effect 77 Problems 72 References 75 3. Capacitance of Transmission Lines 76 3.1 Introduction 76 3.2 Electric Field of a Long Straight Conductor 76 Contents 3.3 Potential Difference between two Conductors of a Group of Parallel Conductors 77 3.4 Capacitance of a Two-Wire Line 78 3.5 Capacitance of a Three-Phase Line with Equilateral Spacing 80 3.6 Capacitance of a Three-Phase Line with Unsymmetrical Spacing 81 3.7 Effect of Earth on Transmission Line Capacitance 83 3.8 Method of GMD (Modified) 91 • 3.9 Bundled Conductors 92 Problems 93 References 94 4. Representation of Power System Components 95 4.1 Introduction 95 4.2 Single-phase Solution of Balanced Three-phase Networks 95 4.3 One-Line Diagram and Impedance or Reactance Diagram 98 4.4 Per Unit (PU) System 99 4.5 Complex Power 105 4.6 Synchronous Machine 108 4.7 Representation of Loads 121 Problems 125 References 127 5. Characteristics and Performance of Power Transmission Lines 128 5.1 Introduction 128 5.2 Short Transmission Line 129 5.3 Medium Transmission Line 737 5.4 The Long Transmission Line—Rigorous Solution 139 5.5 Interpretation of the Long Line Equations 7¥3 5.6 Ferranti Effect 750 5.7 Tuned Power Lines 757 5.8 The Equivalent Circuit of a Long Line 752 5.9 Power Flow through a Transmission Line 158 5.10 Methods of Voltage Control 7 73 Problems 180 References 183 6. Load Flow Studies 184 6.1 Introduction 184 6.2 Network Model Formulation 755 Contents 1 xi 6.3 Formation of 7BUS by Singular Transformation 189 6.4 Load Flow Problem 196 6.5 Gauss-Seidel Method 204 6.6 Newton-Raphson (NR) Method 273 6.7 Decoupled Load Flow Methods 222 6.8 Comparison of Load Flow Methods 228 6.9 Control of Voltage Profile 230 Problems 236 References 239 7. Optimal System Operation 242 7.1 Introduction 242 7.2 Optimal Operation of Generators on a Bus Bar 243 7.3 Optimal Unit Commitment (UC) 250 1.4 Reliability Considerations 253 7.5 Optimum Generation Scheduling 259 7.6 Optimal Load Flow Solution 270 1.1 Optimal Scheduling of Hydrothermal System 276 Problems l284 References 286 8. Automatic Generation and Voltage Control 290 8.1 Introduction 290 8.2 Load Frequency Control (Single Area Case) 297 8.3 Load Frequency Control and Economic Despatch Control 305 8.4 Two-Area Load Frequency Control 307 8.5 Optimal (Two-Area) Load Frequency Control 370 8.6 Automatic Voltage Control 375 8.7 Load Frequency Control with Generation Rate Constraints (GRCs) 320 8.8 Speed Governor Dead-Band and Its Effect on AGC 327 8.9 Digital LF Controllers 322 8.10 Decentralized Control 323 Problems 324 References 325 9. Symmetrical Fault Analysis 327 9.1 Introduction 327 9.2 Transient on a Transmission Line 328 9.3 Short Circuit of a Synchronous Machine (On No Load) 330 9.4 Short Circuit of a Loaded Synchronous Machine 339 9.5 Selection of Circuit Breakers 344 xii I Contents 9.6 Algorithm for Short Circuit Studies 349 9.7 ZBUS Formulation 355 Problems 363 References 368 10. Symmetrical Components 369 10.1 Introduction 369 10.2 Symmetrical Component Transformation 370 10.3 Phase Shift in Star-Delta Transformers 377 10.4 Sequence Impedances of Transmission Lines 379 10.5 Sequence Impedances and Sequence Network of Power System 357 10.6 Sequence Impedances and Networks of Synchronous Machine 357 10.7 Sequence Impedances of Transmission Lines 355 10.8 Sequence Impedances and Networks of Transformers 356 10.9 Construction of Sequence Networks of a Power System 389 Problems 393 References 396 11. Unsymmetrical Fault Analysis 397 11.1 Introduction 397 11.2 Symmetrical Component Analysis of Unsymmetrical Faults 398 11.3 Single Line-To-Ground (LG) Fault 399 11.4 Line-To-Line (LL) Fault 402 11.5 Double Line-To-Ground (LLG) Fault 404 11.6 Open Conductor Faults 414 11.7 Bus Impedance Matrix Method For Analysis of Unsymmetrical Shunt Faults 416 Problems 427 References 432 12. Power System Stability 433 12.1 Introduction 433 12.2 Dynamics of a Synchronous Machine 435 12.3 Power Angle Equation 440 12.4 Node Elimination Technique 444 12.5 Simple Systems 457 ' 12.6 Steady State Stability 454 12.7 Transient Stability 459 12.8 Equal Area Criterion 461 Contents I xiii 12.9 Numerical Solution of Swing Equation 480 12.10 Multimachine Stability 487 12.11 Some Factors Affecting Transient Stability 496 Problems 506 References 508 13. Power System Security 510 13.1 Introduction 570 13.2 System State Classification 572 13.3 Security Analysis 572 13.4 Contingency Analysis 576 13.5 Sensitivity Factors 520 13.6 Power System Voltage Stability 524 References 529 14. An Introduction to State Estimation of Power Systems 531 14.1 Introduction 531 14.2 Least Squares Estimation: The Basic Solution 532 14.3 Static State Estimation of Power Systems 535 14.4 Tracking State Estimation of Power Systems 544 14.5 Some Computational Considerations 544 14.6 External System Equivalencing 545 14.7 Treatment of Bad Data 546 14.8 Network Observability and Pseudo-Measurements 549 14.9 Application of Power System State Estimation 550 Problems 552 References 553 15. Compensation in Power Systems 556 15.1 Introduction 556 15.2 Loading Capability 557 15.3 Load Compensation 557 15.4 Line Compensation 555 15.5 Series Compensation 559 15.6 Shunt Compensators 562 15.7 Comparison between STATCOM and SVC 565 15.8 Flexible AC Transmission Systems (FACTS) 566 15.9 Principle and Operation of Converters 567 15.10 Facts Controllers 569 References 574 xiv I Contents 16. Load Forecasting Technique 575 16.1 Introduction 575 16.2 Forecasting Methodology 577 16.3 Estimation of Average and Trend Terms 577 16.4 Estimation of Periodic Components 557 16.5 Estimation of ys (k): Time Series Approach 552 16.6 Estimation of Stochastic Component: Kalman Filtering Approach 553 16.7 Long-Term Load Predictions Using Econometric Models 557 16.8 Reactive Load Forecast 557 References 589 17. Voltage Stability 591 17.1 Introduction 591 17.2 Comparison of Angle and Voltage Stability 592 17.3 Reactive Power Flow and Voltage Collapse 593 17'.4 Mathematical Formulation of Voltage Stability Problem 593 17.5 Voltage Stability Analysis 597 17.6 Prevention of Voltage Collapse 600 17.7 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