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TRANSIENTS IN ELECTRICAL SYSTEMS
ANALYSIS, RECOGNITION, AND MITIGATION
J. C. Das
Mc Graw Hill
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CONTENTS
Preface xiii
CHAPTER 1 INTRODUCTION TO TRANSIENTS
1-1 Classification of Transients 1
1-2 Classification with Respect to Frequency Groups 1
1-3 Frequency-Dependent Modeling 2
1-4 Other Sources of Transients 3
1-5 Study of Transients 3
1-6 TNAs—Analog Computers 3
1-7 Digital Simulations, EMTP/ATP, and Similar Programs 3
References 4
Further Reading 4
CHAPTER 2 TRANSIENTS IN LUMPED CIRCUITS
2-1 Lumped and Distributed Parameters 5
2-2 Time Invariance 5
2-3 Linear and Nonlinear Systems 5
2-4 Property of Decomposition 6
2-5 Time Domain Analysis of Linear Systems 6
2-6 Static and Dynamic Systems 6
2-7 Fundamental Concepts 6
2-8 First-Order Transients 11
2-9 Second-Order Transients 15
2-10 Parallel RLC Circuit 18
2-11 Second-Order Step Response 21
2-12 Resonance in Series and Parallel RLC Circuits 21
2-13 Loop and Nodal Matrix Methods for Transient Analysis 24
2-14 State Variable Representation 25
2-15 Discrete-Time Systems 28
2-16 State Variable Model of a Discrete System 30
2-17 Linear Approximation 30
Problems 31
Reference 32
Further Reading 32
CHAPTER 3 CONTROL SYSTEMS
3-1 Transfer Function 33
3-2 General Feedback Theory 35
3-3 Continuous System Frequency Response 38
3-4 Transfer Function of a Discrete-Time System 38
3-5 Stability 39
3-6 Block Diagrams 41
3-7 Signal-Flow Graphs 41
3-8 Block Diagrams of State Models 44
3-9 State Diagrams of Differential Equations 45
3-10 Steady-State Errors 47
3-11 Frequency-Domain Response Specifications 49
3-12 Time-Domain Response Specifications 49
3-13 Root-Locus Analysis 50
3-14 Bode Plot 55
3-15 Relative Stability 58
3-16 The Nyquist Diagram 60
3-17 TACS in EMTP 61
Problems 61
References 63
Further Reading 63
CHAPTER 4 MODELING OF TRANSMISSION LINES AND CABLES
FOR TRANSIENT STUDIES
4-1 ABCD Parameters 65
4-2 ABCD Parameters of Transmission Line Models 67
4-3 Long Transmission Line Model-Wave Equation 67
4-4 Reflection and Transmission at Transition Points 70
4-5 Lattice Diagrams 71
4-6 Behavior with Unit Step Functions at Transition Points 72
4-7 Infinite Line 74
4-8 Tuned Power Line 74
4-9 Ferranti Effect 74
4-10 Symmetrical Line at No Load 75
4-11 Lossless Line 77
4-12 Generalized Wave Equations 77
4-13 Modal Analysis 77
V
v i CONTENTS
4-14 Damping and Attenuation 79
4-15 Corona 79
4-16 Transmission Line Models for Transient Analysis 81
4-17 Cable Types 85
Problems 89
References 89
Further Reading 89
CHAPTER 5 LIGHTNING STROKES, SHIELDING,
AND BACKFLASHOVERS
5-1 Formation of Clouds 91
5-2 Lightning Discharge Types 92
5-3 The Ground Flash 92
5-4 Lightning Parameters 94
5-5 Ground Flash Density and Keraunic Level 98
5-6 Lightning Strikes on Overhead lines 99
5-7 BIL/CFO of Electrical Equipment 100
5-8 Frequency of Direct Strokes to Transmission Lines 102
5-9 Direct Lightning Strokes 104
5-10 Lightning Strokes to Towers 104
5-11 Lightning Stroke to Ground Wire 107
5-12 Strokes to Ground in Vicinity of Transmission Lines 107
5-13 Shielding 108
5-14 Shielding Designs 110
5-15 Backflashovers 113
Problems 117
References 121
Further Reading 121
CHAPTER 6 TRANSIENTS OF SHUNT CAPACITOR BANKS
6-1 Origin of Switching Transients 123
6-2 Transients on Energizing a Single Capacitor Bank 123
6-3 Application of Power Capacitors with Nonlinear Loads 126
6-4 Back-to-Back Switching 133
6-5 Switching Devices for Capacitor Banks 134
6-6 Inrush Current Limiting Reactors 135
6-7 Discharge Currents Through Parallel Banks 136
6-8 Secondary Resonance 136
6-9 Phase-to-Phase Overvoltages 139
6-10 Capacitor Switching Impact on Drive Systems 140
6-11 Switching of Capacitors with Motors 140
6-12 Interruptions of Capacitance Currents 144
6-13 Control of Switching Transients 147
6-14 Shunt Capacitor Bank Arrangements 150
Problems 152
References 153
Further Reading 153
CHAPTER 7 SWITCHING TRANSIENTS AND
TEMPORARY OVERVOLTAGES
7-1 Classification of Voltage Stresses 155
7-2 Maximum System Voltage 155
7-3 Temporary Overvoltages 156
7-4 Switching Surges 157
7-5 Switching Surges and System Voltage 157
7-6 Closing and Reclosing of Transmission Lines 158
7-7 Overvoltages Due to Resonance 164
7-8 Switching Overvoltages of Overhead Lines and Underground Cables 165
7-9 Cable Models 166
7-10 Overvoltages Due to Load Rejection 168
7-11 Ferroresonance 169
7-12 Compensation of Transmission Lines 169
7-13 Out-of-Phase Closing 173
7-14 Overvoltage Control 173
7-15 Statistical Studies 175
Problems 179
References 180
Further Reading 180
CHAPTER 8 CURRENT INTERRUPTION IN AC CIRCUITS
8-1 Arc Interruption 181
8-2 Arc Interruption Theories 182
8-3 Current-Zero Breaker 182
8-4 Transient Recovery Voltage 183
8-5 Single-Frequency TRV Terminal Fault 186
8-6 Double-Frequency TRV 189
8-7 ANSLTEEE Standards for TRV 191
8-8 IEC TRV Profiles 193
8-9 Short-Line Fault 195
8-10 Interruption of Low Inductive Currents 197
8-11 Interruption of Capacitive Currents 200
8-12 Prestrikes in Circuit Breakers 200
8-13 Breakdown in Gases 200
CONTENTS v i i
8-14 Stresses in Circuit Breakers 204
Problems 205
References 206
Further Reading 206
CHAPTER 9 SYMMETRICAL AND UNSYMMETRICAL
SHORT-CIRCUIT CURRENTS
9-1 Symmetrical and Unsymmetrical Faults 207
9-2 Symmetrical Components 208
9-3 Sequence Impedance of Network Components 210
9-4 Fault Analysis Using Symmetrical Components 211
9-5 Matrix Methods of Short-Circuit Current Calculations 221
9-6 Computer-Based Calculations 224
9-7 Overvoltages Due to Ground Faults 224
Problems 232
References 233
Further Reading 233
CHAPTER 10 TRANSIENT BEHAVIOR OF SYNCHRONOUS
GENERATORS
10-1 Three-Phase Terminal Fault 235
10-2 Reactances of a Synchronous Generator 237
10-3 Saturation of Reactances 238
10-4 Time Constants of Synchronous Generators 238
10-5 Synchronous Generator Behavior on Terminal Short-Circuit 239
10-6 Circuit Equations of Unit Machines 244
10-7 Park's Transformation 246
10-8 Park's Voltage Equation 247
10-9 Circuit Model of Synchronous Generators 248
10-10 Calculation Procedure and Examples 249
10-11 Steady-State Model of Synchronous Generator 252
10-12 Symmetrical Short Circuit of a Generator at No Load 253
10-13 Manufacturer's Data 255
10-14 Interruption of Currents with Delayed Current Zeros 255
10-15 Synchronous Generator on Infinite Bus 257
Problems 263
References 264
Further Reading 264
CHAPTER 11 TRANSIENT BEHAVIOR OF INDUCTION AND
SYNCHRONOUS MOTORS
11-1 Transient and Steady-State Models of Induction Machines 265
11-2 Induction Machine Model with Saturation 270
11-3 Induction Generator 271
11-4 Stability of Induction Motors on Voltage Dips 271
11-5 Short-Circuit Transients of an Induction Motor 274
11-6 Starting Methods 274
11-7 Study of Starting Transients 278
11-8 Synchronous Motors 280
11-9 Stability of Synchronous Motors 284
Problems 288
References 291
Further Reading 291
CHAPTER 12 POWER SYSTEM STABILITY
12-1 Classification of Power System Stability 293
12-2 Equal Area Concept of Stability 295
12-3 Factors Affecting Stability 297
12-4 Swing Equation of a Generator 298
12-5 Classical Stability Model 299
12-6 Data Required to Run a Transient Stability Study 301
12-7 State Equations 302
12-8 Numerical Techniques 302
12-9 Synchronous Generator Models for Stability 304
12-10 Small-Signal Stability 317
12-11 Eigenvalues and Stability 317
12-12 Voltage Stability 321
12-13 Load Models 324
12-14 Direct Stability Methods 328
Problems 331
References 331
Further Reading 332
CHAPTER 13 EXCITATION SYSTEMS AND POWER
SYSTEM STABILIZERS
13-1 Reactive Capability Curve (Operating Chart) of a Synchronous Generator 333
13-2 Steady-State Stability Curves 336
13-3 Short-Circuit Ratio 336
13-4 Per Unit Systems 337
13-5 Nominal Response of the Excitation System 337
v i i i CONTENTS
13-6 Building Blocks of Excitation Systems 339
13-7 Saturation Characteristics of Exciter 340
13-8 Types of Excitation Systems 343
13-9 Power System Stabilizers 352
13-10 Tuning a PSS 355
13-11 Models of Prime Movers 358
13-12 Automatic Generation Control 358
13-13 On-Line Security Assessments 361
Problems 362
References 362
Further Reading 363
CHAPTER 14 TRANSIENT BEHAVIOR OF TRANSFORMERS
14-1 Frequency-Dependent Models 365
14-2 Model of a Two-Winding Transformer 365
14-3 Equivalent Circuits for Tap Changing 367
14-4 Inrush Current Transients 368
14-5 Transient Voltages Impacts on Transformers 368
14-6 Matrix Representations 371
14-7 Extended Models of Transformers 373
14-8 EMTP Model FDBIT 380
14-9 Sympathetic Inrush 382
14-10 High-Frequency Models 383
14-11 Surge Transference Through Transformers 384
14-12 Surge Voltage Distribution Across Windings 389
14-13 Duality Models 389
14-14 GIC Models 391
14-15 Ferroresonance 391
14-16 Transformer Reliability 394
Problems 395
References 396
Further Reading 396
CHAPTER 15 POWER ELECTRONIC EQUIPMENT
AND FACTS
15-1 The Three-Phase Bridge Circuits 397
15-2 Voltage Source Three-Phase Bridge 401
15-3 Three-Level Converter 402
15-4 Static VAR Compensator (SVC) 405
15-5 Series Capacitors 408
15-6 FACTS 414
15-7 Synchronous Voltage Source 414
15-8 Static Synchronous Compensator 415
15-9 Static Series Synchronous Compensator 416
15-10 Unified Power Flow Controller 419
15-11 NGH-SSR Damper 422
15-12 Displacement Power Factor 423
15-13 Instantaneous Power Theory 424
15-14 Active Filters 425
Problems 425
References 426
Further Reading 426
CHAPTER 16 FLICKER, BUS TRANSFER, TORSIONAL
BYNAMICS, AND OTHER TRANSIENTS
16-1 Flicker 429
16-2 Autotransfer of Loads 432
16-3 Static Transfer Switches and Solid-State Breakers 438
16-4 Cogging and Crawling of Induction Motors 439
16-5 Synchronous Motor-Driven Reciprocating Compressors 441
16-6 Torsional Dynamics 446
16-7 Out-of-Phase Synchronization 449
Problems 451
References 451
Further Reading 452
CHAPTER 17 INSULATION COORDINATION
17-1 Insulating Materials 453
17-2 Atmospheric Effects and Pollution 453
17-3 Dielectrics 455
17-4 Insulation Breakdown 456
17-5 Insulation Characteristics—BIL and BSL 459
17-6 Volt-Time Characteristics 461
17-7 Nonstandard Wave Forms 461
17-8 Probabilistic Concepts 462
17-9 Minimum Time to Breakdown 465
17-10 Weibull Probability Distribution 465
17-11 Air Clearances 465
17-12 Insulation Coordination 466
17-13 Representation of Slow Front Overvoltages (SFOV) 469
17-14 Risk of Failure 470
17-15 Coordination for Fast-Front Surges 472
17-16 Switching Surge Flashover Rate 473
17-17 Open Breaker Position 474
CONTENTS ix
17-18 Monte Carlo Method 474
17-19 Simplified Approach 474
17-20 Summary of Steps in Insulation Coordination 475
Problems 475
References 476
Further Reading 476
CHAPTER 18 GAS-INSULATED SUBSTATIONS—VERY FAST
TRANSIENTS
18-1 Categorization of VFT 477
18-2 Disconnector-Induced Transients 477
18-3 Breakdown in GIS—Free Particles 480
18-4 External Transients 481
18-5 Effect of Lumped Capacitance at Entrance to GIS 482
18-6 Transient Electromagnetic Fields 483
18-7 Breakdown in SF6 483
18-8 Modeling of Transients in GIS 484
18-9 Insulation Coordination 487
18-10 Surge Arresters for GIS 488
Problems 493
References 493
Further Reading 494
CHAPTER 19 TRANSIENTS AND SURGE PROTECTION
IN LOW-VOLTAGE SYSTEMS
19-1 Modes of Protection 495
19-2 Multiple-Grounded Distribution Systems 495
19-3 High-Frequency Cross Interference 498
19-4 Surge Voltages 499
19-5 Exposure Levels 499
19-6 Test Wave Shapes 500
19-7 Location Categories 502
19-8 Surge Protection Devices 505
19-9 SPD Components 508
19-10 Connection of SPD Devices 512
19-11 Power Quality Problems 516
19-12 Surge Protection of Computers 517
19-13 Power Quality for Computers 520
19-14 Typical Application of SPDs 520
Problems 523
References 523
Further Reading 524
CHAPTER 20 SURGE ARRESTERS
20-1 Ideal Surge Arrester 525
20-2 Rod Gaps 525
20-3 Expulsion-Type Arresters 526
20-4 Valve-Type Silicon Carbide Arresters 526
20-5 Metal-Oxide Surge Arresters 529
20-6 Response to Lightning Surges 534
20-7 Switching Surge Durability 537
20-8 Arrester Lead Length and Separation Distance 539
20-9 Application Considerations 541
20-10 Surge Arrester Models 544
20-11 Surge Protection of AC Motors 545
20-12 Surge Protection of Generators 547
20-13 Surge Protection of Capacitor Banks 548
20-14 Current-Limiting Fuses 551
Problems 554
References 555
Further Reading 555
CHAPTER 21 TRANSIENTS IN GROUNDING SYSTEMS
21-1 Solid Grounding 557
21-2 Resistance Grounding 560
21-3 Ungrounded Systems 563
21-4 Reactance Grounding 564
21-5 Grounding of Variable-Speed Drive Systems 567
21-6 Grounding for Electrical Safety 569
21-7 Finite Element Methods 577
21-8 Grounding and Bonding 579
21-9 Fall of Potential Outside the Grid 581
21-10 Influence on Buried Pipelines 583
21-11 Behavior Under Lightning Impulse Current 583
Problems 585
References 585
Further Reading 586
CHAPTER 22 LIGHTNING PROTECTION OF STRUCTURES
22-1 Parameters of Lightning Current 587
22-2 Types of Structures 587
22-3 Risk Assessment According to IEC 588
22-4 Criteria for Protection 589
22-5 Protection Measures 592
22-6 Transient Behavior of Grounding System 594
x CONTENTS
22-7 Internal LPS Systems According to IEC 594
22-8 Lightning Protection According to NFPA Standard 780 594
22-9 Lightning Risk Assessment According to NFPA 780 595
22-10 Protection of Ordinary Structures 596
22-11 NFPA Rolling Sphere Model 597
22-12 Alternate Lightning Protection Technologies 598
22-13 Is EMF Harmful to Humans? 602
Prohlems 602
References 603
Further Reading 603
CHAPTER 23 DC SYSTEMS, SHORT CIRCUITS, DISTRIBUTIONS, AND HVDC
23-1 Short-Circuit Transients 605
23-2 Current Interruption in DC Circuits 615
23-3 DC Industrial and Commercial Distribution Systems 617
23-4 HVDC Transmission 618
Problems 627
References 628
Further Reading 629
CHAPTER 24 SMART GRIDS AND WIND POWER GENERATION
24-1 WAMS and Phasor Measurement Devices 631
24-2 System Integrity Protection Schemes 632
24-3 Adaptive Protection 633
24-4 Wind-Power Stations 634
24-5 Wind-Energy Conversion 635
24-6 The Cube Law 636
24-7 Operation 638
24-8 Wind Generators 639
24-9 Power Electronics 640
24-10 Computer Modeling 642
24-11 Floating Wind Turbines 645
References 645
Further Reading 645
APPENDIX A DIFFERENTIAL EQUATIONS
A-l Homogeneous Differential Equations 647
A-2 Linear Differential Equations 648
A-3 Bernoulli's Equation 648
A-4 Exact Differential Equations 648
A-5 Clairaut's Equation 649
A-6 Complementary Function and Particular Integral 649
A-7 Forced and Free Response 649
A-8 Linear Differential Equations of the Second Order (With Constant Coefficients) 650
A-9 Calculation of Complementary Function 650
A-10 Higher-Order Equations 651
A-ll Calculations of Particular Integrals 651
A-12 Solved Examples 653
A-13 Homogeneous Linear Differential Equations 654
A-14 Simultaneous Differential Equations 655
A-15 Partial Differential Equations 655
Further Reading 658
APPENDIX D LAPLACE TRANSFORM
B-l Method of Partial Fractions 659
B-2 Laplace Transform of a Derivative of/(t) 661
B-3 Laplace Transform of an Integral 661
B-4 Laplace Transform of t/(t) 662
B-5 Laplace Transform of {lit) f(t) 662
B-6 Initial-Value Theorem 662
B-7 Final-Value Theorem 662
B-8 Solution of Differential Equations 662
B-9 Solution of Simultaneous Differential Equations 662
B-10 Unit-Step Function 663
B-ll Impulse Function 663
B-12 Gate Function 663
B-13 Second Shifting Theorem 663
B-14 Periodic Functions 665
B-15 Convolution Theorem 666
B-l 6 Inverse Laplace Transform by Residue Method 666
B-l 7 Correspondence with Fourier Transform 667
Further Reading 667
APPENDIX С /-TRANSFORM
C-l Properties of z-Transform 670
C-2 Initial-Value Theorem 671
C-3 Final-Value Theorem 672
C-4 Partial Sum 672
C-5 Convolution 672
C-6 Inverse z-Transform 672
C-7 Inversion by Partial Fractions 674
C-8 Inversion by Residue Method 674
CONTENTS x i
C-9 Solution of Difference Equations 675
C-10 State Variable Form 676
Further Reading 676
APPENDIX D SEQUENCE IMPEDANCES OF TRANSMISSION
LINES AND CABLES
D-l AC Resistance of Conductors 677
D-2 Inductance of Transmission Lines 678
D-3 Transposed Line 678
D-4 Composite Conductors 679
D-5 Impedance Matrix 680
D-6 Three-Phase Line with Ground Conductors 680
D-7 Bundle Conductors 681
D-8 Carson's Formula 682
D-9 Capacitance of Lines 684
D-10 Cable Constants 685
D-ll Frequency-Dependent Transmission Line Models 688
References 688
APPENDIX E ENERGY FUNCTIONS AND STABILITY
E-l Dynamic Elements 691
E-2 Passivity 691
E-3 Equilibrium Points 691
E-4 State Equations 692
E-5 Stability of Equilibrium Points 692
E-6 Hartman-Grobman Linearization Theorem 692
E-7 Lyapunov Function 692
E-8 LaSalle's Invariant Principle 692
E-9 Asymptotic Behavior 692
E-10 Periodic Inputs 693
References 693
Further Reading 693
APPENDIX F STATISTICS AND PROBABILITY
F-l Mean, Mode, and Median 695
F-2 Mean and Standard Deviation 695
F-3 Skewness and Kurtosis 696
F-4 Curve Fitting and Regression 696
F-5 Probability 698
F-6 Binomial Distribution 699
F-7 Poisson Distribution 699
F-8 Normal or Gaussian Distribution 699
F-9 Weibull Distribution 701
Reference 702
Further Reading 702
APPENDIX G NUMERICAL TECHNIQUES
G-l Network Equations 703
G-2 Compensation Methods 703
G-3 Nonlinear Inductance 704
G-4 Piecewise Linear Inductance 704
G-5 Newton-Raphson Method 704
G-6 Numerical Solution of Linear Differential Equations
G-7 Laplace Transform 706
G-8 Taylor Series 706
G-9 Trapezoidal Rule of Integration 706
G-10 Runge-Kutta Methods 707
G-ll Predictor-Corrector Methods 708
G-12 Richardson Extrapolation and Romberg Integration 708
References 709
Further Reading 709
Index 711