Short-Circuits in AC and DC Systems

ANSI, IEEE, and IEC Standards Volume 1

J.C. Das

G& CRC Press Taylor Si Francis Croup

^ > ~ — ^ Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Croup, an informa business

Contents

Series Preface xv Preface to Volume 1: Short-Circuits in AC and DC Systems xvii Author xix

1. Design and Analyses Concepts of Power Systems 1 1.1 Static and Dynamic Systems 2 1.2 State Variables 3 1.3 Linear and Nonlinear Systems 5

1.3.1 Property of Decomposition 6 1.4 Linearizing a Nonlinear System 6 1.5 Time-Invariant Systems 9 1.6 Lumped and Distributed Parameters 11 1.7 Optimization 12 1.8 Planning and Design of Electrical Power Systems 12 1.9 Electrical Standards and Codes 14 1.10 Reliability Analyses 15

1.10.1 Availability 16 1.10.1.1 Exponential Distribution 17

1.10.2 Data for Reliability Evaluations 18 1.10.3 Methods of Evaluation 18 1.10.4 Reliability and Safety 22

1.11 Extent of System Modeling 25 1.11.1 Short-Circuit Calculations 26 1.11.2 Load Flow Calculations 28 1.11.3 Harmonic Analysis 28

1.12 Power System Studies 28 1.13 Power System Studies Software 29 1.14 System of Units 30 Problems 30 References 32

2. Modern Electrical Power Systems 35 2.1 Classification 35

2.1.1 Utility Companies in the USA 36 2.1.2 North American Power System Interconnections 37

2.2 Deregulation of Power Industry 38 2.2.1 Generation Company (GENCO) 38 2.2.2 Transmission Company (TRANSCO) 39 2.2.3 Distribution Company (DISTCO) 39

2.3 The New Energy Platform 39 2.3.1 Sustainable, Renewable, and Green Energy 40 2.3.2 Green Energy 41 2.3.3 Hydroelectric Plants 41 2.3.4 Pumped Storage Hydroelectric Plants 43

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vi Contents

2.3.5 Nuclear Power 43 2.3.5.1 Breeder Reactors 47 2.3.5.2 Nuclear Fusion 47 2.3.5.3 Nuclear Power around the Globe 48 2.3.5.4 Is Nuclear Power Green Energy? 48

2.3.6 Geothermal Plants 49 2.3.7 Solar and Wind Energy 50 2.3.8 Biofuels and Carbon-Neutral Fuels 50 2.3.9 Local Green Energy Systems 51 2.3.10 Fuel Cells 51 2.3.11 Reducing Caron Emissions 52

2.4 Large Power Stations of the World 53 2.5 Smart Grid 57

2.5.1 Legislative Measures 58 2.5.2 Technologies Driving Smart Grid 58

2.6 Microgrids and Distributed Generation 59 2.7 Energy Storage 63

2.7.1 Fly wheel Storage 64 2.7.2 Superconductivity 67

2.7.2.1 Applications in Electrical Systems 68 2.8 Transmission Systems 68 2.9 Industrial Systems 69 2.10 Distribution Systems 71

2.10.1 The Radial System 72 2.10.2 The Parallel or Loop System 73 2.10.3 Network or Grid System 73 2.10.4 Primary Distribution System 75

2.11 Future Load Growth 77 2.12 Underground versus OH Systems 77

2.12.1 Spot Network 78 2.13 HVDC Transmission 80

2.13.1 HVDC Light 80 2.13.2 HVDC Configurations and Operating Modes 80

Problems 82 Bibliography 82 IEEE Color Books 84

3. Wind and Solar Power Generation and Interconnections with Utility 85 3.1 Prospective of Wind Generation in the USA 85 3.2 Characteristics of Wind Power Generation 87

3.2.1 Maximum Transfer Capability 91 3.2.2 Power Reserves and Regulation 92 3.2.3 Congestion Management 93

3.3 Wind Energy Conversion 93 3.3.1 Drive Train 94 3.3.2 Towers 96 3.3.3 Rotor Blades 96

3.4 The Cube Law 97 3.5 Operation 99

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3.5.1 Speed Control 101 3.5.2 Behavior under Faults and Low-Voltage

Ride Through 102 3.6 Wind Generators 103

3.6.1 Induction Generators 103 3.6.2 Direct Coupled Induction Generator 105 3.6.3 Induction Generator Connected to Grid through

Full Size Converter 105 3.6.4 Doubly Fed Induction Generator 106 3.6.5 Synchronous Generators 107

3.7 Reactive Power and Wind Turbine Controls 107 3.8 Power Electronics and Harmonics I l l

3.8.1 Power Electronics I l l 3.8.2 Harmonics 112

3.9 Computer Modeling 113 3.9.1 A Wind Turbine Controller 113

3.10 Solar Power 115 3.11 CSP Plants 116

3.11.1 Solar Energy Collectors 116 3.11.1.1 Parabolic Dish Concentrators 117 3.11.1.2 Solar Tower 118

3.11.2 Trackers 118 3.11.2.1 Photovoltaic Trackers 119

3.12 Direct Conversion of Solar Energy through PV Cells 120 3.12.1 Cells, Modules, Panels, and Systems 120

3.12.1.1 PV Module 120 3.12.1.2 PV Panel 121 3.12.1.3 PV Array 121 3.12.1.4 PV Array Subfield 121

3.13 Classification of Solar Cells 121 3.14 Utility Connections of Distributed Resources 123

3.14.1 Voltage Control 123 3.14.2 Grounding 123 3.14.3 Synchronizing 123 3.14.4 Distribution Secondary Spot

Networks 124 3.14.5 Inadvertent Energization 124 3.14.6 Metering 124 3.14.7 Isolation Device 124 3.14.8 EMI Interference 124 3.14.9 Surge Withstand 125 3.14.10 Paralleling Device 125 3.14.11 Area Faults 125 3.14.12 Abnormal Frequencies 125 3.14.13 Reconnection 125 3.14.14 Harmonics 125

Problems 126 References 127

viii Contents

4. Short-Circuit Currents and Symmetrical Components 131 4.1 Nature of Short-Circuit Currents 132 4.2 Symmetrical Components 135 4.3 Eigenvalues and Eigenvectors 138 4.4 Symmetrical Component Transformation 139

4.4.1 Similarity Transformation 139 4.4.2 Decoupling a Three-Phase Symmetrical System 141 4.4.3 Decoupling a Three-Phase Unsymmetrical System 145 4.4.4 Power Invariance in Symmetrical Component

Transformation 146 4.5 Clarke Component Transformation 146 4.6 Characteristics of Symmetrical Components 150 4.7 Sequence Impedance of Network Components 153

4.7.1 Construction of Sequence Networks 153 4.7.2 Transformers 155

4.7.2.1 Delta-Wye or Wye-Delta Transformer 155 4.7.2.2 Wye-Wye Transformer 157 4.7.2.3 Delta-Delta Transformer 158 4.7.2.4 Zigzag Transformer 158 4.7.2.5 Three-Winding Transformers 159

4.7.3 Static Load 163 4.7.4 Synchronous Machines 163

4.8 Computer Models of Sequence Networks 168 Problems 170 Bibliography 171

5. Unsymmetrical Fault Calculations 173 5.1 Line-to-Ground Fault 173 5.2 Line-to-Line Fault 175 5.3 Double Line-to-Ground Fault 177 5.4 Three-Phase Fault 179 5.5 Phase Shift in Three-Phase

Transformers 180 5.5.1 Transformer Connections 180 5.5.2 Phase Shifts in Winding Connections 180 5.5.3 Phase Shift for Negative Sequence

Components 183 5.6 Unsymmetrical Fault Calculations 186 5.7 System Grounding 193

5.7.1 Solidly Grounded Systems 195 5.7.2 Resistance Grounding 196

5.7.2.1 High-Resistance Grounded Systems 197 5.7.2.2 Coefficient of Grounding 203

5.8 Open Conductor Faults 204 5.8.1 Two-Conductor Open Fault 204 5.8.2 One-Conductor Open 204

Problems 209 Bibliography 211 References 211

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6. Matrix Methods for Network Solutions 213 6.1 Network Models 213 6.2 Bus Admittance Matrix 214 6.3 Bus Impedance Matrix 219

6.3.1 Bus Impedance Matrix from Open-Circuit Testing 220 6.4 Loop Admittance and Impedance Matrices 221

6.4.1 Selection of Loop Equations 223 6.5 Graph Theory 223 6.6 Bus Admittance and Impedance Matrices by Graph

Approach 226 6.6.1 Primitive Network 226 6.6.2 Incidence Matrix from Graph Concepts 228 6.6.3 Node Elimination in Y-Matrix 232

6.7 Algorithms for Construction of Bus Impedance Matrix 233 6.7.1 Adding a Tree Branch to an Existing Node 234 6.7.2 Adding a Link 236 6.7.3 Removal of an Uncoupled Branch 238 6.7.4 Changing Impedance of an Uncoupled Branch 238 6.7.5 Removal of a Coupled Branch 238

6.8 Short-Circuit Calculations with Bus Impedance Matrix 246 6.8.1 Line-to-Ground Fault 246 6.8.2 Line-to-Line Fault 246 6.8.3 Double Line-to-Ground Fault 247

6.9 Solution of Large Network Equations 256 Problems 257 Bibliography 258

7. Current Interruptions in AC Networks 259 7.1 Rheostatic Breaker 259 7.2 AC Arc Interruption 261

7.2.1 Arc Interruption Theories 261 7.2.1.1 Cassie's Theory 261 7.2.1.2 Mayr's Theory 262 7.2.1.3 Cassie-Mayr Theory 262

7.3 Current-Zero Breaker 263 7.4 Transient Recovery Voltage 264

7.4.1 First Pole to Clear Factor 266 7.5 The Terminal Fault 269

7.5.1 Four-Parameter Method 269 7.5.2 Two-Parameter Representation 270

7.6 The Short-Line Fault 271 7.7 Interruption of Low Inductive Currents 273

7.7.1 Virtual Current Chopping 275 7.8 Interruption of Capacitance Currents 276 7.9 TRV in Capacitive and Inductive Circuits 278 7.10 Prestrikes in Circuit Breakers 279 7.11 Overvoltages on Energizing HV Lines 280

7.11.1 Overvoltage Control 282 7.11.2 Synchronous Operation 283

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7.11.3 Synchronous Capacitor Switching 283 7.11.4 Shunt Reactors 284

7.11.4.1 Oscillation Modes 287 7.12 Out-of-Phase Closing 288 7.13 Resistance Switching 289 7.14 Failure Modes of Circuit Breakers 293 7.15 Stresses in Circuit Breakers 295 7.16 Classification of Circuit Breakers according to Interrupting

Medium 295 7.16.1 SF6 Circuit Breakers 296

7.16.1.1 Electronegativity of SF6 297 7.16.2 Operating Mechanisms 299 7.16.3 Vacuum Interruption 300

7.16.3.1 Current Chopping and Multiple Ignitions 301 7.16.3.2 Switching of Unloaded Dry-Type Transformers 303

7.17 Part Winding Resonance in Transformers 304 7.17.1 Snubber Circuits 306

7.18 Solid-State Circuit Breakers 306 Problems 308 Bibliography 309 References 310

8. Application and Ratings of Circuit Breakers and Fuses according to ANSI Standards 313 8.1 Total and Symmetrical Current Basis 314 8.2 Asymmetrical Ratings 316

8.2.1 Contact Parting Time 316 8.3 Voltage Range Factor K 317 8.4 Circuit Breaker Timing Diagram 320 8.5 Maximum Peak Current 321 8.6 Permissible Tripping Delay 322 8.7 Service Capability Duty Requirements and Reclosing

Capability 322 8.7.1 Transient Stability on Fast Reclosing 323

8.8 Shunt Capacitance Switching 326 8.8.1 Switching of Cables 331

8.9 Line Closing Switching Surge Factor 335 8.9.1 Switching of Transformers 336

8.10 Out-of-Phase Switching Current Rating 337 8.11 Transient Recovery Voltage 337

8.11.1 Circuit Breakers Rated Below 100 kV 338 8.11.2 Circuit Breakers Rated 100 kV and Above 338 8.11.3 Short-Line Faults 342 8.11.4 Oscillatory TRV 344

8.11.4.1 Exponential (Overdamped) TRV 344 8.11.5 Initial TRV 345 8.11.6 Adopting IEC TRV Profiles in IEEE Standards 345 8.11.7 Definite-Purpose TRV Breakers 350

8.11.8 TRV Calculation Techniques 350

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8.12 Generator Circuit Breakers 353 8.13 Specifications of High-Voltage Circuit Breakers 358 8.14 Low-Voltage Circuit Breakers 358

8.14.1 Molded Case Circuit Breakers 358 8.14.2 Insulated Case Circuit Breakers (ICCBs) 359 8.14.3 Low-Voltage Power Circuit Breakers (LVPCBs) 359

8.14.3.1 Single-Pole Interrupting Capability 361 8.14.3.2 Short-Time Ratings 361 8.14.3.3 Series Connected Ratings 362

8.15 Fuses 363 8.15.1 Current-Limiting Fuses 364 8.15.2 Low-Voltage Fuses 365 8.15.3 High-Voltage Fuses 365 8.15.4 Interrupting Ratings 366

Problems 367 References 368

9. Short Circuit of Synchronous and Induction Machines and Converters 371 9.1 Reactances of a Synchronous Machine 372

9.1.1 Leakage Reactance Xl 372 9.1.2 Subtransient Reactance X"ä 372 9.1.3 Transient Reactance Xd 372 9.1.4 Synchronous Reactance Xd 372 9.1.5 Quadrature Axis Reactances Xq, X'q, and Xq 373 9.1.6 Negative Sequence Reactance X2 374 9.1.7 Zero Sequence Reactance X0 374 9.1.8 Potier Reactance Xp 374

9.2 Saturation of Reactances 375 9.3 Time Constants of Synchronous Machines 375

9.3.1 Open-Circuit Time Constant Tdo 375 9.3.2 Subtransient Short-Circuit Time Constant Td* 375 9.3.3 Transient Short-Circuit Time Constant Td 375 9.3.4 Armature Time Constant Ta 375

9.4 Synchronous Machine Behavior on Short Circuit 375 9.4.1 Equivalent Circuits during Fault 380 9.4.2 Fault Decrement Curve 383

9.5 Circuit Equations of Unit Machines 386 9.6 Park's Transformation 390

9.6.1 Reactance Matrix of a Synchronous Machine 390 9.6.2 Transformation of Reactance Matrix 393

9.7 Park's Voltage Equation 395 9.8 Circuit Model of Synchronous Machines 397 9.9 Calculation Procedure and Examples 399

9.9.1 Manufacturer's Data 406 9.10 Short Circuit of Synchronous Motors and Condensers 408 9.11 Induction Motors 409 9.12 Capacitor Contribution to the Short-Circuit Currents 413 9.13 Static Converters Contribution to the Short-Circuit Currents 414

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9.14 Practical Short-Circuit Calculations 417 Problems 418 References 419 Bibliography 419

10. Short-Circuit Calculations according to ANSI Standards 421 10.1 Types of Calculations 421

10.1.1 Assomptions 422 10.1.2 Maximum Peak Current 422

10.2 Accounting for Short-Circuit Current Decay 423 10.2.1 Low-Voltage Motors 424

10.3 Rotating Machine Model 425 10.4 Type and Severity of System Short Circuits 426 10.5 Calculation Methods 427

10.5.1 Simplified Method X/R < 17 427 10.5.2 Simplified Method X/R > 17 427 10.5.3 E/X Method for AC and DC Decrement Adjustments 427 10.5.4 Fault Fed from Remote Sources 428 10.5.5 Fault Fed from Local Sources 430 10.5.6 Weighted Multiplying Factors 435

10.6 Network Reduction 435 10.6.1 E/X or E/Z Calculation 436

10.7 Breaker Duty Calculations 437 10.8 Generator Source Asymmetry 437 10.9 Calculation Procedure 439

10.9.1 Necessity of Gathering Accurate Data 439 10.9.2 Calculation Procedure 440 10.9.3 Analytical Calculation Procedure 441 10.9.4 Hand Calculations 441 10.9.5 Dynamic Simulation 441 10.9.6 Circuit Breakers with Sources on Either Side 441 10.9.7 Switching Devices without Short-Circuit

Interruption Ratings 443 10.9.8 Adjustments for Transformer Taps and Ratios 443

10.10 Examples of Calculations 444 10.10.1 Calculation of Short-Circuit Duties 444 10.10.2 K-Rated 15 kV Breakers 448 10.10.3 4.16kV Circuit Breakers and Motor Starters 452 10.10.4 Transformer Primary Switches and Fused Switches 452 10.10.5 Low-Voltage Circuit Breakers 452 10.10.6 Bus Bracings 452 10.10.7 Power Cables 455 10.10.8 Overhead Line Conductors 456 10.10.9 Generator Source Symmetrical Short-Circuit Current 460 10.10.10 Generator Source Asymmetrical Current 461 10.10.11 System Source Symmetrical Short-Circuit Current 461 10.10.12 System Source Asymmetrical Short-Circuit Current 462 10.10.13 Required Closing Latching Capabilities 462 10.10.14 Selection of the Generator Breaker 463

Contents xiii

10.11 Deriving an Equivalent Impedance 464 10.12 Thirty-Cycle Short-Circuit Currents 469 10.13 Fault Current Limiters 470

10.13.1 Superconducting Fault Current Limiters 473 Problems 474 References 477

11. Short-Circuit Calculations according to IEC Standards 479 11.1 Conceptual and Analytical Differences 479

11.1.1 Breaking Capability 479 11.1.2 Rated Restriking Voltage 480 11.1.3 Rated Making Capacity 480 11.1.4 Rated Opening Time and Break Time 480 11.1.5 Initial Symmetrical Short-Circuit Current 480 11.1.6 Peak Making Current 481 11.1.7 Breaking Current 481 11.1.8 Steady-State Current 481 11.1.9 Highest Short-Circuit Currents 482

11.2 Prefault Voltage 483 11.3 Far-From Generator Faults 483

11.3.1 Nonmeshed Sources 485 11.3.2 Meshed Networks 487

11.3.2.1 Method A: Uniform Ratio R/X or X/R Ratio Method 487

11.3.2.2 Ratio R/X or X/R at the Short-Circuit Location 487 11.3.2.3 Method C: Equivalent Frequency Method 488

11.4 Near-to-Generator Faults 489 11.4.1 Generators Directly Connected to Systems 489 11.4.2 Generators and Unit Transformers of Power

Station Units 490 11.4.3 Motors 491 11.4.4 Short-Circuit Currents Fed from One Generator 491

11.4.4.1 Breaking Current 491 11.4.4.2 Steady-State Current 492

11.4.5 Short-Circuit Currents in Nonmeshed Networks 493 11.4.6 Short-Circuit Currents in Meshed Networks 494

11.5 Influence of Motors 495 11.5.1 Low-Voltage Motor Groups 496 11.5.2 Calculations of Breaking Currents of

Asynchronous Motors 496 11.5.3 Static Converter Fed Drives 497

11.6 Comparison with ANSI/IEE Calculation Procedures 497 11.7 Examples of Calculations and Comparison with

ANSI Methods 499 11.8 Electromagnetic Transients Program Simulation of a

Generator Terminal Short Circuit 513 11.8.1 The Effect of PF 513

Problems 517 References 519

XIV Contents

12. Calculations of Short-Circuit Currents in Direct Current Systems 521 12.1 DC Short-Circuit Current Sources 521 12.2 Calculation Procedures 523

12.2.1 IEC Calculation Procedure 523 12.2.2 Matrix Methods 525

12.3 Short-Circuit of a Lead Acid Battery 525 12.4 Short-Circuit of DC Motors and Generators 531 12.5 Short-Circuit of a Rectifier 537 12.6 Short-Circuit of a Charged Capacitor 543 12.7 Total Short-Circuit Current 544 12.8 DC Circuit Breakers 545 12.9 DC Rated Fuses 548 12.10 Protection of the Semi-Conductor Devices 548 12.11 High-Voltage DC Circuit Breakers 550 Problems 553 References 553

Appendix A: Matrix Methods 555

Appendix B: Sparsity and Optimal Ordering 587

Appendix C: Transformers and Reactors 595

Appendix D: Solution to the Problems 629

Index 709