Short-Circuits in AC and DC Systems Volume 1 J.C. Das

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Short-Circuits in AC and DC Systems Volume 1 J.C. Das

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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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 Flywheel 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
Contents vii
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……………………………………………………………….. 111
3.8.1 Power Electronics……………………………………………………………………………… 111
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
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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
Contents ix
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
x Contents
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
Contents xi
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 XI……………………………………………………………………….. 372
9.1.2 Subtransient Reactance X d……………………………………………………………….. 372
9.1.3 Transient Reactance X d …………………………………………………………………….. 372
9.1.4 Synchronous Reactance Xd……………………………………………………………….. 372
9.1.5 Quadrature Axis Reactances X q, 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
xii Contents
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 4.16 kV 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

Series Preface

This handbook on power systems consists of four volumes. These are carefully planned
and designed to provide state-of-the-art material on the major aspects of electrical power
systems, short-circuit currents, load flow, harmonics, and protective relaying.
An effort has been made to provide a comprehensive coverage, with practical applications,
case studies, examples, problems, extensive references, and bibliography.
The material is organized with sound theoretical base and its practical applications. The
objective of creating this series is to provide the reader with a comprehensive treatise that
could serve as a reference and day-to-day application guide for solving the real-world
problem. It is written for plasticizing engineers and academia at the level of upper-undergraduate
and graduate degrees.
Though there are published texts on similar subjects, this series provides a unique
approach to the practical problems that an application engineer or consultant may face in
conducting system studies and applying it to varied system problems.
Some parts of the work are fairly advanced on a postgraduate level and get into higher
mathematics. Yet the continuity of the thought process and basic conceptual base are
maintained. A beginner and advanced reader will equally benefit from the material covered.
An underground level of education is assumed, with a fundamental knowledge of
electrical circuit theory, rotating machines, and matrices.
Currently, power systems, large or small, are analyzed on digital computers with appropriate
software. However, it is necessary to understand the theory and basis of these calculations
to debug and decipher the results.
A reader may be interested only in one aspect of power systems and may choose to purchase
only one of the volumes. Many aspects of power systems are transparent between
different types of studies and analyses—for example, knowledge of short-circuit currents
and symmetrical component is required for protective relaying and fundamental frequency
load flow is required for harmonic analysis. Though appropriate references are
provided, the material is not repeated from one volume to another.
The series is a culmination of the vast experience of the author in solving real-world
problems in the industrial and utility power systems for more than 40 years.
Another key point is that the solutions to the problems are provided in Appendix D.
Readers should be able to independently solve these problems after perusing the contents
of a chapter and then look back to the solutions provided as a secondary help. The problems
are organized so these can be solved with manual manipulations, without the help of
any digital computer power system software.
It is hoped the series will be a welcome addition to the current technical literature.
The author thanks CRC Press editor Nora Konopka for her help and cooperation throughout
the publication effort.