1 Introduction to Distribution Systems .......................... 1
1.1 The Distribution System ................................... 2
1.2 Distribution Substations .................................. 2
1.3 Radial Feeders ............................................ 5
1.4 Distribution Feeder Map ................................... 6
1.5 Distribution Feeder Electrical Characteristics ............ 8
1.6 Summary ................................................... 9
2 The Nature of Loads .......................................... 11
2.1 Definitions .............................................. 11
2.2 Individual Customer Load ................................. 13
2.2.1 Demand ............................................. 13
2.2.2 Maximum Demand ..................................... 13
2.2.3 Average Demand ..................................... 14
2.2.4 Load Factor ........................................ 14
2.3 Distribution Transformer Loading ......................... 15
2.3.1 Diversified Demand ................................. 16
2.3.2 Maximum Diversified Demand ......................... 17
2.3.3 Load Duration Curve ................................ 18
2.3.4 Maximum Noncoincident Demand ....................... 18
2.3.5 Diversity Factor (DF) .............................. 19
2.3.6 Demand Factor ...................................... 19
2.3.7 Utilization Factor ................................. 20
2.3.8 Load Diversity ..................................... 21
2.4 Feeder Load .............................................. 21
2.4.1 Load Allocation .................................... 21
2.4.1.1 Application of DFs ......................... 22
2.4.1.2 Load Survey ................................ 22
2.4.1.3 Transformer Load Management ................ 26
2.4.1.4 Metered Feeder Maximum Demand .............. 26
2.4.1.5 What Method to Use? ........................ 28
2.4.2 Voltage Drop Calculations Using Allocated Loads .... 28
2.4.2.1 Application of DFs ......................... 28
2.4.2.2 Load Allocation Based upon Transformer
Ratings .................................... 32
2.5 Summary .................................................. 34
3 Approximate Method of Analysis ............................... 41
3.1 Voltage Drop ............................................. 41
3.2 Line Impedance ........................................... 43
3.3 K Factors ................................................ 45
3.3.1 Kdrop Factor ........................................ 45
3.3.2 Krise Factor ........................................ 48
3.4 Uniformly Distributed Loads .............................. 49
3.4.1 Voltage Drop ....................................... 49
3.4.2 Power Loss ......................................... 52
3.4.3 The Exact Lumped Load Model ........................ 54
3.5 Lumping Loads in Geometric Configurations ................ 57
3.5.1 The Rectangle ...................................... 57
3.5.2 The Triangle ....................................... 62
3.5.3 The Trapezoid ...................................... 68
3.6 Summary .................................................. 73
References ................................................... 73
4 Series Impedance of Overhead and Underground Lines ........... 79
4.1 Series Impedance of Overhead Lines ....................... 79
4.1.1 Transposed Three-Phase Lines ....................... 80
4.1.2 Untransposed Distribution Lines .................... 81
4.1.3 Carson's Equations ................................. 84
4.1.4 Modified Carson's Equations ........................ 85
4.1.5 Primitive Impedance Matrix for Overhead Lines ...... 88
4.1.6 Phase Impedance Matrix for Overhead Lines .......... 89
4.1.7 Sequence Impedances ................................ 92
4.2 Series Impedance of Underground Lines ................... 102
4.2.1 Concentric Neutral Cable .......................... 103
4.2.2 Tape Shielded Cables .............................. 109
4.2.3 Parallel Underground Distribution Lines ........... 113
4.3 Summary ................................................. 117
References .................................................. 118
5 Shunt Admittance of Overhead and Underground Lines .......... 125
5.1 General Voltage Drop Equation ........................... 126
5.2 Overhead Lines .......................................... 127
5.2.1 The Shunt Admittance of Overhead Parallel Lines .... 131
5.3 Concentric Neutral Cable Underground Lines .............. 134
5.4 Tape-Shielded Cable Underground Lines ................... 138
5.5 Sequence Admittance ..................................... 140
5.6 The Shunt Admittance of Parallel Underground Lines ...... 141
5.7 Summary ................................................. 142
References .................................................. 143
6 Distribution System Line Models ............................. 145
6.1 Exact Line Segment Model ................................ 145
6.2 The Modified Line Model ................................. 152
6.2.1 The Three-Wire Delta Line ......................... 153
6.2.2 The Computation of Neutral and Ground Currents .... 154
6.3 The Approximate Line Segment Model ...................... 157
6.4 The General Matrices for Parallel Lines ................. 162
6.4.1 Physically Parallel Lines ......................... 166
6.4.2 Electrically Parallel Lines ....................... 172
6.5 Summary ................................................. 178
References .................................................. 178
7 Voltage Regulation .......................................... 185
7.1 Standard Voltage Ratings ................................ 185
7.2 Two-Winding Transformer Theory .......................... 187
7.3 Two-Winding Autotransformer ............................. 191
7.3.1 Autotransformer Ratings ........................... 196
7.3.2 Per-Unit Impedance ................................ 198
7.4 Step-Voltage Regulators ................................. 202
7.4.1 Single-Phase Step-Voltage Regulators .............. 204
7.4.1.1 Type A Step-Voltage Regulator ............. 204
7.4.1.2 Type B Step-Voltage Regulator ............. 204
7.4.1.3 Generalized Constants ..................... 208
7.4.1.4 The Line Drop Compensator ................. 209
7.4.2 Three-Phase Step-Voltage Regulators ............... 215
7.4.2.1 Wye Connected Regulators .................. 215
7.4.2.2 Closed Delta Connected Regulators ......... 221
7.4.2.3 Open Delta Connected Regulators ........... 224
7.5 Summary ................................................. 234
References .................................................. 235
8 Three-Phase Transformer Models .............................. 241
8.1 Introduction ............................................ 241
8.2 Generalized Matrices .................................... 242
8.3 The Delta-Grounded Wye Step-Down Connection ............. 243
8.3.1 Voltages .......................................... 244
8.3.2 Currents .......................................... 248
8.4 The Ungrounded Wye-Delta Step-Down Connection ........... 254
8.5 The Grounded Wye-Grounded Wye Connection ................ 264
8.6 The Delta-Delta Connection .............................. 266
8.7 Open Wye-Open Delta ..................................... 278
8.4 Thevenin Equivalent Circuit ............................. 284
8.9 Summary ................................................. 287
9 Load Models ................................................. 293
9.1 Wye Connected Loads ..................................... 293
9.1.1 Constant Real and Reactive Power Loads ............ 294
9.1.2 Constant Impedance Loads .......................... 294
9.1.3 Constant Current Loads ............................ 295
9.1.4 Combination Loads ................................. 295
9.2 Delta Connected Loads ................................... 298
9.2.1 Constant Real and Reactive Power Loads ............ 299
9.2.2 Constant Impedance Loads .......................... 299
9.2.3 Constant Current Loads ............................ 300
9.2.4 Combination Loads ................................. 300
9.2.5 Line Currents Serving a Delta Connected Load ...... 300
9.3 Two-Phase and Single-Phase Loads ........................ 300
9.4 Shunt Capacitors ........................................ 300
9.3.1 Wye Connected Capacitor Bank ...................... 301
9.3.2 Delta Connected Capacitor Bank .................... 301
9.5 The Three-Phase Induction Machine ....................... 302
9.5.1 The Induction Machine Model ....................... 303
9.5.2 The Equivalent T Circuit .......................... 308
9.5.3 Computation of Slip ............................... 315
9.5.4 Induction Generator ............................... 318
9.6 Summary ................................................. 319
References .................................................. 319
10 Distribution Feeder Analysis ............................... 323
10.1 Power-Flow Analysis ................................... 323
10.1.1 The Ladder Iterative Technique ................. 324
10.1.1.1 Linear Network ........................ 324
10.1.1.2 Nonlinear Network ..................... 325
10.1.2 General Feeder ................................. 328
10.1.3 The Unbalanced Three-Phase Distribution
Feeder ......................................... 330
10.1.3.1 Series Components ..................... 330
10.1.3.2 Shunt Components ...................... 332
10.1.4 Applying the Ladder Iterative Technique ........ 333
10.1.5 Let Us Put It All Together ..................... 334
10.1.6 Load Allocation ................................ 343
10.1.7 Summary of Power-Flow Studies .................. 343
10.2 Short-Circuit Studies ................................. 344
10.2.1 General Theory ................................. 344
10.2.2 Specific Short Circuits ........................ 348
10.3 Summary ............................................... 353
References ............................................ 353
11 Center-Tapped Transformers and Secondaries ................. 357
11.1 Center-Tapped Single-Phase Transformer Model .......... 358
11.1.1 Matrix Equations ............................... 359
11.1.2 Center-Tapped Transformer Serving Loads
through a Triplex Secondary .................... 365
11.2 Ungrounded Wye-Delta Transformer Bank with
a Center-Tapped Transformer .................... 370
11.2.1 Backward Sweep Equations ....................... 370
11.2.2 Forward Sweep Equations ........................ 375
11.3 Leading Open Wye-Open Delta Transformer Connection .... 382
11.4 Lagging Open Wye-Open Delta Connection ................ 385
11.5 Four-Wire Secondary ................................... 388
11.6 Putting It All Together ............................... 392
11.6.1 Computation of Line-to-Neutral Transformer
Voltages ....................................... 393
11.7 Summary ............................................... 404
Reference ............................................. 404
Appendix A .................................................... 407
Appendix B .................................................... 411
Index ......................................................... 413
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