Index

A

Allocation factor (AF), 26

Approximate line segment model, 155–160

Approximate method of analysis

“K” factors

Kdrop factor, 43–45

Krise factor, 45–47

line impedance, 41–42

lumping loads, geometric configurations

rectangle, 55–60

trapezoid, 65–71

triangle, 60–65

uniformly distributed loads

exact lumped load model, 52–55

power loss, 50–52

voltage drop, 47–50

voltage drop, 39–41

Average demand, 12

B

Backfeed ground fault currents

complete three-phase circuit analysis, 426–435

one downstream transformer bank, 423–426

C

Carson’s equations, see also Modified Carson’s equations

concentric neutral cable, 100

overhead lines, 81–83

parallel distribution lines, 97

triplex secondary, 460

Center-tapped transformers

backward sweep, 490–493

forward sweep, 486–489

four-wire secondary, 493–497

IEEE 4 Node Text Feeder, 497–498

lagging open wye–open delta connection, 485

leading open wye–open delta transformer connection, 484

open wye–delta connections, 503–507

secondary winding, 449–450

serving constant impedance loads, 456

single-phase transformer model diagram, 450

matrix equations, 453–460

transformer impedances, 451–453

triplex secondary, 460–466

three-winding, 449–450

ungrounded wye–delta connection, 497–502

ungrounded wye–delta transformer bank with

basic transformer equations, 466–483

transformer connection diagram, 467

voltage and current unbalances, comparisons of, 508

Closed delta–connected regulators, 226–229

Concentric neutral 15 kV cable, 517

Concentric neutral cable, 100–106

Conductors, 513–515

D

Delta-connected capacitor bank, 335–336

Delta-connected loads

combination loads, 334

constant current loads, 333

constant impedance loads, 333

constant real and reactive power loads, 332

diagram, 332

line currents serving, 334

specified complex powers and voltages, 332

Delta–delta connection, three-phase transformer models, 299–309

Delta–grounded wye step-down connection

currents, 256–258

voltages, 251–256

Delta–grounded wye step-up connection

diagram for, 267

generalized matrices, 267–269

Demand, 11

Demand factor, 12, 19

Distribution feeder analysis

power-flow analysis

application, 381

general feeder, 386–387

ladder iterative technique, 382–386

load allocation, 398–399

loop flow, 399–412

modified ladder iterative technique, 389–390

unbalanced three-phase distribution feeder, 387–389

short-circuit studies

backfeed ground fault currents, 422–435

general theory, 413–417

specific, 417–422

Distribution feeder map, 6–8

Distribution substations

layout, 4

major components, 2–4

one-line diagram, 2

Distribution system

distribution feeder map, 6–8

distribution substations

layout, 4

major components, 2–4

one-line diagram, 2

feeder electrical characteristics, 8–9

line models

approximate line segment model, 155–160

electrically parallel lines, 172–177

equivalent Pi circuits, 163

exact line segment model, 141–149

modified “ladder” iterative technique, 160–162

modified line model, 150–155

neutral and ground currents, computation of, 152–155

parallel lines, general matrices for, 163–177

parallel three-phase lines, 163–166

physically parallel lines, 166–172

three-wire delta line, 150–151

power system components, 1–2

radial feeders, 5–6

Distribution transformer loading

demand factor, 19

diversified demand, 16

diversity factor, 18–19

individual customer load characteristics, 16

load curves, 15–16

load diversity, 20

load duration curve, 17–18

maximum diversified demand, 16–17

maximum noncoincident demand, 18

utilization factor, 20

Diversified demand, 12, 16

Diversity factor, 12, 18–19

E

Electrically parallel lines, 172–177

Equivalent T circuit, three-phase induction machine, 354–360

Exact line segment model, 141–149

F

Feeder

demand curve, 20–21

electrical characteristics, 8–9

load allocation, 20–27

voltage drop calculations, 28–33

G

General voltage drop equation, 121–122

Grounded wye–delta step-down connection

diagram, 283

line-to-line voltages, 285–289

line-to-neutral voltages, 285

turn’s ratio, 284

Grounded wye–grounded wye connection, 296–298

H

High-side and low-side switching, 2

I

IEEE 13 Bus Test Feeder, 405–412

IEEE 4 Node Text Feeder, 497–498

Induction generator, 362–364

Induction machine Thevenin equivalent circuit, 365–368

Induction motor, phase analysis of, 346–352

K

Kdrop factor

application of, 44–45

definition, 43

determination of, 43

voltage drop computation, 44

Krise factor, 45–47

L

Ladder iterative technique, 382–386

linear network, 382–383

modified, 389–390

nonlinear network, 383–386

Lagging open wye–open delta connection, 485

Leading open wye–open delta transformer connection, 484

Line drop compensator

analog circuit of, 209

current transformer turns ratio, 210

purpose of, 210

table of base values, 210–211

Line impedance, 41–42

Load diversity, 12, 20

Load factor, 12

Load models

delta-connected loads

combination loads, 334

complex powers and voltages, 332

constant current loads, 333

constant impedance loads, 333

constant real and reactive power loads, 332

diagram, 332

line currents serving, 334

shunt capacitors

delta-connected capacitor bank, 335–336

wye-connected capacitor bank, 334–335

three-phase induction machine

equivalent T circuit, 354–360

induction generator, 362–364

induction machine model, 337–340

induction machine Thevenin equivalent circuit, 365–368

motor starting current, 354

phase analysis of induction motor, 346–352

slip computation, 361–362

symmetrical component analysis of motor, 340–345

ungrounded wye–delta transformer bank with induction motor, 368–375

voltage and current unbalance, 353–354

two-phase and single-phase loads, 334

wye-connected loads

combination loads, 328–331

constant current loads, 327

constant impedance loads, 327

constant real and reactive power loads, 326

diagram, 326

Loads

definitions, 11–12

distribution transformer loading

demand factor, 19

diversified demand, 16

diversity factor, 18–19

individual customer load characteristics, 16

load curves, 15–16

load diversity, 20

load duration curve, 17–18

maximum diversified demand, 16–17

maximum noncoincident demand, 18

utilization factor, 20

feeder

demand curve, 20–21

load allocation, 20–27

voltage drop calculations, 28–33

individual customer, 13–14

lumping, in geometric configurations

rectangle, 55–60

trapezoid, 65–71

triangle, 60–65

uniformly distributed

exact lumped load model, 52–55

power loss, 50–52

voltage drop, 47–50

Lumping loads, geometric configurations

rectangle, 55–60

trapezoid, 65–71

triangle, 60–65

M

Maximum demand, 11

Maximum diversified demand, 12

Maximum noncoincident demand, 12

Maximum system voltage, 185

Metered feeder maximum demand, 26–27

Metering, distribution substation, 3–4

Modified Carson’s equations

in grounded wye systems, 88

parallel overhead distribution lines, 97

series impedance of overhead lines, 83–85

tape-shielded cables, 106

Modified “ladder” iterative technique, 160–162

Modified line model

neutral and ground currents, computation of, 152–155

three-wire delta line, 150–151

Motor starting current, 354

N

Neutral and ground currents, computation of, 152–155

Node test feeder, 7

Nominal system voltage, 185

Nominal utilization voltage, 185

O

Open delta–connected regulators, 229–241

Open delta–open delta, three-phase transformer models, 309–314

Open wye–open delta, three-phase transformer models, 290–296

Overhead lines

series impedance of

Carson’s equations, 81–83

magnetic fields, 77–78

modified Carson’s equations, 83–85

mutual inductance, 78

parallel overhead distribution lines, 96–99

phase impedance matrix, 86–89

primitive impedance matrix, 86

self-inductance, 78

sequence impedances, 89–96

transposed three-phase lines, 78–79

untransposed distribution lines, 79–81

shunt admittance of

conductors and images, 123

general voltage drop equation, 123

of overhead parallel lines, 127–130

primitive potential coefficient matrix, 124–125

relative permittivity, 124

P

Parallel lines, general matrices for

electrically parallel lines, 172–177

equivalent Pi circuits, 163

parallel three-phase lines, 163–166

physically parallel lines, 166–172

Parallel overhead distribution lines, 96–99

Parallel underground distribution lines, 109–113

Phase analysis of induction motor, three-phase induction machine, 346–352

Phase frame admittance matrix, 349

Phase impedance matrix, for overhead lines, 86–89

Physically parallel lines, 166–172

Power-flow analysis

application, 381

general feeder, 386–387

ladder iterative technique

linear network, 382–383

nonlinear network, 383–386

load allocation, 398–399

loop flow

IEEE 13 Bus Test Feeder, 405–412

single-phase feeder, 399–404

modified ladder iterative technique, 389–390

unbalanced three-phase distribution feeder

one-line diagram, 388

shunt components, 388–389

Power system components, 1–2

Primitive impedance matrix, for overhead lines, 86

Protection, distribution substation, 3

R

Radial feeders, 5–6

S

Sequence admittance, three-phase line, 136

Series impedance

overhead lines

Carson’s equations, 81–83

magnetic fields, 77–78

modified Carson’s equations, 83–85

mutual inductance, 78

parallel overhead distribution lines, 96–99

phase impedance matrix, 86–89

primitive impedance matrix, 86

self-inductance, 78

sequence impedances, 89–96

transposed three-phase lines, 78–79

untransposed distribution lines, 79–81

underground lines

concentric neutral cable, 100–106

parallel underground distribution lines, 109–113

tape-shielded cables, 106–109

three-phase underground cables, 99–100

Service voltage, 185

Shunt admittance

of overhead lines

conductors and images, 123

general voltage drop equation, 123

of overhead parallel lines, 127–130

primitive potential coefficient matrix, 124–125

relative permittivity, 124

of underground lines

concentric neutral cable underground lines, 130–134

of parallel underground lines, 137–138

tape-shielded cable underground lines, 134–136

Shunt capacitors

delta-connected capacitor bank, 335–336

wye-connected capacitor bank, 334–335

Single-phase step-voltage regulators

generalized constants, 208–209

line drop compensator, 209–216

type A step-voltage regulator, 204–205

type B step-voltage regulator, 206–208

Single-phase transformer model

diagram, 450

matrix equations, 453–460

transformer impedances, 451–453

triplex secondary, 460–466

Slip computation, three-phase induction machine, 361–362

Standard voltage ratings, 185–187

Step-voltage regulators

control circuit, 202, 203

settings, 202–204

single-phase

generalized constants, 208–209

line drop compensator, 209–216

type A step-voltage regulator, 204–205

type B step-voltage regulator, 206–208

three-phase

closed delta–connected regulators, 226–229

connections, 216

open delta–connected regulators, 229–241

wye-connected regulators, 216–226

type B, 202, 203

types, 187, 202

System voltage, 185

T

Tape-shielded 15 kV cable, 518

Tape-shielded cables, 106–109

Thevenin equivalent circuit

short-circuit currents, balanced three-phase system, 414–415

three-phase transformer models, 314–317

Three-phase induction machine

equivalent T circuit, 354–360

induction generator, 362–364

induction machine model, 337–340

induction machine Thevenin equivalent circuit, 365–368

motor starting current, 354

phase analysis of induction motor, 346–352

slip computation, 361–362

symmetrical component analysis of motor, 340–345

ungrounded wye–delta transformer bank with induction motor, 368–375

voltage and current unbalance, 353–354

Three-phase line

configuration, 41

line-to-neutral equivalent circuit, 39, 40

phasor diagram, 39, 40

segment model, 142

Three-phase step-voltage regulators

closed delta–connected regulators, 226–229

connections, 216

open delta–connected regulators, 229–241

wye-connected regulators, 216–226

Three-phase transformer models

delta–delta connection, 299–309

delta–grounded wye step-down connection

currents, 256–258

voltages, 251–256

delta–grounded wye step-up connection

diagram for, 267

generalized matrices, 267–269

generalized matrices, 250–251

general three-phase transformer bank, 249–250

grounded wye–delta step-down connection

diagram, 283

line-to-line voltages, 285–289

line-to-neutral voltages, 285

turn’s ratio, 284

grounded wye–grounded wye connection, 296–298

open delta–open delta, 309–314

open wye–open delta, 290–296

Thevenin equivalent circuit, 314–317

ungrounded wye–delta step-down connection

characterization, 269

generalized constant matrix, 273–274

line currents, 272

line-to-line voltages, 271

positive sequence current phasor diagrams, 270

standard diagram, 270

voltage phasor diagrams, 270

ungrounded wye–delta step-up connection

diagram, 281

turn’s ratio, 281

Three-phase underground cables, 99–100

Three-wire delta line, 150–151

Transformer load management program, 25–26

Transposed three-phase lines, 78–79

Two-phase and single-phase loads, 334

Two transformer substation, 4

Two-winding autotransformer

autotransformer ratings, 196–199

per-unit impedance, 199–202

step-down autotransformer, 194–196

step-up autotransformer, 192–193

Two-winding transformer theory

approximate equivalent circuit, 188

exact equivalent circuit, 187

input voltage and current, 189

total “leakage” impedance, 187

Type A step-voltage regulator, 204–205

Type B step-voltage regulator, 206–208

U

Unbalanced three-phase distribution feeder, 387–389

Underground cables

concentric neutral 15 kV cable, 517

tape-shielded 15 kV cable, 518

Underground lines

series impedance of

concentric neutral cable, 100–106

parallel underground distribution lines, 109–113

tape-shielded cables, 106–109

three-phase underground cables, 99–100

shunt admittance of

concentric neutral cable underground lines, 130–134

of parallel underground lines, 137–138

tape-shielded cable underground lines, 134–136

Ungrounded wye–delta step-down connection

characterization, 269

generalized constant matrix, 273–274

line currents, 272

line-to-line voltages, 271

positive sequence current phasor diagrams, 270

standard diagram, 270

voltage phasor diagrams, 270

Ungrounded wye–delta step-up connection

diagram, 281

turn’s ratio, 281

Ungrounded wye–delta transformer bank

basic transformer equations, 466–483

with induction motor, 368–375

transformer connection diagram, 467

Uniformly distributed loads

exact lumped load model, 52–55

power loss, 50–52

voltage drop, 47–50

Untransposed distribution lines, 79–81

Utilization factor, 12, 20

Utilization voltage, 185

V

Voltage drop, 39–41

Voltage regulation, 2–3

standard voltage ratings

ANSI standards, 185–186

step-voltage regulators, 187

step-voltage regulators

control circuit, 202, 203

settings, 202–204

single-phase, 204–216

three-phase, 216–241

types, 202

two-winding autotransformer

autotransformer ratings, 196–199

per-unit impedance, 199–202

step-down autotransformer, 194–196

step-up autotransformer, 192–193

two-winding transformer theory, 187–191

Voltage transformation, 2

Voltage unbalance, 146, 186, 353–354, 508

W

Wye-connected capacitor bank, 334–335

Wye-connected loads

combination loads, 328–331

complex powers and voltage, 326

constant current loads, 327

constant impedance loads, 327

constant real and reactive power loads, 326

diagram, 326

Wye-connected regulators, 216–226