EE544 Distribution 1

1
EE544 Distribution 1

• Distribution Feeder Analysis
• Models and Parameters Part II

2
Distribution Feeder

• Detailed Performance of distribution feeders
• -Power flow loading, voltage drop, loss
• -Short Circuit
• - Motor Start

Single phase lateral
Three phase primary
Regulator Or LTC
M
3
Distribution Feeder Impedance

• Electric Circuit Models for Distribution Lines
• Basic definition of Inductance/mutual inductance
• General flux linkage equation
• Single phase Line X 0.1213 ln(D/r) ohm/mi_at_ 60
  Hz
• Carsons Equations
• Using Carsons equation to define impedance and
  impedance matrices
• Mechanism of impedance calculation

4
Carsons Equations
1. What do they model? a. The earth modifies the
magnetic field intensity from conductor b. In
single-phase or unbalanced three phase case some
current returns along ground
5
Carsons Equations
Carson gives formulas for Zik and Zkk
6
Carsons Equations
7
Carsons Equations
8
Carsons Equations-Impedance Matrix
Single-phase line
Model Equations
V1g-V1'g Z11 I1 Z12 I2 V2g-V2'g Z21 I1
Z22 I2 I1I2Ig0 V2g0 and V2'g0
Carsons Equations give the Phase Impedance
matrix ZP
9
Carsons Equations-Impedance Matrix
GS Linnet 336400, 26/7, 60 Hz, 50 deg. C
r0.306 ohm/mi 0.058 ohm/1000 VB Linnet
336400, 26/7, 60 Hz, 50 deg. C r0.300 ohm/mi
0.057 ohm/1000 K
10
Carsons EquationsCircuit Impedance
Single-phase line
Model Impedance
The circuit impedance Z Voltage Drop/
Current Z (V1g-V1g) (V2g-V2g)/I1
V1g-V1'g Z11 I1 Z12 I2 V2g0 and
V2'g0 V2g-V2'g Z21 I1 Z22 I2 So I2 -Z21
I1/Z22 and V1g-V1'g (Z11-Z12 Z21/Z22)I1 Line
impedance Z Z11-(Z12 Z21/Z22)
Ignoring earth and grounding
Z0.612 j 1.291
11
Carsons Equations
Single-phase line
Model Impedance
The circuit impedance Z Voltage Drop/
Current Z (V1g-V1g) (V2g-V2g)/I1
V1g-V1'g Z11 I1 Z12 I2 I1-I2 V2g-V2'g Z21
I1 Z22 I2 V1g-V1'g (V2g-V2'g) (Z11 2 Z12
Z22) I1 Line impedance Z Z11 2 Z12 Z22
Ignoring earth and grounding
Z0.612 j 1.291
Z0.612 j 1.291
12
Circuit impedance and Neutral Grounding
Earth provides parallel return All current
returns on neutral Path for current Z0.401 j
1.413 Z0.612 j 1.291 ohm/mi What path
does Earth current Is neutral voltage
zero? take? One form of stray currents One
form of stray voltage
I1
Line model
Z L
V1n
V1n
13
Earth/Dirt/Stray Currents/Voltages
If I1 100 Find V2g
The basic equations are V1g-V1'g Z11 I1 Z12
I2 I1-I2 V2g 0 V2g-V2'g Z21 I1 Z22 I2
We expect neutral Voltages to be 0!
14
Earth/Dirt/Stray Currents/Voltages
If I1 100 I2 and Ig
The basic equations are V1g-V1'g Z11 I1 Z12
I2 V2g0 V2g 0 V2g-V2'g Z21 I1 Z22 I2 0
Z21 I1 Z22 I2 gt I2 -(Z21/Z22)I1 Ig-I1-I2
Leads to stray current
15
Carsons Equations
dik
De
Fictitious conductor at distance De Models earth
return
16
Carsons Equations
Can model series impedance Matrix for arbitrary
configuration - No transposition assumption -
Currents can be unbalanced
EHV line 3 phase conductors or bundle 2 shield
wires
17
Carsons Equations 3 phase line
2.5
7
3
4
Linnet 336400 26/7 ACSR
18
Carsons Equations 3 phase line
19
Carsons Equations 3 phase line-Kron Reduction
Voltage drop model
20
Carsons Equations 3 phase line- phase
impedance matrix
21
Carsons Equations-phase domain model
The receiving end voltage is balanced positive
sequence, 4160V And current is balanced positive
sequence 100A Find sending end voltage
Note unbalance
22
Carsons Equations Transposed Line
Transposed Lines phases are rotated through each
position on the pole The terms of the Zp matrix
would each average out Diagonal terms
(ZaaZbbZcc)/3 Off Diagonal terms
(ZabZbcZca)/3
23
Carsons Equations Transposed Line
Transposed Lines Positive(negative) Sequence
Impedance
VDa Zaa Ia Zab Ib Zac Ic IaIbIc 0 and
ZacZab VDa (Zaa-Zab) Ia gt Z1 Zaa- Zab
Z1 0.306 j 0.609 ohm/mi
Balanced Conditions can be anlyzed By single
phase model
Ia
Z1
Van
Van
VDa Ia Z1 69.7/64deg V
24
Carsons Equations Transposed LineNeutral
Ignored
X1 (1/3) 0.1213 ln(1/gmra) 0.1213
ln(1/gmrb) 0.1213 ln(1/gmrc) 37.934
- (1/3) 0.1213 ln(1/Dab) 0.1213 ln(1/Dbc)
0.1213 ln(1/Dca) 37.934
X1 (1/3) 0.1213 ln(3v(Dab Dbc Dca))/gmra)
The Power II Formula!!!!
25
Inductance Calculation- Transposed Line
Conventional Method
GMR r

a
Three phase equilateral line Balanced
Positive Sequence( or Negative Sequence)
Current IaIbIc0 Total Flux linkage to Phase
a ? 2 10 7 Ia ln(1/r) Ib ln(1/D)Ic
ln(1/D 2 10 7 Ia ln(1/r)-Ialn(1/D)
because IbIc-Ia   L1 2 10 7 ln(D/r)
H/m/phase Applies for positive or negative
sequence Equilateral
D
D
D
c
b
L1
a
n
26
Inductance Calculation-Non Equilateral
Line is transposed. Over the three Sections,
each conductor moves through the left,center
and right positions. The flux linkages ?a ?b
and ?c (and induced voltages) are unequal in
each section. Flux linkages to Phase a are as
follows ?a 2 10 7 Ia ln(1/r) Ib
ln(1/D12)Ic ln(1/D13 Section 1 ?a 2 10 7
Ia ln(1/r) Ib ln(1/D23)Ic ln(1/D12 Section
2 ?a 2 10 7 Ia ln(1/r) Ib ln(1/D13)Ic
ln(1/D23 Section 3 With balanced currents, the
average linkages and induced voltages become
balanced three-phase quantities ?a avg 2 10
7 Ia ln(1/r)(IbIc)(1/3) ln(1/D121/D131/D2
3) 2 10 7 Ia ln (3v (D12 D13
D23) /r

A B C
1 2 3
27
Inductance Calculation-Non Equilateral Transposed
LinesGeometric Mean Distance
Ia Ib Ic
So we get L1 2 10 7 ln 3v (D12 D13 D23)
/ r H/m/phase L1 2 10 7 ln Deq
/ r H/m/phase We define the Phase
Geometric Mean Distance (GMD or DSL), for
Inductance Calculations as Deq 3v (D12 D13
D23)

A B C
C A B
B C A
D12 D23
D13
28
Inductance Calculation-Non Equilateral Transposed
LinesGeometric Mean Distance
Ia Ib Ic
GMD between a pair of things M v
(Product of all M possible distances between the
two things) Distance between phase a and other
phases 6v (D12 D13) ( D23 D12) (D13
D23) AB AC AB AC AB AC
Section 1 Section 2 Section
2 Phase-Phase Distance Phase
A------------------------------ Phase
B------------------------------ Phase
C------------------------------ 18v (D12 D13) (
D23 D12) (D13 D23) (D12 D23) ( D23 D13) (D13 D12)
(D13 D23) ( D12 D13) (D12 D23)

A B C
C A B
B C A
D12 D23
D13
29
Inductance Calculation-Bundled ConductorsGeometr
ic Mean Radius
GMR r d

GMR v (rd)
A A B B C
C

Ib
Ia/2
Ic
Ia
At EHV, each phase consists of multiple
conductors. This reduces surface electric fields
by charge division, and thus, Corona The concept
of GMD appears in a different form here. The
contribution of phase A current to phase A flux
linkages is ?aa 2 10 7 (Ia/2) ln(1/r)
(Ia/2) ln(1/d)) 2 10 7 Ia ln1/v
(rd) Phase A thus appears to have a larger
GMR of Dsl v (rd) Rewriting this a 4v
(rd)(r d) we see this is the GMD between
conductors in phase A we include the distance
from a conductor to itself, i.e. the
GMR Bundled conductors are modeled by an
equivalent GMR
30
Inductance Calculation-Summary

• Positive sequence Inductance for Transposed Line

L1 2 10 7 ln ( Deq /DsL) H/m Deq
Geometric mean distance between phases DsL
Geometric mean radius of phases Conductor
distance to itself conductor GMR We usually
use inductive reactance in ohms/mile X 0.1213
ln ( Deq /DsL) ohm/mi
31
Positive Sequence Inductance Calculation-Three
phase line
4.5
2.5
336,400 26/7 ACSR
R0.306 ohm/mi at 60 Hx, 50 deg.C Deq
3v((2.5 4.5 7) 4.3 GMR 0.0244 X1 0.1213
ln(Deq/GMR) 0.306 j 0.627 ohm Z1 0.306 j
0.627 ohm /mi
32
Carsons Equations- Impedance Calculation

• General form accounts for
• Modification of magnetic field by earth current
• Return current in earth
• Simplified or full form gives primitive impedance
  matrix
• Can reduce to phase impedance matrix
• Allows detailed analysis
• Reduces to standard positive sequence formula for
  transposed balanced case