EE544 Distribution 1

1
EE544 Distribution 1

• Distribution Feeder Analysis
• Models and Parameters

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
Conductors

• ACSR

• XLPE

Jacket
Shield
Cross-linked Poly(XLPE)
Steel Core
Three,counter spiraled Aluminum strand layers
Aluminum
Semicon
4
Distribution FeederIncreased use of cable,
particularly for lateralsDifferent geometric
arrangement than transmission

• ACSR

• XLPE

Jacket
Shield
Cross-linked Poly(XLPE)
Steel Core
Three,counter spiraled Aluminum strand layers
Aluminum
Semicon
5
Distribution FeederIncreased use of cable,
particularly for lateralsDifferent geometric
arrangement than transmission
A B N C
Sag
D25
D25
6
Distribution FeederIncreased use of cable,
particularly for lateralsDifferent geometric
arrangement than transmission
Layout affects - impedance -AMPACITY
7
Transmission Lines Electrical Models

• Transmission Lines are characterized by their
  Resistance, Inductance and Capacitance
• Distributed, coupled, RLC Circuit
• Frequency, and possibly voltage dependent,
  parameters

Indicates coupling
GS 4.7 and 4.11
8
Transmission Lines Electrical Models

• General electrical model goes back to the wave
  equation
• L and C and R are inductance, capacitance and
  resistance matrices respectively
• V is a vector of phase-ground voltages
• Such models are used in transient studies
  involving lightning and switching surge
  propagation

GS Ch.12
9
Transmission Lines Electrical Models

• In the balanced positive or negative sequence
  case, when lines are transposed, and for a
  specific(low) frequency, a per-phase distributed
  parameter model suffices
• And can be reduced to an equivalent PI model
• This model is used in steady state analysis

Z
Y/2
Y/2
Y/2
Y/2
GS Ch.5
10
Part 2 Transmission Line ParametersResistance

• The dc resistance of a solid, uniform conductor
  is given by
• Rdc ? l/A
• ? resistivity
• l length
• A area
• Resistance is affected by
• Temperature RT2 RT1(TT2)/(TT1)
• T material temperature constant
• Frequency Skin Effect-Current density increases
  towards the surface
• Structure of composite conductor
• Steel core
• Spiraling
• Resistance is obtained from Manufacturers Tables

11
Conductor Tables (P.647)

• Contain physical information
• Conductor name Falcon
• Area 1 590 000 circular mil
• Resistance 0.0684 ohm/mi
• 50 deg C
• 75 current
• GMR at 60 Hz 0.052
• Diameter 1.545 in
• Ampacity 1380 A

12
Inductance Calculation-- Definition of Inductance
Area S

• Current(i) in coil produces a magnetic field ( B
  or H)
• In free space the field strength is proportional
  to current i
• Resulting flux linkage(?) to the coil ,
• is also proportional to current
• ? L i
• The voltage induced is given by Faradays
  law
• e d? /dt (ignoring negative sign)
• Equivalently e Ldi/dt
• Self inductance L ? /i can be defined as
  the flux linkage to a circuit per ampere of
  current in the circuit

- e
i
B
i
L
- e
13
Inductance CalculationBasic Approach

• Assume current
• Calculate magnetic Field
• Calculate Flux Linkage
• Take ratio of flux linkage to current
• Can develop a general formula for flux linkage
• From this formula we derive an inductance formula

14
Inductance Calculation-Magnetic Field of a
Infinitely Long Straight Conductor

• Magnetic field intensity has circular symmetry
• Amperes law for magnetic field intensity H
• H 2p x i gt H i/ 2p x A/m
• In free space Flux density
• B µo i/ 2p x T ( 1 Tesla10000 Gauss)
• µo 4p 10 7
• More generally say conductor lays in z-direction,
  B as a vector has components Bx and By.
• Bx -(µo I / 2p) (y-y1) /(x-x1)2(y-y1) 2
•   By (µo I / 2p) (x-x1) /(x-x1)2(y-y1) 2
• These equation can be used to calculate magnetic
  fields from lines

i
H(x)
H(x)
x
x
o
H(x)
y
(x1,y1)
i
(x,y)
By
Bx
x
15
Single phase line, perfect conductors
I

• Field due to current in conductor 1, at distance
  x from conductor 1
• B1(x) µo I /2 p x out of page
• Field due to current in conductor 1, at distance
  x from conductor 1
• B2(x) µo I/ 2 p (d-x) out of page
• Flux linkage, per meter, due to current in
  conductor 1
• WbT/m
• Flux linkage, per meter, due to current in
  conductor 2
• WbT/m

B1,B2
I
x
1m
1 D 2
Radius r Resistivity 0
16
Inductance CalculationSingle phase line, perfect
conductors
I
B1,B2
I
x
Circuit Self Inductance L ?/I (µo / p)
ln (D/r) H/m Per conductor Self Inductance
L1 (µo / 2p) ln (D/r) H/m
1m
1 D 2
Radius r Resistivity 0
17
Inductance Calculation Digression Conductor
Geometric Mean radius

• An ac magnetic field also exists within
  alternating current carrying conductors with
  finite conductivity
• Inductance is slightly larger
• This is effect was ignored in previous formula
• For a solid conductor with uniform current
  density we can show
• Conductor of radius r meters with internal
  magnetic field
• is equivalent to
• Ideal Conductor of radius r r e (-1/4) with no
  internal magnetic field
• r is called the Geometric Mean Radius (GMR)
• For stranded and more complicated conductors GMR
  is obtained from tables
• From this point forward we will replace
  conductor radius with GMR for Inductance
  Calculations
• GMR is obtained from conductor tables

B
r
r
18
Inductance Calculation-General flux linkage
equation

• Given m parallel conductors
• Currents add up to zero
• Whats the flux linking an area, 1 meter long,
  bounded at one edge by conductor k and at the
  other by infinity?
• Result can be used to quantify flux linkage to
  any circuit in an arbitrary transmission line
  consisting of long parallel conductors.

1 k . .m M
Dkm
?
1 m
8
I1
Ik
Im
19
Inductance Calculation-General flux linkage
equation

• Flux linkage to conductor k in region from
  conductor to infinity
• Dkm center-center distance from conductor k
  to m
• Dkk distance from a conductor to itself
• conductor GMR

i1
ik
iM
im
20
Inductance Calculation-General flux linkage
equation

• Kersting interprets this as Self and Mutual
  inductance
• Lkk µo ln(1/Dkk) /2p
• Self inductance(H/m) conductor k
• Lkm µo ln(1/Dkm) /2p
• Mutual inductance (H/m) conductor m to k
• This corresponds to writing
• Vkk Lkk dik/dt Lkm dim/dt

i1
ik
iM
im
21
Inductance Calculation-Putting it all together

I
Single phase line( notice we use r) Total Flux
linkage to conductor 1 ? 2 10 7 I ln(1/r)
(-I) ln(1/D) 2 10 7 I ln(D/r)   L1
2 10 7 ln(D/r) H/m/conductor L 4
10 7 ln(D/r) H/m
B1,B2
I
x
1 D 2
L1
Gmr r
1
1
2
2
L1
GS 4.5
22
Inductance Calculation-Single Phase Line Example
23
Inductance Calculation-Putting it all together
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
24
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
25
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
26
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
27
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
28
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
29
Inductance Calculation-Three phase line Example 1
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
30
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
31
Carsons Equations
Carson gives formulas for Zik and Zkk
32
Carsons Equations
33
Carsons Equations
34
Carsons Equations
35
Carsons Equations
Can model series impedance Matrix for arbitrary
configuration - No transposition assumption -
Currents can be unbalanced
36
Inductance Calculation- Three phase line Example
2

7
32 32
954000 ACSR Falcon 20 Square
37
Inductance Calculation- Three phase line Example 2

7
32 32
954000 ACSR Falcon 20 Square
v2 d
d
d
GMR
38
Inductance Calculation- Three phase line Example 2

7
32 32
D12
D23
D13
7
32 32
39
Inductance Calculation- Three phase line Example 2

7
32 32
40
Capacitance Calculation- Basic Ideas

A voltage is applied to a two-conductor Line Curr
ent i will flow to establish a surface charge on
the conductors The charge results in an electric
field E that balances the applied voltage v, such
that If the voltage v is a dc the line charges
and current eventually goes to zero With ac
voltage a charging current, i, is
established idq/dt
i
1
q


E
v
-
-q
2
GS 4.8
41
Capacitance Calculation- Definition of Capacitance

In free space the electric field E is
proportional to charge q Thus voltage v is also
proportional to charge q q C v and i C
dv/dt Capacitance C q/v Capacitance is
charge acquired by line per unit voltage
i
1
q


E
v
-
-q
2
i

C
v
42
Capacitance Calculation

• Assume a charge distribution (q)
• Calculate Electric Field (E)
• Calculate potential difference ( voltage v)
• 4. Cq/v

i
1

q


E
v
-
-q
2
i

C
v
43
Capacitance Calculation- Electric Field
Infinitely Long straight conductor with charge q
Coulomb/meter Charge is on surface. Electric
field within is zero Electric Field has Radial
Symmetry Field at distance x meters from
conductor is radially directed and has
magnitude(Gauss Law) E q/ 2p eo x V/m eo
8.85 10 12 for air For multiple conductors
add vector contributions to E
E


GS 4.9
44
Capacitance Calculation- Potential Equation
The potential difference between two points P1
and P2

V12 (1/2p eo) q ln(D2/D1) V General
potential difference equation for an array of
M charged conductors q1q2q3qM0
P2

D2
q C/m
D1
P1
q1
qi
q2
Dki
qk
qM
GS 4.10
45
Capacitance Calculation- Single Phase Line

V (1/2p eo) q ln(D/r)- q ln(D/r) Volts
Cq/V p eo / ln(D/r) F/m C1q/V 2p eo /
ln(D/r) F/m/Conductor
r
q

D
V
-
-q
C
C12C
C12C
46
Capacitance Calculation- Three Phase Line
Equilateral vab (1/2p eo) qa ln(D/r)qb
ln(D/r)qcln(D/D) Volts Balanced positive
sequence qaqbqc0 Correction
9/22/02 In phasor terms Vab v 3Van/30o Qa-Qbv
3Qa/30o Van (1/2p eo) Qa ln(D/r) Vbc, Vca
have similar equation and equal
magnitude Per-phase ( phase-ground) capacitance
(positive/negative sequence) Can C1 2p eo/
ln(D/r) F/m

r
qa
qa

qc
D
V
-
qb
a
C
n
47
Capacitance Calculation- Three Phase Line
Applying the principles in the previous
derivation To general, 3-phase, transposed,
configurations, can show Distance D is replaced
by GMD Radius r is replaced by equivalent GMR,
Dsc The c subscript on Ds is meant to remind
us that it is Conductor radius r, and not GMR
r, that is used in capacitance calculations

r
qa
qa

qc
D
V
-
qb
a
Can
n
48
Capacitance Calculation-Summary

C1 2p eo/ ln(Deq/DSc) F/m Deq
Geometric mean distance between phases DsC
Geometric mean radius of phases for
capacitance Conductor distance to itself
conductor radius We usually use capacitive
susceptance in ohms/mile Bc 33.745 /ln (
Deq /DsC) micro-mho/mi
49
Capacitance Calculation- Final Example

Step 1 Bundle GMR Dsl
7
32 32
954000 ACSR Falcon 20 Square
vd
d
d
GMR
50
Capacitance Calculation- Final Example

7
32 32
D12
D23
D13
7
32 32
51
Capacitance Calculation- Final Example

Step 3 Capacitance/Susceptance
7
32 32
52
Part 3 Steady State Performance
Analysis-Introduction

• The three-phase 500 kV line in our examples is
  200 miles long.
• It is required to supply 600MW at a power factor
  of 0.95 lagging with rated voltage at the
  receiving end.
• What is the voltage at the sending end?
• Calculate losses.
• 2. With sending end voltage as above the load is
  removed.
• What is the voltage at the receiving end?
• 3. Is performance acceptable?
• In the following we will answer these questions
  and introduce performance
• metrics

53
Steady State Performance Analysis-Models

• Possible models ( per-phase, positive sequence)
• Short line (lt50 Mi)
• Ignores capacitance
• Vs Sending end ( phasor) voltage
• Vr Receiving end ( phasor) voltage
• Is Sending end ( phasor) current
• Ir Receiving end ( phasor) current
• z line impedance in ohm/length
• y line shunt (capacitive) admittance in
  mho/length
• l line length

Ir
Is
Vs -
Zzl
Vr -
GS 5.1
54
Steady State Performance Analysis-Models

• Possible models ( per-phase, positive sequence)
• Medium line (lt150 Mi)
• Lumps half the capacitance
• at each end
• Long line
• Z and Y based on
• Solution of wave equation
• (Next Lecture)

Ir
Is
Zzl
Vs -
Vr -
Y2yl/2
Y2yl/2
Z
Vs -
Vr -
Y/2
Y/2
55
Steady State Performance Analysis-Models-A Comment

• We use computer programs for analysis
• Good programs support comprehensive models
• Dont use approximations if you dont have to!
• Resulting database is useful in its own right as
  electronic documentation!
• Simplified models are useful for
  back-of-the-envelope calculation and in
• understanding phenomena

56
Steady State Performance Analysis-Models

• Lets work the problem posed using the medium
  line model
• Problem 1
• Given
• Receiving end Load Sr 600 MW _at_ 0.95 pf lag
• Receiving end voltage Vr 500 kV
• Find Sending end voltage

Is
Ir
Zzl
Vs -
Vr -
Sr
Y2yl/2
Y2yl/2
57
Steady State Performance Analysis-Models
I

• Our model is a per phase model

58
Steady State Performance Analysis-Full Load
I

59
Steady State Performance Analysis-Full Load
I

IC2
Corrected
60
Steady State Performance Analysis-Full Load
I

IC2
Corrected
61
Steady State Performance Analysis-No Load
I
Is
Ir
Zzl
Vs -
Vr -

IC2
Y2yl/2
Y2yl/2
62
Steady State Performance Analysis-Models

I
63
Steady State Performance Analysis-Comments
We can expect voltage to vary about 11, as load
varies from no load to full load. We will
formally define this as "Voltage Regulation" and
like to keep it around 10. Formally, voltage
regulation is defined as follows VR 100
(Vnl-Vfl)/Vfl where Vfl Magnitude of
receiving end voltage at full- load(rated
voltage) Vnl Magnitude of receiving end
voltage at no-load, with sending end voltage
set as needed to obtained rated receiving end
voltage at full load In our case Vfl 500
kV Vnl 560.5 kV VR 12.1 This figure can
be improved by adding a shunt reactor (
compensation)

I
64
Steady State Performance Analysis-Comments

I
65
Summary

• Reviewed basic ideas about transmission lines
• Discussed basic modeling approaches
• Reviewed and illustrated parameter calculation
• For positive or negative sequence
• Xl 0.1213 ln(Deq/Ds) ohm/mi.
• Bc33.745 /ln(Deq/Dsc) micro mho/mi.
• R conductor resistance / no of conductors/phase
• Deq GMD between phases
• Ds,Dsc GMR for inductance and capacitance,
  respectively

66
Summary

• Reviewed performance analysis with simple, medium
  PI model.
• Voltage drop with inductive load
• No Load voltage rise due to capacitance
• Line angle
• Losses
• Ampacity

67
Some Closing Comments- Inductance-Carsons Formula

• Weve talked only about positive/negative
  sequence models
• More general models account for
  neutrals/shield/earth effects and are necessary
  in transient/unbalanced cases.

Carsons Equations provide formulas for R and L
to account for Small effect of earth on magnetic
field Effect of earth return current v r I L
di/dt V voltage vector I current vector
Ia

Va


Vb
-
-
Earth
Ia
68
Some Closing Comments-Capacitance and Earth

• Earth and other conductors affect capacitance

The method of images is used to account for
the Effect of equi-potential earth
surface Advanced models also account for
grounded shield wires And neutral(Method of
potential coefficients)
69
Some Closing Comments

• Advanced models can be systematically reduced to
  the balanced steady-state model.
• We did not discuss cable impedance calculations
• the basic principles can be used.
• Manufacturers tables

Gmr core
Permittivity ?
Gmr neutral
Deq
70
Some Closing Comments-Magnetic and Electric Fields

• Magnetic/Electric Field derivations form the
  basis of line electromagnetic field assessment
• Conductor Surface fields, corona, RFI
• Ground level fields, EMF guidelines
• Induction into nearby objects, safety

71
Transmission Lines 2

• Outline Transmission Lines 2
• Distributed Parameter Model
• Analysis using ABCD parameters
• Line loadability
• Introduction to Compensation

72
Transmission Lines 1 Homework
Slides and Answers www.ece.nmsu.edu\sranade
under classes

• 3 rd Ed.
• 4.19,4.24
• 5.7 Skip part a, just use basic circuit analysis
• 2nd Ed
• For cable shown, find GMR,GMD and write formula
  for Inductance

73
Transmission Lines 1 Homework
Slides and Answers www.ece.nmsu.edu\sranade
under classes

• 3 rd Ed.
• 4.19,4.24
• 5.7 Skip part a, just use basic circuit analysis
• For cable shown, find GMR,GMD and write formula
  for Inductance