NATIONAL ELECTRIFICATION ADMINISTRATION U. P. NATIONAL ENGINEERING CENTER Certificate in Power System Modeling and Analysis Competency Training and Certification Program in Electric Power Distribution System Engineering U. P. NATIONAL ENGINEERING CENTER U. P. NATIONAL ENGINEERING CENTER Competency Training and Certification Program in Electric Power Distribution System Engineering Short Circuit Analysis Training Course in
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NATIONAL ELECTRIFICATION ADMINISTRATIONU. P. NATIONAL ENGINEERING CENTER
Certificate in
Power System Modeling and Analysis
Competency Training and Certification Program in Electric Power Distribution System Engineering
U. P. NATIONAL ENGINEERING CENTERU. P. NATIONAL ENGINEERING CENTER
Competency Training and Certification Program in Electric Power Distribution System Engineering
Short Circuit Analysis
Training Course in
2
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Course Outline
1. Analysis of Faulted Power System by Symmetrical Components
2. Bus Impedance Matrix Method
3. Short Circuit Analysis of Unbalanced Distribution Feeders
4. Short Circuit Studies
3
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
� Sources of Short Circuit Currents
� Types of Fault
� The Fault Point
� Three-Phase Fault
� Single-Line-to-Ground Fault
� Line-to-Line Fault
� Double-Line-to-Ground Fault
Analysis of Faulted Power System by Symmetrical Components
4
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
FaultMV
LV
UtilityG
Fault Current Contributors
Sources of Short Circuit Currents
5
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Shunt Fault: Unintentional Connection between phases or between phase and ground.
1. Single Line-to-Ground Fault
2. Line-to-Line Fault3. Double Line-to-Ground Fault 4. Three Phase Fault
Series Fault: Unintentional Opening of phase conductors
Simultaneous Fault
Types of Fault
6
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Shunt Faults
Line-to-Line
Double Line-to-Ground Single Line-to-Ground
Three Phase
Types of Fault
7
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
The system is assumed to be balanced, with regards to impedances, except at one point called the fault point.
Fault Currents
ab
c
aVr
bVr
cVr
Ground
aIr
bIr
cIrLine-to-
ground voltages
F
The Fault Point
8
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Since we mentioned that various power system components behave/respond differently to the flow of the currents’ sequence components, it follows that the there will be a unique power system model for each of the sequence component. These are called the sequence networks.
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
The Thevenin equivalent of the power system at the fault point is called the sequence network.
F1
N1
thf VVr
=
Z11aIr
1aVr
+
-
+
-
F2
N2
Z2
2aIr
2aVr
+
-
0aVr
F0
N0
Z0
0aIr
+
-
11ath1a ZIVVrrr
−= 22a2a ZIVrr
−= 00a0a ZIVrr
−=
Positive Sequence
Negative Sequence
Zero Sequence
The Fault Point
10
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
� On a balanced three phase system, the same magnitude of fault currents will flow in each phase of the network if a three phase fault occurs.
� Since faults currents are balanced, the faulted system can, therefore, be analyzed using the single phase representation.
Three Phase Fault
11
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Three-Phase Fault
Note: The system is still balanced. Currents and voltages are positive sequence only. The ground current is zero.gI
r
ab
c
aVr
bVr
cVr
Ground
bIr
cIr
gZ
aIr
fZ fZ
gIr
fZ
12
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
F2
N2
Z2
2aIr
2aVr+
-
F1
fV
Z1 1aIr
1aVr+
-
+
-
N1 N0
F0
Z0
0aIr
0aVr+
-
Sequence currents
f
fa ZZ
VI
+=
11
r020 == aa II
rr
Zf
13
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Three Phase Fault Currents:
f
faaaa ZZ
VIIII
+=++=
1210
f
faaab ZZ
VaaIIaII
+=++=
1
2
212
0
f
faaac ZZ
aVIaaIII
+=++=
12
210
14
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Example:
Determine the fault current for a three phase bolted fault in each bus for the 4 bus system below.
G
1
432
Line 1Line 3
Line 5 Line 4
Line 2
4-bus system
LINE FB TB Z(p.u.)
Line1 1 4 j0.2
Line2 1 3 j0.4
Line3 1 2 j0.3
Line4 3 4 j0.5
Line5 2 3 j0.6
The generator is rated 100 MVA, 6.9 kV and has a subtransient reactance of 10%. Base Values: 100 MVA, 6.9 kV
15
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Solution:
Draw the impedance diagram
E 1.0
0.1
0.20.3
0.6 0.5
0.4
43
2
1
16
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
E 1.0
0.1
0.20.3
0.6 0.5
0.4
43
2
1If
-
+
a) Fault @ Bus 4= +
= +
=
=+
=+
=
= +
= +
=
a1 2 23
b 13
13
c34
R edu ce th e n e tw o rk
X
0 .3 0 .6 0 .9
X
(0 .9 )(0 .4 )
0 .9 0 .4 0 .2 76 9 2 3
X
0 .2 7 6 92 3 0 .5 0 .7 7 6 92 3
a
a
b
X X
X XX X
X X
17
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
cd 14
c14
X XX
X X
(0.776923) (0.2)
0.776923 0.2 0.159055
=+
=+
=
dequiv genX X X
0.1 0.159055 0.259055
= +
= +
=
If
-
+E 1.0
0.25905
f
1.0I
0.259055 3.860184 p.u.
=∂
=
base
f
100 1000I 8367.64 A
3(6.9)I 3.860184 x 8367.64
= 32,300.63 A
x= =
=
18
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
b) Fault @ Bus 3
E 1.0
0.1
0.20.3
0.6 0.5
0.4
432
1If
-
+
a23 12X X X
0.3 0.6 0.9
= +
= +
=
b14 34X X X
0.2 0.5 0.7
= +
= +
=
a bequiv 13X (X ||X ) ||X
0.198425
=
=
19
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
gen equivX X X
= 0.1 0.198425= 0.298425
= +
+
f
1.0I
0.298425= 3.350923 p.u.
=
If
-
+E 1.0
0.298425
20
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
c) Fault @ Bus 2
E 1.0
0.1
0.20.3
0.6 0.5
0.4
432
1If
-
+
a14 34X X X
0.2 0.50.7
= +
= +
=
ab 13
a13
X XX
X X
(0.7)( 0.4)
0.7 0.40.254545
=+
=+
=
21
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
= +
= +
=
c b23X X X
0.254545 0.6 0.854545
=+
=+
=
cd 12
c12
X XX
X X
(0.854545)( 0.3)
0.854545 0.3 0.222047
= +
=
dgenX X X
0.322047
=
=
f
1.0I
0.322047 3.095525 p.u.
If
-
+E 1.0
0.322047
22
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
d) Fault @ Bus 1
E 1.0
0.1
0.20.3
0.6 0.5
0.4
432
1
If
-
+
=
=
genX X
0.1
=
=
f
1.0I
0.1 10.0 p.u.
23
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Example: A three-phase fault occurs at point F. Assuming zero fault impedance, find the fault currents at fault point F. Determine the phase currents in the line and the generator. Assume Eg= 1.0 p.u.
Note: All reactances are in per-unit of a common MVA base.
24
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Positive-Sequence Network:
+
-
j0.6 1AIr
1.0
N1
F1+
-
1AVr
j0.05
j0.4+
gEr
-
Openj0.05
j0.15 1AIr
L1AIr
g1aIr
F1
N1
25
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
Zf
The sequence fault currents
=
=
=+
=
0
2
11
a
a
f
fa
I
I
ZZ
VI
r
r
r
+
-
j0.6 1AIr
1.0
N1
F1+
-
1AVr
The phase fault currents
=
=
=
c
b
a
I
I
I
26
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Single Line-to-Ground FaultAssuming the fault is in phase a,
Boundary Conditions: (1) afa IZVrr
=
ab
c
aVr
bVr
cVr
Ground
aIr bI
rcIr
fZ
(2) 0== cb IIrr
27
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
abcIAIrr
1012
−=
Transformation: From (2), we get
aa
aa2
2
1
1
111
3
1
2
1
0
a
a
a
I
I
I
r
r
r
=
0
0aIr
31
=
a
a
a
I
I
I
r
r
r
which means a31
2a1a0a IIIIrrrr
===
From (1), we get
)( 210210 aaafaaa IIIZVVVrrrrrr
++=++
or0210 3 afaaa IZVVV
rrrr=++
28
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
F1
N1
fVr
Z1 1aIr
1aVr+
-
+
-
F2
N2
Z2
2aIr
2aVr+
-
0aVr
F0
N0
Z0
0aIr+
-
3Zf
The sequence fault currents
f
faaa ZZZZ
VIII
3210210
+++===
rrr
29
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Single-Line-to-Ground Phase Fault Currents:
f
faaaa ZZZZ
VIIII
3
3
021210
+++=++=
0=bI
0=cI
30
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Example: A single line-to-ground fault occurs at point F. Assuming zero fault impedance, find the phase currents in the line and the generator. Assume Eg = 1.0 p.u.
Note: All reactances are in per-unit of a common MVA base.
42
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
N1 N2
F1
J0.6 1AIr
+
-
F2
J0.62AI
r
1.0
N0
F0
J0.044
0AIr
43
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Fault Currents:
44
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Phase Fault Currents:
45
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
ab
c
aVr
bVr
cVr
Ground
bIr
cIr
gZ
aIr
fZ fZ
cb IIrr
+
Double-Line-to-Ground FaultAssuming the fault is in phases b and c,
Boundary Conditions: 0=aIr
cgbgfb IZIZZVrrr
++= )(
bgcgfc IZIZZVrrr
++= )(
(1)
(2)
(3)
46
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Transformation: From (1), we get
2100 aaaa IIIIrrrr
++==
From
212
0 aaab VaVaVVrrrr
++=
212
0 aaab IaIaIIrrrr
++=
we get2
210 aaac VaVaVV
rrrr++=
22
12 )()( aacb VaaVaaVV
rrrr−+−=−
Likewise, from
22
10 aaac IaIaIIrrrr
++=
47
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
22
12 )()( aacb IaaIaaII
rrrr−+−=−
we get
From boundary conditions (2) and (3), we get
)( cbfcb IIZVVrrrr
−=−
Substitution gives
])()[( 22
12
aaf IaaIaaZrr
−+−=2
21
2 )()( aa VaaVaarr
−+−
Simplifying, we get
2211 afaafa IZVIZVrrrr
−=−
48
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
From boundary conditions (2) and (3), we get
))(2( cbgfcb IIZZVVrrrr
++=+
Substitution gives
2102 aaacb VVVVVrrrrr
−−=+
We can also show
2102 aaacb IIIIIrrrrr
−−=+
)2(2 210210 aaafaaa IIIZVVVrrrrrr
−−=−−
)2(2 210 aaag IIIZrrr
−−+
49
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Rearranging terms, we get
11000 422 afaagafa IZVIZIZVrrrrr
−=−−
)(2 2122 aagafa IIZIZVrrrr
+−−+
Earlier, we got
2211 afaafa IZVIZVrrrr
−=−
021 aaa IIIrrr
−=+
Substitution gives
)(2622 11000 afaagafa IZVIZIZVrrrrr
−=−−
( ) 1100 3 afaagfa IZVIZZVrrrr
−=+−
50
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
F1
fV
Z1 1aIr
1aVr+
-
+
-
N1
F2
N2
Z2
2aIr
2aVr+
-
N0
F0
Z0
0aIr
0aVr+
-
ZfZf Zf+3Zg
Let gfT ZZZZ 300 ++=
fT ZZZ += 11
fT ZZZ += 22
51
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
The sequence fault currents
TT
TTT
fa
ZZ
ZZZ
VI
20
201
1
++
=r
210 aaa IIIrrr
−−=
From current division, we get
From KCL, we get
120
02 a
TT
Ta I
ZZ
ZI
rr
+−=
120
20 a
TT
Ta I
ZZ
ZI
rr
+−=or
52
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Double-Line-to-Ground Phase Fault Currents:
212
0 aaab aIIaII ++=
0=aI
( )
TTTTTT
TTf
ZZZZZZ
aZZVj
020121
203++
−−=
22
10 aaac IaaIII ++=
( )TTTTTT
TTf
ZZZZZZ
ZaZVj
020121
22
03++
−+=
53
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Example: A double-line-to-ground fault occurs at point F. Assuming zero fault impedance, find the fault currents at fault point F. Assume Eg = 1.0 p.u.
Note: All reactances are in per-unit of a common MVA base.
54
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Network Interconnection:
N1 N2
F1
J0.6 1AIr
+
-
F2
J0.62AI
r
1.0
N0
F0
J0.044
0AIr
55
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Sequence Fault Currents:
56
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Phase Fault Currents:
57
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
� Development of the Model
� Rake Equivalent
� Formation of Zbus
� Analysis of Shunt Fault
Bus Impedance Matrix Method
58
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Observations on Manual Network Solution
The procedure is straight forward, yet tedious and could be prone to hand-calculation error.
Is there a way for a computer to implement this methodology?
Development of the Model
59
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Development of the ModelConsider the three-bus system shown below. Let us analyze the system for a three-phase fault in any bus.
G1, G2 : X1=X2=0.2 X0=0.1
G1G2
L1
L2
1 2
3
L1 : X1=X2=0.6 X0=1.2L2 : X1=X2=0.24 X0=0.5
60
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Positive-Sequence Network:
j0.2
1GEr +
-
+
-2GE
rj0.2
j0.6j0.24
1 2
3
Combine the sources and re-draw. Assume EG = 1.0 per unit.
j0.6
3
j0.2j0.2
j0.24
+
-
GEr
1 2
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For a three-phase fault in bus 1 (or bus 2), we get the positive-sequence impedance.
16.0j)]6.02.0//(2.0[jZ1 =+=
25.6jZ1
ZE
I11
GF −===r
For a three-phase fault in bus 3, we get
4.0j)]6.02.0//(2.024.0[jZ1 =++=
5.2jZ1
ZE
I11
GF −===r
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Next, use loop currents to describe the circuit with all fault switches closed. Since there are four loops, we need to define four loop currents.
Let us connect a fault switch to each bus. In order to simulate a three-phase fault in any bus, close the fault switch in that bus.
j0.6
3
j0.2j0.2
j0.24
+
-
GEr
1 2
1Ir
3Ir
2Ir
4Ir
4
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The loop equations are)(2.00.1 431 IIIj
rrr−+=loop 1:
loop 2:
loop 4:
loop 3:
)(2.06.0)(2.00 314442 IIIjIjIIjrrrrrr
−−+++=
or
0.12.02.02.02.044.002.0
2.002.002.02.002.0
−−
−
−
00.10.10.1
4
3
2
1
IIII
r
r
r
r
= j
)(2.00.1 42 IIjrr
+=
3431 24.0)(2.00.1 IjIIIjrrrr
+−+=
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Current I4 is not a fault current. It can be eliminated using Kron’s reduction. We get
31
421)1(
bus ZZZZZ −−=
IZV )1(bus
rr=
where
44.002.002.002.002.0
Z1 = j
2.02.02.0
−
−
Z2 = j
Z3 = j[-0.2 0.2 -0.2 ] Z4 = j[1.0]
and
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Substitution gives
40.004.016.004.016.004.016.004.016.0
= j
0.10.10.1
3
2
1
III
r
r
r
IZV )1(bus
rr=
)1(busZ
(1) The equation can be used to analyze a three-phase fault in any bus (one fault at a time).
(2) is called the positive-sequence bus-impedance matrix, a complex symmetric matrix.
Note:
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One possible equivalent circuit is shown. This circuit is called a rake-equivalent.
+
-
Z11 Z22 Z33
Z12 Z23
Z13
1.0
1Ir
2Ir
3Ir
Consider the matrix voltage equation
332313
232212
131211
ZZZZZZZZZ
=
0.10.10.1
3
2
1
III
r
r
r
Suppose we are asked to find a circuit that satisfies the matrix equation.
Rake Equivalent
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Consider again the three-bus system. The circuit is described by the matrix equation
40.004.016.004.016.004.016.004.016.0
= j
0.10.10.1
3
2
1
III
r
r
r
The rake equivalent is shown. The diagonal elements of the matrix are self impedances while the off-diagonal elements are mutual impedances.
+
-
j0.16
1.0
1Ir
2Ir
3Ir
j0.16 j0.4j0.04j0.04
j0.16
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For the three-bus system, assume a fault in bus 3. The equation for bus 3 is
321 I4.0jI04.0jI16.0j0.1rrr
++=
Since only bus 3 is faulted, I1=I2=0. We get
3I4.0j0.1r
=or
5.2j4.0j
1I3 −==r
+
-
j0.16
1.0
3Ir
j0.16 j0.4j0.04j0.04
j0.16
2Vr+
-1Vr+
-
From KVL, we get the voltage in bus 1.
6.0ZZ
0.1IZ0.1V33
133131 =−=−=rr
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In general, for a three-phase fault in bus k of a system with n buses, the fault current is
kkk Z
1I =r
k=1,2,…n
Similarly from KVL, we get the voltage in bus 2.
9.0ZZ
0.1IZ0.1V33
233232 =−=−=rr
Note: Once the voltages in all the buses are known, the current in any line can be calculated.
The voltage in any bus j is given by
kk
jkj Z
Z0.1V −=
rj=1,2,…n
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The current in any line, which is connected from bus m to bus n, can be found using
mn
nmmn z
VVI
rrr −
=
where zmn is the actual impedance of the line.
j0.2
1GEr +
-
+
-2GE
rj0.2
j0.6j0.24
1 2
3-j2.0
-j2.5
-j0.5
5.0j6.0j
6.09.0z
VVI
21
1221 −=
−=
−=
rrr
For example, the current in the line between buses 2 and 1 is
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Zbus can be built, one step at a time, by adding one branch at a time until the entire network is formed.
The first branch to be added must be a generator impedance. This is necessary in order to establish the reference bus.Subsequent additions, which may be done in any order, fall under one of the following categories:
(1) Add a generator to a new bus;
(2) Add a generator to an old bus;
(3) Add a branch from an old bus to a new bus;
(4) Add a branch from an old bus to an old bus.
Formation of Zbus
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+
-
Z11 Z22 Zkk
Z12 Z2k
1.0
1Ir
2Ir
kIr
Znn
Zkn
nIr1 2 k n
Let us examine each category in the addition of a new branch.
Assume that at the current stage, the dimension of Zbus is n.
nn2n1n
n22221
n11211
ZZZ
ZZZZZZ
0.1
0.10.1
n
2
1
I
II
r
r
r
=
……
…
… … …
oldbusZ
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+
-
Z11 Z22 Zkk
Z12 Z2k
1.0
1Ir
2Ir
kIr
Znn
Zkn
nIr1 2 k n
Type 1: Add a generator to a new bus
Let Zg be the impedance of the generator to be added.
Zg
1nI +
r n+1
g
nn2n1n
n22221
n11211
Z0000ZZZ0ZZZ0ZZZ
0.10.10.10.1
1n
n
2
1
IIII
+
r
r
r
r
=
…
…
…The dimension is (n+1).
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Type 2: Add a generator to an old bus k
Let Zg be the impedance of the generator to be added.
+
-
Z11 Z22 Zkk
Z12 Z2k
1.0
1Ir
2Ir
kIr
Znn
nIr1 2 k n
The new current in impedance Zkk is (Ik+Iw). The new equations for buses 1 to n are
nn1wkk1212111 IZ...)II(Z...IZIZ0.1rrrrr
++++++=
nn2wkk2222121 IZ...)II(Z...IZIZ0.1rrrrr
++++++=
nnnwknk22n11n IZ...)II(Z...IZIZ0.1rrrrr
++++++=
Zg
wIr
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For the added generator loop, we get
wgnknwkkk22k11k IZIZ...)II(Z...IZIZ0rrrrrr
+++++++=
In matrix form, we get
wknkk2k1k
nknnnk2n1n
k2n2k22221
k1n1k11211
ZZZZZZZZZZ
ZZZZZZZZZZ
00.1
0.10.1
=
w
n
2
1
II
II
r
r
r
r… …… …
… …… …
…… … …
where Zw=Zkk+Zg. The last row is eliminated using Kron’s reduction. The dimension remains as n.
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Type 3: Add a branch from an old bus k to a new bus +
-
Z11 Z22 Zkk
Z12 Z2k
1.0
1Ir
2Ir
kIr
Znn
Zkn
nIr1 2 k n
The new current in impedance Zkk is (Ik+In+1). The new equations for buses 1 to n are
nn11nkk1212111 IZ...)II(Z...IZIZ0.1rrrrr
++++++= +
nn21nkk2222121 IZ...)II(Z...IZIZ0.1rrrrr
++++++= +
nnn1nknk22n11n IZ...)II(Z...IZIZ0.1rrrrr
++++++= +
Zb
1nI +
r n+1
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where Zw=Zkk+Zb. Kron’s reduction is not required. The dimension increases to (n+1).
In matrix form, we get
… …
wknkk2k1k
nknnnk2n1n
k2n2k22221
k1n1k11211
ZZZZZZZZZZ
ZZZZZZZZZZ
0.10.1
0.10.1
=
1n
n
2
1
II
II
+
r
r
r
r
… …
… …… …
…… … …
For the new bus, we get...)II(Z...IZIZ0.1 1nkkk22k11k +++++= +
rrrr
1nbnkn IZIZ +++rr
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Type 4: Add a branch from an old bus j to an old bus k
+
-
Z11 Z22 Zjj
Z12 Z2j
1.0
1Ir
2Ir
kIr
Zkk
Zkn
nIr1 2
jk
Znn
jIr n
The new current in impedance Zjj is (Ij+Iw). The new current in impedance Zkk is (Ik-Iw). The new equations for buses 1 to n are
)II(Z...IZIZ0.1 wjj1212111
rrrr++++=
nn1wkk1 IZ...)II(Zrrr
++−+
wIr
Zb
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)II(Z...IZIZ0.1 wjj2222121
rrrr++++=
)II(Z...IZIZ0.1 wjnj22n11n
rrrr++++=
nn2wkk2 IZ...)II(Zrrr
++−+
nnnwknk IZ...)II(Zrrr
++−+
For the added loop, we get
)II(Z)II(Z...IZIZ0 wkjkwjjj22j11j
rrrrrr−+++++=
...IZIZ[IZIZ... 22k11kwbnjn ++−+++rrrr
]IZ...)II(Z)II(Z nknwkkkwjkj
rrrrr++−+++
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In matrix form, we get
vknjn2k2j1k1j
nknjnn2n1n
k2j2n22221
k1j1n11211
ZZZZZZZZZZZZ
ZZZZZZZZZZ
−−−
−
−
−
00.1
0.10.1
w
n
2
1
II
II
r
r
r
r
=
…
…
…
…
…… …
where Zv=Zjj+Zkk-2Zjk+Zb. The last row is eliminated using Kron’s reduction. The dimension remains as n.
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Example: For the network shown, use the step-by-step building algorithm to form the bus impedance matrix.
Step 1. Add generator G1 to bus 1.
1Xbus = [0.2]1
Step 2. Add generator G2 to bus 2.
Xbus =2.00
02.0
1 2
1
2
j0.2+
-
+
-
j0.2
j0.6j0.24
1 2
3 1.01.0
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Step 3. Add the line from bus 1 to bus 2.
0.12.02.02.02.00
2.002.0
−
−Xnew =
1 2
1
2
*
*
Apply Kron’s reduction to eliminate the last row and column. We get
=−3
142 XXX
2.02.0
−[0.2 -0.2]
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04.004.004.004.0
−
−=−
31
42 XXX
We get
31
421bus XXXXX −−= =
1 2
1
2 16.004.004.016.0
Step 4. Finally, add the line from bus 1 to bus 3.
4.004.016.004.016.004.016.004.016.0
Xbus =
1 2
1
2
3
3
No Kronreduction is required.
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Example: Determine the positive-sequence bus-impedance matrix for the four-bus test system shown.
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Three-Phase Line Segment Model
zaa
zbb
zcc
zab
zbc
zca
½ Yabc½ Yabc
clineI
blineI
alineI
cmI
bmI
amI
cnI
bnI
anI
mcgV
mbgV
magV
ncgV
nbgV
nagV
abcmc,Iabc
nc,I
abc
abcabc
abcm
abcmLG
abcnLG
Zb
YZUa
IbVaV
=
⋅⋅+=
⋅+⋅=
21
,,
abcabc
abcabcabcabc
abcm
abcmLG
abcn
YZUd
YZYYc
IdVcI
⋅⋅+=
⋅⋅⋅+=
⋅+⋅=
21
41
,
Voltages & currents at node n in terms of the voltages & currents at node m:
Node n Node m
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Three-Phase Line Segment Model
abcn
abcnLG
abcm
abcn
abcnLG
abcmLG
IdVcI
IbVaV
⋅+⋅−=
⋅−⋅=
,
,,
Voltages & currents at node m in terms of the voltages & currents at node n:
Voltages at node m as a function of voltages at node n and currents entering node m:
baB
aA
IBVAV
⋅=
=
⋅−⋅=
−
−
1
1
,,abcm
abcmLG
abcmLG
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Transformer Generalized Matrices
abctabcLNtABC
abctabcLNt
ABCLN
IdVcI
IbVaV
⋅⋅⋅⋅++++⋅⋅⋅⋅====
⋅⋅⋅⋅++++⋅⋅⋅⋅====
abctABCLNt
abcLN IBVAV ⋅⋅⋅⋅−−−−⋅⋅⋅⋅====
Generalized three-phase transformer bank:
H1
H2
H3
H0
X1
X2
X3
X0
aI
bI
cI
nI
AI
BI
CI
NI
ANV
BNV
CNV
anV
bnV
cnV
H1
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Transformer Generalized Matrices
Delta – Grounded Wye Step-Down Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−
−−−−
−−−−
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
⋅⋅⋅⋅−−−−
====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅
−−−−====
====
101
110
0111
d
000
000
000
c
side voltage-low the to referred are ,,
02
20
20
3b
012
201
120
3a
sidelow rated ,
side high rated ,
ttt
ct
bt
at
bt
at
ct
at
ct
bt
tt
tt
LN
LLt
n
ZZZ
ZZ
ZZ
ZZnn
V
Vn
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Transformer Generalized Matrices
Delta – Grounded Wye Step-Down Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−
−−−−
−−−−
⋅⋅⋅⋅====ct
bt
at
tt
t
Z
Z
Z
n00
00
00
B
110
011
1011
A
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Ungrounded Wye – Delta Step-Down Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−
−−−−
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−
−−−−
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−
−−−−
−−−−
⋅⋅⋅⋅====
====
012
021
011
31
d
000
000
000
c
side voltage-low the to referred are ,,
02
02
0
3b
101
110
011
a
sidelow rated ,
side high rated ,
ttt
cat
bct
abt
cat
cat
bct
bct
abt
abt
tttt
LL
LNt
n
ZZZ
ZZ
ZZ
ZZn
n
V
Vn
131
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
Transformer Generalized Matrices
Ungrounded Wye – Delta Step-Down Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−−−−−
−−−−−−−−
−−−−++++
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
024
0422
0222
91
B
201
120
012
31
A
cat
abt
cat
abt
cat
bct
cat
bct
abt
bct
bct
abt
t
tt
ZZZZ
ZZZZ
ZZZZ
n
132
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Transformer Generalized Matrices
Grounded Wye – Gounded Wye Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
====
100
010
0011
d
000
000
000
c
side voltage-low the to referred are ,,
00
00
00
b
100
010
001
a
sidelow rated ,
side high rated ,
ttt
ct
bt
at
ct
bt
at
tttt
LN
LNt
n
ZZZ
Z
Z
Z
nn
V
Vn
133
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Transformer Generalized Matrices
Grounded Wye – Grounded Wye Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
ct
bt
at
tt
t
Z
Z
Z
n00
00
00
B
100
010
0011
A
134
Competency Training & Certification Program in Electric Power Distribution System Engineering
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Transformer Generalized Matrices
Delta – Delta Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−
−−−−−−−−
−−−−−−−−
⋅⋅⋅⋅====
====
cat
bct
abt
abcttV
tt
LL
LLt
Z
Z
Z
n
n
V
Vn
00
00
00
Z
100
010
001
A
201
120
012
31
W
211
121
112
3a
sidelow rated ,
side high rated ,
135
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Transformer Generalized Matrices
Delta – Delta Connection
1
1
1
GZWB
221
121
112
31
A
100
010
0011
d
GZAWb
0
0
01
G
⋅⋅⋅⋅⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−
−−−−−−−−
−−−−−−−−
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡⋅⋅⋅⋅====
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−−−−−
++++
−−−−
⋅⋅⋅⋅++++++++
====
abctt
tt
tt
abctVt
bct
bct
abt
cat
abt
cat
bct
cat
cat
bct
abt
nn
ZZZ
ZZZ
ZZ
ZZZ
136
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Transformer Generalized Matrices
Open Wye – Open Delta Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡−−−−⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡−−−−
−−−−
⋅⋅⋅⋅====
====
000
100
0011
d
000
000
000
c
side voltage-low the to referred are ,
000
00
00
b
000
110
011
a
sidelow rated ,
side high rated ,
ttt
bct
abt
bct
abt
tttt
LL
LNt
n
ZZ
Z
Z
nn
V
Vn
137
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Transformer Generalized Matrices
Open Wye – Open Delta Connection
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−
−−−−−−−−
−−−−
⋅⋅⋅⋅====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−
−−−−⋅⋅⋅⋅====bct
abt
bct
abt
bct
abt
tt
t
ZZ
ZZ
ZZ
n20
0
02
31
B
021
011
012
31
A
138
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
Short-Circuit Analysis of Unbalanced Feeders
ABCsysZ ABC
subZ ABCeqSZ abc
xfmZ abceqLZ
SystemVoltageSource
EquivalentSystem
Impedance
SubstationTransformer
TotalPrimary
LineSegment
Impedance
In-lineFeeder
Transformer
TotalSecondary
LineSegment
Impedance
1 2 3 4 5
System: (((( )))) (((( ))))
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
++++−−−−−−−−
−−−−++++−−−−
−−−−−−−−++++
⋅⋅⋅⋅====
ΩΩΩΩ−−−−====ΩΩΩΩ====φφφφφφφφ
011010
100110
101001
)(
11
2
03
2
1
2
2
2
31
Z
23
Z
ZZZZZZ
ZZZZZZ
ZZZZZZ
ZMVAkV
MVAkV
Z
ABCapproxsys
LLLL
139
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� Thevenin equivalent voltages at points 2 and 3: computed by multiplying the system voltages by the generalizedtransformer matrix At of the substation transformer.
� Thevenin equivalent voltages at points 4 and 5: the voltage at node 3 multiplied by the generalized transformermatrix At of the in-line transformer.
� Thevenin equivalent phase impedance matrices:sum of the phase impedance matrices of each device between thesystem voltage source and the point of fault.
Short-Circuit Analysis of Unbalanced Feeders
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
Define abcabcP EYI ⋅⋅⋅⋅====
Substituting & rearranging,
xgabcxabcf
abcP VYVYII ⋅⋅⋅⋅++++⋅⋅⋅⋅++++====
Expanding,
(((( ))))
(((( ))))
(((( )))) xga
Scxaabxcbaxcacf
cP
xga
Scxbcbxbbaxbabf
bP
xga
Scxacbxabaxaaaf
aP
xg
xg
xg
cccbca
bcbbba
acabaa
cx
bx
ax
cccbca
bcbbba
acabaa
cf
bf
af
cP
bP
aP
VYVYVYVYII
VYVYVYVYII
VYVYVYVYII
V
V
V
YYY
YYY
YYY
V
V
V
YYY
YYY
YYY
I
I
I
I
I
I
++++++++++++++++====
++++++++++++++++====
++++++++++++++++====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡++++
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡++++
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
Short-Circuit Analysis of Unbalanced Feeders
143
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Training Course in Short Circuit Analysis
where
acabaaa
S
bcbbbab
S
acabaaa
S
YYYY
YYYY
YYYY
++++++++====
++++++++====
++++++++====
3 equations, 7 unknowns - xgcxbxaxcf
bf
af VVVVIII ,,,,,,
cP
bP
aP III ,, are functions of the total impedance &
the Thevenin voltages and are known
Needed: 4 additional equations
Short-Circuit Analysis of Unbalanced Feeders
144
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
Three-Phase Faults:
0
0
====++++++++
============
cba
cxbxax
III
VVV
Three-Phase-to-Ground Faults:
0
0
====++++++++
================
cba
xgcxbxax
III
VVVV
Line-to-Line Faults (assume i-j fault with phase k unfaulted):
0
0
0
====++++
====
========
jf
if
kf
jxix
II
I
VV
Short-Circuit Analysis of Unbalanced Feeders
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
Line-to-Ground Faults (assume phase k fault withphases i and j unfaulted):
0
0
========
========
jf
if
xgkx
II
VV
Short-Circuit Analysis of Unbalanced Feeders
146
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Competency Training & Certification Program in Electric Power Distribution System Engineering
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Training Course in Short Circuit Analysis
7 equations in matrix form:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−====
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎦⎦⎦
⎤⎤⎤⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎣⎣⎣
⎡⎡⎡⎡
xg
cx
bx
ax
cf
bf
af
S
S
S
cP
bP
aP
V
V
V
V
I
I
I
YYYY
YYYY
YYYY
I
I
I
3333231
2232221
1131211
100
010
001
0
0
0
0
In condensed form:
XCI ⋅⋅⋅⋅====sP
Short-Circuit Analysis of Unbalanced Feeders
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Solving,sPICX 1 ⋅⋅⋅⋅==== −−−−
Example: 3-phase fault
1
1 1 1
737271
665544
============
============
CCC
CCC
All of the other elements in the last 4 rows of C will be set to zero.
Short-Circuit Analysis of Unbalanced Feeders
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ExampleInfinite
bus
1 2 3 4
ABCI abcI
ABCeqSZ abc
eqLZ
Compute the short-circuit currents for a bolted line-to-line fault between phases a and b at node 4.
SLNtth EAE 4, ⋅⋅⋅⋅====
Line-to-neutral Thevenin voltage at node 4:
Thevenin equiv. impedance at secondary terminals (node 3):
abctt
ABCeqStthZ ZdZA3, ++++⋅⋅⋅⋅⋅⋅⋅⋅====
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Total Thevenin impedance at node 4:
abceqLthTOTth ZZZZ 3.4, ++++========
Equivalent admittance matrix at node 4:1
4, ZY −−−−==== TOTeq
Equivalent injected currents at point of fault:
4,4, EYI theqP ⋅⋅⋅⋅====
For the a-b fault at node 4,
0
0
0
========
====
====++++
bxax
cf
bf
af
VV
I
II
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Unknowns are computed as
SPICX 1 ⋅⋅⋅⋅==== −−−−
Suppose that the phase impedance matrices for the 2 line segments are:
Ω
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
Ω
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
4970.01921.01931.00614.01751.00598.0
1931.00614.04885.01939.02302.00607.0
1751.00598.02302.00607.05035.01907.0
5353.01414.02955.00361.02752.00361.0
2953.00361.05353.01414.03225.00361.0
2752.00361.03225.00361.05353.01414.0
jjj
jjj
jjj
jjj
jjj
jjj
abceqL
ABCeqS
Z
Z
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The transformer bank consists of three single-phase transformers eachrated: 2000 kVA, 12.47-2.4 kV, Z = 1.0 + j6.0 %
Source line segment:
[ ]0
5353.01414.02955.00361.02752.00361.0
2953.00361.05353.01414.03225.00361.0
2752.00361.03225.00361.05353.01414.0
100
010
001
1
1
11
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡===
c
Zb
Uda
jjj
jjj
jjj
ABCeqS
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⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
−
−
5353.01414.02955.00361.02752.00361.0
2953.00361.05353.01414.03225.00361.0
2752.00361.03225.00361.05353.01414.0
100
010
001
11
11
111
jjj
jjj
jjj
baB
aA
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[ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
4970.01921.01931.00614.01751.00598.0
1931.00614.04885.01939.02302.00607.0
1751.00598.02302.00607.05035.01907.0
100
010
001
0
4970.01921.01931.00614.01751.00598.0
1931.00614.04885.01939.02302.00607.0
1751.00598.02302.00607.05035.01907.0
100
010
001
2
2
2
2
22
jjj
jjj
jjj
jjj
jjj
jjj
B
A
c
b
da
Load line segment:
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Transformer:Transformer impedance in ohms referenced to the low-voltage winding
Ω+=⋅+=
Ω=⋅
=
1728.00288.088.2)06.001.0(Z
88.22000
10004.2 2
jj
Z
lowt
base
Transformer phase impedance matrix
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
+
+
=
1728.00288.000
01728.00288.00
001728.00288.0
j
j
jabctZ
Turns ratio:
1958.54.247.12
==tn
Transformer ratio:
9998.24.23
47.12=
⋅=ta
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Generalized matrices are:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
−−−−
−−−−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−
−−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
−=
000
000
000
02993.00499.05986.00998.0
5986.00998.002993.00499.0
2993.00499.05986.00998.00
02
20
20
3
07319.14639.3
4639.307319.1
7319.14639.30
012
201
120
3
t
tt
tt
ttt
t
tt
jj
jj
jj
ZZ
ZZ
ZZn
n
c
b
a
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⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
+
+
==
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
⋅=
1728.00288.000
01728.00288.00
001728.00288.0
1925.01925.00
01925.01925.0
1925.001925.0
110
011
1011
1925.001925.0
1925.01925.00
01925.01925.0
101
110
0111
j
j
j
n
n
abctt
tt
tt
ZB
A
d
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The infinite bus balanced line-to-line voltages are 12.47 kV, which leads tobalanced line-to-neutral voltages at 7.2 kV:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∠
−∠
∠
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∠
−∠
∠
=o
o
o
sLNo
o
o
sLL1206.7199
1206.7199
06.7199
V
150470,12
90470,12
30470,12
,, EE
The line-to-neutral Thevenin circuit voltages at node 4 are:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∠
−∠
−∠
=⋅=o
o
o
sLNtth902400
1502400
302400
,4, EAE
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The Thevenin equivalent impedance at the secondary terminals (node 3)of the transformer consists of the primary line impedances referredacross the transformer, plus the transformer impedances:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−−−−
−−+−−
−−−−+
=
+⋅⋅=+⋅⋅=
1906.00366.00071.00039.00106.00039.0
0071.00039.01886.00366.00086.00039.0
0106.00039.00086.00039.01921.00366.0
3,
jjj
jjj
jjj
abctt
ABCeqSttt
ABCeqStth ZdZABdZAZ
Total Thevenin impedance at node 4:
Ω
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+++
+++
+++
=
+==
6876.02287.01860.00575.01645.00559.0
1860.00575.06771.02305.02216.00568.0
1645.00559.02216.00568.06955.02273.0
3,4,
jjj
jjj
jjj
abceqLthTOTth ZZZZ
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Equivalent admittance matrix at node 4:
S
4532.14843.03133.01145.02510.00688.0
3133.01148.05280.15501.03907.01763.0
2510.00688.03907.01763.04771.15031.0
14,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−+−
+−−+−
+−+−−
=
= −
jjj
jjj
jjj
TOTeq ZY
The equivalent injected currents at the fault point:
A
4.169.4440
0.1389.4878
4.968.4466
4,4,⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∠
∠
−∠
=⋅=o
o
o
theqp EYI
Sums of each row of the equivalent admittance matrix:
S
8889.03007.0
8240.02590.0
8353.02580.03
1,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
== ∑= j
j
j
Yk
ikeqsY
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For the a-b fault at node 4,
0 0
0 0
==
==+
bxax
cf
bf
af
VV
III
The coefficient matrix
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−+−−−
−+−−+−
−+−+−−
=
0001000
0010000
0000100
0000011
889.0301.0452.1484.0313.0115.0251.0069.0100
824.0259.0314.0115.0528.1550.0390.0176.0010
835.0258.0252.0069.0390.0176.0477.1501.0001
jjjj
jjjj
jjjj
C
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The injected current matrix:
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∠
∠
−∠
=
0
0
0
0
4.169.4440
0.1389.4878
4.968.4466
o
o
o
spI
The unknowns are computed by:
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∠
−∠
∠
−∠
=⋅= −
o
o
o
o
sp
1.899.2587
0
0
6.904.7740
0
6.1717.8901
4.87.8901
1 ICX
The interpretation of the results are:
oxg
cxcf
bxob
f
oax
oaf
XV
XVXI
XVXI
XVXI
1.899.2587
0 0
0 6.1717.8901
6.904.7740 4.87.8901
7
63
52
41
∠==
====
==∠==
−∠==−∠==
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Fault Current at Different Times
Clearing Time of High Voltage
Breakers
Contact Opening Time
of High Voltage Breakers
Clearing Time of
Fuse
Clearing Time of Molded Breakers
Fault Current that upstream overcurrent
devices must withstand while downstream devices
isolate the fault
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Fault Current at Different Times
� First (1/2) Cycle Fault Current� Short circuit ratings of low voltage equipment
� Ratings of High Voltage (HV) switch and fuse
� Close & Latch (Making) capacity or ratings of HV Circuit Breakers
� Maximum Fault for coordination of instantaneous trip of relays
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� 1.5 to 4 Cycles Fault Current� Interrupting (breaking) duties of HV circuit
breakers
� Interrupting magnitude and time of HV breakers for coordination
� 30 Cycles Fault Current� For time delay coordination
Fault Current at Different Times
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Application of Short Circuit Analysis
� Comparison of Closed-and-Latch (Momentary or Making) and Interrupting (Breaking) Duties of Interrupting Devices
� Comparison of Short-time or withstand rating of system components
� Selection of rating or setting of short circuit protective devices
� Evaluation of current flow and voltage levels in the system during fault
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Characteristic of Short Circuit Currents
( )sindi
Ri L E tdt
ω φ+ = +
( )2 2 2 2
sin sin( ) R tXE t E
i eR X R X
ωω θ φ θ φ −+ − −= +
+ +
R L
E sin (ωωωωt+φφφφ)
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Characteristic of Short Circuit Currents
( ) ωω θ φ θ φ −+ − −= +
+ +2 2 2 2
sin sin( ) R tXE t E
i eR X R X
FactoralAsymmetricII RMS lsymmetricaRMStotal •=,
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ANSI/IEEE and IEC Standards
� ANSI/IEEE: American National Standards Institute/ Institute of Electrical and Electronics Engineers
� IEC: International ElectrotechnicalCommission
Prescribes Test Procedures and Calculation Methods
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1.5-4 Cycle Network: the network used to calculate interrupting short-circuit current and protective device duties 1.5-4 cycles after the fault.
N/ARelay
N/ASwitchgear and MCC
N/AFuse
N/ALow Voltage CB
Interrupting CapabilityHigh Voltage CB
DutyType of Device
ANSI/IEEE Calculation Method
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½ Cycle Network: also known as the subtransientnetwork because all rotating machines are represented by their subtransient reactances
ANSI/IEEE Calculation Method
Type of Machine XscUtility X”Turbo generator Xd”Hydro-generator with amortisseur windings Xd”Hydro-generator without amortisseur windings 0.75 Xd’Condenser Xd”Synchronous motor Xd”Induction Machine > 1000 hp @ 1800 rpm or less Xd” > 250 hp @ 3600 rpm Xd” All other ≥ 50 hp 1.2 Xd” < 50 hp 1.67 Xd”
Xd” of induction motor = 1/(per-unit locked-rotor current at rated voltage)
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30 Cycle Network: also known as the steady-state network
InfinityInduction Machine
InfinitySynchronous Motor
InfinityCondenser
Xd’Hydro-generator w/o Amortisseur Winding
Xd’Hydro-generator w/ Amortisseur Winding
Xd’Turbo Generator
X’’Utility
XscType of Machine
ANSI/IEEE Calculation Method
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ANSI Multiplying Factor: determined by the equivalent X/R ratio at a particular fault location. The X and the R are calculated separately.
Local and Remote Contributions
A local contribution to a short-circuit current is the portion of the short-circuit current fed predominantly from generators through no more than one transformation, or with external reactance in series which is less than 1.5 times the generator subtransient reactance. Otherwise the contribution is defined as a remote contribution.
ANSI/IEEE Calculation Method
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Momentary (1/2 Cycle) Short-Circuit Current
Peak Momentary Short-Circuit Current
π−
= ⋅
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠
, , ,
2 1
mom peak p mom rms symm
X Rp
I MF I
MF e
ANSI/IEEE Calculation Method
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Momentary (1/2 Cycle) Short-Circuit Current
Asymmetrical RMS value of Momentary Short-Circuit Current
π
−
−
=
= ⋅
= +
, ,
, , , ,
2
3
1 2
pre faultmom rms symm
eq
mom rms asymm m mom rms symm
X Rm
VI
Z
I MF I
MF e
ANSI/IEEE Calculation Method
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High Voltage Circuit Breaker Interrupting Duty (1.5-4 Cycle)
Adjusted RMS value of Interrupting Short-Circuit Current (for total current basis CBs)
−=int, ,
3pre fault
rms symm
eq
VI
Z
ANSI/IEEE Calculation Method
symmrmsiadjrms IAMFI ,int,,int, ⋅=
( )lrli MFMFNACDMFAMF −+=where
178
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
No AC Decay (NACD) Ratio
The NACD ratio is defined as the remote contributions to the total contributions for the short-circuit current at a given location
ANSI/IEEE Calculation Method
total
remote
IINACD =
• Total short circuit current Itotal = Iremote + Ilocal• NACD = 0 if all contributions are local• NACD = 1 if all contributions are remote
179
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
High Voltage Circuit Breaker Interrupting Duty (1.5-4 Cycle)
π−
= +
4
1 2t
X RrMF e
ANSI/IEEE Calculation Method
Circuit BreakerRating in Cycles
Contact PartingTime ( t ) in Cycles
8 45 33 22 1.5
180
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
High Voltage Circuit Breaker Interrupting Duty (1.5-4 Cycle) Calculation
ANSI/IEEE Calculation Method
Multiplying factors (total current basis CBs) MFr for 3-phase & line-to-ground faults.
181
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
High Voltage Circuit Breaker Interrupting Duty (1.5-4 Cycle)
ANSI/IEEE Calculation Method
Multiplying factors (total current basis CBs) MFl for 3-phase faults.
182
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
ANSI/IEEE Calculation Method
High Voltage Circuit Breaker Interrupting Duty (1.5-4 Cycle)
Adjusted RMS value of Interrupting Short-Circuit Current (for symmetrically rated CBs)
SIAMF
I symmrmsiadjrms
,int,,int,
⋅=
Circuit Breaker ContactParting Time (Cycles)
S Factor
4 1.03 1.12 1.2
1.5 1.3
183
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Low Voltage Circuit Breaker Interrupting Duty (1/2 Cycle) Calculation
Adjusted asymmetrical RMS value of Interrupting Short-Circuit Current
−=int, ,
3pre fault
rms symm
eq
VI
Z
ANSI/IEEE Calculation Method
symmrmsadjrms IMFI ,int,,int, ⋅=
184
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Low Voltage Circuit Breaker Interrupting Duty (1/2 Cycle) Calculation
π
π
π
π
−
−
−
−
+=
+
+=
+
( )
( )
2(1 )
2(1 )
1 2
1 2
test
test
X R
X R
X R
X R
eMF
e
eMF
e
Unfused power breakers
Fused power breakers &
Molded Case
ANSI/IEEE Calculation Method
Note: If calculated MF < 1.0, set MF = 1.0
185
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Low Voltage Circuit Breaker Interrupting Duty (1/2 Cycle) Calculation
ANSI/IEEE Calculation Method
Circuit Breaker Type (X/R)testPower Breaker (Unfused) 6.59Power Breaker (Fused) 4.90Molded Case (> 20 kA) 4.90Molded Case (10.001 – 20 kA) 3.18
Molded Case (10 kA) 1.73
186
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Fuse Interrupting Short-Circuit Current Calculation
- same procedure as Circuit Breaker Interrupting
Duty calculation.
ANSI/IEEE Calculation Method
187
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
IEC Calculation Method
An equivalent voltage source at the fault location replaces all voltage sources. A voltage factor c is applied to adjust the value of the equivalent voltage source for minimum and maximum current calculations.
All machines are represented by internal impedances
Line capacitances and static loads are neglected, except for the zero-sequence network.
Calculations consider the electrical distance from the fault location to synchronous generators.
188
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Initial Symmetrical Short-Circuit Current (I’’k) RMS value of the AC symmetrical component of an available short-circuit current applicable at the instant of short-circuit if the impedance remains at zero time value.
Peak Short-Circuit Current (ip)Maximum possible instantaneous value of the available short-circuit current.
Symmetrical Short-Circuit Breaking Current (Ib)RMS value of an integral cycle of the symmetrical AC component of of the available short-circuit current at the instant of contact separation of the first pole of a switching device
IEC Calculation Method
189
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Steady-state Short Circuit Current (Ik)RMS value of the short-circuit current which remains after the decay of the transient phenomena.
Subtransient Voltage (E’’) of a Synchronous MachineRMS value of the symmetrical internal voltage of a synchronous machine which is active behind the subtransientreactance Xd’’ at the moment of short circuit.
Far-from-Generator Short-Circuit Short-circuit condition to which the magnitude of the symmetrical ac component of the available short-circuit current remains essentially constant
IEC Calculation Method
190
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Near-to-Generator Short-Circuit Short-circuit condition to which at least one synchronous machine contributes a prospective initial short-circuit current which is more than twice the generator’s rated current or a short-circuit condition to which synchronous and asynchronous motors contribute more than 5% of the initial symmetrical short-circuit current (I”k) without motors.
Subtransient Reactance (Xd’’) of a Synchronous MachineEffective reactance at the moment of short-circuit. MS value of the symmetrical internal voltage of a synchronous machine which is active behind the subtransient reactance Xd’’ at the moment of short circuit.
( )
ϕ
= +
=+
''
max''1 sin
K G d
nG
r d r
Z K R jX
ckVK
kV x
kVn = nominal voltage of the terminal buskVr = motor rated voltageXd” = subtransient reactanceϕr = machine rated power factor
IEC Calculation Method
191
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Minimum Time Delay (Tmin) of a Circuit Breaker
Shortest time between the beginning of the short-circuit current and the first contact separation of one pole of the switching device
Voltage Factor (c) Factor used to adjust the value of the equivalent voltage sourcefor the minimum and maximum current calculations
1.001.10> 35 KV to 230 KV
1.001.10> 1 kV to 35 kV
1.001.05Other LV up to 1 KV
0.951.00230/400 V
Min SC CalculationMax SC Calculation
Voltage FactorVoltage Factor
IEC Calculation Method
192
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
IEC Calculation Method
k
nk Z
UcI3
=′′ Zk = equiv. Impedance at fault point
kp Iki ′′= 2 k = function of system R/X at fault location
kb II ′′= for far-from-generator fault
kb II ′′μ=for synch. machines, for near-to-generator
faults
kb IqI ′′μ= for ind. machines, for near-to-generator
faults
193
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
IEC Calculation Method
HV CB asymmetrical breaking & dc current rating
⎟⎠⎞
⎜⎝⎛ π
−=
⎟⎠⎞
⎜⎝⎛ π−+=
RXtfII
RXtfII
symmbdc
symmbasymmb
/ 2exp2
/ 4exp21
min,
min,,
f = system frequencytmin = minimum delay timeIb,symm = AC breaking current
X/R = calculated based on testing PF of 7% at 50 Hz
194
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
IEC Calculation Method
LV CB asymmetrical breaking current rating
⎟⎠⎞
⎜⎝⎛ π−+=
RXtfII symmbasymmb / 4exp21 min
,,
f = system frequencytmin = minimum delay timeIb,symm = AC breaking currentX/R = calculated based on testing PF given by IEC
195
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
IEC Calculation Method
Fuse asymmetrical breaking current rating
⎟⎠⎞
⎜⎝⎛ π−+=
RXtfII symmbasymmb / 4exp21 min
,,
f = system frequencytmin = assumed to be a half cycleIb,symm = AC breaking currentX/R = calculated based on testing PF of 15%
196
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Training Course in Short Circuit Analysis
Selection of Device Duties
40.277428013.8 KV13.8 – 1000
30.45813.56013.8 KV13.8 – 750
19.637214013.8 KV13.8 – 500
46.97848.6804.16 KV4.16 – 350
33.25835604.16 KV4.16 – 250
10.11910.5204.16 KV4.16 – 75
Short-Circuit Capability
(Symmetrical RMS Current at 3-Cycle Parting
Time
Closing and Latching
Capability (Total First Cycle RM
Current)
Interrupting Rating (Total
RMS Current at 4-cycle Contact-
Parting Time
Momentary Rating (Total 1st-Cycle RMS
Current
Example Maximum System
Operating Voltage
Circuit Breaker Nominal Size Identification
8-Cycle Total-Rated Circuit Breakers (KA)
5-Cycle Symmetrical-Rated Circuit Breakers (KA)
198
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration
Competency Training & Certification Program in Electric Power Distribution System Engineering
U. P. National Engineering CenterNational Electrification Administration