SHORT CIRCUIT CALCULATIONS
REVISITED
FORTUNATO C. LEYNES
FORTUNATO C. LEYNES
FORTUNATO C. LEYNES
FORTUNATO C. LEYNES
Chairman
Professional Regulatory
Board of Electrical Engineering
Short Circuit (Fault) Analysis
FAULT-PROOF SYSTEM
not practical
neither economical
faults or failures occur in any power system
In the various parts of the electrical network
under short circuit or unbalanced condition,
the determination of the magnitudes and phase
angles
Application of Fault Analysis
1. The determination of the required mechanical
strength of electrical equipment to withstand
the stresses brought about by the flow of high
short circuit currents
2. The selection of circuit breakers and switch
ratings
3. The selection of protective relay and fuse
ratings
4. The setting and coordination of protective
devices
5. The selection of surge arresters and insulation
ratings of electrical equipment
6. The determination of fault impedances for use
in stability studies
7. The calculation of voltage sags caused resulting
from short circuits
8. The sizing of series reactors to limit the short
circuit current to a desired value
9. To determine the short circuit capability of
series capacitors used in series compensation of
long transmission lines
10. To determine the size of grounding
transformers, resistances, or reactors
Three-phase Systems
Φ
−
Φ
−
=
×
=
3
2
3
2
)
,
(
1000
)
,
(
MVA
base
kV
voltage
base
Z
kVA
base
kV
voltage
base
Z
L
L
B
L
L
B
Per Unit Quantities
)
(
)
(
)
(
B
pu
B
pu
impedance
actual
Z
kV
Voltage
Base
kV
voltage
actual
V
I
Current
Base
current
actual
I
=
=
=
Changing the Base of Per Unit
Quantities
(
)
(
)
(
)
]
[
2
]
[
]
[
]
[
2
]
[
]
[
]
[
2
]
[
]
[
)
(
1000
1000
)
(
1000
)
(
,
new
new
new
B
old
old
old
pu
old
old
old
pu
Z
Z
kVA
base
kV
base
Z
kVA
base
kV
base
Z
Z
kVA
base
kV
base
Z
impedance
actual
Z
Ω
=
×
=
×
=
Ω
×
Ω
=
Changing the Base of Per
Unit Quantities
(
)
(
)
(
)
2
]
[
]
[
2
]
[
]
[
]
[
2
]
[
]
[
1000
1000
)
(
1000
)
(
,
new
old
old
old
pu
old
old
old
pu
kV
base
Z
kVA
base
kV
base
Z
Z
kVA
base
kV
base
Z
impedance
actual
Z
×
=
×
=
Ω
×
Ω
=
kVA Base for Motors
Induction < 100 hp
1.00
Synchronous 1.0 pf
0.80
Induction > 1000 hp
0.90
Induction 100 < 999 hp
0.95
Synchronous 0.8 pf
1.00
hp rating
kVA/hp
SYMMETRICAL
COMPONENTS
1 c
V
V
a1 1 bV
Positive Sequence
2 aV
2 cV
2 bV
Negative Sequence
Unbalanced Phasors
Zero Sequence
0 0 0 b c aV
V
V
=
=
cV
aV
bV
0 c V 1 c V 2 c V 0 b V 2 b V 1 b V 1 a V 2 a V 0 a VSymmetrical Components of
Unbalanced Three-phase Phasor
(
)
(
)
(
a
b
c
)
a
c
b
a
a
c
b
a
a
aV
V
a
V
V
V
a
aV
V
V
V
V
V
V
+
+
=
+
+
=
+
+
=
2
2
2
1
0
3
1
3
1
3
1
2
2
1
0
2
1
2
0
2
1
0
a
a
a
c
a
a
a
b
a
a
a
a
V
a
aV
V
V
aV
V
a
V
V
V
V
V
V
+
+
=
+
+
=
+
+
=
In matrix form:
Symmetrical Components of
Unbalanced Three-phase Phasor
=
2
1
0
2
2
1
1
1
1
1
a
a
a
c
b
a
V
V
V
a
a
a
a
V
V
V
=
c
b
a
a
a
a
V
V
V
a
a
a
a
V
V
V
2
2
2
1
0
1
1
1
1
1
3
1
Sequence Networks
Power System Short Circuit
Calculations
Fault Point
The fault point of a system is that point to
which the unbalanced connection is
Definition of Sequence
Networks
Positive-sequence Network
E
a1
=
Thevenin’s equivalent
voltage as seen at the fault
point
Z
1
=
Thevenin’s equivalent
impedance as seen from
the fault point
Z
I
E
V
=
−
Z
1E
a1I
a1V
a1+
Negative-sequence Network
Z
2
=
Thevenin’s equivalent
negative-sequence
impedance as seen at
the fault point
Definition of Sequence
Networks
2
2
2
I
Z
V
a
=
−
a
Z
2I
a2V
a2+
-Zero-sequence Network
Z
0
=
Thevenin’s equivalent
zero-sequence impedance
as seen at the fault point
Definition of Sequence
Networks
0
0
0
I
Z
V
a
=
−
a
Z
0I
a0V
a0+
Sequence Network Models of
Power System Components
Power System Short Circuit
Calculations
Synchronous Machines
(Positive Sequence Network)
d
jx
Synchronous Machines
(Negative Sequence Network)
2
''
''
2
q
d
x
x
j
Z
=
+
Where:
Z
2
I
a2
V
a2
+
-x”
d
= direct-axis sub-transient reactance
Synchronous Machines
(Zero Sequence Network)
Solidly-Grounded Neutral
jX
0
I
0
n
Impedance-Grounded Neutral
Synchronous Machines
(Zero Sequence Network)
jX
0
I
0
n
3Z
g
Ungrounded-Wye or Delta Connected
Generators
Synchronous Machines
(Zero Sequence Network)
jX
0
I
0
n
Two-Winding Transformers
(Positive Sequence Network)
Standard
Symbol
P
S
Z
PS
Equivalent
Positive-seq. Network
P
S
Three-Winding Transformers
(Positive Sequence Network)
s p ps
Z
Z
Z
=
+
Z
p=
(
Z
ps+
Z
pt−
Z
st)
2
1
P
T
Standard
Symbol
Z
pP
Z
SS
Z
tT
Equivalent Positive-Sequence
Network
S
Transformers
(Negative Sequence Network)
The negative-sequence network of
two-winding and three-two-winding transformers
are modeled in the same way as the
positive-sequence network since the
positive-sequence and negative-sequence
impedances of transformers are equal.
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
P
Q
S1 = 1 and S3 = 0
or
S2 = 1 or S4 = 0
Grounded wyeS1 = 0 and S3 = 1
or
DeltaSimplified Derivation of Transformer
Zero-Sequence Circuit Modeling
S1 = 1
S2 = 1
S3 = 0
S4 = 0
P
Q
Z
S1
S3
S4
S2
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
S1 = 1
S2 = 0
S3 = 0
S4 = 0
P
Q
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
S1 = 1
S2 = 0
S3 = 0
S4 = 1
P
Q
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
S1 = 0
S2 = 0
S3 = 0
S4 = 0
P
Q
Delta – Delta
Transformers
(Zero-Sequence Circuit Model)
Transformer
Connection
Zero-Sequence
Circuit Equivalent
P
Q
Z
PQP
Q
P
Q
Z
PQP
Q
Transformer
Connection
Q
P
Z
PQP
Q
Transformers
(Zero-Sequence Circuit Model)
Transformers
(Zero-Sequence Circuit Model)
Zero-Sequence
Circuit Equivalent
Transformer
Connection
Zero-Sequence
Circuit Equivalent
P
Q
R
Z
PZ
QZ
RP
R
Q
P
Q
Z
PQP
Q
Transformers
(Zero-Sequence Circuit Model)
Transformers
Transformer
Connection
Zero-Sequence
Circuit Equivalent
Z
PZ
QZ
RP
R
Q
P
Q
R
Z
PZ
QP
Q
Transformers
(Zero-Sequence Circuit Model)
Transformers
Transmission Lines
(Positive Sequence Network)
r
a
jX
L
Transmission Lines
(Negative Sequence Network)
The same model as the positive-sequence
network is used for transmission lines
inasmuch as the positive-sequence and
negative-sequence impedances of
Transmission Lines
(Zero Sequence Network)
The zero-sequence network model for a
transmission line is the same as that of the
positive- and negative-sequence networks.
The sequence impedance of the model is of
course the zero-sequence impedance of the
line. This is normally higher than the
positive- and negative-sequence impedances
because of the influence of the earth’s
Power System Short Circuit
Calculations
Classification of Power System
Short Circuits
Shunt Faults
Single line-to-ground faults
Double line-to-ground faults
Line-to-line faults
Series Faults
One-line open faults
Two-line open faults
Combination of Shunt and
Series Faults
Single line-to-ground and one-line open
Double line-to-ground and one-line open faults
Line-to-line and one-line open faults
Single line-to-ground and two-line open faults
Double line-to-ground and two-line open faults
Line-to-line and two-line open faults
Three-phase and two-line open faults
Combination of Shunt and
Series Faults
Balanced Faults
Symmetrical or
Three-Phase Faults
Derivation of Sequence Network Interconnections
Boundary conditions:
Zf Zf Vf System A System B A B C Va Vb Vc Ia Ib Ic ZfZ0 Ia0 Va0 + -Z1 Ea1 Ia1 Va1 + -Zf Z2 Ia2 Va2 + -Zf 1 1 1 2 1 0 1 1 1
,
0
Z
Z
Z
E
I
I
I
I
I
I
Z
Z
E
I
f a a f a a a f f a a f=
+
=
=
+
+
=
+
=
Unbalanced Faults
0
0
=
=
=
=
f a a c bZ
I
V
I
I
Zf System A System B A B C Va Vb Vc Ia Ib IcDerivation of Sequence Network Interconnections
f a a a a
Z
Z
Z
Z
E
I
I
I
3
2 1 0 1 2 1 0+
+
+
=
=
=
Z1 Ea1 Ia1 Va1 + -Z2 Ia2 Va2 + -Ia0 + 3Zf 0 1 2 1 0 1 0 1 2 1 0 2 1 2 1 0 1 2 1 00
3
3
2
0
Z
Z
and
Z
If
I
I
I
I
I
I
I
Z
Z
E
I
I
I
Z
Z
Z
Z
Z
E
I
I
I
Z
If
a a a a a a f a a a a a a a a f=
=
=
=
+
+
=
=
+
=
=
=
=
+
+
=
=
=
=
Unbalanced Faults
I
=
0
Zf System A System B A B C Va Vb Vc Ia Ib IcDerivation of Sequence Network Interconnections
1 1 1 2 1 2 1 1 1
2
0
Z
E
I
Z
Z
and
Z
If
Z
Z
Z
E
I
a a f f a a=
=
=
+
+
=
(
)
1
1
2
2
1
2
1
2
0
3
;
0
a
a
f
a
a
ao
a
a
a
c
b
f
I
j
I
a
a
I
I
I
I
aI
I
a
I
I
I
I
current
fault
The
−
=
−
=
−
=
=
+
+
=
−
=
=
Z1 Ea1 Ia1 Va1 + -Zf Z2 Ia2 Va2 + -Z0 Ia0 Va0 +-1
]
3
[
1
1
1
1
1
1
2
1
2
3
2
3
0
if
2
3
with
,
thus
φ
a
f
a
a
f
f
f
a
f
Z
E
I
Z
E
Z
E
j
I
Z
Z
Z
E
j
I
Z
Z
=
=
−
=
=
+
−
=
=
Unbalanced Faults
I
= 0
Eq’n BC-1
Zf System A System B A B C Va Vb Vc Ia Ib Ic Zf Vf Zg (Ib+Ic)Derivation of Sequence Network Interconnections
(
)(
)
g f g f f f a aZ
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
E
I
3
2
3
0 2 0 2 1 1 1+
+
+
+
+
+
+
+
=
Z0 Ia0 Va0 + -Z2 Ia2 Va2 + -Zf Zf Zf+3Zg Z1 Ea1 Ia1 Va1 +-Negative-sequence Component:
+
+
+
+
+
−
=
g f g f a aZ
Z
Z
Z
Z
Z
Z
I
I
3
2
3
0 2 0 1 2Zero-sequence Component:
The fault current
(
) (
)
(
)
(
)
( )
( )
(
)
2 2 1 2 0 2 2 1 0 2 1 2 02
a a a f a a a a a a c b fI
a
a
I
a
a
I
I
I
a
aI
I
aI
I
a
I
I
I
I
+
−
=
−
+
−
+
=
+
+
+
+
=
+
+
+
+
+
=
+
=
+
+
+
+
−
=
g f f a aZ
Z
Z
Z
Z
Z
I
I
3
2
0 2 2 1 0(
)
0
1
2
1
1
0
0
1
0
0
1
0
1
1
1
0
1
0
1
2
0
1
2
1
1
0
1
0
1
0
1
1
1
1
2
1
2
2
0
Z
Z
Z
E
Z
Z
Z
Z
Z
Z
Z
Z
Z
E
Z
Z
Z
I
I
Z
Z
Z
E
Z
Z
Z
Z
Z
Z
Z
E
I
Z
Z
and
Z
Z
If
a
a
a
a
a
a
a
g
f
+
−
=
+
+
+
−
=
+
−
=
+
+
=
+
+
=
=
=
=
0 1 1 0 1 1 1 0 1 1 1 0
Z
Z
Z
Z
Z
Z
E
Z
Z
Z
I
I
a a a
+
+
+
=
+
−
=
(
)
0 1 2 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 2 0 1 2 1 1 0 1 0 1 0 1 1 1 1 2 12
2
0
Z
Z
Z
E
Z
Z
Z
Z
Z
Z
Z
Z
Z
E
Z
Z
Z
I
I
Z
Z
Z
E
Z
Z
Z
Z
Z
Z
Z
E
I
Z
Z
and
Z
Z
If
a a a a a a a g f+
−
=
+
+
+
−
=
+
−
=
+
+
=
+
+
=
=
=
=
Voltage Rise Phenomenon
Unfaulted Phase B Voltage During Single
Line-to-Ground Faults
Unfaulted Phase B Voltage During Single
Line-to-Ground Faults
+
−
−
=
+
−
−
=
+
−
−
=
+
−
+
−
+
+
−
=
+
+
=
1 0 1 0 2 1 1 0 1 0 1 1 2 1 0 1 1 0 2 1 1 0 1 1 1 0 1 1 1 2 0 0 1 1 2 1 2 02
1
2
1
2
2
2
2
Z
Z
Z
Z
a
E
V
Z
Z
Z
Z
Z
Z
a
E
Z
Z
Z
Z
a
E
V
Z
Z
Z
E
a
Z
Z
Z
E
E
a
Z
Z
Z
E
V
aV
V
a
V
V
a b a a b a a a a b a a a bUnfaulted Phase B Voltage During
Single Line-to-Ground Faults
Neglecting resistances R0 & R1
0 5 10 15 20 25 30 35 40 45 50 -10 -6 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 0 1 2 5 9 25 45 65 85 P h a s e B V o lt a g e ( p .u .)
