SHORT CIRCUIT CALCULATIONS
REVISITED
FORTUNATO C. LEYNES
Chairman
Professional Regulatory
Board of Electrical Engineering
Professional Regulation Commission
Short Circuit Calculations
IIEE Presentation
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
Currents
Voltages
Impedances
Short Circuit Calculations
IIEE Presentation
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
Short Circuit Calculations
IIEE Presentation
Application of Fault Analysis
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
Short Circuit Calculations
IIEE Presentation
Application of Fault Analysis
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
Short Circuit Calculations
IIEE Presentation
Per Unit Calculations
Short Circuit Calculations
IIEE Presentation
Three-phase Systems
(base voltage, kVL− L ) × 1000
ZB =
base kVA3Φ
2
(base voltage, kVL− L )
ZB =
base MVA3Φ
Short Circuit Calculations
IIEE Presentation
2
Per Unit Quantities
I pu
actual current
=
Base Current ( I B )
actual voltage (kV )
V pu =
Base Voltage ( kVB )
Z pu
Short Circuit Calculations
IIEE Presentation
actual impedance
=
Base impedance (Z B )
Changing the Base of Per Unit
Quantities
Z pu[ old ]
actual impedance, Z (Ω)
=
2
(base kV[old ] ) ×1000
base kVA[ old ]
Z pu[ old ] (base kV[ old ] ) ×1000
2
Z (Ω ) =
base kVA[ old ]
(
base kV ) ×1000
=
2
Z B[ new]
[ new ]
base kVA[ new]
Z pu[ new]
Short Circuit Calculations
IIEE Presentation
Z (Ω )
=
Z B[ new]
Changing the Base of Per
Unit Quantities
Z pu [ old ]
actual impedance, Z (Ω)
=
(base kV[old ] )2 × 1000
base kVA[ old ]
Z pu[ old ] (base kV[ old ] ) ×1000
2
Z ( Ω) =
base kVA[ old ]
(
base kV ) ×1000
=
2
Z B[ new]
[ new ]
base kVA[ new]
Z pu[ new]
Short Circuit Calculations
IIEE Presentation
Z ( Ω)
=
Z B[ new]
kVA Base for Motors
kVA/hp
1.00
Induction < 100 hp
1.00
Synchronous 0.8 pf
0.95
Induction 100 < 999 hp
0.90
Induction > 1000 hp
0.80
Synchronous 1.0 pf
Short Circuit Calculations
IIEE Presentation
hp rating
SYMMETRICAL
COMPONENTS
Short Circuit Calculations
IIEE Presentation
Va 2
Va1
Vc1
Vb 2
Vb1
Positive Sequence
Vc 2
Negative Sequence
Va 0
Va 0 = Vb 0 = Vc 0
Va
Vc 0
Vc1
Vc
Va 2
Va1
Vc 2
Vb0
Vb
Vb1
Zero Sequence
Short Circuit Calculations
IIEE Presentation
Vb2
Unbalanced Phasors
Symmetrical Components of
Unbalanced Three-phase Phasor
Va = Va 0 + Va1 + Va 2
Vb = Va 0 + a Va1 + aVa 2
2
Vc = Va 0 + aVa1 + a Va 2
2
Short Circuit Calculations
IIEE Presentation
1
Va 0 = (Va + Vb + Vc )
3
1
Va1 = (Va + aVb + a 2Vc )
3
1
Va 2 = (Va + a 2Vb + aVc )
3
Symmetrical Components of
Unbalanced Three-phase Phasor
In matrix form:
Va  1 1 1  Va 0 
 
  
2
V
=
1
a
a
V
b
  a1 
  
Vc  1 a a 2  Va 2 
Short Circuit Calculations
IIEE Presentation
1 1 1  Va 
Va 0 
V  = 1 1 a a 2  V 
 b 
 a1  3 
1 a 2 a  Vc 
Va 2 


Power System Short Circuit
Calculations
Sequence Networks
Short Circuit Calculations
IIEE Presentation
Fault Point
The fault point of a system is that point to
which the unbalanced connection is
attached to an otherwise balanced system.
Short Circuit Calculations
IIEE Presentation
Definition of Sequence
Networks
Positive-sequence Network
Ea1 =
Thevenin’s equivalent
voltage as seen at the fault
point
Z1 =
Thevenin’s equivalent
impedance as seen from
the fault point
Va1 = Ea1 − I a1Z1
Short Circuit Calculations
IIEE Presentation
Ia1
+
Z1
Ea1
Va1
-
Definition of Sequence
Networks
Negative-sequence Network
Z2 =
Thevenin’s equivalent
negative-sequence
impedance as seen at
the fault point
Ia2
+
Va 2 = − I a 2 Z 2
Short Circuit Calculations
IIEE Presentation
Z2
Va2
-
Definition of Sequence
Networks
Zero-sequence Network
Z0 =
Thevenin’s equivalent
zero-sequence impedance
as seen at the fault point
Va 0 = − I a 0 Z 0
Ia0
+
Z0
Va0
-
Short Circuit Calculations
IIEE Presentation
Power System Short Circuit
Calculations
Sequence Network Models of
Power System Components
Short Circuit Calculations
IIEE Presentation
Synchronous Machines
(Positive Sequence Network)
Z1 = jx' 'd
Short Circuit Calculations
IIEE Presentation
Synchronous Machines
(Negative Sequence Network)
Ia2
x ' 'd + x ' 'q
Z2 = j
2
+
Z2
Va2
-
Where:
x”d = direct-axis sub-transient reactance
x”q = quadrature-axis sub-transient reactance
Short Circuit Calculations
IIEE Presentation
Synchronous Machines
(Zero Sequence Network)
Solidly-Grounded Neutral
jX0
n
I0
ground
Short Circuit Calculations
IIEE Presentation
Synchronous Machines
(Zero Sequence Network)
Impedance-Grounded Neutral
jX0
n
3Zg
ground
Short Circuit Calculations
IIEE Presentation
I0
Synchronous Machines
(Zero Sequence Network)
Ungrounded-Wye or Delta Connected
Generators
jX0
n
I0
ground
Short Circuit Calculations
IIEE Presentation
Two-Winding Transformers
(Positive Sequence Network)
P
S
P
S
ZPS
Standard
Symbol
Short Circuit Calculations
IIEE Presentation
Equivalent
Positive-seq. Network
Three-Winding Transformers
(Positive Sequence Network)
ZS
Zp
S
P
T
Standard
Symbol
Z ps = Z p + Z s
Z pt = Z p + Z t
Z st = Z s + Z t
Short Circuit Calculations
IIEE Presentation
S
P
Zt
Equivalent Positive-Sequence
Network
1
Z p = (Z ps + Z pt − Z st )
2
1
Z s = (Z ps + Z st − Z pt )
2
1
Z t = (Z pt + Z st − Z ps )
2
T
Transformers
(Negative Sequence Network)
The negative-sequence network of twowinding and three-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.
Short Circuit Calculations
IIEE Presentation
Simplified Derivation of Transformer ZeroSequence Circuit Modeling
(Thanks to Engr. Antonio C. Coronel, Retired VP, Meralco, an d form er m em ber, Board of Electrical Engineering)
P
Q
Grounded wye
S1 = 1 and S3 = 0
or
S2 = 1 or S4 = 0
Delta
S1 = 0 and S3 = 1
or
S2 = 0 and S4 = 1
Ungrounded wye
S1 = 0 and S3 = 0
or
S2 = 0 and S4 = 0
Short Circuit Calculations
IIEE Presentation
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
Grounded wye – Grounded wye
S1 = 1
S2 = 1
S3 = 0
S4 = 0
Z
P
S1
S2
S3
Short Circuit Calculations
IIEE Presentation
S4
Q
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
Grounded wye – Ungrounded wye
S1 = 1
S2 = 0
S3 = 0
S4 = 0
P
Short Circuit Calculations
IIEE Presentation
Q
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
Grounded wye – Delta
S1 = 1
S2 = 0
S3 = 0
S4 = 1
P
Short Circuit Calculations
IIEE Presentation
Q
Simplified Derivation of Transformer
Zero-Sequence Circuit Modeling
Delta – Delta
S1 = 0
S2 = 0
S3 = 0
S4 = 0
P
Short Circuit Calculations
IIEE Presentation
Q
Transformers
(Zero-Sequence Circuit Model)
Zero-Sequence
Circuit Equivalent
Transformer
Connection
P
P
Q
Q
ZPQ
P
P
Short Circuit Calculations
IIEE Presentation
Q
Q
ZPQ
Transformers
(Zero-Sequence Circuit Model)
Zero-Sequence
Circuit Equivalent
Transformer
Connection
P
P
Q
Q
ZPQ
P
P
Short Circuit Calculations
IIEE Presentation
Q
Q
ZPQ
Transformers
(Zero-Sequence Circuit Model)
Zero-Sequence
Circuit Equivalent
Transformer
Connection
P
P
Q
ZPQ
P
P
IIEE Presentation
ZP
Q
R
Short Circuit Calculations
Q
ZQ
ZR
R
Q
Transformers
(Zero-Sequence Circuit Model)
Zero-Sequence
Circuit Equivalent
Transformer
Connection
ZP
P
P
ZQ
Q
ZQ
Q
Q
ZR
R
R
P
P
Q
R
Short Circuit Calculations
IIEE Presentation
ZP
ZR
R
Transmission Lines
(Positive Sequence Network)
Z1
ra
Short Circuit Calculations
IIEE Presentation
jXL
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 are the same
Short Circuit Calculations
IIEE Presentation
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
resistivity and the ground wire/s.
Short Circuit Calculations
IIEE Presentation
Power System Short Circuit
Calculations
Classification of Power System
Short Circuits
Short Circuit Calculations
IIEE Presentation
Shunt Faults
Single line-to-ground faults
Double line-to-ground faults
Line-to-line faults
Three-phase faults
Short Circuit Calculations
IIEE Presentation
Series Faults
One-line open faults
Two-line open faults
Short Circuit Calculations
IIEE Presentation
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
Three-phase and one-line open faults
Short Circuit Calculations
IIEE Presentation
Combination of Shunt and
Series 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
Short Circuit Calculations
IIEE Presentation
Balanced Faults
Symmetrical or
Three-Phase Faults
Short Circuit Calculations
IIEE Presentation
Derivation of Sequence Network Interconnections
A
Ia
B
Ib
C
Ic
System A
Zf
Va
Vb
Zf
System B
Zf
Vc
Vf
Boundary conditions:
Ia + Ib + Ic = 0
VF = Va − I a Z f = Vb − I b Z f = Vc − I c Z f
Short Circuit Calculations
IIEE Presentation
Eq’n (1)
Eq’n (2)
Zf
Zf
Ia1
Ia0
I a2
+
Z1
Ea1
+
Z2
Va1
Va2
-
-
Ea1
I a1 =
Z1 + Z f
I f = I a 0 + I a1 + I a 2
Ea1
I f = I a1 =
Z1 + Z f
Zf = 0,
Short Circuit Calculations
IIEE Presentation
Ea 1
I f = I a1 =
Z1
+
Z0
Va0
-
Unbalanced Faults
Single Line-to-Ground Faults
Short Circuit Calculations
IIEE Presentation
Derivation of Sequence Network Interconnections
A
Ia
B
Ib
Ic
System A
Zf
Va
Vb
Vc
Boundary conditions:
Ib = Ic = 0
Va = I a Z f = 0
Short Circuit Calculations
IIEE Presentation
C
System B
Ia1
I a 0 = I a1 = I a 2
+
Z1
Ea1
Va1
If Z f = 0
-
Ia2
+
Z2
Ea1
=
Z 0 + Z1 + Z 2 + 3Z f
3Zf
I a 0 = I a1 = I a 2
E a1
=
Z 0 + Z1 + Z 2
I a 0 = I a1 = I a 2
Ea1
=
Z 0 + 2 Z1
Z1 = Z 2
Va2
-
Ia0
I f = I a = I a 0 + I a1 + I a 2 = 3I a1 = 3I a 0
+
Z0
Va0
If Z f = 0 and Z1 = Z 2
-
3Ea1
I f = Ia =
Z 0 + 2 Z1
Short Circuit Calculations
IIEE Presentation
Unbalanced Faults
Line-to-Line Faults
Short Circuit Calculations
IIEE Presentation
Derivation of Sequence Network Interconnections
A
Ia
B
Ib
Ic
System A
Va
Vb
Vc
Zf
Boundary conditions:
Ia = 0
Ib = −Ic
Vb − I b Z f = Vc ; or Vb − Vc = I b Z f
Short Circuit Calculations
IIEE Presentation
C
System B
Zf
Ia1
Ia2
Ia0
+
Z1
Ea1
+
Va1
Z2
+
Z0
Va2
-
Va0
-
-
E a1
I a1 =
Z1 + Z 2 + Z f
If Z f = 0 and Z1 = Z 2
E a1
I a1 =
2 Z1
Short Circuit Calculations
IIEE Presentation
The fault current
I f = Ib = −I c = I a 0 + a 2 I a1 + aI a 2
I ao = 0; I a1 = − I a 2
(
)
I f = a 2 − a I a1 = − j 3I a1
thus, with Z1 = Z 2
 Ea 1
I f = − j 3
 2Z1 + Z f
if Z f = 0



3 Ea1  3  Ea1

If = − j
= 
 Z
2Z 1
2

 1
Ea1
I f [3φ ] =
Z1
I f [ L −L ]
Short Circuit Calculations
IIEE Presentation
3
=
I f [3φ ]
2
Unbalanced Faults
Double-to-line Ground Fault
Short Circuit Calculations
IIEE Presentation
Derivation of Sequence Network Interconnections
A
Ia
B
Ib
C
Ic
System A
Zf
Va
Vb
Zf
Vc
Vf
Zg
(Ib+Ic)
Boundary conditions:
Ia = 0
Vb = I b Z f + (I b + I c )Z g
Eq’n BC-2
Vc = I c Z f + (I b + I c )Z g
Eq’n BC-3
Short Circuit Calculations
IIEE Presentation
Eq’n BC-1
System B
Zf
Zf
Z f+3Z g
Ia1
Ia2
Ia0
+
Z1
+
Z2
Ea1
Va1
Va2
-
-
I a1 =
Z1 + Z f +
Short Circuit Calculations
IIEE Presentation
Ea1
(Z2 + Z f )(Z 0 + Z f + 3Z g )
Z 2 + Z 0 + 2Z f + 3Z g
+
Z0
V a0
-
Negative-sequence Component:
 Z 0 + Z f + 3Z g


= − I a1 

Z
+
Z
+
2
Z
+
3
Z
0
f
g 
 2
Ia 2
Zero-sequence Component:
Ia0


Z2 + Z f

= − I a1 
 Z + Z + 2 Z + 3Z 
0
f
g 
 2
The fault current
I f = I b + I c = (I a 0 + a 2 I a1 + aI a 2 ) + (I a 0 + aI a1 + a 2 I a 2 )
I f = 2 I a 0 + (a 2 + a )I a1 + (a + a 2 )I a 2
I f = 2 I a 0 + (− 1)I a1 + (− 1)I a 2 = 2 I a 0 − (I a1 + I a 2 )
but I a 0 + I a1 + I a 2 = 0; or I a 0 = −(I a1 + I a 2 )
thus, I f = 3I a 0
Short Circuit Calculations
IIEE Presentation
If Z f = Z g = 0 and Z1 = Z 2
Ea 1
(
Z1 + Z 0 )Ea1
I a1 =
= 2
Z1Z 0
Z
+
2
Z
Z
1
1
0
Z1 +
Z1 + Z 0
I a2


 Z0 
Ea1

 = −
= − I a1 

Z1Z 0
 Z1 + Z 0 
 Z1 + Z + Z

1
0
Z 0 Ea1
− 2
Z1 + 2 Z1Z 0
Short Circuit Calculations
IIEE Presentation


 Z 0  =
 Z1 + Z 0 


If Z f = Z g = 0 and Z1 = Z 2
I a1 =
Ia2
Ea 1
(Z + Z 0 )Ea1
= 12
ZZ
Z1 + 2 Z1Z 0
Z1 + 1 0
Z1 + Z 0


 Z0 
Ea1



= − I a1
=−


Z1Z0
 Z1 + Z0 
 Z1 + Z + Z
1
0

Z 0 Ea 1
− 2
Z1 + 2 Z1Z 0


 Z 0  =
 Z1 + Z 0 






 Z1  
Ea1
 Z1 
I a 0 = −I a1 
 =


 Z1 + Z 0   Z + Z1 Z0  Z1 + Z 0 
 1 Z +Z 

1
0 
ZE
Ea1
= 2 1 a1 =
Z1 + 2 Z1 Z 0 Z1 + 2 Z 0
I f = 3I a 0 =
Short Circuit Calculations
IIEE Presentation
3 Ea1
Z1 + 2 Z 0
Voltage Rise Phenomenon
Single-to-line Ground Fault
Short Circuit Calculations
IIEE Presentation
Unfaulted Phase B Voltage During Single
Line-to-Ground Faults
Vb = Va0 + a 2Va1 + aVa 2

 Ea 1 
 E a1    Ea1 
2
Z 0 + a  Ea1 − 
Z1  − a
Z1
Vb = −
 2 Z1 + Z 0 
 2 Z1 + Z 0    2 Z1 + Z 0 


 Z0

−
1



 2  Z 0 − Z1  


Z
Z

 = Ea1 a 2 −  1  1
Vb = Ea1 a − 


 2 Z1 + Z 0  
 Z1  2 + Z 0 




Z

1 


 Z0

−
1



Z

Vb = Ea1 a 2 −  1

Z 0 

2
+



Z

1


neglecting resistances, R 1 and R 0 ;

 X0

−
1



X

Vb = Ea1 a 2 −  1

X 0 

Short Circuit Calculations
 2 + X 

IIEE Presentation

1 

Unfaulted Phase B Voltage During
Single Line-to-Ground Faults
Neglecting resistances R0 & R1
50
45
40
Phase B Voltage (p.u.)
35
30
25
20
15
10
5
0
-10
-6
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
X0/X1
Short Circuit Calculations
IIEE Presentation
-1
0
1
2
5
9
25
45
65
85
Fault MVA
MVAF = I F × MVAbase
where, for E
a1
= 1.0 p.u.;
for three − phase fault in p.u. :
I F ( 3φ )
1
=
Z1
for singe line - to - ground fault in p.u. :
I F ( SLG )
Short Circuit Calculations
IIEE Presentation
3
=
Z 0 + 2Z1
Fault MVA
Three − phase fault MVA :
MVAF ( 3φ ) = I F (3φ ) ( p.u.) × MVAbase
Z1 =
1
p.u.
I F (3φ )
Single line − to − ground fault MVA :
MVAF ( SLG ) = I F ( SLG ) ( p.u.) × MVAbase
2 Z1 + Z 0 =
Z0 =
Short Circuit Calculations
IIEE Presentation
3
I F ( SLG )
3
I F ( SLG )
− 2Z 1
p.u.
Assumptions Made to Simplify
Fault Calculations
1. Pre-fault load currents are neglected.
2. Pre-fault voltages are assumed equal to 1.0
per unit.
3. Resistances are neglected (only for 115kV &
up).
4. Mutual impedances, when not appreciable are
neglected.
5. Off-nominal transformer taps are equal to 1.0
per unit.
6. Positive- and negative-sequence impedances
are equal.
Short Circuit Calculations
IIEE Presentation
Outline of Procedures for
Short Circuit Calculations
1 Setup the network impedances expressed in per
unit on a common MVA base in the form of a
single-line diagram
2 Determine the single equivalent (Thevenin’s)
impedance of each sequence network.
3 Determine the distribution factor giving the
current in the individual branches for unit total
sequence current.
Short Circuit Calculations
IIEE Presentation
Outline of Procedures for
Short Circuit Calculations
4 Interconnect the three sequence networks for the
type of fault under considerations and calculate
the sequence currents at the fault point.
5 Determine the sequence current distribution by
the application of the distribution factors to the
sequence currents at the fault point
6 Synthesize the phase currents from the sequence
currents.
Short Circuit Calculations
IIEE Presentation
Outline of Procedures for
Short Circuit Calculations
7 Determine the sequence voltages throughout the
networks from the sequence current distribution
and branch impedances
8 Synthesize the phase voltages from the sequence
voltage components
9 Convert the pre unit currents and voltages to
actual physical units
Short Circuit Calculations
IIEE Presentation
CIRCUIT BREAKING SIZING
(Asymmetrical Rating Factors)
Momentary Rating
Multiplying Factor = 1.6
Interrupting Rating
Multiplying Factor
8 cycles
5 cycles
3 cycles
1 ½ cycles
Short Circuit Calculations
IIEE Presentation
= 1.0
= 1.1
= 1.2
= 1.5
EXAMPLE PROBLEM
In the power system
shown, determine the
momentary and
interrupting ratings for
primary and secondary
circuit breakers of
transformer T2.
Short Circuit Calculations
IIEE Presentation
Solution:
Equivalent 69kV system@ 100MVA :
T1 :
 800 
I F (3 φ ) = 
= 8 pu

100 
1000 
I F ( SLG ) = 
= 10 pu

 100 
1
x1 = = 0.125 pu
8
3
x0 = − 2 × 0.125 = 0.05 pu
10
100 
x = 0 .08
= 0.2667 pu

 30 
G1 :
100 
x" = 0.10 
= 0.40 pu

 25 
100 
x0 = 0.06 
= 0.24 pu

 25 
T2, T3, T3 :
100 
x = 0 .06 
= 0.80 pu

 75 
M1, M2 : kVAB = 0.9 × 5000 = 4500
or MVAB = 4.5
Short Circuit Calculations
IIEE Presentation
100 
x = 0 .15
= 3 .3333 pu

 4.5 
Solution:
Positive sequence network:
Short Circuit Calculations
IIEE Presentation
Solution:
Zero sequence network:
Short Circuit Calculations
IIEE Presentation
Improper Sequence Network Models
Short Circuit Calculations
IIEE Presentation
Improper Sequence Network Models
Short Circuit Calculations
IIEE Presentation
Improper Sequence Network Models
Short Circuit Calculations
IIEE Presentation
Incorrect Sequence Network Models
Short Circuit Calculations
IIEE Presentation
QUESTIONS?
Short Circuit Calculations
IIEE Presentation
Short Circuit Calculations
IIEE Presentation