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4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 1
4.1. Types of electrical fault Electrical fault is the deviation of voltages and currents from nominal values or states.
Under normal operating conditions, power system equipment or lines carry normal
voltages and currents which results in a safer operation of the system. But when fault
occurs, it causes excessively high currents to flow which causes the damage to
equipments and devices.
Causes of electrical faults are
o Weather conditions
Lightning strikes
Heavy rains
Heavy winds
Salt deposition
Snow and ice accumulation
o Equipment failures
Malfunctioning
Ageing
Insulation failure of cables and winding
o Human errors
Improper rating of equipment or devices
Forgetting metallic or electrical conducting parts after servicing or maintenance
Switching the circuit while it is under servicing
o Smoke of fires
Ionization of air, due to smoke particles, surrounding the overhead lines results in
spark between the lines or between conductors to insulator
Electrical faults are classified as
o Short circuit faults: A short circuit fault is an abnormal connection of very low
impedance between two points of different potential. These are the most common and
severe kind of faults, resulting in the flow of abnormal high currents through the
equipment or transmission lines. Short circuit faults are also called as shunt faults.
Symmetrical faults: These are very severe faults and occur infrequently in the
power systems. It affects each of the three phases equally. In transmission line
faults, roughly 5% are symmetric.
Line to line to line to ground (L-L-L-G)
Line to line to line (L-L-L)
Unsymmetrical faults: These are very common and less severe than symmetrical
faults. It does not affect each of the three phases equally. In transmission line
faults, roughly 95% are unsymmetrical.
Line to ground (L-G)
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 2
Line to line (L-L)
Line to line to ground (L-L-G)
o Open circuit faults: These faults occur due to the failure of one or more conductors.
Open circuit faults are also called as series faults. These are unsymmetrical or
unbalanced type of faults except three phase open fault.
One conductor open
Two conductor open
o Transient fault: A transient fault is a fault that is no longer present if power is
disconnected for a short time and then restored. Basically, transients are momentary
changes in voltage or current that occur over a short period of time i.e. interval is
usually described as approximately one sixteenth of a voltage cycle.
Electrical faults on power system has very ill effect such as
o Abnormal operation of the system
o Danger fatal accidents
o Insulation failures of the equipment
o Fire and explosion
o Equipment overheating
o Low system voltages
o Restricted power flow
o Complete blackout
4.2. Transient on a transmission line
In power system, selection of switchgear not only depends on rated current but it also
depends on momentary maximum current that supposed to flow under abnormal or
short circuit condition.
Hence it becomes important to find out initial maximum current at the time of short
circuit.
Consider a line fed from constant voltage source and short circuit takes place when line
is unloaded.
For simplification line capacitance is neglected hence analysis of the initial maximum
current is the study of transients in R-L series circuit.
Let, a series R-L circuit is connected to the source at the instant t=0.
( )
( )
m
m
div Ri L
dt
diV Sin t Ri L
dt
Vdi Ri Sin t
dt L L
The corresponding characteristic equation is 0R
sL
and the solution of this equation
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 3
has two parts, complementary function and particular integral.
The complementary function is,Rt
Lci ke
The particular integral is, ( ) ( )pi ACos t BSin t
The constants A and B undefined and can be found out by putting the value of ip in original
equation.
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
m
m
m
dV Sin t R ACos t BSin t L ACos t BSin t
dt
V Sin t RACos t RBSin t LASin t LBCos t
V Sin t RB LA Sin t RA LB Cos t
Comparing the equation on both the side
& 0mV RB LA RA LB
Solving both the equation, value of A and B can be
2 2 2 2 2 2&m mLV RV
A BR L R L
Substituting the value of A and B in the equation
2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 2 2
2 2 2
( ) ( )
( ) ( )
( ) ( )
( )
m mp
m
m
m
LV RVi Cos t Sin t
R L R L
V L RCos t Sin t
R L R L R L
VSin t Cos Cos t Sin
R L
VSin t
R L
Hence total solution is
2 2 2( )
Rt
mLc p
Vi i i ke Sin t
R L
The constant k can be evaluated by knowing the initial condition. When switch is closed
at t=0, the initial current in inductor is zero i.e. i(0-) = i(0)=i(0+)=0.
0
2 2 2
2 2 2
0 ( 0 )
( )
R
mL
m
Vke Sin
R L
Vk Sin
R L
By putting the value of k, the solution of R-L series circuit is
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 4
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2
( ) ( )
( ) ( )
( ) ( )
Rt
m mL
Rt
m m L
Rt
m L
V Vi Sin e Sin t
R L R L
V VSin t Sin e
R L R L
VSin t Sin e
R L
In above equation has two responses
( ) Steady state response
( ) Transient responseRt
L
Sin t
Sin e
Sinusoidal steady state current is called symmetrical short circuit current and
unidirectional transient current is called DC off set current.
2 2 2
2 2 2
( )
( )
ms
Rt
m Lt
Vi Sin t
R L
Vi Sin e
R L
The maximum momentary short circuit current at the instant of first peak (decay of
transient current in this short time is neglected and transmission line resistance is very
small i.e. =90°) is
2 2 2 2 2 2
m mmm
V Vi Cos
R L R L
The maximum possible value of momentary short circuit current occurs when voltage
wave is passing through zero i.e. α=0.
max 2 2 2 2 2 2
2 2 2
2 2 2
0
2
22
m mmm
m
V Vi Cos
R L R L
V
R L
V
R L
Above equation states that maximum possible momentary short circuit current value is
twice the maximum of symmetrical short circuit current i.e. doubling effect.
Usually selection of switchgear ratings are carried out on the bases of maximum possible
momentary short circuit current as safe choice.
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 5
Vis
t
t
it
i = is + it
t
Maximum Momentary Current, imm
Modern day switchgears are designed to interrupt the current in first few cycle it means
when short circuit current is interrupted, DC off set current has not deceased and so it
contributes to the current to be interrupted.
For the contribution, amount of DC off set current is not calculated but symmetrical short
circuit current alone is calculated and then the value is increased by multiplying factor to
account for DC off set current.
Switchgear operation speed Multiplying factor
8 cycles or slower 1.0
5 cycles 1.1
3 cycles 1.2
2 cycles 1.4
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 6
4.3. Short circuit on synchronous machine when unloaded Under steady state short circuit condition, the armature reaction of a synchronous
generator produces a demagnetizing flux and this effect is modelled as a reactance Xa in
series with the induced emf.
The total reactance leakage reactance Xl and armature reactance Xa of the machine is
called synchronous reactance Xd.
When sudden three phase short circuit occurs at the unloaded synchronous generator i.e.
open circuit condition, it undergoes a transient in all three phases by gradually reaching
to steady state condition.
The circuit breaker has to interrupt the short circuit current before steady state
condition.
When fault occurs, immediately a DC off set current appears in all three phase, each with
different magnitude because each voltage wave has different magnitude at the time of
occurrence of fault.
At the instant of short circuit, current is limited by leakage reactance of machine. Since
airgap flux cannot change instantaneously to counter the demagnetization of armature
short circuit current, current appears in the field winding as well as in damper winding
in the direction of main flux.
This current decays depending upon the time constants of winding. Time constant of
damper wing is much less than the field winding, thus during initial short circuit, damper
winding reactance and field winding reactance appears in parallel with armature
reactance.
During middle period of short circuit damper winding current dies out and it becomes
open circuit. At this instant only field winding reactance appears in parallel with
armature reactance.
During finally steady state period of short circuit, field winding current dies out and it
also becomes open circuit. At this instant only armature reactance appears.
o The reactance of synchronous machine appeared in initial short circuit is called sub
transient reactance (Xd’’).
o The reactance of synchronous machine appeared in middle of short circuit is called
transient reactance (Xd’).
o The reactance of synchronous machine appeared in end of short circuit is called
synchronous reactance (Xd).
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 7
Xf
+
Eg
Xl
Direct axis subtransient reactance
Xdw
Xa
+
Eg
Xl
Direct axis transient reactance
Xf
Xa
Xa
+
Eg
Xl
Xd
Synchronous Reactance
" "
"
' '
'
1
1 1 1
1
1 1
1
1
g
d l
d
a f dw
g
d l
d
a f
g
d l l a
d
a
EX X I
XX X X
EX X I
XX X
EX X X X I
XX
Synchronous machine is offering time varying reactance from Xd” to Xd’ and finally Xd such
that Xd” < Xd’ < Xd.
4.4. Short circuit on synchronous machine when loaded Analysis of short circuit on synchronous machine when loaded is too complicated, hence
several methods are used to carry out short circuit analysis.
Let, synchronous generator operating under steady conditions.
0
0
"
'
Load current
Bus terminal voltage
Induced emf
Direct axis synchronous reactance
Voltage behind subtransient reactance
Voltage behind transient reactance
g
d
g
g
I
V
E
X
E
E
When short circuit occurs at the synchronous generator terminal, induced emf can be
modelled as
" 0 0 "
' 0 0 '
g d
g d
E V jI X
E V jI X
For synchronous motor internal emf and reactances are similar to that of synchronous
generator except that the direction of current is reversed.
" 0 0 "
' 0 0 '
m d
m d
E V jI X
E V jI X
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 8
In case of short circuit of an interconnected system, the synchronous machines i.e.
generators and motors are replaced by their corresponding circuit models having voltage
behind sub transient/ transient reactance in series with sub transient/transient
reactance. The rest of the network being passive remains unchanged.
4.5. Short circuit current calculation using Thevenin Theorem
Analysis of short circuit on synchronous machine through Thevenin theorem is faster and
easy for large network.
Consider a synchronous generator feeding a synchronous motor through transmission
line under steady load condition. A symmetrical fault is made at F, at the motor terminals.
For fault analysis, synchronous machines are represented by their sub transient
reactances or transient reactances if transient currents are of interest in series with
voltages behind sub transient reactances. This change does not disturb the prefault
current I0 and prefault voltage V0 at F.
+
I0
+
X
V0
F
G
+
I0
+
X
V0
F
G
Xdg
Eg
Xdm
Em
X dg
E g
X dm
E m
Thevenin equivalent circuit looking from FG
XF
G
V0
XF
G
V0+
X dg X dm X dg X dm
DIg DIm
If
Zf
In a case of fault at F through an impedance Zf.
0 0
'' ''
'' ''
f
fTH dm dg f
dm dg
V VI
jX Z X X Xj Z
X X X
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 9
Current through generator and motor circuit
''
'' ''
''
'' ''
fdmg
dg dm
dg fm
dg dm
XI I
X X X
X XI I
X X X
D
D
Postfault currents and voltages
'' 0
'' 0
0 0
g g
m m
f fTH
I I I
I I I
V V jX I V V
D
D
D
So, the prefault current flowing out of the fault point F is always zero and the postfault
current out of F is independent of load for a given prefault voltage at F.
Steps for solving short circuit current by thevenin theorem approach
o Obtain the steady state solution using load flow and formulate the circuit model.
o Replace reactance of synchronous machine by their transient or sub transient values.
o Short Circuit all emf sources and find the value of or TH THZ X .
o Compute the required currents using Thevenin’s theorem.
Assumptions
o All prefault voltage equal to 1pu.
o Load current neglected as it is very less as compared to short circuit current.
4.6. Criteria for selection of circuit breaker Usually selection of switchgear ratings are carried out on the bases of maximum possible
momentary short circuit current as safe choice.
Modern day switchgears are designed to interrupt the current in first few cycle it means
when short circuit current is interrupted, DC off set current has not deceased and so it
contributes to the current to be interrupted.
For the contribution, amount of DC off set current is not calculated but symmetrical short
circuit current alone is calculated and then the value is increased by multiplying factor to
account for DC off set current.
Switchgear operation speed Multiplying factor
8 cycles or slower 1.0
5 cycles 1.1
3 cycles 1.2
2 cycles 1.4
If short circuit MVA is greater than 500, than above value is increase by 0.1 each.
Current that a circuit breaker interrupts is inversely proportional to the operating voltage
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 10
over a certain range. Of course, operating voltage is not allowed to exceed maximum
design value.
No matter how voltage is low, rated interrupting current should not exceed rated
maximum interrupting current i.e. over this range of voltage, product of operating voltage
and interrupting current is constant.
Rated voltageAmperes at operating voltage = Ampere at rated voltage ×
Operating voltage
line linerated rated interrupting currentRated interrupting MVA = 3 V I
Short circuit MVA = 3 × Pre fault line voltage (kV) × Short circuit current (kA)
For the selection of a circuit breaker, a maximum possible short circuit MVA to be
interrupted with respect to type and location of fault and generating capacity connected
to the system is required to calculate.
A three phase fault i.e. very rare is generally the one that has highest short circuit MVA
and a circuit breaker must be capable to interrupt it.
4.7. Algorithm for short circuit studies It is always necessary to develop a systematic approach to carry out short circuit
calculation of large system.
Consider, n-bus system operating under steady load. It is required to first calculate,
prefault voltages at all buses and current through line and rth bus gets faulty through fault
impedance Zf and hence the voltage at each bust is about to change.
G2G1 Gr Gn
System
1
2
n
r
System
1
2
n
r
X dnX drX d2X d1
V0r
+
-
Zf
If
0 Prefault bus voltages vector
Postfault bus voltages vector
V Changes in bus voltages vector due to fault
Bus impedance matrix of Thevenin's network
Bus current injection vector
Faul
f
f
BUS
fBUS
BUS
V
V
Z
J
Z
D
0
t impedance vector
Prefault current in line vector
Postfault current in line vector
BUS
fBUS
I
I
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 11
Step-1: Calculate prefault voltages and postfault bus voltages at all buses
01
00 2
0
0
BUS
n
fBUS BUS
V
VV
V
V V V
D
Step-2: Draw Thevenin’s equivalent circuit i.e. short circuit all emf sources and replace all
reactances by their respective transient reactances transient or sub transient reactances.
Step-3: Calculate DV
1
211 1
1
11 1
1
0
0
0
f
BUS
f
f
n
fr
n nn
fn
n
fr
n nn
frr
V Z J
I
IZ Z
I
Z Z
I
Z Z
V I
Z Z
Z I
D
D
Step-4: Calculate the postfault voltage for the rth bus
0 0
0
0
0
fr r r
fr rr
f f fr rr
f rf
rr
V V V
V Z I
Z I V Z I
VI
Z Z
D
Step-5: Calculate the postfault voltage for any ith bus
0 0
0
00
0 0
fi i i
fi ir
ri ir f
rr
iri rf
rr
V V V
V Z I
VV Z
Z Z
ZV V
Z Z
D
For i = r, post fault voltage at ith bus
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 12
0 0
0 0
0
0
f irr i rf
rr
rrr rf
rr
frr rr
rfrr
f
rfrr
ZV V V
Z Z
ZV V
Z Z
Z Z ZV
Z Z
ZV
Z Z
Step-6: Find the post fault current in lines
f f fij ij i jI Y V V
4.8. ZBUS formation by step by step method It is a step by step modelling technique that proceeds branch by branch. It is more
advantages because any modification in the network does not require complete
reconstruction of ZBUS
Let us consider that a ZBUS of a certain system is already constructed and at later stage
another branch is added.
Due to addition of a new branch, below mentioned situation can be possible.
Type – 1 modification: Branch is added between new bus and reference bus
It is the modification where Zb is added from a new bus to the reference bus i.e. a new
branch is added and dimension of ZBUS goes up by one.
Consider a passive n-bus network in which branch with impedance Zb is added to the new
bus k and reference bus r.
Passive Linear n-bus
network
1
n
i
j
k
rVk
Ik
Zb
0; 1,2,3, ,
k k k
ki ik
kk b
V Z I
Z Z i n
Z Z
0
0 0
BUS
BUS
b
Z old
Z new
Z
Type – 2 modification: Branch is added between new bus and old bus
It is the modification where Zb is added from a new bus to an old bus i.e. a new branch is
added and dimension of ZBUS goes up by one.
Consider a passive n-bus network in which branch with impedance Zb is added from new
bus k to old bus j.
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 13
Passive Linear n-bus
network
1
n
i
j
k
r
Ij
Ik
Vk
Ij+Ik
1 1 2 2
1 1 2 2
k b k j
b k j j jj j k jn n
j j jj j jn n b jj k
V Z I V
Z I Z I Z I Z I I Z I
Z I Z I Z I Z I Z Z I
1
2
1 2
j
BUS j
BUS
jn
j j jn jj b
Z
Z old Z
Z new
Z
Z Z Z Z Z
Type – 3 modification: Branch is added between old bus and reference bus
It is the modification where Zb is added from old bus to reference bus i.e. a new loop is
formed but dimension of ZBUS does not change.
Consider a passive n-bus network in which branch with impedance Zb is added from old
bus j to reference bus r by setting Vk = 0
Passive Linear n-bus
network
1
n
i
j
r
Zb
1 1 1
2 2 2
1 20
j
BUS j
n jn n
j j jn jj b k
V Z I
V Z old Z I
V Z I
Z Z Z Z Z I
1 1 2 2
1 1 2 2
1 1 2 2
0
1
k b k
b k j j jj j k jn n
j j jj j jn n b jj k
k j j jj j jn n
b jj
V Z I
Z I Z I Z I Z I I Z I
Z I Z I Z I Z I Z Z I
I Z I Z I Z I Z IZ Z
1 1 2 2
1 2
1 1 2 2
i i i in n ij j
ij j ij j ij jn
i i in n
b jj b jj b jj
V Z I Z I Z I Z I
Z Z Z Z Z ZZ I Z I Z I
Z Z Z Z Z Z
1
2
1 2
1
j
j
BUS BUS j j jn
jj b
jn
Z
ZZ new Z old Z Z Z
Z Z
Z
Type – 4 modification: Branch is added between two old buses
It is the modification where Zb connects two old bus i.e. a new loop is formed but
dimension of ZBUS does not change.
4 Symmetrical Fault Analysis
Shital Patel, EE Department Power System - II (3150911) 14
Consider a passive n-bus network in which branch with impedance Zb connects two old
buses.
Passive Linear n-bus
network
1
n
i
j
r
Ii+Ik
Ij-Ik
Zb
Ik
1 1 2 2
1 1 2 2
1 1 2 2
1 1 1 2 2 2
...
0
i i i ii i k ij j k in n
j b k i
j j ji i k jj j k jn n
b k i i ii i k ij j k in n
i j i j ii ji i ij jj j in jn n b i
V Z I Z I Z I I Z I I Z I
V Z I V
Z I Z I Z I I Z I I Z I
Z I Z I Z I Z I I Z I I Z I
Z Z I Z Z I Z Z I Z Z I Z Z I Z Z
i jj ij ji kZ Z Z I
11 1 1
22 2 2
1 1 2 20 2
i j
BUS i j
nn in jn
i j i j jn jn b ii jj ij j
IV Z Z
IV Z old Z Z
IV Z Z
Z Z Z Z Z Z Z Z Z Z I
1 1
2 2
1 1 2 2
1
2
i j
i j
BUS BUS i j i j in jn
b ii jj ij
in jn
Z Z
Z ZZ new Z old Z Z Z Z Z Z
Z Z Z Z
Z Z