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9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 1
9.1. Types of system transients
Power system transients are an abnormal situation, where the power system is in
dynamic state with large scale fear caused by fault, operations of switches or other
disturbances.
Transient persists for a very short duration i.e. few μs to 1 s but during its occurrence,
system is subjected to great stress from over voltage or over current. In some of the case
severity of transients are such large that it leads to complete shutdown of plant or black
out of entire area.
The main cause of transients in a system are lightning, switching, short circuit and
resonance.
Usually the classifications of transients is carried out based on the speed of its occurrence
such as
(a) Surge phenomena
This type of transients are caused by lightning and switching. It is extremely fast in terms
of its commencement to conclusion speed.
Surges initiates an electromagnetic waves travelling with the speed of light i.e. 3108 m/s
on transmission line. Transient associated with the surges occurs in few millisecond and
dies out after a few reflection.
The reflections of surges builds up an excessive rate of rise of voltage at open end line or
transformer which may damage insulation of equipment.
Selection of insulation level of line equipment and transformer is based on the
overvoltage caused by surge phenomena.
(b) Short circuit
Short circuits results from symmetrical as well as unsymmetrical faults. Occurrence of
symmetrical fault brings the power transfer across the line to zero immediately, whereas
the partially in case of unsymmetrical fault.
It is moderately fast in terms of its commencement to conclusion speed and its speed
depends upon the time constant of the generator winding. The time range of short circuit
transient is from 10 to 100 ms i.e. first few cycles of short circuit current.
Short circuit current attains a very high value that may result in thermal breakdown of
equipment. It does not cause permanent damage as faulty section is quickly isolated form
healthy section by breaker operation.
(c) Transient stability
Due to the short circuit at any part of power system, there is an instantaneous total or
partial drop in bus voltage. This results in reduction of generator power output.
Initially for some instant, turbine tries to keep power output constant, but if this condition
sustained for more time than it causes most sever type of transient i.e. mechanical
oscillation of synchronous machine rotor.
This electromechanical oscillation leads to the loss of synchronism for some or all
machine. Once it happens then it takes hours to resynchronize such black out system.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 2
Transient stability is slow in terms of its commencement to conclusion speed as rotor
swing is quite low and its time ranges from few milliseconds to one minute.
9.2. Three phase sudden short circuit of an alternator
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).
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 3
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.
If the oscillation of short circuit current of a synchronous machine after the DC off set
current is removed, it is observed that current has three time periods – initial
subtransient period when current is large as machine offers subtransient reactance,
middle transient period when machine offers transient reactance and finally steady state
period when machine offers synchronous reactance.
a
b
c
θ
Extrapolation of transient envelope
Extrapolation of steady envelope
Actual envelope
t
Subtransient period
Transient period
Steady state period
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 4
If the transient envelop is extrapolated backward in time, the difference between
transient and subtransient envelope decays fast depending on the damper winding time
constant. Similarly difference between steady state and transient envelop decays in
accordance with the time constant of field winding.
Let,
'
"
Steady state current
' Transient current excluding DC component
" Subtransient current excluding DC component
Direct axis synchronous reactance
Direct axis transient reactance
Direct axis su
d
d
d
I
I
I
X
X
X
"
'
btransient reactance
No load voltage
Difference between transient and subtransient current
Difference between steady state and transient current
Damper winding time constant
Field winding
g
dw
f
E
i
i
time constant
The interrupt ob for finding transient reactance is determined by means of logarithmic
plot. Both current differences decrease exponentially as
//" " ' '0 0and fdw
tti i e i i e
At time t >> τdw the current Δi” dies out and hence
" ' " "0 0log( ) log
dwtf f
t ti i i i
The plot of log (Δi” + Δi’) versus time for t >> τdw becomes straight line with slop of (-1/
τf). As straight line portion of the plot is extrapolated taking inverse log of intercept
corresponding to t = 0.
a
b
c
θ
Extrapolation of transient envelope
Steady state current amplitude
Δi
Δi
t
1 ' ' "0 00
0
log ( ) log expt
ft
tab i i i ab
Though the machine reactances are dependent upon magnetic saturation, the value of
machine reactances are used to calculate fault current and stability limits.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 5
9.3. Restriking voltage after removal of short circuit
Restriking voltage is the transient voltage that appears at or in close immediacy to reach
zero current pause during arcing time i.e. voltage that appears across the breaking
contact at the instant of arc extinction.
L
C
CB
Fault
ie
Restriking Voltage
Recovery Voltage
Time
earc
Vmax
2Vmax
When an alternator connected to a busbar and the load is removed after a short circuit
occurs, it is required to determine the voltage across the circuit breaker during the
opening period.
Let, consider the circuit having L inductance per phase up to fault point and C capacitance
to earth of circuit breaker terminal.
When circuit breaker operates under fault condition, before current interruption the
capacitance C is short circuited by fault and this short circuited current is limited by
inductance L of the system.
When breaker contacts are opened and the arc finally extinguishes at current zero, a
voltage v is suddenly applied across capacitor and so across breaker contacts. The current
i that flows through the fault is not injected to capacitor and inductor.
When breaker contacts are open and arc is extinguished, the current i is diverted through
C so that the voltage v, which is so far been effective only across L is suddenly applied to
inductance L and capacitance C in series.
This L-C series combination forms an oscillatory circuit having natural frequency in order
of 10 Hz to 10 kHz depending on the value of L and C.
2
12
21
n nf
LC
LC
Initial charging current surges tends to carry the voltage across the capacitor and hence
across circuit breaker contacts to double voltage i.e. 2Vmax. This restriking voltage
transient re-establishes the arc.
Assume short circuit current i lags behind the system voltage by 90° and zero time at zero
current.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 6
2
2
maxmax
max
max
2max
2
2
max 2
max
1
Before opening of circuit breaker
sin sin
cos
, 0,
,
…………(i)
Where,
Peak value of reco
L Ci i i
dvi vdt C
L dt
di v d vC
dt L dt
Vi I t t
L
Vdit
dt L
VdiAt t
dt L
V v d vSo C
L L dt
d vV v LC
dt
V
very voltage (phase- to-neutral)
Restricking voltage (phase- to-neutral)
t = Time (sec)
v
The solution of equation (i)
2max
2max
2 2maxmax max max
2 2 22
2
max
( ) ( )
( ) 1
11
( )1 1
,
…………(ii)
Multiply equation (ii) by and pu
n n
n nn
n
n n n n
Vv s LCs v s
s
Vv s LCs
s
VV V VLCv ss LCs s s j s js s
s sLCOr
V A B C
ss s j s j s j s j
s
2
maxmax
t 0
n
n n
s
VA V
j j
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 7
2
max max
2
max max
max max maxmax
Multiply equation (ii) by and put
22
Multiply equation (ii) by and put
22
,
1 1 1( )
2 2 2 2
n n
n
n n
n n
n
n n
n n n
s j s j
V VB
j j
s j s j
V VC
j j
Hence
V V Vv s V
s ss j s j s j s
max max max
( ) 1 1 1 cos2 2 2
n n n n
n
j t j t j t j t
n
j
e e e ev t V V V t
Maximum value of restriking voltage occurs when, dv(t)/dt=0
max
max
1 cos 0
sin 0
sin 0
n
n n
n
n
n
dV t
dt
V t
t
t
t
LC
Note
o Only solid fault is considered i.e. no arcing.
o Magnitude of the positive sequence impedance is constant for the period for which
overvoltage is to be determined.
o Effects of saturation and corona are neglected
o Charging current of the transmission line before the fault and load currents are
neglected.
o Current interruption takes place at current zero when the voltage passes through
maximum value.
o System is lossless.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 8
9.4. Classification of restriking voltage transient
The circuit that suffers from voltage transients can be placed under two category.
(a) Single frequency oscillatory transient
L
C
CB
Fault
ie
Restriking Voltage
Recovery Voltage
Time
earc
Vmax
2Vmax
When breaker contacts are open, an arc is formed which generally extinguished at a
current zero. At that time circuit voltage is at peak and it appears across the breaker
terminal but it delays in doing so because of presence of capacitance C i.e. first capacitance
gets charged and then it establishes the voltage across breaker terminal.
Restriking transient voltage at natural frequency of L-C series circuit superimposed on
system voltage i.e. recover voltage.
The natural frequency of oscillation is in the order of 1000 to 10,000 Hz.
(b) Double frequency oscillatory transient
L1
C1
CB
FaultC2
L2
ie
Restriking Voltage
Recovery Voltage
Time
earc
Vmax
2Vmax
In some the case, circuit breaker has section of line on its both side i.e. L and C on both
side of breaker contacts. Before fault clearing, both circuit breaker contacts are at same
voltage.
Once fault has been cleared and arc is extinguished, both side of the circuit oscillates at
their own natural frequencies. Hence a composite double frequency transient appears
across breaker terminals.
When the circuit breaker operates in such case, the load is completely isolated from the
generator and the two halves of the circuit behave independently. Before the switch
operates, the voltage across the capacitors is
2
1 2
c
LV V
L L
Usually L2 > L1 hence the capacitor voltage is a little less than the source voltage at any
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 9
time. When the current passes through zero value, the voltage is at its maximum.
When the circuit breaker interrupts the current at its zero, the capacitor C2 oscillates with
L2 at a natural frequency f2 and C1 oscillates with L1 at a natural frequency f1.
1 2
1 1 2 2
1 1and
2 2f f
L C L C
Hence, opening of the circuit breaker results in double frequency transients in the system.
9.5. Travelling waves on transmission lines
Transmission line parameter R, L and C are uniformly distributed over the length of line
and for steady state analysis it is represented by lumped parameter. But for transient
analysis transmission line needs to be represented by distributed parameter.
To understand the travelling wave phenomenon over transmission line, let L and C are
the inductance and capacitance per unit length of the line.
LOAD
VSC1
L1 L2 L3 Ln
C2 C3 Cn
S
When switch S is closed, the inductance L1 acts as an open circuit and C1 as short circuit
instantaneously.
At the same instant the next section cannot be charged because the voltage across the
capacitor C1 is zero and hence charging of the capacitor C2 through L2 is not possible. It
takes take some finite time to charge all successive section of line.
Voltage at the consecutive sections builds up gradually and this gradual building up of
voltage over the transmission line is viewed as a voltage wave travelling from one end to
the other end.
Let, wave after time t has travelled through a distance x and a distance dx is travelled by
the waves in time dt.
Current in the transmission line is the rate at which the charge flows into and out of the
line and the velocity of the travelling wave over the transmission line is dx/dt.
(VCx) VC (i)dq d dx
I VCvdt dt dt
Voltage is the rate at which the flux linkages link around the transmission line due to the
current flowing in up to distance x
(ii)
d d LIx dxV LI LIv
dt dt dt
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 10
Taking the ratio of equation (ii) and (i)
2
2
C
V LIv
I VCv
V I L
I V C
V L
CI
V LZ
I C
The ratio of voltage to current has the dimensions of impedance i.e. surge
(characteristics) impedance of the line. It is also known as the natural impedance because
it has nothing to do with the load impedance i.e. purely a characteristic of the
transmission line.
The value of this impedance is about 400 ohms for overhead transmission lines and 40
ohms for cables.
The vast difference in the surge impedance of overhead transmission line and cable is
because of the fact that in a cable conductors are much closer to each other compared to
overhead transmission line so it possesses much smaller L and much larger C.
Now, multiplying equation (ii) and (i)
2
7 0
7
0
12 7
8
1
1
1
22 10 ln
ln
1
4 10
1
4 8.852 10 1 10
3 10 m/s
=Velocity of light
rm
ms
s
r
VI LIv VCv
vLC
vLC
D
DD
D
Above equation suggest that, velocity of propagation of the travelling waves over the
overhead transmission lines is equals to the velocity of light.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 11
In actual practice because of the resistance and reactance of the lines the velocity of the
travelling wave is slightly less than the velocity of light i.e. approximately 250 m/μs.
Equation also concludes that the velocity of propagation of the travelling waves over the
cables is smaller than the velocity of propagation of the travelling wave over the overhead
lines because of the permittivity i.e. for overhead lines εr = 1 and for cable εr > 1.
9.6. Reflection and refraction of travelling wave
Vi , ii Vt , it
Vr , ir
ZC1 ZC2
Discontinuity (Junction)
When travelling wave arrives at a discontinuity in a line, where Zc of the line changes
abruptly.
Some adjustments must take place, if proportionality between voltage and current wave
is to be maintained. This adjustment takes the form of initiation of two new wave pairs.
The reflected voltage wave and its companion current wave travelled back down the line
and are superimposed on the incident wave.
The refracted voltage wave and its companion current wave penetrates and travels
beyond the discontinuity.
Let,
1 2
, Incident wave pair
, Reflected wave pair
, Refracted(transmitted) wave pair
, Characteristic impedances
Reflection coefficient (for voltage)
Refraction(transmission) coefficient (fo
i i
r r
t t
C C
v i
v i
v i
Z Z
r voltage)
Applying Kirchhoff’s voltage and current law at junction.
1 1 2
(i)
(ii)
ti ri r t
C C C
i r t
vv vi i i
Z Z Z
v v v
Solving equation (i) and (ii)
2 1 2
1 2 1 2
2andC C C
r i i t i i
C C C C
r i
Z Z Zv v v v v v
Z Z Z Z
i i
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 12
Note
o Open circuited line : ZC2 = ∞ gives α = 1, β = 2
o Short circuited line : ZC2 = 0 gives α = -1, β = 0
o Matched termination : ZC2 = ZC1 gives α = 0, β = 1
It is observed that for matched termination, there is no reflection and line acts as
infinitely long. This thought is important in communication lines but not in power line.
9.7. Variation in voltage and current with the receiving end open circuited
VS
Sdx
When switch S is closed, a voltage and current wave of magnitudes vi and ii travels
towards the receiving open circuited end.
If Zc is the characteristic impedance of the line, than wave is going to see a change in
impedance from Zc to infinite at last element dx of the line.
As the current at the open circuited end is zero, the electromagnetic energy vanishes and
it is transformed into electrostatic energy. As a result, let the change in voltage and
current is
2 21 1
2 2i r
r i c i i
Li Cv
Lv i Z i v
C
vi
ii
1
vi
ii
2
t = t2
t = t1
vr
vi
t = t3
vr
t = t4
vr
vr
ir
t = t5
vr
ir
Also,
1 1
1
r i
r i
C C
r i
r i
r i
v v
v v
Z Z
i i
i i
i i
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 13
For a transmission line with an open circuited end, voltage and current wave pair reflects
back fully i.e. voltage positively and current negatively with reflection coefficient α = 1.
9.8. Variation in voltage and current with the receiving end short circuited
VS
Sdx
When switch S is closed, a voltage and current wave of magnitudes vi and ii travels
towards the receiving short circuited end.
If Zc is the characteristic impedance of the line, than wave is going to see a change in
impedance from Zc to zero at last element dx of the line.
As the voltage at the short circuited end is zero, the electrostatic energy vanishes and it
is transformed into electromagnetic energy. As a result, let the change in voltage and
current is
2 21 1
2 2i r
ir i i
c
Cv Li
vCi v i
L Z
vi
ii
1
vi
ii
2
t = t2
t = t1
ir
ii
t = t3ir
t = t4
iiir
vr
t = t5
ir
vr
Also,
1 1
r i
r i
C C
r i
i i
v v
Z Z
v v
For a transmission line with a short circuited end, voltage and current wave pair reflects
back fully i.e. voltage negatively and current positively with reflection coefficient α = 1.
9.9. Variation in voltage and current with the receiving end connected to
cable
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 14
LineCable
(vi,ii) (vt,it) When switch S is closed, a voltage and current wave of magnitudes vi and ii travels
towards the receiving end connected to the cable.
If Zc1 is the characteristic impedance of the line, than wave is going to see a change in
impedance from Zc1 to ZC2 at last element dx of the line.
When wave travels over the line and enters the cable looks into a different impedance,
hence it suffers reflection and refraction at the junction. As a result, let the change in
voltage and current is
2
1 2
2 Ct i
C C
Zv v
Z Z
The impedance of the overhead line and cable are approximately 400 Ω and 40 Ω. With
these values, voltage entering the cable becomes vt = 0.181vi. It means voltage wave
travelling along the line reduces its strength when enters into the cable.
For this reason, an overhead line terminated near a station is connected to the station
equipment through a short length of underground cable.
9.10. Variation in voltage and current with the receiving end terminated
through capacitor
V C
Z
Let, DC surge of infinite length travels over the line of surge impedance ZC1 = Z and is
incident on the capacitor then the voltage across the capacitor i.e. refracted voltage can
be
2
1 2
1 12
2 ( ) ( ) ( )2( ) 2 ( )
1 11
Or,
1
…………(i)1 1
Multiply equation (i) by and put 0
1
C i i it i
C C
Z v s v s v sCs ZCv s v sZ Z s s ZCs sZ s s
Cs ZC
A BZCs
s s sZC ZC
s s
A
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 15
1 1Multiply equation (i) by and put
1
Hence,
1 1( ) 2 ( ) 2 ( )
1 1
( ) 2 ( ) 1t
ZC
t i i
t i
s sZC ZC
B
A Bv s v s v s
s ss sZC ZC
v t v t e
It is to be noted that, terminating impedance is not a transmission line, therefore, v t(t) is
not a travelling wave but it is the voltage across the capacitor C.
Capacitor connection at T-junction
V C
Z1 Z2
V C
Z1
Z2
Let, DC surge of infinite length travels over the line of surge impedance ZC1 = Z1 and is
incident on the capacitor then the voltage across the capacitor i.e. refracted voltage can
be
2 2
22 1 2 1
2 1 21 21
2 1 2
1 2
1 2
1
2 12
12 ( ) ( ) ( )( ) 2 ( )
1
1 '
, '
Or,
1
…………(i)1 1
' '
Multiply equ
C i i it i
C C
Z Z
sCZZ v s v s Z Z C v s Z Cv s v s
Z Z ZZ Z s s sZ s s ssCZ Z Z C Z C
Z ZWhere Z
Z Z
Z C A B
ss s s
Z C Z C
1
1
ation (i) by and put 0
'
1 1Multiply equation (i) by and put
' '
'
s s
ZA
Z
s sZ C Z C
ZB
Z
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 16
'
2
1 1 2
2
1 2
Hence,
2' 1 1( ) 2 ( ) 2 ( ) 1 ( ) 1
1 1 1
' ' '
2( ) ( ) 1
tZ C
t i i i
t i
ZA B Zv s v s v s v s
s Z Z Zs s sZ C Z C Z C
Zv t v t e
Z Z
It is to be noted that, terminating impedance is not a transmission line, therefore, v t(t) is
not a travelling wave but it is the voltage across the capacitor C.
9.11. Variation in voltage and current with the receiving end terminated
through inductor
V
Z1 Z2
L
v1 v2i1
V
Z1
Z2
L
v1 v2i1
Let, DC surge of infinite length travels over the line of surge impedance ZC1 = Z1 and is
incident on the inductor then the voltage across the inductor i.e. refracted voltage can be
221
1 2 1 2
211
2 2 1 2 1 2
2
22 2 1
1 21 2
2
1 2
22 ( ) ( )( )
2( ) ( ) 2 ( )1( )
2 ( ) 2 ( )( ) ( )
Or,
C i it
C C
t i it
i it t
Z sLZ v s v sv s
Z Z s Z Z sL s
Z sLv s v s v si s
Z sL Z sL Z Z sL s s Z Z sL
ZZ v s v sLv s Z i s
Z Zs Z Z sL ss
L
ZAL
Z Z ss s
L
1 2
2
1 1
…………(i)
Multiply equation (i) by and put 0
B
Z Zs
L
s s
ZA
Z Z
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 17
1 2
1 2 1 2
2
1 1
22
1 2 1 21 1
22
1 2
Multiply equation (i) by and put
Hence,
1( ) 2 ( ) 2 ( ) 1
2( ) ( ) 1
t i i
Z Zt
Lt i
Z Z Z Zs s
L L
ZB
Z Z
ZA Bv s v s v s
Z Z Z Zs Z Zs sL L
Zv t v t e
Z Z
It is to be noted that, when rate of rise of incoming traveling wave is infinite, the outgoing
wave beyond L rises much more gradually.
9.12. Attenuation and distortion of travelling waves
Behaviour of traveling wave on multiconductor transmission line is more complex than
lossless simple twin conductor transmission line.
For multiconductor transmission lien, travelling wave’s attenuation and distortion is
required to address.
Distortion is caused by losses and multiconductor arrangement and attenuation is
accompanied by distortion.
Attenuation is caused by series resistance and leakage resistance and determination of
attenuation is generally empirical.
X = 0
X
Δx
v0,i0 v,iRΔx LΔx
CΔxGΔx
Δxv,i
Let,
Magnitude of surge at distance x from point of origination
Resistance per unit length
Inductance per unit length
Capacitance per unit length
Conductance per unit length
Power loss in the diff
xv
R
L
C
G
dp
erential element
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 18
The power loss,
2 2
2 2 2
2 2
22
2
2
0 0
2
0
0
0
2
2
2
2
ln2
At 0, ln
ln2
ln
Where,2
C
C
C C
C
C
C
C
C
C
x
C
dp i Rdx v Gdx
d i Z i Rdx v Gdx
iZ di i R v G dx
iZ di i R iZ G dx
R Z Gdidx
i Z
R Z Gi x A
Z
x i i A i
R Z Gix
i Z
ix i i e
i
R Z G
Z
C
2 2
22 2
2 2
2
2
2
2
0 0
2
0
0
0
2
2
2
ln2
At 0, ln
ln2
ln
C
C
C C
C
C
C
C
C
C
dp i Rdx v Gdx
vd i Rdx v Gdx
Z
vdv i R v G dx
Z
v vdv R v G dx
Z Z
R Z Gdvdx
v Z
R Z Gv x A
Z
x v v A v
R Z Gvx
v Z
vx v v
v
2
Where,2
x
C
C
e
R Z G
Z
Above equation concludes that current and voltage waves get attenuated exponentially
as they travel over the line and the magnitude of attenuation depends upon the
parameters of the line. Since the value of resistance depends not only on the size of the
conductors but also on the shape and length of the waves. An empirical relation for
calculating the voltage and current at any point on the line after it has travelled through
a distance x units and is
0
01
vv
kxv
When x is in km, v and V0 are in kV than the value of attenuation constant k is
k = 0.00037 for chopped waves
= 0.00019 for short waves
= 0.0001 for long waves
In the estimation of transmission line transients, the effect of attenuation and distortion
are normally ignored for simplicity in analysis. The results so obtained are pessimistic i.e.
voltage computed are higher than that obtained actually.
9.13. Capacitance switching of transmission line
Switching of a capacitance i.e. disconnecting a line or a cable or capacitor bank creates
serious problems in power systems in terms of abnormally high voltages across the
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 19
circuit breaker contacts.
L
C1
CB
C2V Vg Vc
Vc=Vg=V
Vc
t1
ir
t2 t3
5Vg
3Vg
Consider the equivalent circuit of an unloaded transmission line carrying only capacitive
current i. Line is opened by a circuit breaker at the instant t1 when capacitive current is
passing through zero.
Under this situation the current leads the voltage by about 90°. If current interruption
takes place when it passes through zero then the capacitor is charged to negative
maximum voltage.
After instant t1, the breaker contacts is subjected to potential difference of Vg and Vc. At
instant t2, the voltage across breaker contacts becomes twice the maximum value of Vg.
Within such interval of half cycle, beaker contacts are subjected to severe voltage change
with the result that breaker may restrike.
If such condition occurs, the voltage across breaker contacts falls almost instantaneously
from twice the maximum value of Vg to zero.
In doing so, high frequency oscillations are setup that builds up the voltage three times
the maximum value of Vg. The line is charged to three times the maximum value of Vg to
earth after interruption of restriking current.
At this instant, voltage across breaker contacts is twice the maximum value of Vg and
generated voltage is itself Vg. The voltage across breaker is continuously increases and
reaches the value four times the maximum value of Vg.
If breaker restrikes again, than whole cycle repeats and voltage reaches to eight times the
maximum value of Vg.
The extreme condition of capacitance switching is very rare, but if that occurs it causes
serious damage to the system.
9.14. Overvoltage due to arcing ground
Under balanced conditions and complete transposed transmission lines, the potential of
the neutral is near the ground potential and the currents in various phases through the
shunt capacitors are leading their corresponding voltages by 90° and displaced from each
other by 120° i.e. net sum of the three currents is zero.
9 Transients in Power Systems
Shital Patel, EE Department Power System - II (3150911) 20
When a line to ground fault occurs in any of one phase, voltage across the shunt capacitor
of that phase reduces to zero whereas those of the healthy phases become line to line
voltages displaced by 60° rather than 120°.
Net charging current is three times the phase current under balanced condition. These
currents flow through the fault and the windings of the alternator. The magnitude of this
current is sufficient to sustain an arc.
Therefore an arcing ground occurs. This is due to a flashover of a support insulator i.e.
flashover acts as a switch.
If the arc extinguishes when the current is passing through zero value, the capacitors in
healthy phases are charged to line voltages. The voltage across the line and the grounded
points of the post insulator superposition the capacitor voltage and the generator voltage.
This voltage is good enough to cause flashover that is equivalent to restrike in a circuit
breaker.
Because of the presence of the inductance of the generator winding, the capacitances
forms an oscillatory circuit and builds up to still higher voltages. Arc may reignite causing
further transient disturbances that may finally lead to complete rupture of the post
insulators.