3 reactive power and voltage control

REACTIVE POWERREACTIVE POWER
ANDAND
VOLTAGE CONTROLVOLTAGE CONTROL
Punit Sompura

(i)(i) It is a system instability ORIt is a system instability OR
(ii)(ii) Reduction in voltageReduction in voltage
It involves many power system componentsIt involves many power system components
and variables of a particular section of aand variables of a particular section of a
power system.power system.
What is voltage collapse?What is voltage collapse?

(i)(i) Heavily loaded transmission linesHeavily loaded transmission lines
(ii)(ii) Faulted lineFaulted line
(iii)Reactive power shortage(iii)Reactive power shortage
The nature/behavior of voltage collapse canThe nature/behavior of voltage collapse can
be studied by examining generation,be studied by examining generation,
transmission and consumption of reactivetransmission and consumption of reactive
power.power.
Where Voltage Collapse occurs?Where Voltage Collapse occurs?

Generation of reactive power is limited byGeneration of reactive power is limited by
reactive power compensator limits.reactive power compensator limits.
Transmission of real/active power is limitedTransmission of real/active power is limited
due todue to
(i)(i) high reactive power losses on heavilyhigh reactive power losses on heavily
loaded lines.loaded lines.
(ii)(ii) Limited action of AVRLimited action of AVR
Reactive power demand of load increases due toReactive power demand of load increases due to
(i)(i) Increase in loadIncrease in load
(ii)(ii) Motor stallingMotor stalling
Where Voltage Collapse occurs?Where Voltage Collapse occurs?

(i)(i) Increase in inductive loadingIncrease in inductive loading
(ii)(ii) OLTC operationOLTC operation
(iii)(iii)Line outageLine outage
(iv)(iv)Generator outageGenerator outage
(v)(v) Load recovery dynamicsLoad recovery dynamics
(vi)(vi)Limit of reactive power compensator andLimit of reactive power compensator and
generatorsgenerators
Why Voltage Collapse Occurs?Why Voltage Collapse Occurs?

(i)(i) Switching of shunt capacitors (by increasingSwitching of shunt capacitors (by increasing
limit of reactive power compensators)limit of reactive power compensators)
(ii)(ii) Blocking of OLTC operationBlocking of OLTC operation
(iii)(iii)Generation reschedulingGeneration rescheduling
(iv)(iv)Strategic load sheddingStrategic load shedding
(v)(v) Increasing reactive power of generatorsIncreasing reactive power of generators
Possible measures to reduce voltage collapsePossible measures to reduce voltage collapse

If change in load is gradual then the re-If change in load is gradual then the re-
stabilisation is faster and the system can comestabilisation is faster and the system can come
back to stable operating point.back to stable operating point.
However, sudden change in load results inHowever, sudden change in load results in
dynamic fall of voltage and the system candynamic fall of voltage and the system can
reach a stable operating point.reach a stable operating point.
Voltage collapseVoltage collapse

Power System Stability (PSS): A characteristicPower System Stability (PSS): A characteristic
of power system to remain in a state ofof power system to remain in a state of
equilibrium during normal conditions and alsoequilibrium during normal conditions and also
to restore an acceptable state of equilibriumto restore an acceptable state of equilibrium
after a disturbance.after a disturbance.
PSS is related to rotor angle stability which isPSS is related to rotor angle stability which is
related to synchronous operation.related to synchronous operation.
Instability occur due to loss of synchronism.Instability occur due to loss of synchronism.
It also occurs due to voltage instability.It also occurs due to voltage instability.
Definition of Voltage StabilityDefinition of Voltage Stability

11stst
DefinitionDefinition
Voltage Instability (VIS): The ability of a powerVoltage Instability (VIS): The ability of a power
system to become unstable during voltagesystem to become unstable during voltage
reduction due to outage of many equipments.reduction due to outage of many equipments.
(i)(i) Outage of GeneratorOutage of Generator
(ii)(ii) Outage of TransformerOutage of Transformer
(iii)(iii) Outage of BusbarOutage of Busbar
(iv)(iv) Weakening of voltage controlWeakening of voltage control
(v)(v) Decrement of production of reactive generationDecrement of production of reactive generation
Definition of Voltage InstabilityDefinition of Voltage Instability

22ndnd
DefinitionDefinition
Voltage Instability (VIS): When the attempt ofVoltage Instability (VIS): When the attempt of
load dynamics to restore power consumption isload dynamics to restore power consumption is
just beyond the capability of the combinedjust beyond the capability of the combined
transmission and generation system, voltagetransmission and generation system, voltage
instability occurs.instability occurs.
Definition of Voltage InstabilityDefinition of Voltage Instability

Classification of power system stabilityClassification of power system stability
Time-Scale Generator-Driven Load-Driven
Short-term Rotor angle stability Short-term voltage stability
Steady
State
Transient
Long-term Frequency stability Long-term voltage stability
Small
disturbances
Long
disturbances

Steady state stability present for smallSteady state stability present for small
disturbances in the form of un-dampeddisturbances in the form of un-damped
electromechnical oscillations.electromechnical oscillations.
Transient stability is due to lack ofTransient stability is due to lack of
synchronizing torque and is initiated by largesynchronizing torque and is initiated by large
disturbances.disturbances.
The time-frame of rotor angle stability is due toThe time-frame of rotor angle stability is due to
electromechanical dynamics of power system.electromechanical dynamics of power system.
This time-frame is called short-term time scaleThis time-frame is called short-term time scale
because the dynamics last for a few seconds.because the dynamics last for a few seconds.

The voltage stability can be classified asThe voltage stability can be classified as
(i)(i) short-term voltage stabilityshort-term voltage stability
(ii)(ii) Long-term voltage stabilityLong-term voltage stability
Short-term voltage stability occurs due toShort-term voltage stability occurs due to
(i)(i) IMsIMs
(ii)(ii) Excitation of synchronous generators orExcitation of synchronous generators or
(iii)(iii)CPD (TCR, TCSC, SVC, UPFC)CPD (TCR, TCSC, SVC, UPFC)

Long-term voltage stability comes under longLong-term voltage stability comes under long
term time scale.term time scale.
It lasts for several minutes.It lasts for several minutes.
Long-term voltage stability occurs due toLong-term voltage stability occurs due to
(i)(i) OLTCOLTC
(ii)(ii) Delayed load restorationDelayed load restoration
(iii)(iii) Delayed corrective action of shuntDelayed corrective action of shunt
compensation device.compensation device.

Long-term voltage stability can be classifiedLong-term voltage stability can be classified
intointo
(i)(i) Small disturbancesSmall disturbances
(ii)(ii) Long disturbancesLong disturbances
(i)(i) Small disturbance voltage stability:Small disturbance voltage stability:
It is the ability of power system to controlIt is the ability of power system to control
voltage after small disturbances.voltage after small disturbances.
Example: Change in loadExample: Change in load

(ii) Large disturbance voltage stability:(ii) Large disturbance voltage stability:
It is the ability of power system to controlIt is the ability of power system to control
voltage after large disturbances.voltage after large disturbances.
Examples:Examples:
(i)(i) FaultsFaults
(ii)(ii) SwitchingSwitching
(iii)(iii)Loss of loadLoss of load
(iv)(iv)Loss of generationLoss of generation

Mechanism of Voltage CollapseMechanism of Voltage Collapse
Voltage collapse usually involves largeVoltage collapse usually involves large
disturbances.disturbances.
(including rapid increase in load or power(including rapid increase in load or power
transfer)transfer)
It is mostly associated with reactive powerIt is mostly associated with reactive power
deficit.deficit.
Table shows the time-frame of the components
causing voltage instability.

Factors affecting transient voltage stability
in time-scale
Factors affecting long-term
voltage stability in time-scale
Sr. No. Example Time
(second)
Example Time
(minute)
1 Static VAR
compensator
1 OLTC operation 2
2 Switched capacitors 2 Generation
readjustment
2
3 Generator excitation 1.5 Line overload 5
4 IM dynamics 1 Distribution voltage
regulation
3
5 Under voltage load
shedding
10 - -
6 HVDC operation 1-2 - -

(i)(i) Transient voltage stability: It is 0-10 secondTransient voltage stability: It is 0-10 second
in time-scale.in time-scale.
(a)(a) IM and dc converter may lead to voltageIM and dc converter may lead to voltage
collapse. Hence, reactive power demand ofcollapse. Hence, reactive power demand of
IM increases which further leads to voltageIM increases which further leads to voltage
collapse.collapse.
(a)(a) If one IM is not able to accelerate duringIf one IM is not able to accelerate during
post-disturbance period then it leads topost-disturbance period then it leads to
stalling of adjoining IM.stalling of adjoining IM.

(b)(b) Islanding: Electrical islanding and underIslanding: Electrical islanding and under
frequency load shedding may lead tofrequency load shedding may lead to
voltage collapse particularly when powervoltage collapse particularly when power
imbalance between the areas is more thanimbalance between the areas is more than
50%.50%.
(c)(c) The use of HVDC links may affect theThe use of HVDC links may affect the
transient voltage stability.transient voltage stability.
(ii) Steady-state Voltage Stability(ii) Steady-state Voltage Stability
It is for several minutes.It is for several minutes.

It involves high loads and high power importsIt involves high loads and high power imports
from neighboring areas following a largefrom neighboring areas following a large
disturbance and involving high reactive powerdisturbance and involving high reactive power
loss and voltage dip in the receiving side.loss and voltage dip in the receiving side.
(a)(a) Tap changing transformer and distributionTap changing transformer and distribution
voltage transformer sense this low voltagevoltage transformer sense this low voltage
and act to restore the distribution voltageand act to restore the distribution voltage
restoring load power. This load restorationrestoring load power. This load restoration
causes further voltage sag in thecauses further voltage sag in the
transmission voltage.transmission voltage.

(b)(b) If adjacent generators are overexcited andIf adjacent generators are overexcited and
overloaded then it leads to voltage collapseoverloaded then it leads to voltage collapse
due to increase in reactive power loss.due to increase in reactive power loss.
(c)(c) Due to large load demand or large rapidDue to large load demand or large rapid
magnitude of power transfer, reactive powermagnitude of power transfer, reactive power
demand increases and voltage reduces.demand increases and voltage reduces.
Remedy:Remedy:
Strategic load shedding and fast acting reactiveStrategic load shedding and fast acting reactive
compensators can reduce voltage instability.compensators can reduce voltage instability.

Analytical concept of voltage stability for a two-Analytical concept of voltage stability for a two-
bus systembus system
*
*
*
*
*
,
X
VV
X
VE
S
X
VE
IwhereVIS
−=
−
==
Taking V as reference vector and afterTaking V as reference vector and after
simplification of above equation,simplification of above equation,
X
V
X
EV
Q
X
EV
P
X
V
X
EV
j
X
VE
S
2
2
cos
sin
)cos(sin
−=
=
−−=
δ
δ
δδ

Elimination ofElimination of results in the steady-stateresults in the steady-state
receiving end voltage equation given by,receiving end voltage equation given by,
This is a quadratic equation given by,This is a quadratic equation given by,
Since imaginary value of V carries no physicalSince imaginary value of V carries no physical
significance, the positive real root is given by,significance, the positive real root is given by,
0)()2( 222224
=++−+ QPXEQXVV






+−−±
+−
= )(4)2(
2
1
2
2 22222
2
2
QPXEQX
EQX
V
δ

For UPF, the above equation is given by,For UPF, the above equation is given by,
Assuming E=1.0 pu at sending end,Assuming E=1.0 pu at sending end,
2
1
22222
2
)(4)2(
2
1
2
2






+−−±
+−
= QPXEQX
EQX
V
2
1
224
2
4
2
1
2






−±= PXE
E
V
2
1
22
41
2
1
2
1






−±= PXV

Both the real roots of V are equal when theBoth the real roots of V are equal when the
expression under the radical sign is zero. Thisexpression under the radical sign is zero. This
is whenis when
Thus, the final equation is given by,Thus, the final equation is given by,
criX
P
X ==
2
1
2
1
2
)(1
2
1
2
1






−±=
criX
X
V

(i)(i) For X<Xcri, roots are real.For X<Xcri, roots are real.
(ii)(ii) For X>Xcri, roots are imaginary.For X>Xcri, roots are imaginary.
(iii)(iii)For X=Xcri, the value of V is known asFor X=Xcri, the value of V is known as
Critical receiving end voltage (Vcri).Critical receiving end voltage (Vcri).
It’s value is given by Vcri=0.7 pu for anIt’s value is given by Vcri=0.7 pu for an
uncompensated, loss less line at UPF.uncompensated, loss less line at UPF.
2
1
2
)(1
2
1
2
1






−±=
criX
X
V

(i)The critical value of the receiving end voltage(i)The critical value of the receiving end voltage
(Vcri) is obtained when X=Xcri.(Vcri) is obtained when X=Xcri.
(ii) This state represents voltage stability limit of(ii) This state represents voltage stability limit of
a loss less transmission line.a loss less transmission line.
(iii)Mathematically, voltage stability limit is(iii)Mathematically, voltage stability limit is
obtained when the two real roots of theobtained when the two real roots of the
system voltage equation converge to asystem voltage equation converge to a
particular point and the Jacobian of the LFparticular point and the Jacobian of the LF
equation becomes singular.equation becomes singular.

(iv) Therefore, the voltage stability limit can(iv) Therefore, the voltage stability limit can
be defined as the limiting stage in abe defined as the limiting stage in a powerpower
system beyond which no amount ofsystem beyond which no amount of
reactive power injection will elevate thereactive power injection will elevate the
system voltage to its normal state.system voltage to its normal state.
(v)(v) The system voltage can only be adjustedThe system voltage can only be adjusted
by reactive power injection till the systemby reactive power injection till the system
voltage stability is maintained.voltage stability is maintained.

Expression for critical system reactance atExpression for critical system reactance at
voltage stability limit for any PFvoltage stability limit for any PF
2
42422
4222
22222
8
16164
044
)(4)2(
P
EPEQQE
X
EXQEPX
QPXEQX
+±−
=
=−+
+=−
2
1
22222
2
)(4)2(
2
1
2
2






+−−±
+−
= QPXEQX
EQX
V
The expression for Xcri at voltage stability limitThe expression for Xcri at voltage stability limit
for any PF can be obtained by equating thefor any PF can be obtained by equating the
above equation with the radical sign to zero.above equation with the radical sign to zero.

Expression for critical system reactance atExpression for critical system reactance at
voltage stability limit for any PFvoltage stability limit for any PF
)sectan(
2
2
θθ +−=
P
E
Xcri
Using, Q=P*tanUsing, Q=P*tanθθ, the final equation is given by,, the final equation is given by,
Method-IIMethod-II
In a loss less line, the expression of reactiveIn a loss less line, the expression of reactive
power flow is given by,power flow is given by,
0cos2
=++ QVEBBV δ






−±=
B
PE
E
V
1
tan4cos
2
1
cos
2
22
θδδ
)sectan(
2
2
θθ +−=
P
E
Xcri

But the receiving end voltage at voltageBut the receiving end voltage at voltage
stability limit is given by,stability limit is given by,
The above equation gives the value of powerThe above equation gives the value of power
transfer angle at voltage stability limit in termstransfer angle at voltage stability limit in terms
of E, P, B and PF.of E, P, B and PF.










−
= −
2
1
2
1
)
tan4
2(
cos
B
P
E
E
cri
θ
δ

Graphical representationGraphical representation
Fig. shows the characteristic of receiving endFig. shows the characteristic of receiving end
voltage (V) for varying system reactance (X).voltage (V) for varying system reactance (X).

(i)(i) The receiving end voltage falls with theThe receiving end voltage falls with the
increase in X at any fixed value of P till theincrease in X at any fixed value of P till the
voltage stability limited is attained.voltage stability limited is attained.
(ii)(ii) For X=Xcri, V=Vcri beyond which real powerFor X=Xcri, V=Vcri beyond which real power
demand can not be increased as otherwisedemand can not be increased as otherwise
voltage collapse and voltage instabilityvoltage collapse and voltage instability
occurs.occurs.
(iii)(iii)For X≠Xcri, load requires large current fromFor X≠Xcri, load requires large current from
the source and voltage instability occurs.the source and voltage instability occurs.

(iv)Higher value of receiving end voltage(iv)Higher value of receiving end voltage
indicates “voltage stable state”.indicates “voltage stable state”.
(v)(v) At knee point of the curve, there is a sharpAt knee point of the curve, there is a sharp
increase in transmission line current whichincrease in transmission line current which
leads to heavy series reactive loss of theleads to heavy series reactive loss of the
line that may lead to voltage collapse.line that may lead to voltage collapse.

Fig. shows the characteristic of receiving endFig. shows the characteristic of receiving end
voltage (V) v/s X at fixed value of P for varyingvoltage (V) v/s X at fixed value of P for varying
PF.PF.

Fig. shows the characteristic of cri v/s X atFig. shows the characteristic of cri v/s X at
different PFs.different PFs.
δ

As X increases, cri decreases. This offersAs X increases, cri decreases. This offers
anan
inherent limitation of operation of PS. This isinherent limitation of operation of PS. This is
because voltage stability limited is attained atbecause voltage stability limited is attained at
much lower value of at higher value of Xmuch lower value of at higher value of X
δ
δ

SummarySummary
(i)(i) Vcri at voltage stability limit is governed byVcri at voltage stability limit is governed by
Xcri at any specific amount of power flow.Xcri at any specific amount of power flow.
(ii)(ii) Due to low short-circuit capacity (high X),Due to low short-circuit capacity (high X),
Vcri and power angle ( ) are low at stabilityVcri and power angle ( ) are low at stability
limit.limit.
(iii)(iii)As PF is low (lagging), the stability limit andAs PF is low (lagging), the stability limit and
critical power angle ( cri) reduces.critical power angle ( cri) reduces.
(iv)(iv) Cri is limited by X and load PF.Cri is limited by X and load PF.
δ
δ
δ

SummarySummary
(v)(v) Shunt capacitor compensation increasesShunt capacitor compensation increases
thethe
cri and receiving end voltage magnitudecri and receiving end voltage magnitude
as it injects capacitive reactive power atas it injects capacitive reactive power at
load bus.load bus.
(v)(v) However, this may lead to decrease inHowever, this may lead to decrease in
voltage stability margin.voltage stability margin.
(vi)(vi)Proper selection of shunt compensationProper selection of shunt compensation
device is desired to achieve an acceptabledevice is desired to achieve an acceptable
voltage profile.voltage profile.
δ

Expression for Vcri and cri at voltage stabilityExpression for Vcri and cri at voltage stability
limit for two-bus systemlimit for two-bus system
The basic power flow equations in a two-busThe basic power flow equations in a two-bus
system being given bysystem being given by
The Jacobian can be obtained asThe Jacobian can be obtained as
δsin
X
EV
P =
X
V
X
EV
Q
2
cos −= δ






+−
=
δδ
δδ
cos2-sin
sincos1
EVEV
EEV
X
J
δ

0sincoscos 22222
=+− δδδ VEEVVE
Voltage stability limit is obtained when theVoltage stability limit is obtained when the
Jacobian becomes singular the determinant ofJacobian becomes singular the determinant of
J is zero.J is zero.
(1)(1)
0cos2 22
=− δEVVE
δcos2
E
V =
δ ,
cri
δ
In this equation V represents Vcri at voltageIn this equation V represents Vcri at voltage
stability limit when =stability limit when =
(a)(a)θδθ tansintan
X
EV
PQ ==

(b)(b)
Comparing both equations (a) and (b) we get,Comparing both equations (a) and (b) we get,
(2)(2)
Comparing equation (1) and (2), we finally get,Comparing equation (1) and (2), we finally get,
X
V
X
EV
X
EV 2
costansin −= δφδ
( )φδδ tansincos −= EV
X
V
X
EV
Q
2
cos −= δ
24
θπ
δ −=

Since represents the power angle at criticalSince represents the power angle at critical
state of voltage stability. At = andstate of voltage stability. At = and
V=VcriV=Vcri
δ
,
cri
δδ
24
where,
cos2
,
θπ
δ
δ
−== cri
cri
cri
E
V

At unity power factor (UPF) operation,At unity power factor (UPF) operation,
we findwe find and withand with
deterioration in power factordeterioration in power factor and wouldand would
further reduce.further reduce.
[ ]°=°= 045, θδ cri
p.u.707.0, =cri
V
,
cri
δ ,
cri
V

Relation between Voltage Stability and rotorRelation between Voltage Stability and rotor
angle stabilityangle stability
(i)(i) Rotor angle stability and Voltage stabilityRotor angle stability and Voltage stability
are affected by reactive power control.are affected by reactive power control.
(ii)(ii) The small disturbance voltage stabilityThe small disturbance voltage stability
increases rotor angle.increases rotor angle.

Difference between Voltage Stability and rotorDifference between Voltage Stability and rotor
angle stabilityangle stability
Sr.
No.
Rotor Angle Stability Voltage Stability
1 It is located near to the
generator.
It is located in
the load area.
2 It may also involve voltage
collapse.
It may or may
not involve rotor
angle stability.
3 It is for Generator Stability It is for load
voltage stability.

Factors affecting Voltage StabilityFactors affecting Voltage Stability
(i)(i) Reactive Power Capability of a SynchronousReactive Power Capability of a Synchronous
GeneratorGenerator
Synchronous generators are the primary deviceSynchronous generators are the primary device
for voltage and reactive power control in PS.for voltage and reactive power control in PS.
In voltage stability studies, active and reactiveIn voltage stability studies, active and reactive
power capability of generator is consideredpower capability of generator is considered
accurately to achieve the best results.accurately to achieve the best results.
The limit of active and reactive power ofThe limit of active and reactive power of
generator are commonly shown on P-Qgenerator are commonly shown on P-Q
diagram.diagram.

Active power limits are due to design of theActive power limits are due to design of the
turbine and boiler. They are constant.turbine and boiler. They are constant.
Reactive power limits are voltage dependent
and
have a circular shape.
Reactive power limits should be taken into
account in these studies.
The limitation of reactive power has three
different causes.

(i)(i) Stator currentStator current
(ii)(ii) Overcurrent excitationOvercurrent excitation
(iii)(iii)Under excitationUnder excitation
When the excitation current is limited toWhen the excitation current is limited to
maximum value, the terminal voltage is themaximum value, the terminal voltage is the
maximum excitation voltage minus the voltagemaximum excitation voltage minus the voltage
drop in Xs.drop in Xs.
The PS becomes weaker as the constantThe PS becomes weaker as the constant
voltagevoltage
moves away from the load.moves away from the load.

The voltage dependent limit of excitation current
is calculated by,
Where, Pg=active power of generator
Emax= the maximum electromotive force
Xd=synchronous reactance
V=terminal voltage
2
2
2
max22
max g
dd
s P
X
EV
X
V
I −+−=

The reactive power limit corresponding to stator
current limit can be calculated by,
Reactive power capability increases when
terminal voltage decreases.
The stator current limiter is used to limit reactive
power output in order to avoid stator
overloading. However, at the same time, it also
reduces voltage.
22
max
2
max gss PIVQ −=

The generator reactive power capability is
generally much less than that indicated by
manufacturer’s reactive capability curve.
This is due to constraints imposed by power
plant auxiliaries. It’s operation is threatened
when system voltage is low.

(ii) Automatic Voltage Control of Synchronous
Generator
The automatic voltage controllers maintain
constant voltage when generators are operated
inside P-Q diagrams.
AVC also includes the excitation current limiters
(Over and under) and stator current limiter.

(ii) Automatic Voltage Control of Synchronous
Generator
Due to overheating of the excitation circuit, the
excitation current must be limited after a few
seconds.
The overloading capability of generator may be
improved by making the cooling of generator
more effective.

Fig. shows the action of automatic rotor and
stator current limiters of generator.

When the generator is regulating the voltage,
the curves for the constant terminal voltage (Vt)
are flat.
This indicates large change in Q.
If network voltage becomes sufficiently low,
either rotor current limit or stator current limit is
hit. This will change the generator characteristic
drastically.

Since the slope of rotor current limit is almost
vertical, it is clear that the generator is on the
verge of losing the voltage control capability if
rotor current limit is hit.
In this situation, the reactive power from the
generator reduces fast which ultimately leads to
voltage instability.

Role of Transformer on Voltage control of a PSRole of Transformer on Voltage control of a PS
By changing transformation ratio, the voltage on
secondary side of any bus can be changed.
Change in transformation ratio is manual or
automatic.
Automatic change is done by OLTC.

Objective:
To determine the tap-changing ratio needed to
completely or partially compensate the voltage
drop in line so that desired voltage control at
receiving end is obtained.

1:ts = sending end tap ratio
tr:1 = receiving end tap ratio
The receiving end current is given by,
BIEAtEt rs += 21
θtan
22 E
P
j
E
P
IR −=

The line current is given by,
Put
)tan1(
)tan1(
2
21
2
θ
θ
j
Et
P
BEAtEt
j
Et
P
t
I
I
r
rs
rr
R
−+=
−==
βα ∠=∠= BBAA ,
}sintancossin{
}tansincoscos{
)sin)(costan1()sin(cos
22
2
22
21
2
21
βθβα
θββα
ββθαα
Et
P
B
Et
P
BEAt
j
Et
P
B
Et
P
BEAtEt
jj
Et
P
BjEAtEt
rr
r
rr
rs
r
rs
+−
+++=
+−++=

Equating only magnitudes and squaring both
sides
The minimum transformer tap ratio for
coordination between two transformers is
achieved when ts.tr =1
ts=1/tr
2
22
2
2
22
2
2
1
2
}sintancossin{
}tansincoscos{
βθβα
θββα
Et
P
B
Et
P
BEAt
j
Et
P
B
Et
P
BEAtEt
rr
r
rr
rs
+−
+++=

2
22
2
2
22
22
2
1
}sintancossin{
}tansincoscos{
βθβα
θββα
Et
P
B
Et
P
BEAt
j
Et
P
B
Et
P
BEAt
t
E
rr
r
rr
r
r
+−
+++=
βαθββ
θβαθβα
θβββα
θββα
θββα
sinsin2tansincos2
tancossin2tansincos2
tansincos2coscos2
tancossinsin
tansincoscos
2
2
222
2
2
22
2
2
222
2
2
222222224
2
42
222222224
2
422
2
2
1
EABPtPB
EABPtEABPt
PBEABt
PBPBEtA
PBPBEtAEE
r
rr
r
r
r
+−
−
++
++
+++=

θβαβα
θ
tan)sin(2)cos(2
)tan1(
2
2
22
2
2
2224
2
422
2
2
1
−−−
+++=
EABPtEABPt
PBEtAEE
rr
r
[ ]
2
2
2
1
222
3
2
22
4
2
2
1
3
2
2
4
1
sec
tan)sin()cos(2
0
EEPBC
ABPEC
EAC
CtCtC rr
−=
−−−=
=
=++
θ
θβαβα
2
1
1
2
1
31
2
22
2
)4(







 −±−
=
C
CCCC
tr

Conclusion:
(i) It is to be noted that the transformer doesn’t
improve the reactive power flow position and
only redistributes it.
(ii) The current in transmission line increases as
transformation ratio increases.

Effect of OLTC on Voltage StabilityEffect of OLTC on Voltage Stability
The secondary voltage of a transformer is
maintained near to nominal value by tap
changer when primary system voltage drops.
This is possible when the system does not have
shortage of reactive power (system is capable to
provide sufficient reactive power).
During heavy load demand, the secondary
voltage may become unstable even with tap
changing.

Effect of OLTC on Voltage StabilityEffect of OLTC on Voltage Stability
In this situation, raising tap position in order to
raise secondary voltage will not work and the
bus voltage will gradually collapse.

The secondary voltage is given by,
To obtain sensitivity of the voltage with tap
change we have to take rate of change of
V with respect to a.
aE
XaR
R
V
222
)(+
=
E
XaR
XaRXaRR
a
V
2
3
222
22
))((
))((
+
−+
=
∂
∂

To have a stable voltage state,
The above condition is true only and only if
Hence, the secondary voltage drops if the tap
position is raised in order to boost up the load
bus voltage.
The voltage stability is lost when
0>
∂
∂
a
V
XaR 2
>
XaR 2
<

P-V characteristic with static impedance load atP-V characteristic with static impedance load at
the receiving end busthe receiving end bus

WorkingWorking
Initially the system is operated at A0 (a=1).
Raising tap position (a=1.015), shifts the
operating point to A’0.
At this point, the voltage is enhanced and the
system is capable to transmit more power
(PA1 to PA1). New stable operating point is A’1.
If tap position is further raised, (a=1.02),
voltage is further enhanced and the system is
capable to transmit more power (PA2 to PA3).
New stable operating point is A’.

WorkingWorking
If tap changing operation is done is stable zone
of characteristic then the system may settle at a
new operating position with higher voltage at
load bus.
If it is carried out in unstable zone then bus
voltage collapses even though tap position is
increased and system becomes unstable.

WorkingWorking
If dynamic loading (IM) is assumed at load bus,
P-V characteristic is shown below.
Stable zone
Unstable zone

3 reactive power and voltage control

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