(i) It is a system instability OR (ii) Reduction in ... · (i) It is a system instability OR (ii)...

(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 componentsand variables of a particular section of aand variables of a particular section of apower 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 canbe studied by examining generation,be studied by examining generation,transmission and consumption of reactivetransmission and consumption of reactivepower.power.

Where Voltage Collapse occurs?Where Voltage Collapse occurs?

Generation of reactive power is limited byGeneration of reactive power is limited byreactive power compensator limits.reactive power compensator limits.

Transmission of real/active power is limitedTransmission of real/active power is limiteddue to due 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 AVRReactive 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 and Limit of reactive power compensator and

generatorsgenerators

Why Voltage Collapse Occurs?Why Voltage Collapse Occurs?

(i)(i) Switching of shunt capacitors (by increasing Switching 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 comeback to stable operating point.back to stable operating point.

However, sudden change in load results inHowever, sudden change in load results indynamic fall of voltage and the system candynamic fall of voltage and the system canreach a stable operating point.reach a stable operating point.

Voltage collapseVoltage collapse

Power System Stability (PSS): A characteristicPower System Stability (PSS): A characteristicof power system to remain in a state ofof power system to remain in a state ofequilibrium during normal conditions and alsoequilibrium during normal conditions and alsoto restore an acceptable state of equilibriumto restore an acceptable state of equilibriumafter a disturbance.after a disturbance.

PSS is related to rotor angle stability which isPSS is related to rotor angle stability which isrelated 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 Definition DefinitionVoltage Instability (VIS): The ability of a powerVoltage Instability (VIS): The ability of a powersystem to become unstable during voltagesystem to become unstable during voltagereduction 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 Definition Definition

Voltage Instability (VIS): When the attempt ofVoltage Instability (VIS): When the attempt ofload dynamics to restore power consumption isload dynamics to restore power consumption isjust beyond the capability of the combinedjust beyond the capability of the combinedtransmission and generation system, voltagetransmission and generation system, voltageinstability 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

Classification of power system stabilityClassification of power system stabilitySteady state stability present for smallSteady state stability present for smalldisturbances in the form of un-dampeddisturbances in the form of un-dampedelectromechnical oscillations.electromechnical oscillations.

Transient stability is due to lack ofTransient stability is due to lack ofsynchronizing torque and is initiated by largesynchronizing torque and is initiated by largedisturbances.disturbances.

The time-frame of rotor angle stability is due toThe time-frame of rotor angle stability is due toelectromechanical 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 scalebecause the dynamics last for a few seconds.because the dynamics last for a few seconds.

Classification of power system stabilityClassification of power system stabilityThe voltage stability can be classified as The 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)

Classification of power system stabilityClassification of power system stabilityLong-term voltage stability comes under longLong-term voltage stability comes under longterm 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 shunt Delayed corrective action of shunt

compensation device.compensation device.

Classification of power system stabilityClassification of power system stabilityLong-term voltage stability can be classified Long-term voltage stability can be classified

into into (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 controlvoltage after small disturbances.voltage after small disturbances.

Example: Change in loadExample: Change in load

Classification of power system stabilityClassification of power system stability(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 controlvoltage 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 CollapseVoltage collapse usually involves largeVoltage collapse usually involves largedisturbances. disturbances. (including rapid increase in load or power(including rapid increase in load or powertransfer)transfer)

It is mostly associated with reactive powerIt is mostly associated with reactive powerdeficit.deficit.

Table shows the time-frame of the componentscausing voltage instability.

Mechanism of Voltage CollapseMechanism of Voltage CollapseFactors affecting transient voltage stability

in time-scaleFactors 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 - -

Mechanism of Voltage CollapseMechanism of Voltage Collapse(i)(i) Transient voltage stability: It is 0-10 second Transient voltage stability: It is 0-10 second

in time-scale.in time-scale.

(a)(a) IM and dc converter may lead to voltage IM and dc converter may lead to voltage collapse. Hence, reactive power demand of collapse. Hence, reactive power demand of IM increases which further leads to voltage IM increases which further leads to voltage collapse.collapse.

(a)(a) If one IM is not able to accelerate during If one IM is not able to accelerate during post-disturbance period then it leads to post-disturbance period then it leads to stalling of adjoining IM.stalling of adjoining IM.

Mechanism of Voltage CollapseMechanism of Voltage Collapse(b)(b) Islanding: Electrical islanding and under Islanding: Electrical islanding and under

frequency load shedding may lead to frequency load shedding may lead to voltage collapse particularly when power voltage collapse particularly when power imbalance between the areas is more than imbalance between the areas is more than 50%.50%.

(c)(c) The use of HVDC links may affect the The use of HVDC links may affect the transient voltage stability.transient voltage stability.

(ii) Steady-state Voltage Stability(ii) Steady-state Voltage StabilityIt is for several minutes.It is for several minutes.

Mechanism of Voltage CollapseMechanism of Voltage CollapseIt involves high loads and high power importsIt involves high loads and high power importsfrom neighboring areas following a largefrom neighboring areas following a largedisturbance and involving high reactive powerdisturbance and involving high reactive powerloss and voltage dip in the receiving side.loss and voltage dip in the receiving side.

(a)(a) Tap changing transformer and distribution Tap changing transformer and distribution voltage transformer sense this low voltage voltage transformer sense this low voltage and act to restore the distribution voltage and act to restore the distribution voltage restoring load power. This load restoration restoring load power. This load restoration causes further voltage sag in the causes further voltage sag in the transmission voltage.transmission voltage.

Mechanism of Voltage CollapseMechanism of Voltage Collapse(b)(b) If adjacent generators are overexcited and If adjacent generators are overexcited and

overloaded then it leads to voltage collapse overloaded 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 rapid Due to large load demand or large rapid magnitude of power transfer, reactive power magnitude 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 reactivecompensators 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

VES

X

VEIwhereVIS

−=

−==

Taking V as reference vector and afterTaking V as reference vector and aftersimplification of above equation,simplification of above equation,

X

V

X

EVQ

X

EVP

X

V

X

EVj

X

VES

2

2

cos

sin

)cos(sin

−=

=

−−=

δ

δ

δδ


Elimination of Elimination of results in the steady-stateresults in the steady-statereceiving 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 physicalsignificance, 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 222222

2 QPXEQXEQX

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,

21

222222

)(4)2(2

1

2

2

+−−±+−= QPXEQX

EQXV

21

2242

42

1

2

−±= PXE

EV

21

22412

1

2

1

−±= PXV


Both the real roots of V are equal when theBoth the real roots of V are equal when theexpression under the radical sign is zero. Thisexpression under the radical sign is zero. Thisis when is when

Thus, the final equation is given by,Thus, the final equation is given by,

criXP

X ==2

1

21

2)(12

1

2

1

−±=

criX

XV


(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 anuncompensated, loss less line at UPF.uncompensated, loss less line at UPF.

21

2)(12

1

2

1

−±=

criX

XV


(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 the obtained when the two real roots of the

system voltage equation converge to a system voltage equation converge to a particular point and the Jacobian of the LF particular 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 a be defined as the limiting stage in a power power system beyond which no amount of system beyond which no amount of reactive power injection will elevate the reactive 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 adjusted The system voltage can only be adjusted by reactive power injection till the system by reactive power injection till the system voltage stability is maintained.voltage stability is maintained.

Expression for critical system reactance at Expression 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

EPEQQEX

EXQEPX

QPXEQX

+±−=

=−+

+=−

21

222222

)(4)2(2

1

2

2

+−−±+−= QPXEQX

EQXV

The expression for Xcri at voltage stability limitThe expression for Xcri at voltage stability limitfor any PF can be obtained by equating thefor any PF can be obtained by equating theabove equation with the radical sign to zero.above equation with the radical sign to zero.

Expression for critical system reactance at Expression for critical system reactance at voltage stability limit for any PFvoltage stability limit for any PF

)sectan(2

2

θθ +−=P

EX cri

Using, Q=P*tanUsing, Q=P*tanθθ, the final equation is given by,, the final equation is given by,

Method-IIMethod-IIIn a loss less line, the expression of reactiveIn a loss less line, the expression of reactivepower flow is given by,power flow is given by,

0cos2 =++ QVEBBV δ

−±=

BPE

EV

1tan4cos

2

1cos

222 θδδ

)sectan(2

2

θθ +−=P

EX cri


But the receiving end voltage at voltageBut the receiving end voltage at voltagestability limit is given by,stability limit is given by,

The above equation gives the value of powerThe above equation gives the value of powertransfer angle at voltage stability limit in termstransfer angle at voltage stability limit in termsof E, P, B and PF.of E, P, B and PF.

−= −

212

1

)tan4

2(cos

BP

E

Ecri θδ

Graphical representationGraphical representation

Fig. shows the characteristic of receiving endFig. shows the characteristic of receiving endvoltage (V) for varying system reactance (X).voltage (V) for varying system reactance (X).


(i)(i) The receiving end voltage falls with the The receiving end voltage falls with the increase in X at any fixed value of P till the increase 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 power For X=Xcri, V=Vcri beyond which real power demand can not be increased as otherwise demand can not be increased as otherwise voltage collapse and voltage instability voltage collapse and voltage instability occurs.occurs.

(iii)(iii)For X≠Xcri, load requires large current from For 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 sharp At knee point of the curve, there is a sharp increase in transmission line current which increase in transmission line current which leads to heavy series reactive loss of the leads 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 endvoltage (V) v/s X at fixed value of P for varyingvoltage (V) v/s X at fixed value of P for varyingPF.PF.


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

δ


As X increases, cri decreases. This offers As X increases, cri decreases. This offers anan

inherent limitation of operation of PS. This isinherent limitation of operation of PS. This isbecause voltage stability limited is attained atbecause voltage stability limited is attained atmuch 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 by Vcri 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 stability Vcri and power angle ( ) are low at stability limit.limit.

(iii)(iii)As PF is low (lagging), the stability limit and As 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

δδ

SummarySummary

(v)(v) Shunt capacitor compensation increases Shunt capacitor compensation increases the the

cri and receiving end voltage magnitude cri and receiving end voltage magnitude as it injects capacitive reactive power at as it injects capacitive reactive power at load bus.load bus.

(v)(v) However, this may lead to decrease in However, this may lead to decrease in voltage stability margin. voltage stability margin.

(vi)(vi)Proper selection of shunt compensation Proper selection of shunt compensation device is desired to achieve an acceptable device is desired to achieve an acceptable voltage profile.voltage profile.

δ

Expression for Vcri and cri at voltage stability Expression 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-bussystem being given bysystem being given by

The Jacobian can be obtained as The Jacobian can be obtained as

δsinX

EVP =

X

V

X

EVQ

2

cos −= δ

+−

=δδ

δδcos2- sin

sin cos 1

EVEV

EEV

XJ

δ

0sincoscos 22222 =+− δδδ VEEVVE

Voltage stability limit is obtained when the Voltage stability limit is obtained when the Jacobian becomes singular the determinant of Jacobian becomes singular the determinant of J is zero. J is zero.

(1)(1)

0cos 2 22 =− δEVVE

δcos2

EV =

δ ,criδ

In this equation V represents Vcri at voltage In this equation V represents Vcri at voltage stability limit when =stability limit when =

(a)(a)θδθ tansintanX

EVPQ ==

(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

EVQ

2

cos −= δ

24

θπδ −=

Since represents the power angle at critical Since represents the power angle at critical state of voltage stability. At = and state of voltage stability. At = and V=VcriV=Vcri

δ,cri

δδ

24 where,

cos2,

θπδδ

−== cricri

cri

EV

At unity power factor (UPF) operation, At unity power factor (UPF) operation, we find we find and with and with deterioration in power factor deterioration in power factor and would and would further reduce. further reduce.

[ ]°=°= 045, θδ cri

p.u.707.0, =criV

,criδ ,cri

V

Relation between Voltage Stability and rotor Relation between Voltage Stability and rotor angle stabilityangle stability

(i)(i) Rotor angle stability and Voltage stability Rotor angle stability and Voltage stability are affected by reactive power control.are affected by reactive power control.

(ii)(ii) The small disturbance voltage stability The small disturbance voltage stability increases rotor angle.increases rotor angle.

Difference between Voltage Stability and rotor Difference 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 Synchronous Reactive Power Capability of a Synchronous GeneratorGenerator

Synchronous generators are the primary deviceSynchronous generators are the primary devicefor 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 reactivepower capability of generator is consideredpower capability of generator is consideredaccurately 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 ofgenerator are commonly shown on P-Qgenerator are commonly shown on P-Qdiagram.diagram.


Active power limits are due to design of theActive power limits are due to design of theturbine 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 intoaccount in these studies.

The limitation of reactive power has threedifferent causes.


(i)(i) Stator currentStator current(ii)(ii) Overcurrent excitationOvercurrent excitation(iii)(iii)Under excitationUnder excitationWhen the excitation current is limited toWhen the excitation current is limited tomaximum value, the terminal voltage is themaximum value, the terminal voltage is themaximum excitation voltage minus the voltagemaximum excitation voltage minus the voltagedrop in Xs.drop in Xs.

The PS becomes weaker as the constant The PS becomes weaker as the constant voltagevoltage

moves away from the load.moves away from the load.


The voltage dependent limit of excitation currentis calculated by,

Where, Pg=active power of generatorEmax= the maximum electromotive forceXd=synchronous reactanceV=terminal voltage

2

2

2max22

max g

dds P

X

EV

X

VI −+−=


The reactive power limit corresponding to statorcurrent limit can be calculated by,

Reactive power capability increases when terminal voltage decreases.

The stator current limiter is used to limit reactivepower output in order to avoid statoroverloading. However, at the same time, it alsoreduces voltage.

22max

2max gss PIVQ −=


The generator reactive power capability isgenerally much less than that indicated bymanufacturer’s reactive capability curve.

This is due to constraints imposed by powerplant auxiliaries. It’s operation is threatenedwhen system voltage is low.


(ii) Automatic Voltage Control of Synchronous Generator

The automatic voltage controllers maintainconstant voltage when generators are operatedinside 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, theexcitation current must be limited after a fewseconds.

The overloading capability of generator may beimproved by making the cooling of generatormore effective.

Factors affecting Voltage StabilityFactors affecting Voltage StabilityFig. shows the action of automatic rotor andstator 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 ishit. This will change the generator characteristicdrastically.


Since the slope of rotor current limit is almostvertical, it is clear that the generator is on theverge of losing the voltage control capability ifrotor current limit is hit.

In this situation, the reactive power from thegenerator reduces fast which ultimately leads tovoltage instability.

Role of Transformer on Voltage control of a PSRole of Transformer on Voltage control of a PS

By changing transformation ratio, the voltage onsecondary side of any bus can be changed.

Change in transformation ratio is manual orautomatic.

Automatic change is done by OLTC.


Objective:

To determine the tap-changing ratio needed tocompletely or partially compensate the voltagedrop in line so that desired voltage control atreceiving end is obtained.


1:ts = sending end tap ratiotr:1 = receiving end tap ratio

The receiving end current is given by,BIEAtEt rs += 21

θtan22 E

Pj

E

PIR −=


The line current is given by,

Put

)tan1(

)tan1(

221

2

θ

θ

jEt

PBEAtEt

jEt

P

t

II

rrs

rr

R

−+=

−==

βα ∠=∠= BBAA ,

}sintancossin{

}tansincoscos{

)sin)(costan1()sin(cos

222

2221

221

βθβα

θββα

ββθαα

Et

PB

Et

PBEAt

jEt

PB

Et

PBEAtEt

jjEt

PBjEAtEt

rrr

rrrs

rrs

+−

+++=

+−++=


Equating only magnitudes and squaring bothsides

The minimum transformer tap ratio forcoordination between two transformers isachieved when ts.tr =1

ts=1/tr

2

222

2

222

21

2

}sintancossin{

}tansincoscos{

βθβα

θββα

Et

PB

Et

PBEAt

jEt

PB

Et

PBEAtEt

rrr

rrrs

+−

+++=


2

222

2

2222

21

}sintancossin{

}tansincoscos{

βθβα

θββα

Et

PB

Et

PBEAt

jEt

PB

Et

PBEAt

t

E

rrr

rrr

r

+−

+++=

βαθββ

θβαθβα

θβββα

θββα

θββα

sinsin2tansincos2

tancossin2tansincos2

tansincos2coscos2

tancossinsin

tansincoscos

22

222

22

222

2

2222

2

2222222242

42

2222222242

4222

21

EABPtPB

EABPtEABPt

PBEABt

PBPBEtA

PBPBEtAEE

r

rr

r

r

r

+−

−

++

++

+++=


θβαβα

θ

tan)sin(2)cos(2

)tan1(22

222

2

22242

4222

21

−−−

+++=

EABPtEABPt

PBEtAEE

rr

r

[ ]22

21

2223

222

42

21

32

24

1

sec

tan)sin()cos(2

0

EEPBC

ABPEC

EAC

CtCtC rr

−=

−−−=

=

=++

θ

θβαβα

21

1

21

31222

2

)4(

−±−=C

CCCCtr


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 ismaintained near to nominal value by tapchanger when primary system voltage drops.

This is possible when the system does not haveshortage of reactive power (system is capable toprovide sufficient reactive power).

During heavy load demand, the secondaryvoltage may become unstable even with tapchanging.

Effect of OLTC on Voltage StabilityEffect of OLTC on Voltage Stability

In this situation, raising tap position in order toraise secondary voltage will not work and thebus voltage will gradually collapse.


The secondary voltage is given by,

To obtain sensitivity of the voltage with tapchange we have to take rate of change of V with respect to a.

aEXaR

RV

222 )(+=

EXaR

XaRXaRR

a

V

23222

22

))((

))((

+

−+=∂∂


To have a stable voltage state,

The above condition is true only and only if

Hence, the secondary voltage drops if the tapposition is raised in order to boost up the loadbus voltage.

The voltage stability is lost when

0>∂∂a

V

XaR 2>

XaR 2<

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

WorkingWorkingInitially the system is operated at A0 (a=1).Raising tap position (a=1.015), shifts theoperating point to A’0.

At this point, the voltage is enhanced and thesystem 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 iscapable to transmit more power (PA2 to PA3).New stable operating point is A’.

WorkingWorkingIf tap changing operation is done is stable zoneof characteristic then the system may settle at anew operating position with higher voltage atload bus.

If it is carried out in unstable zone then busvoltage collapses even though tap position isincreased and system becomes unstable.

WorkingWorkingIf dynamic loading (IM) is assumed at load bus,P-V characteristic is shown below.

Stable zone

Unstable zone

(i) It is a system instability OR (ii) Reduction in ... · (i) It is a system instability OR (ii)...

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