4 voltage stability analysis

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Power System Dynamics and Stability (EG 804EE)

Instructor:

Ramesh Paudel

Institute of EngineeringPulchowk Campus

Power System Engineering

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Voltage Stability

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IZVV 12

jQPIV *1

*1V

jQPI

I

V1

jX

P+jQ V2

load

Load connected to source through a line

XV

jQPjVV

1

12

XV

PjX

V

QVV

11

12

(QX)/V1 V1

(PX)/V1

V2

Phasor diagram

Relation between Q and V. Sources and Sinks of Reactive Power

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Voltage Stability and Reactive Power Transmission

• As real power is the main variable to rotor anglestability analysis, reactive power is key to voltagestability analysis.

• Reactive power is easier to generate compared to realpower, but more difficult to transmit.

• Deficit of reactive power either locally or globally leadsto poor voltage profile and with the increasing inloading, it may lead to voltage collapse.

• The transmission of reactive power becomes difficultin heavily loaded lines. This can be illustrated throughan example given below.

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• This table clearly indicates that at higher loading conditionstransmission lines are unable to transfer reactive power.

Case Vs Vr Angle Qs Qr

a. Lightly loaded 1.10 1.00 10 0.845 0.555 b. Moderately loaded 1.05 0.90 20 1.894 0.056 c. Heavily loaded 1.00 0.80 50 3.238 -1.048

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It has also been observed that greater the distance of thereactive power sources from the reactive power demand:

The more difficult to control the voltage level.

The greater the required amount of reactive powercompensation.

The greater are the voltage gradient on line supplyingreactive power.

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Voltage Collapse

• The phrase “Voltage collapse” implies a non-viablevoltage which magnitude is decreasing fast in time.

• Voltage instability is the absence of voltage stability,and results in progressive voltage decrease (orincrease).

• Following voltage instability, a power systemundergoes voltage collapse if the post-disturbanceequilibrium voltages are below acceptable limits.

• Voltage collapse may be total (blackout) or partial.

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Major Voltage Collapse Incidents

Date Location

30, 31 July 2012 India

2 January 2001 India

1 November 2014 Bangladesh

26 January 2015 Pakistan

18 Aug 2005 Indonesia

11 March 1999 Brazil

10–11 Nov 2009 Brazil, Paraguay

31 March 2015 Turkey

14–15 Aug 2003 United States, Canada

28 Sep 2003 Italy, Switzerland, Austria, Slovenia, Croatia

18 Mar 1978 Thailand

9 Nov 1965 United States, Canada

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Causes of Voltage Collapse

Typically power blackouts are not caused by a singleevent but by a combination of several deficiencies.

There is no outage known where a faultless gridcollapsed completely due to a single cause.

The following preconditions are the basis for a highpower outage risk:

• High grid utilisation or high power demand

• Defects due to material ageing

• High power plant utilisation 10

Causes of Voltage CollapseIf the following events occur in combination with the above mentionedconditions there is a very high likelihood for a power blackout to occur:

• Cyber attacks

• Power plant shutdown

• Simultaneous grid interruption

• Sudden simultaneous high power demand

• Unforeseen simultaneous interruptions of several power plants

• Human failure during maintenance work or switching operations

• Insufficient communication between transmission/distributionsystem operators and power suppliers

• Power line collapse or electrical equipment breakdown due tonatural hazards

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Phenomenon of Voltage Collapse

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Voltage Stability

• Voltage Stability is the ability of a system to maintain voltage sothat when load admittance is increased, load power will increase,and so that both power and voltage are controllable.

• Voltage instability is a dynamic phenomenon.

• The voltage decay may take just a few seconds or ten to twentyminutes.

• If the decay continues unabated, steady-state angular instability orvoltage collapse will occur.

• A successful avoidance of system collapse is based on accuracy ofmethod and its low computation time.

• Fast response and accurate voltage stability indications in powersystems are still a challenging task to achieve, particularly whenpower systems operated close to its transmission capacity limits.

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Voltage Stability

Voltage instability may arise due many reasons, but somesignificant contributors are:

• Increase in loading.

• Load recovery dynamics.

• Line tripping or generator outages.

• Action of tap changing transformers.

• Generators, synchronous condensers, or SVC reachingreactive power limits.

• System changes causes voltage to drop quickly or driftdownward.

• Operators and automatic system controls fail to halt thedecay.

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Classification of Voltage Stability

Large disturbance voltage instability:

– Large disturbance such as faults, loss of generation orcontingencies.

– Requires long-term transient simulation.

Small disturbance voltage instability:

– Small disturbance such as incremental changes in loads etc.

– Static analysis used to determine margin and effect ofvarious factors.

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Classification of Voltage Stability

Other Classification:

(a) Long term stability (few minutes to hours):

The study of long term voltage stability involves the dynamicsof slower acting equipment such as tap changingtransformers, thermostatically controlled loads andgenerator current limiters.

(b) Short term or transient voltage stability (several seconds).

It involves dynamics of fast acting load components such asinduction motors, electronically controlled loads and HVDCconverters.

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Static Voltage Stability Analysis using P-V and V-Q curves

• Most of the analyses in voltage stability area have beenconsidered it as “static phenomenon”.

• This is due to the fact that in most of the incidences ofvoltage instability, it has been observed as slowphenomenon characterized by slow variation of voltageover long period of time (several minutes) followed byrapid change when it reaches very close to the instability.

• Some of the concerns of operating engineers are:

– To know the steady state loadability limit

– Effect of control actions

– Effect of system parameter changes on the system stability orits loadability.

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• Two curves that have been popularly used are P-V curveand V-Q curve (or Q-V curve).

• These are especially useful for conceptual analysis ofvoltage stability of radial systems or longitudinaltransmission lines.

• Once again consider a simple radial system having a load ofP+jQ; E< V<0

P+jQjx

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• The load voltage equation can be written as:

• For a given values of loading conditions (i.e. P,Q, and E),there are four values for V; only two of them are real.

222

2

E

VQX

E

PXV

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• For a fixed load power factor and different values of P, thecurve obtained is known as P-V curve.

• When both solutions coalesce with each other, the systemloading corresponds to maximum real power loading(loading margin of the system) and the voltage is known as“Critical Voltage”.

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PV curve or Nose curve

Vcritical

Total load PL (p.u)

Voltage (p.u.)

Maximum loading point

High voltage solution

Low voltage solution

Stable

Unstable

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• There are three regions related to the parameter P: In the first region, the power flow has two distinct solutions for each choice of

P; one is the desired stable voltage and the other is the unstable voltage.

As P is increased, the system enters the second region, where the twosolutions intersect to form one solution for P, which is the maximum.

If P is further increased, the power-flow equations fail to have a solution.

• Each value of the transmissible power corresponds to a valueof the voltage at the bus until V=Vcrit after which furtherincrease in power results in deterioration of bus voltage.

• The risk of voltage collapse is much lower if the bus voltage isfurther away, by an upper value, from the critical voltagecorresponding to Pmax.

• Hence, the P-V curve can be used to determine the system’scritical operating voltage and collapse margin.


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• The P-V curves are used to determine the MW distance from theoperating point to the critical voltage.

• The curve is constructed through the variation of the active power, with aconstant power factor, in a particular bus where to infer about the stabilityis desired.

• These curves are built carrying out load flow calculations with a gradualincrease in the active power and observing the voltage variation in the busunder analysis.

• The nose of the PV curve represents the voltage instability critical point. Itis the maximum power (Pmax) point that can be delivered to the bus underanalysis . If the system demands a bigger load, it will enter into a conditionof voltage instability.

• Through the PV curves it is possible to estimate the maximum voltageloadability conditions and the critical solutions.

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Maximum loading point shifts with different power factor.

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The maximum loading point shifts with contingencyin the system.

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• For a given P, variation of voltage with Q can be plotted andV-Q curve can be obtained.

• In an interconnected system, P-V and Q-V curves can beused to study the voltage variation at a bus for change inloading at that bus.

• These curves can be generated by running several loadflow solutions at different loading conditions.

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A typical Q-V curve

Q

V

StableUnstable

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• The curve can be used as an index for voltage instability. Nearthe nose of a Q-V curve, sensitivities get very large and thenreverse sign.

• It can be seen that the curve shows two possible values ofvoltage for the same value of power.

• The power system operated at lower voltage value wouldrequire very high current to produce the power. That is whythe left portion of the curve is classified as an unstable region;the system cannot be operated, in steady state, in this region.

• The right portion of the curve represents the stability regionwhile the bottom portion from the stability limit indicates theunstable operating region. It is preferred to keep theoperating point far from the stability limit.


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• The Q-V curves are used to determine the Mvar distance fromthe operating point to the critical voltage.

• It shows the sensitivity and variation of bus voltages withrespect to reactive power injections or absorptions.

• They are used by utilities as a workhorse for voltage stabilityanalysis to determine the proximity to voltage collapse and toestablish system design criteria based on Q and V marginsdetermined from the curves.

• Q axis shows the reactive power that needs to be added orremoved from the bus to maintain a given voltage at a givenload.

• The reactive power margin is the Mvar distance from theoperating point to the bottom of the curve.


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Q-V curves and Reactive Margins

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The P-V and V-Q curves can also be used to studyimpacts of load variations, load type and controls onsystem maximum loadability and voltage collapse.

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Impact of Load Type

• Load characteristics have a profound effect on voltageinstability and collapse.

• Usually, three Static load models are used in thestability analysis:

– constant PQ

– constant current

– constant impedance.

• Their relative severity to the voltage instability is in thesame order, in other words, the constant PQ load isthe most onerous load in the system followed byconstant current and then constant impedance.

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Impact of Load Type

Illustration of the Impact of Loads on voltage stability:

Normal case (Intact case)

Constant PQ load

Constant impedance

Q limit violated

Stressed condition

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P

V

Reverse action of transformer OLTC

V

P

n1

n2 > n1

E<0 V1 V

RnE

XnR

RV

222 )(

2

3222

22

})({

))((

XnR

XnRXnRR

n

V

jX

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Voltage Stability Analysis

The analysis of voltage stability involves– The examination of proximity to voltage instability and

– Mechanism of voltage instability

Proximity to voltage instability:

How close the system is to voltage instability?

• This involves measuring the proximity of instability in termsof measurable quantities such as real and reactive power.

• The most appropriate measures for any given situationdepends on the specific system and the intended use ofmargin, e.g. planning vs. operating margin.

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• Consideration must be given to possible contingencies e.g.line outages, loss of generating units or other reactivepower source etc.

• Mechanism of voltage instability:

How and why does instability occur?

What are the factors contributing to instability?

What are the voltage weak-areas?

What measures are the most effective in improving voltagestability?

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• Time-domain simulations, in which appropriate modeling isincluded, capture the events and their chronological eventsleading instability.

• The problems with this kind of simulations arecomputationally time consuming and do not providesensitivity information and the degree of stability, readily.

• System dynamics influencing voltage instability are usuallyslow.

• Many aspects of the problem can be effectively analyzed byusing static analysis methods

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• The static analysis techniques allow examination of a widerange of system condition of power system and providemuch insight of the nature of the problem and various

contributing factors.

• Dynamic analysis, on the other hand, is useful for detailedstudy of voltage collapse situations.

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Modeling Requirement

• The following components, which have significant impacton voltage stability, have to be taken into account in theanalysis:

– Loads (Voltage and frequency dependence of load,induction machines, expansion of sub-transmissionnetwork, including OLTC, compensators and regulators).

– Generator excitation control (excitation limits)

– FACTS controllers

– Automatic generation control

– Protection and control

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Dynamic Analysis

• The general structure used in the dynamic voltage stabilityanalysis is similar to that for transient stability studies.

• The overall system equations include a set of first-orderdifferential equation and a set of algebraic equations.

• Since we include the representation of transformer tap-changer and phase-shift angle controls, the elements of thenetwork node admittance matrix changes as a function ofbus voltage and time.

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Dynamic Analysis

• The set of ODE and algebraic equations can be solved intime-domain by using any of the numerical integrationmethods.

• The study period is typically in the order of severalminutes.

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Static Analysis

• For static voltage stability analysis of an interconnectedsystem, power flow models have been considered.

• It has been proved that near system maximum loadingpoint the load flow diverges and Power flow Jacobian(Newton Raphson Load Flow, NRLF) becomes singular.

• The NRLF equations in polar coordinates can be written as

VJJ

JJ

Q

P

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Static Analysis

• Singularity of the reduced Jacobian, JR defining Q-V has alsobeen used to determine static voltage instability.

Where,

• At system maximum loadability or the point of staticvoltage instability, the determinant of full Jacobian J or thereduced Jacobian JR becomes zero or the NRLF diverges.

• Divergence of load flow can be due to some other reasonsas well.

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134 JJJJJR

VJQ R

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Static Analysis

• Some of the other methods which establish the singularityof Jacobian are as following.

Singular value decomposition Technique:

• The Jacobian J is decomposed into three matrices as givenbelow:

Where,

U and V are left and right eigenvectors, respectively.

D is a diagonal matrix contains singular values of J.

Tnn VDUJ 44

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Static Analysis

• When J become singular, the minimum singular value ofthe matrix J assumes zero.

• Ratio of the maximum to minimum singular values of theJacobian is know as the condition number of J.

• The condition number assumes a larger number (infinite)when J becomes singular.

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Static Analysis

Q-V modal Analysis

• Voltage stability characteristics of the system can beidentified by computing the eigenvalues and eigenvectorsof the reduced Jacobian matrix JR.

Where,

v=V is the vector of modal voltage variations and

q=Q is the vector of modal reactive power variation .

RJ

qv 1

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Static Analysis• The above equation can be written in the following form

• If i > 0, the ith modal voltage and the ith modal reactivepower variations are along the same direction, indicatingthat the system voltage is stable.

• If i < 0, the ith modal voltage and the ith modal reactivevariation are along in the opposite directions, indicatingthe system voltage is unstable.

i

i

i qv

1

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Static Analysis

• The magnitude of each modal voltage variation is equal tothe inverse of i times the magnitude of the modalreactive power variation.

• Therefore the magnitude i determines the degree ofstability of the ith modal voltage.

• The smaller the magnitude of positive i, the closer the ithmodal voltage is to being unstable.

• When i = 0, the ith modal voltage collapse because anychange in that modal reactive power causes infinite changein the modal voltage.

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Static Analysis

Continuation Power Flow (CPF)

• CPF methods are typically employed to trace the upper andlower part of P-V curve, including the maximum loadingpoint of the system.

• The technique is computationally demanding for largersystems, however, it provide additional information such assensitivity with respect to parameter variation, which areuseful in analyzing the system further.

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Static Analysis


• The CPF technique uses an iterative process involvingpredictor and corrector steps.

• In some cases additional parameterization step is used toavoid certain convergence problem.

• Additional technique like “step cutting” can also be used tosolve the divergence problem.

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Static Analysis


PredictorCorrector

Critical voltage

Voltage

Total power P or

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Static Analysis

• Continuation Power Flow (CPF)

• From a known initial point A, a tangent predictor step isused to estimate the solution point B for a given loaddirection defined by the parameter .

• A corrector step is then used to determine the exactsolution C using a power flow with an additional equationto find out the proper value of .

• This process repeated until the desired bifurcation or P-Vcurve is obtained.

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Static Analysis

• Mathematical Formulation of CPF

• In this technique, the basic equations are similar to theones used in standard power flow analysis, except that isadded as a parameter.

• The re-formulated power flow equations, with provision forincreasing generation as the load is increased, may beexpressed as:

KVF ),( (1)

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Static Analysis


• Where

= the load parameter,

= the vector of bus voltage angles,

V= the vector of bus voltage magnitudes,

K= the vector representing percentage load change at eachbus.

• The above set of nonlinear equations is solved byspecifying a value for such that lie between 0 and itscritical value.

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Static Analysis


• Where =0 represents the base load condition, and =critical represents the critical or maximum loading conditionfor the given load and generation direction.

• The equation (1) can be rearranged and written in thefollowing form:

0),,( VF (2)

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Static Analysis


Predictor Step:

• A linear approximation is used to estimate the nextsolution for a change in one of the state variables (i.e. , V,or ).

• Taking the derivatives of both sides of Equation (2), withthe state variables corresponding to the initial solution, willresult in the following set of linear equations:

0 dFdVFdF v

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Static Analysis


Predictor Step:

• or the equation can be written in the following form:

• Since the insertion of in the power flow equation addedan unknown variable, one more equation is needed tosolve the above equations.

0

d

dV

d

FFF V

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Static Analysis


Predictor Step:

• This is carried out by setting one of the components of thetangent vector to +1 or –1, i.e. tk= 1.

• This component is referred to as the continuationparameter. Now the above equation can be re-written as:

1

0

d

dV

d

e

FFF

k

V

where ek=(0,0,….,1,0,0).58

Static Analysis


Predictor Step:

• Once the tangent vector has been found, the prediction ofstates can be done as:

• Where is the step size, chosen to make sure the powerflow solution exist with the specified continuationparameter

d

dV

d

VV

0

0

0

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Static Analysis


Predictor Step:

• If for a given step size a solution cannot be found in thecorrector step, the step size is reduced and the correctorstep is repeated until a successful solution is obtained.

• This process is popularly known as the “step cutting”method.

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Static Analysis


Corrector Step:

• In the corrector step, the original set of equationsF(,V,)=0 is augmented by one equation that specifies thestate variable selected as the continuation parameter.

• xk is the state variable selected as the continuationparameter and is equal to the predicted value of xk.

]0[),,(

kx

VF

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Static Analysis


Corrector Step:

• This set of equations can be solved using NR (NewtonRaphson) power flow method.

• The introduction of additional equation specifying xkmakes the jacobian non-singular at the critical point.

• The continuation power flow can be continued up to andbeyond the critical point (i.e. lower portion of the P-Vcurve).

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Proximity Indicators to Static Voltage Stability Limit

• Proximity indicators or performance indices are topredict or detect voltage collapse problems arevery useful for power system operation staffs.

• These indices could be used as online or off-linetools to help operators determine how close thesystem is to collapse or instability point.

• The objective of these indices to define a scalarmagnitude that can be monitored as the systemparameter changes (e.g. loading level).

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• These indices should have a “predictable shape”and smooth behavior so that a possible predictioncan be made well in advance to avoid possiblecollapse by taking appropriate remedial measures.

• Another aspect of these indices is that they shouldbe computationally fast, especially in the on-linesystem monitoring.

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(1) Sensitivity Factors

• Sensitivity factors are used in several utilities throughoutthe world to detect voltage stability problems and to devicecorrective measures.

• Some of the sensitivity information can be obtained fromload flow Jacobian at no cost.

• These indices were first used to predict voltage controlproblems in generator QV curves.

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• As generator i approaches the bottom of its QV curve, thevalue of VSF of the generator becomes large.

• Based on this concept, more general system wide indiceshave been proposed.

i

iiidQ

dVVSF max

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• When SF becomes larger the system turns “insecure” andeventually collapses, due to all entries dzi/d , whenthe system approaches a maximum value of the parameter ( 0).

• As typically represents the load changes, the collapsepoint associated with the maximum value of is usuallyreferred to as maximum loadability point.

d

dzSF

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• If only system voltages V are monitored, asequivalent VSF can be defined as

d

dVVSF

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(2) Singular Value and Eigenvalues

• Singular value or eigenvalue of power flow jacobian can bemonitored with the loading parameter.

• At the collapse or instability point the Singular value indexor eigenvalue index become zero.

• These indices have been used for AC as well as AC-DCnetwork.

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(3) Real and Reactive Power Margin

• Real or reactive power margin can be more practicalindicator to the operators.

• Reactive power margin can be expressed as the first orsecond norm of distance between present reactive powerloading vector say QL

o and nearest extreme loading pointQL*.

• The worst case loading QL* can be determined byminimizing ||QL*- QL

o|| subjected to satisfying the powerbalance and in addition the singularity condition ofJacobian.

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Other Indices

(4) Voltage Collapse Proximity Indicators (VCPI)

(5) System determinant

(6) Voltage controllability index

(7) Center Manifold Based Index

(8) P and Q angles

(9) Energy Functions

(10) Reactive Power Margin

(11) V/Vo Index

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Prevention of Voltage Collapse

Based on system design measures and operating measures,prevention of voltage collapse can be classified as:

System Design measures

a. Control of network voltage and generator reactive output

b. Application of reactive power-compensating devices

c. Coordination of Protection/Controls

d. Control of transformer tap changers

e. Under-voltage load shedding

System Operating Measures

(a) Stability margin

(b) Spinning reserve

(c) Operator’s action72

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System Design measures

(a) Control of network voltage and generator reactiveoutput

• Load compensation of a generator AVR regulatesvoltage on the high-tension side of or partwaythrough, the step-up transformer.

• In many situations this has a beneficial effect onvoltage stability by moving the point of constantvoltage closer to the loads.

• Alternatively, secondary outer loop of generatorexcitation may be used to regulate network sidevoltage.

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(b) Application of reactive power-compensating devices

• Adequate stability margin should be ensured byproper selection of compensation schemes.

• Some well know reactive power compensation devicesare shunt, series capacitors, SVC and STATCOM.

• Section of the size and installation locations of thesecompensation devices should be based on a detailedstudy covering the most onerous system conditions forwhich the system is required to operate satisfactorily.

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(c) Coordination of Protection/Controls

• Adequate coordination should be ensured basedon dynamic simulation studies.

• Tripping of equipment to prevent an overloadcondition should be the last resort.

• Wherever possible, adequate control measures(automatic or manual) should be provided forrelieving the overload condition before isolatingthe equipment from the system.

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(d) Control of transformer tap changers

• Tap changers can be controlled, either locally orcentrally, so as to reduce the risk of voltagecollapse.

• Where tap changing is detrimental, a simplemethod is to block tap changing when the sourceside voltage sags, and unblock when the voltagerecovers .

• There is a potential application of improved UnderLoad Tap Changes (ULTC) control strategies.

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(e) Under-voltage load shedding

• To manage with unplanned or extreme situations,it may be necessary to use under-voltage load-shedding schemes.

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Thank you.

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4 voltage stability analysis

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