Advanced Contingency Selection MethodologyGeorge K. Stefopoulos, Fang Yang, George J. Cokkinides and A. P. Sakis Meliopoulos

Abstract- This paper describes the development andimplementation of contingency ranking and selection algorithms,as part of a power system security assessment program. Thework concentrates on performance-index-based algorithms anduses a contingency control variable for precise contingencyrepresentation. The ranking is based on the value of thesensitivity of the performance index with respect to thecontingency control variable for each outage. The computation ofthe sensitivities is performed using the very efficient co-statemethod. Furthermore an approach for improving the accuracy ofperformance-index-based contingency ranking methods isintroduced. This approach is based on state rather thanperformance index linearization with respect to the contingencyvariable and it provides more accurate results in contingencyranking and selection. The effectiveness of the proposed methodin identifying critical contingencies is illustrated using some smalltest systems. The ultimate goal is to achieve fast and accuratecontingency selection, without having to solve the full load-flowproblem for each contingency (as is the current utility practice).

Index Terms- Contingency Analysis, Contingency Ranking,Performance Index.

I. INTRODUCTIONONE of the main computational issues in power system

steady state security assessment as well as in reliabilitystudies is contingency ranking and selection. Specifically, thecritical contingencies from all possible contingencies need tobe identified and analyzed. For large scale systems, theprocess imposes a substantial computational burden. For thisreason, there have been consistent efforts to invent fastcontingency selection algorithms and subsequent contingencyanalysis. Significant developments in the area of contingencyselection include (a) contingency ranking with performanceindices (PI) [1], (b) local solutions based on concentricrelaxation [2], (c) bounding methods [3,4], etc. In addition,significant developments for contingency analysis include theintroduction of the fast decoupled power flow [5], sparsity-oriented compensation method [6], sparse vector methods [7],partial refactorization [8], etc.

Much attention on this issue has been focused on

This work was supported by the Power System Engineering ResearchCenter (PSERC).

George K. Stefopoulos is with the Georgia Institute of Technology,Atlanta, GA 30332 USA (e-mail: gstefopg ece.gatech.edu).

Fang Yang is with the Georgia Institute of Technology, Atlanta, GA 30332USA (e-mail: gtg65 lj mail.gatech.edu).

George J. Cokkinides is with the Georgia Institute of Technology, Atlanta,GA 30332 USA (e-mail: george.cokkinides@ ece.gatech.edu).

Sakis Meliopoulos is with the Georgia Institute of Technology, Atlanta,GA 30332 USA (e-mail: sakis.meliopoulos@ ece.gatech.edu).

0-7803-9255-8/05/$20.00 2005 IEEE

subnetwork solutions which solve for a subset of statevariables, using sparse vector methods. A typical applicationof subnetwork solutions to contingency selection is thebounding method [3], [4]. The subnetwork solution methodhad been also generalized with the zero mismatch approach[9], which is an iterative AC power flow solution method andis effective for both contingency screening and analysis.

Typical contingency selection methods consist of eitherranking methods using a performance index (PI) or screeningmethods based on approximate power flow solutions. It iswidely recognized that PI based methods are efficient butvulnerable to misrankings, while screening methods are moreaccurate but inefficient. It has been also identified that theinaccuracies of the P1 based methods are mainly due to (a)nonlinearities of the system model [10], and (b) discontinuitiesof the system model arising from generator reactive powerlimits and regulator tap limits [11], [12].A hybrid contingency selection method had been proposed

in the past which takes advantage of the best features of thetwo approaches [11]. This method utilizes a procedure basedon the concept of contingency stiffness index to identify'nonlinear' contingencies and a performance index method toidentify contingencies causing discontinuities. While thehybrid method performs very well, the computational burdenis many times greater than pure PI based methods.

This paper describes the development and implementationof contingency ranking and selection algorithms as part of apower system analysis program. The implementation is basedon PI-based methods and on the use of the quadratized powerflow system model, which is described in the Appendix. Eachcontingency is modeled with the introduction of a contingencyor outage control variable. The ranking is based on the valueof the sensitivity of the performance index with respect to thecontingency control variable for each contingency. Thecomputation of the sensitivities is performed using the veryefficient co-state method. Moreover, the paper introducessome new concepts as a way for improving the accuracy ofthe P1-based contingency ranking methods. The basic conceptis the use of state-linearization rather than performance-index-linearization with respect to the outage variable and it appearsto provide more accurate results in contingency ranking andselection, at the expense of slightly increased computationaltime. However, with the appropriate use of sparsity techniquesthis increase can become minimal. The methodologyimplementation is demonstrated with two simple systems.Numerical experiments are proposed to further investigate andquantify its performance in large scale systems. The Pl-contingency selection is implemented as part of a power

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system analysis program and it is used in combination withcompensation-based contingency analysis method for steadystate security assessment, and as a pre-filter for contingencyranking in power system reliability analysis.

II. PI METHODS FOR CONTINGENCY RANKINGPI-based contingency ranking methods reported in the

literature are based on the evaluation of the PI gradient withrespect to an outage. In [12] a rigorous definition of an outagehas been proposed with the use of the outage (or contingency)control variable Uc that has the following property [12]:

1.0, if the component is in operation (1)10.0, if the component is outaged

The outage control variable is used in the componentmodeling of a power system, as illustrated in Fig I and 2.

BUS k BUS mu. (gkm + jbkm)

jbkms U

Fig. 1. Circuit outage control variable Uc.

Jc =YEwX(-~~'CINwherewj weighting factor, 0 < wj <I,

IN,j : current-based thermal limit of the line,I. : magnitude of actual current through circuit j,

n positive integer parameter defining the exponent.

* Circuit power-based index:

wherewj weighting factor, O < Wj < I,

SN,J : power-based thermal limit of the line,

Si : apparent power through circuit ],n positive integer parameter defining the exponent.

k mYlkmUc

Fig. 3. Representation ofcommon mode outages with control variable Uc.F. G ut oPower System

Fig. 2. Generating unit outage control variable Uc~

Note that in the case of a generator outage the outagecontrol variable not only affects the power produced by theoutaged unit, but also indicates the re-dispatch of its producedpower to the remaining units, via a re-dispatch algorithm (Fig2 illustrates a linear re-dispatch, based on participationfactors). The contingency control variable can not only modelindependent contingencies, like the ones indicated above, butcan be also effectively used to model common modecontingencies, i.e., contingencies that are dependent upon

each other. Such a situation is illustrated in Fig 3.

Depending on the purpose of contingency selection, a

variety of performance indices can be applied to it:* Circuit current-based index:

* Voltage index:

Jv = W V( ) (4)

whereWk : weighting factor, 0 < w < I,

vk mezm nominal bus voltage value (typically I p.u.),(it is in general the mean value in the desired range,

i.e. I (Vma + 7i.

Vk*,,p :voltage deviation tolerance (i.e. I (Vkmax - Vknin )),2

Vk : actual voltage magnitude at bus k,n : positive integer parameter defining the exponent,N : total number ofPQ buses.

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(2)

(3)

* Generation reactive power index:

JQ l QQmeanjJmea

whereWj

Q.i,meanweighting factor, 0 < w1 < 1,

: expected generated reactive power value,This is the mean value is the allowable range for

each generator, i.e., I (Qmax + Qmin )

Qj?,ep : reactive power deviation tolerance,This is half of the allowable range, i.e.,I(Qrmax _ Qrmini)

Qj :actual reactive power generated by unit j,n : positive integer parameter defining the exponent,L : total number of generating units.

Several other performance indices can be defined. In thispaper the circuit current-based index and the voltage index aremainly considered. By including the outage control variable inthe system modeling the defined performance index J can beexpressed as a function of the state vector and of the controlvariables,J(x,uU). The change of the performance index dueto the contingency is:

AJ =J(x ',u. =0.0)-J(x0,UL = 1.0), (6)

where x° is the initial state, prior to the contingency, xnew isthe state after the contingency and the control variable u.changes from 1.0 to 0.0, modeling the component outage. Thefirst order approximation of the performance index variation isprovided by the derivative of the PI with respect to the controlvariable:

dl dil7AJ= - Au =-.(U-l) (7)duc duc

and for a change in uc from 1.0 to 0.0:

A^r=_tJ=(8)duc

It is therefore expected that the derivative of the performanceindex with respect to the outage control variable at the presentoperating point will provide a measure of the severity of adisturbance. Therefore, contingencies are ranked based on thevalues of dJ , which expresses the first order change of the

ducperformance index. The values of these derivatives can becalculated using the co-state method:

dl aJ AT ag(X,uC) (9)d _=u-X * 9duc auc Ouc

(10)-T~~~~~~~~-T 5)J(x, U() ag(x, U,)x=Ax

J(x,u,) is the performance index, g(x,u,) = 0 are the powerflow equations, with the outage control variable incorporatedin them, uC is the vector of the control variables of interest, x

is the state vector, xT is the co-state vector.Note that the performance index may depend explicitly on thecontrol variables and it also depends implicitly on themthrough the power flow equations. The explicit dependence iscaptured by the partial derivative ai and the implicit by the

aucterm rT. ag(x, '0 . Furthermore, the co-state vector is

auc

invariant for all contingencies, therefore it is pre-computed atthe present operating condition, resulting in extremely fastcomputations, even for large scale power systems. After theco-state vector is computed the sensitivity of the performanceindex for each contingency is simply a vector-vectormultiplication, with one of the vectors being very sparse.

The contingency ranking algorithm based on the co-statemethod is efficient and precise as a first order method. As amatter of fact the computational requirements are equivalentto one iteration of the power flow algorithm for the entire setof contingencies (cost of computing the co-state vector).However, for contingencies that trigger severe nonlinearities,the method may lead to misrankings. This is because thebehavior of the performance index around the presentoperating point may be significantly different from thebehavior as we move away from the current operating point.This issue has been addressed with the hybrid method whichseparates contingencies into those that trigger severnonlinearities and those that do not. The former are processedwith more accurate and computationally demandingcontingency selection methods and the latter (which representthe majority of contingencies) are ranked with the abovedescribed PI based method. The processing of a small numberof contingencies via selection methods adds to thecomputational burden. Therefore it is important to be able touse PI based methods on all contingencies. The followingproposed method in this paper provides a promising approachtowards this goal.

III. HIGHER ORDER STATE-LINEARIZATION PI-BASEDCONTINGENCY RANKING

The proposed state linearization approach is a variation ofthe PI-based contingency ranking algorithm. In this method,instead of linearizing the performance indices directly, thesystem states are linearized with respect to the contingencycontrol variable; the performance index J is then calculatedas follows:

J = J(x +-(u - 1)U )cduc

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(1 1)

wherex : present operating condition,X : system state vector,u : contingency control variable.The utilization of the linearized system states in calculating

the system performance index provides higher order terms inTaylor's series. The unique potential of this method has beenproven in some preliminary work by the authors described in[13]. The state-linearization sensitivity method provides thetraces of indices with curvature, which can follow the highlynonlinear variations of the original indices to some extent,while the PI-linearization method provides only the straightline. Therefore, the higher order sensitivity method is superiorto the simple PI-linearization-based method.

The contingency selection is based on the computation ofthe performance index change due to a contingency andsubsequent ranking of the contingencies on the basis of thechange. Mathematically one can view the outage of a circuitas a reduction of the admittance of the circuit to zero. We useagain the outage control variable, uc, as illustrated in Figures 1through 3. Consider the performance index, J. The change ofthe performance index due to the contingency is:

AJ = Jrx + duic - 1),Uc - J(X ,uc = 1.0)du,(12)

where x° is the present operating condition. The sensitivity ofthe state with respect to the control variable can be easilycomputed as:

dx aG(x, u) -aaG(x, u)duc Ax .

x

a)C:(13) 4)1E0t !

Note that aG(x, u) is the Jacobian of the system and thereforeax

it is pre-computed at the present operating condition andremains invariant for all contingencies. Thus for eachcontingency we have to only compute the partial derivativesof the power flow equation G(x,u) with respect to thecontingency control variable. This vector has only fewnonzero entries and therefore the computations are extremelyfast. Taking into account the sparsity of this vector can greatlyimprove the efficiency of the method. It should also be noted

that dx is a vector of the same size as the state vector eachduc

element of which is the derivative of the corresponding statewith respect to the control variable. Once the new state iscomputed via this linear approximation, the calculation of thenew value of the performance index is a straightforwardoperation.

The concept of the approach is illustrated graphically inFigures 4 and 5 based on results obtained from the applicationof the method to a test system. The first order analysis curverepresents the PI-sensitivity linear curve after performing thelinearization of the index with respect to the contingency

control variable. The higher order analysis curve is the statelinearization curve with respect to the contingency controlvariable.

Plot of Jc vS u1.5 l-l ,-r - --

~~I1.4 I

~~I

i ugI el!dlrl-lljl141.2-R-t----|------I----±---r--------------

f, ctua ciurwte,

0. 'ana 1sc I I

1.2 ----- --------- - - - - - -4 - - - 9- - - -

- -F

a) AI

__,_ __ __ I_ __

anlrtbreralysis cru_____________

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-Control Variable u

Fig. 4. Plots of circuit-loading index vs. the contingency control variable uc.

Plot ofJvVs u

2I

aCIa. IB

0.4 0.5 0.6Control Varable u

Fig. 5. Plots of voltage index vs. the contingency control variable uc.

IV. METHODOLOGY IMPLEMENTATIONThe proposed methodology is implemented in a Visual C++

environment using object oriented techniques. It is created asan expansion of existing power system analysis software,developed by the power systems laboratory of Georgia Tech.Both the PI-linearization and the state-linearizationapproaches are integrated in the same environment, and theuser can pick the one to chose. An additional option ofcalculating the sensitivity of the PI wrt to the outage controlvariable numerically, using a secant approximation is alsoavailable and mainly used for debugging purposes. Themethods and data needed to perform the PI-based contingency

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ranking are implemented in a separate class, thus facilitatingthe module implementation and its re-usability from otherparts of the program (like contingency analysis, securityassessment, reliability analysis, etc). The linearization of theperformance indices under consideration are performed byobjects of a different class, the Linearization class, making theprogram more generic and easily expandable.A graphical user interface has been developed to facilitate

the input of data. The user defines the performance index to beused, the contingencies to be considered and the desiredsolution approach (i.e. first order analysis with Pl-linearization, higher-order analysis with state-linearization andif numerical check with secant approximation will beperformed) along with the required solution parameters. Theresults provided are the estimation of the performance indexchange, AJ, based on which the ranking is performed. Theresults are presented to the user in a numerical form. The userinterface of the program is shown in Fig 6.

using the PI-linearization approach, the state-linearizationapproach and the full load flow solution, for the current-basedcircuit loading index. Table II presents similar results for thevoltage index.

n -

Close__-

CX TWBCsed Cxcu Loadingktdex Exponent n

oogs" index2Acietiv Power Ba.d Ciouk Loading IndexCwwation Reactive Power hidex

iSme Device * AM Deyvt

Contmml 3 (spao)Comnat 4 (Spamn

CktIl0 ak.e

goCVdwnafI*,ghw O.drMUelyxe -

Delta u: 'O Q, >

Fig. 6. Main user interface for probabilistic power flow analysis.

V. NUMERICAL EXAMPLESThe method has been applied to two small power systems, a

four-bus system and the IEEE 24-bus reliability test system(RTS). The developed power system analysis software is usedto perform all the computations, including the base-casepower flow analysis. The quadratic power flow (QPF) modelis used in both cases. The proposed state-linearization methodis compared to the PI-linearization contingency selectionalgorithm. The cases and the system loading is chosen so thatthe traditional index-linearization approach results in somemisrankings.

A. Four-bus systemThe four-bus test system is depicted in Fig. 7. It consists of

two constant power loads, two generators and fourtransmission lines represented with pi-equivalent circuits.Table I presents comparative results from contingency ranking

Fig. 7. Test system used for contingency ranking evaluation.

TABLE IRANKING RESULTS FOR CURRENT-BASED CIRCUIT LOADING INDEX FOR FOUR-

BUS SYSTEM.

Outaged Actual P1-linearization State-linearizationLine Ranking Ranking Ranking10-3020-4030-40

23

24

23

TABLE IIRANKING RESULTS FOR VOLTAGE INDEX FOR FOUR-BUS SYSTEM.

Outaged Actual PI-linearization State-linearizationLine Ranking Ranking Ranking10-30 1 4 120-40 2 1 210-20 3 2 330-40 4 3 4

The state-linearization approach shows considerablyimproved performances and provides the correct ranking inboth cases. The traditional PI-sensitivity approach results inone misranking in each case, especially in the voltage index,where it ranks the most severe contingency as least severe (therelative ranking of the rest of the contingencies is correct).This is related to the nature of the index and the system.Doing a homotopy continuation plot of the value of theperformance index for values of the outage variable uC fromone to zero we can see that the value of the index is reducingwhen we are not far away from the base-case operating point(where the sensitivity if computed) and then starts increasingrapidly as uC approaches zero. This effect cannot be capturedby the first order sensitivity, it is, though, partially captured bythe higher order state-linearization. It should also be noted thatthere is no load flow solution for an outage of line 10-30, andthis is the reason why this contingency is ranked as the mostsevere in the actual ranking.

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B. IEEE 24-bus reliability test system (RTS)The IEEE 24-bus test system is shown in Fig 8. System data

can be found in [14].

Fig. 8. The IEEE 24-bus reliability test system.

The eleven most severe contingencies in terms of voltagedeviation (as identified by full power flow solution of thesystem) are presented in Table III, along with the rankingobtained by the PI-linearization and the state-linearizationapproach. The traditional PI-linearization approach results inseveral misrakings. Though most of them are not significant,there are three very important misrankings, with severecontingencies belonging to the top-ten list being ranked asmuch less severe. The most severe contingency is among themand it is in fact identified as the least severe. The reason forthat is again similar, as in the four-bus system and is related tothe nature of the performance index, as explained in theprevious paragraph. Most of the other severe misrankings are,nevertheless, identified as being in the top-ten list. The state-linearization approach performs much better and all the topcontingencies are identified. There is only one severemisranking, with one contingency being ranked significantlylower than it actually is.

TABLE IIIRANKING RESULTS FOR VOLTAGE INDEX FOR THE TOP 10 CONTINGENCIES OF

THE IEEE 24-BUS SYSTEM

Outaged Actual PI-linearization State-linearizationBranch Ranking Ranking Ranking60-100 C 1 39 120-60 2 1 2

10-110 T 3 2 3150-240 4 19 850-100 5 5 6

100-120 T 6 3 480-100 7 4 510-50 8 6 7240-30 9 31 32110-140 10 7 930-90 1 1 8 10

T: Transformer branch, C: Underground cable

VI. CONCLUSIONThis paper presents the basic concepts and an

implementation of a computationally efficient performance-index-based contingency ranking methodology along withexample results. The presented methodology utilizes multipleperformance indices and it can capture various effects on thesystem. Furthermore, the paper introduces some basicconcepts for improving the accuracy of the P1 contingencyselection methods. The basic idea used is linearization of staterather then direct linearization of the PI with respect to thecontingency variable. Preliminary results of this approach arepresented and they indicate satisfactory and promisingbehavior. It should be noted, however, that since both rankingapproaches are based on linearization around a base-caseoperating point, and because of the highly non-linear nature ofthe power systems, some misrankings are always unavoidablein such approaches. A contingency may result in significant,non-linear deviation from the base case which may not beappropriately captured by an approach based on linearization.

The contingency ranking methodology is implemented as amodule of a general power system analysis software. Themodule is tested with some small size test-systems withsatisfactory results in terms of accuracy and efficiency.Further improvements in computational issues will allow theadequate and efficient application of the methodology forlarge scale systems. The module is to be used as contingencyselection pre-filter in the implementation static securityassessment methodologies, where combined withcompensation-method-based algorithms, will be part ofsecurity assessment software. In addition it is also used inreliability studies for contingency selection [15].

APPENDIX: QUADRATIC POWER FLOW[13,161The quadratic power flow (QPF) model is based on

modeling any power system component as a set of equationsof order no greater than 2 (quadratic). This can be alwaysachieved with the introduction of additional state variableswithout any simplifications. Application of connectivityconstraints (Kirchoffs current law) yields the quadratizedpower flow equations:

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G(x,u() = A[X,u, + Ix, u'. k?i [x,u ]I+b = O, (A.1)

wherex : the state vector, in Cartesian (rectangular) form,Uc : the vector of control variables,A ,Q : constant matrices,b : constant vector.The solution to the quadratic equations is obtained usingNewton's method:

xk±1 =k J(xk ) 'G(xk) (A.2)

wherek : the step of iterationsj(xk ) the Jacobian matrix at iteration kThe Iterative procedure terminates when the norm of the QPFequations is less than a certain tolerance.

Therefore the QPF equations G(x, uc) = 0 comprise a

different mathematical system of nonlinear algebraicequations from the traditional power flow (TPF) equations.The state vector consists of the real and imaginary part of thevoltage at each bus and of additional internal state variablesfor each device. The system G(x, u,) = 0 consists of thecurrent balance equations at each bus, plus additional internalequations for each one of the nonlinear devices that exist inthe system. All the equations are of order at most quadratic.

The application of the co-state method in the QPFfornulation is practically identical to the TPF formulation.Assuming the performance index:

J = F(X,u') I (A.3)

the derivative of J with respect to Uc is given by:

dJ dF aF(X, u,) 'TOaG(X,uc) (A.4)dUt, dui auc aucwhere

XT aF(X,u,)a(G(x,c ) -' is the co-state vector of the set ofax ax )

QPF equations and aG(X, uC) is the Jacobian matrix of the setax

of QPF equations.

[5] B Stott and 0. Alsac, "Fast Decoupled Power Flow," IEEE Trans. onPower Apparatus and Systems, vol. PAS-93, No. 3, pp. 859-869,May/June 1974.

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[11] A. P. Sakis Meliopoulos and C. Cheng, "A hybrid contingency selectionmethod," in Proceedings of the 10th Power System ComputationConference, Graz, Austria, Aug. 1990, pp. 605-612.

[12] A. P. Sakis Meliopoulos, Carol S. Cheng, Feng Xia, "PerformanceEvaluation of Static Security Analysis Methods," IEEE Trans. on PowerSystems, vol. 9, No. 3, pp. 1441-1449, Aug. 1994.

[13] Sun Wook Kang, A. P. Meliopoulos, "Contingency Selection viaQuadratized Power Flow Sensitivity Analysis," in Proceedings of theIEEE 2002 Power Engineering Society Summer Meeting, vol.3,pp.1494-1499.

[14] IEEE Committee Report, "IEEE Reliability Test System," IEEE Trans.Power Apparatus and Systems, vol. PAS-98, No. 6, pp. 2047-2054,Nov./Dec. 1979.

[15] Fang Yang, A. P. Sakis Meliopoulos, George J. Cokkinides and GeorgeStefopoulos, "A Bulk Power System Reliability AssessmentMethodology," in Proceedings of the 8th Int. Conf on ProbabilisticMethods Applied to Power Svstems, Ames, IA, Sept. 12-16, 2004, pp.44-49.

[16] S. Kang and A. P. Meliopoulos, "Analytical approach for the evaluationof Actual Transfer Capability in a deregulated environment," inProceedings of the 32nd Annual North American Power Symposium,1999.

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