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A New Approach to Security-ConstrainedOptimal Power Flow Analysis

Pmg Yan, Student Member, IEEE, and Arun. Sekar, Senior Member, IEEE

Abstract: Security-Constrained Optimal Power Flow (SCOPF)is becoming increasingly important in the new deregulatedenvironment of electric power systems. This paper focuses onthe development of an integrated approach to linear-programming-based (LP-based) solution of SCOPF. The powerflow model includes bns voltage magnitudes and line power andreactive power flow directly in the formulation. This permitsgreater flexibility in incorporating voltage and line flowconstraints. A six-bns system example illustrates the method.The resnlts obtained show the power of the new approach.

Keywords: Security-Constrained Optimal Power Flow, LinearPrograms and Load Flow

1. INTRODUCTIONSince the beginning of 1960s, research efforts have been

continuing for developing new methods and for integratingthe optimal power flow (OPF) analysis with the exactformulation of the security constrained economic dispatch. In1974, Podmore [1] introduced the concept of satisfying non-linear network security constraints by linearizing them. In thesame year Wollenberg and Stadlin [2] used linearizednetwork model, and using the security constraints developeda security constrained economic dispatch algorithm byapplying Dantzig-Wolfe decomposition. Linear programmingtechniques were also applied to the Security Dispatchproblems in the 1970s [3]. A milestone was reached in 1987when Sanders and Momroe[4] presented a real-timeconstrained economic dispatch calculation algorithm, whichis called as the CEDC and described a Newton-Raphsonbased/linearized constraint approach in thecontext of activepower security control. In 1988 a method similar to CEDCwas presented by Bather and Van Meeteren [5] using aquadratic programming approach. A new algorithm presentedin Barcelo and Rastgoufards paper [6] solves the ExtendedSecurity Constrained Economic Dispatch (ESCED) problemreliably, and at a speed suitable for real-time economic

Ping Yan is with Center for Electric Power and Electrical andComputer Engineering Department, Tennessee TechnologicalUniversity, Cookeville, TN 28505, USA.

(e-mail: BV8102@.tntcch.cdu)A run. Sekar IS with Electrical and Computer Engineering

Department, Tennessee Technological University, Cookewlle,TN38505, USA.

( c-mall:

dispatch. The algorithm is guaranteed to converge under anytype of constraint situation, including parallel andoverconstrained conditions. It is fast enough that both aconstrained and an unconstrained dispatch can be computedin real-time for accumulation of the cost of the constraints inimproving control area performance.Power system security is classified into both of line flow

security and voltage security. By monitoring the line flowsand bus voltages of the system, doing contingency analysisand making corrective actions, the system security operationis achieved and the system defensive capability will beimplemented alternatively to minimize the inevitable failures.The method of simulating contingencies, and obtaining theSCOPF results is quite time consuming. OPF is solvedwithout considering contingency constraints, with the systemin the correctively secure state. Whenever a contingencytakes place, the predetermined rescheduling results areimplemented to maintain system security. This also ensureseconomic and/or minimum loss operation of the system,when it is intact. Including the security constraints atminimum operation cost or minimum system loss is still achallenge to the researchers and util ities in [7-8].The new approach provides an integrated solution method

based on linear programing with line flow and bus voltageas decision variables. The power flow model is first describedand is followed by the integrated OPF analysis procedure. Asix-bus system used by Wollenberg in [9] is solved byproposed method and the results compared.

2. OPF GENERAL PROBLEM DEFINITION ANDFORMULATIONS

2,1. OPF General Definition and CategoryOptimization problems involve finding the best value,

maximum or minimum, of some performance index (i.e., theobjective fmction, f) which depends on a set of parameters(i.e., state variables, x) by adjusting a set of parameters (i.e.,control variables, u) in a model.Consider the objective function f to be a function of the

controllable variables, u, and of the uncontrollable variables,x. The constrained optimization problem, to be solved, can bestated as:

Min ~ = F(x, u) (1)

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Subject to system equality constraintsg(x, u) = o (2)

And set of inequality constraints/h(x,u)

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.Vz~,kp., ~4TH (9)/,=1If MVA or current limits are considered, AC sensitivity

factors can be used.In order to ensure security of the system, contingency

constraints for few severe contingencies should be identifiedand OPF schedule should be modified (through correctiverescheduling) to eliminate limit violation.In the method proposed here, the conventional approach is

modified by using a line power flow and bus voltagemagnitude as variables in the formulation. The model isdescribed in the next section.

3. LINE FLOW BASED POWER FLOW MODELThe nodal variables of bus voltage magnitudes and phase

angles in the common power flow models do not reflect theultimately required practical knowledge of the line flows. Thepower system operation focuses on maintaining a satisfactorylevel of voltage magnitudes, while dispersing real power andreactive power over transmission lines to the loads. Choice ofline flows as variables will provide greater flexibility inproblem resolution from a practical viewpoint. A newformulation of power flow approach developed by the authorsis described.Line flow based power flow model equations are derived

using graph theory. A and C are called the element-busincidence matrix and loop incidence matiix, respectively.Real and reactive power balance equations at all bussesexcept slack bus can be written using the incidence matrix.Since all shunt connections are excluded in the incidencematrix, their real and reactive power contributions areaccounted for separately in the power balance equations. Realand reactive power loads, shunt capacitors and line chargingsusceptances can be treated as shunt branches. The real andreactive power mismatch equation of each bus except theslack bus are expressed as

A.p P~~ A1=0 (lo)A.q QGL A.m13. v=fl (11)

Where,A and A are defined a incidence matrix and amodified incidence matrix with all +1 in A set tozero, then it is easy to include the relevant branchlosses in the power balance equation equations byusing the vectors of branch real and reactive powerlosses 1and m.H is an n-l diagonal matrix with the sum ofcharging and compensating susceptances at each busas the diagonal element except the slack bus.r CL and Q ~, are the injection power vcxxorsdefined as P~,,,= P~,- P,, and Q.,, = Qc, - Q,,.p and q are real and reactive line flow vectors atreceiving buses of the branches in the network. The

reactive mismatch equations should be deleted at thePV buses.V 2, the unknown voltage vector which contains thevoltages at the PQ buses, is n pus 1 dimension.pvs is the number of PV buses.

Therefore, equation (11) can be rewritten as

A,q-QC., -A~. m- H,. V=O (12)Because the shunt branches of power system such as line

capacitance and shunt susceptance do not affect the incidencematrix and are not described in its graph, the new branchmodel without its shunt branch is defined. Althoughdesignation of sending and receiving ends is arbih-ary, thebranch voltage drop equation can be written as2Rp+2Xq(A;+ +AA; ).VZ = k~AA~V;v (13)

S,?z:where k = is a vector with each branch value. A, isv,incidence matrix corresponding to the PV buses. V~Vs avector of square of voltage of PV buses and the slack bus.A is a diagonal matrix of order lifle number . Al+ and Al_come from Al. Al+ are the positive element part of Al. Al_are the negative element part of Al. R and X are diagonalbranch resistance and reactance matrixes.Bus phase angles across a branch maybe describes as

CXp-CRq=CQ (14)

0 is a vector for branch transformer phase shifting angles.When variable vectors of power flow equations are linebranch real and reactive power and the square voltage values,

(10), (12), (13) and (14) can be assembled into matrix format.

The matrix in (15) has a linear form in k+f iteration asfollows:

A X(+)=Y, +Y2() =Y() (16)pqv2Where A~g,, is a constant matrix that maybe factorized to LUform. The iteration procedure may be used to solve (16). y,is a constant vector and y~ is a variable vector consisting ofbranch losses, and charging and compensating powers.The advantages of the new power flow model are the

following:G Line flows and bus voltage magnitudes reflect the

physical aspects of power system operation.

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G The formulation does not include any longtrigonometric expression as in the traditionalNewton Raphson method.

G The formulation leads to a more comprehensiveapproximate linear model as shown in the nextsections.

4. MODIFIED SCOPFThe optimal power flow problem evolved from the earlier

formulation of e;onomic dispatch (ED). The generation costminimization is achieved through expressing the losses as afunction of generation in ED. Minimization is achieved usingnonlinear optimization procedures such as Lagrange methodsand gradient techniques. LP based methods in literature applylinearization around an operating point such as the Jacobian.The system model developed in the first part leads to a largechange linear mlodel that can be used more effectively withlinear programing techniques.4.1. Strate~fiw Solving LP-Based SCOPFThe purpose of SCOPF is to schedule power system

controls to optimize an objective fimction which satisfies aset of nonlinear equality and inequality constraints on theoperating limits of system variables. The equality constraintsare the power flow equations. Generally, OPF is formulatedas a constrained nonlinear optimization problem inmathematics. Inequality constraints will be difficult to handlein such a formulation. LP is very effective in handlinginequality constraints. The LP-Based SCOPF has thefollowing advantages: reliability, strong recognition ofinfeasibility, high calculation speed, flexibility and accuracy.The LP based SCOPF is first proposed to apply for securityconstraints and contingency constraints. To create SCOPFusing linear programming, the nonlinear cost functions can betreated as before, using multiple segmented piecewiselinear approximations, besides linearized constraints. Thegeneral procedure used in LP-based SCOPF solution is topiecewise linearize the cost function, and linearize the powerflow equation around the operating point by the Jacobianmatrix. Inequality constraints and loss equation are againlinearized around the operating point. Iterative procedurewith LP solution as the inner loop is defined the overallalgorithm. Fig 1. shows the procedure. Using the line flowbased power flow model, the linearization is not required.The line flow based model incorporates voltage and line flowconstraints.4.2. Test ExampleIn the conventional procedure, the power flow analysis is

done fist and the linearization is effected next. With the lineflow based model, the constraints are directly on the flowvariables and there is no need for a separate step. Withvoltages magnitudes and real power flow as variables, theprobability of better convergence and performance is greater.

Start the simulating systemI*

PieceWise linearize the cost functions

9olve Power Flow41:1Conventional ApproachG Jacobian Linearized ModifiedApproachG Line Flow Based ModelsUpdate thevariables Set up the LP and Solve LPNot converged

Converged

Figure 1. The strategy of solving the LP-SCOPF

An example from the six-buses system shown in Figure 2is taken as a testing system. Its bus data and line parametersare shown in Table 1 and Table 2. Table 2 gives real powerline constraints in MW and voltage constraints are taken inrange of So/o changing of normal voltage. The generation costfunction coefficients and their generation limitation are listedin Table 3. The line flow based method has beenimplemented as a program and tested. To exploit its flexibleand extensive characteristics, four different load flowmethods are used to comparison. Then the AC SCOPFmethods are developed by analyzing Economic LoadDispatch (ELD), the k-method based on Jacobian matrix andLinear Programming with conventional load flow method

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such as fast decoupled load flow method. Traditional andmodified forrnul ations are programmed and tested.

-J

bus3bus 2

In()/I bus6

I bus 54 bus 4

Figure2. Thesixbus example systemTABLE 1

THE SIX -BUS EXAMPLE SYSTEM BUS PARAMETERS IN P.U

Bus No. Voltage Phase angle Gen. P Gen. Q Load P Load Q1 1.05 0 0 02 1.05 0.5 0 03 1.07 0.6 0 04 0 0 0.7 0.75 0 0 0.7 0.76 0 0 0.7 0.7

TABLE 2THE SIX BUS TESTING SYSTEM LINE PARAMETERS WITH REALPOWER LINE CONSTRAINTS

R x 1/,B Max line PLine No. From bus I TO bus J in pu in pu in pu in MW

1 1 2 0.1 0.2 0.02 302 1 4 0.05 0.2 0,02 503 1 5 0.08 0.3 0.03 404 2 3 0.05 0.25 0.03 205 2 6 0.07 0.2 0.025 306 2 4 0.05 0.1 0.01 407 2 5 0.1 0.3 0.02 208 3 5 0.12 0.26 0.025 209 3 6 0.02 0.1 0.01 6010 4 5 0.2 0.4 0.04 2011 5 6 0.1 0.3 0.03 20

TABLE 3THE GENERATION COST COEFFICIENTS AND GENERATOR

BOUNDARIES~Min gen in Max gen. in

4.3. Test ResultsThe test system is used with conventional load flow models

in the SCOPF and the line-flow-based model. The resultsobtained for minimum cost for different models are show inTable 4. The bus voltages and line flows at the solution aregiven in Table 5 and Table 6.The proposed method gives results very close to

conventional method. The proposed approach can handle thesecurity constraints in an integrated manner and this will be asignificant advantage in improving the present SCOPFalgorithms.The advantages of the proposed procedure can be

summarized in the following:G The system model via more comprehensive list of

practical variables is directly embedded in the LPformulation,

. Constraints or line flows and voltage levels formpart of the LP problem with no linearization aroundthe operating point.

G The performance of the new procedure can beexpected in the original form are directly applied.

G The procedure can exploit all the advantages of thehighly developed LP algorithm.

TABLE 4.THE RESULTS OF DIFFERENT SCOPF UNDER MIN COST BY

ADJUSTING GENERATOR MW ONLYCase P, P, P, Total 10SS Fuel

IIDecoupled LFLP-the New 50 76.7 1 FkmrtmAApproach

TABLE 5CONVERGED RESULTS FOR VARIOUS METHODS AFTER

CONVERGENCE (CRITIA=O.001)

1 angle angle angle1 I 1.05 I o I o 1.05 I o 1.05 I o I1 1.05 1-3.67081 -2.90243 I 1.07 1-4.27291 -3.1679 1%E%EE%lw::: ~~0044A-628359470.9894 -4.4217 0.9894 -4.19540.9854 -5.5371 0.9854 -5.27640-7803-7031-7/01/$10.00 (C) 2001 IEEE0-7803-7173-9/01/$10.00 2001 IEEE 1466

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TABLE 6 [9]THE REAL POWER LINE FLOWS IN MW FOR VARIOUS METHODSIN FIG 2 AFTER CONVERGENCE (ITERATION CRITIA=O 001)

[10]from to Newton DC load flow Line flow Fastbus I bus J method decoupled1 2 28.69 2533 26.79 28.681 4 4358 41.57 43.03 43.581 5 356 331 3498 3562 3 2.93 185 242 2.932 4 33.09 3248 3128 33.12 5 15.51 16.22 14.91 15.522 6 26.25 24.78 25.56 26253 5 19.12 1693 17.89 19123 6 43.77 4492 42.44 43.774 5 4.08 4.04 4.27 4085 6 1.61 0.3 1.99 1.61

5. CONCLUSIONThe conventional LP-based SCOPF method N modified

using a hne-flow-based power flow model such a procedurewill help integrate the security constraints on bus voltagemagnitudes and line flows with power flow model. Researchis in progress to exploit the power of LP in the newframework. The proposed method compares favorably withthe conventional methods.

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6. REFERENCER. Podmore, Economic Power Dispatch W]th Lme Security Llmlts,IEEE Trans. On PAS, Jan/Feb 1974. p. 289B.F Wolllengerg and W.O. Stadhn, A Real-TlmeOptlmlzer forSeccmty Dispatch, IEEE Trans. On PAS, Jan/Feb 1974. p.1640.B Stott and J, L Mannho, Linear Programming for Power-SystemNetwork Security Applicahons, IEEE Trans On PAS, May/June,1979, p 837.C W. Sanders and C.A Monroe, A Algorithm for Real-Time SecurityConstrained Econom]c Dispatch, IEEE Trans. on Power Systems, Vol.2, No 4, November, 1987, pp. 1068-1076.R Bather and H. P. Van Meetem, Real-Time Optimal Power Flow InAutomahc Generation Control, IEEE Trans. On Power Systems, Vol.3, No 4, November, 1988, pp. 1518-1529.W. R Barcelo and P Rastgoufard, Contril Area PerformanceImprovement By Extened Security constrained Economic Dispatch,IEEE Trans On Power Systems, Vol 12, No 1, February, 1997,pp,120-127Alex D. Papalexopoulos, Challenges to Otz-LineOPF Inlplementatlon,IEEE Tutorial Course, IEEE Power Engureermg Society, 1996.J. A Momoh, R. J. Koessler, M. S. Bond, B. Stott, D Sun, A.Papalexopoulos and P. Rwtanowc, Challenges to Ophmal PowerFlow. IEEE Trans. Power Systems, Vol. 12, No 1, February 1997, pp.444-453

A J Wood and B F Wollenberg, Power Geneurrton OpercrnonamiControl, John Wdey & Sons, 1996.O.Alsac, J. Bright, M. Prals and B Sttot, Further Development m LP-Ophmal Power Flow, IEEE Trans. Power Systems, VOI.5, No 3,August 1990, pp. 697-712

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