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  1. 1. A GENERALIZED QUADRATICBASED MODEL FOR OPTIMAL POWER FLOW James A. Momoh, Senior Member, IEEE Dept. of Electrical Engineering Howard University Washington, DC 20059 ABSTRACT voltage deviation, cost, flows and their combinations. The current approach for solving optimal power flows used by A quadratic form of power systems has been used to researchers is based on the sensitivity method, generalizedformulate a generalized optimum power system flow problem. reduced gradient and, recently, the MINOS augmented methodThe conditions for feasibility, convergence and optimality are [ 8 1. These methods are generally implemented on quiteincluded in the construction of the optimal power flow (OPF) large systems and have drawn considerable attention byalgorithm. It is also capable of using hierarchical structures researchers in the optimal power flow work.to include multiple objective functions and selectableconstraints. The generalized algorithm using sensitivity of A literature review and experience of operators revealsobjective functions with optimal adjustments in the that these methods lack the capability of identifying whichconstraints yields a global optimal solution. The generalized constraints or controls are necessary to achieve a givenalgorithm includes special cases of the OPF that reduces objective. Moreover, the inclusion of various constraintscomputational burden when the number of constraints is which are not adequately prioritized makes it burdensome onequal to the number of variables and can select the constraint the operator to use existing OPF packages efficiently. Anothersets that are binding or nonbinding. The optimal power flow area of concern is the approximation of the models used inpackage developed is flexible and can accommodate other OPF as demonstrated in the so-called quadratic sequentialmodels and solution methods. models. The algorithm has been tested using different objective The power industry is seeking improved, flexible and easyfunctions on actual power systems, and optimal solutions to use models for OPF packages with conditionality checks onreached in relatively few iterations. The potential of the feasibility and optimality. The work reported here utilizes amethod for on-line application is being investigated with quadratic model of OPF with capability to handle generalizedseveral utility companies. sets of constraints, convert two-sided constraints into one to reduce computational burden and criteria for checkingKevwords: Optimal Power Flow, Quadratic Model, Multiple feasible conditions, necessary and sufficient conditions whileObjective Functions. converging to a global optimum. The algorithm incorporates multiple objective functions such as losses, voltage deviation, cost and flows, and selectable constraints on the generation, I. INTRODUCTION loads, transformer taps, flows, etc. The task of operating a power system economically and This paper is organized as follows: formulation ofsecurely is commonly referred to as the OPF problem. An quadratic model OPF, feasibility and necessary conditions forOPF solution gives the optimal active and reactive power quadratic forms, development of generalized OPF algorithm,dispatch, and the optimal setting of all controllable variables special cases of OPF solutions, implementation procedure,for static power system loading conditions. Computationally, scenarios for selecting multiple objective functions, and testthis is a very demanding nonlinear programming problem due cases and conclusions.to the large number of variables, in particular, to the large number and types of limit constraints imposed on the power System by engineering design limits. These constraints, II. FORMULATION OF QUADRATIC MODELwhich define the technical feasibility, can be equalities or OPTIMAL POWER FLOWinequalities. Interest in the OPF problem has been growing since the In general, the OPF problem in standard form is given asearly 1960s and is perhaps at its peak at present [ 1, 2, 3 1because of its potential for real-time economic and secure Min F(x, U) (1)power system dispatch. Several techniques of optimal powerflow models using linear and nonlinear programs have been subject toproposed. Their capability to include complete sets ofconstraints or controls is developed in-house to suit different g o , U) = 0 (2)utilities. To date no generalized algorithm is available to theuser to accommodate various objectives such as losses, h(x, U) f; (3) 261 CH2809-2/89/0000-0261$1.00 1989 IEEE
  2. 2. where TABLE 2 SUMMARY OF DESCRIPTIONS OF OBJECTIVE FUNCTIONSx is the state variable for voltage and U represent thecontrollable generator or transformer taps. The functionF(x,u) denotes costs, voltage deviation, losses or maximumpower transfer. g(x,u) denotes the equality constraints andh(x,u) the inequality constraints. The optimal power flowformulation is expressed in hierarchical structure. Theobjective function in scalar form is represented as:Min Fo(X) = 1/2 XT R X + aTX (4)where the constraints are denoted as Ki = Fi (X) = 1/2 XT Hi X + biT X (5) TABLE 3: Summary of Constraintsand Ci < Ki < Di i = 1, 2, ..., m The matrices R and Hi, vectors a and bi and scalars Ciand Di are described by the hierarchical structure tables1 -71. These hierarchical structure tables are described as 1follows: Table 1, gives the summary of objective functions andconstraints. The relationship between the different objectivefunctions and constraints is listed in this table. For examplefor objective functions OF1, the corresponding constraintsare 1, 2, 3 and 4. The summary of description of objectivefunctions is listed in Table 2. The detail formulae of thedifferent objective functions are given in this table. In Table3, the summary of contraints formulations is given in a list.Table 4 contains the objective function variable description. TABLE 4: Objective Function Variable DescriptionTable 5 provides the objective function parameterdescription, Table 6 gives the constraint variable ~description, while Table 7 details the constraints parameter VariabkS DEScRlpTIONdescription. For any given objective function, theformulation for the objective functions and its constraints Voltage vector, 2n x 1, real and imag;narY part of voltage.are easily obtained from these tables. For example, when lossminimization is desired, the objective function from Table 2 V Expected voltage valw vector. 2n x 1is OF3 and the corresponding constraints are constraints1,2, 3 and 5 in Table 1. Real Power of Generator. g x 1. r Bus number of of a given system.TABLE 1: SUMMARY OF OBJECTIVEFUNCTIONS g Generatornumber of a given system. AND CONSllWNTS I Objanve PmcUm I Constraints I TABLE 5: Objective Function Parameter Description 262
  3. 3. TABLE 6: Constraint Variable Description This formulation of the OPF in quadratic form lends itself to the following questions. 1. Can we derive the optimum solutions using the VpriablcS DESCRIPTION properties of quadratic forms analytically?1 ii 1 1 X The voltage vector, the same as objective function 2. What is required to guarantee an optimum solution when numerical techniques are used for power ~ w t e BU Voltage vector d system dispatch problems? Realgmemtionpowervector 3. What is the scenario for selecting candidate constraint objectives to attain a desired optimal Reactive generationpower vector operation andlor schedule? Real load power vector P Demands power vector for generator 111. FEASIBILITY AND NECESSARY CONDITIONS i Real power for the imBus FOR QUADRATIC FORMSI ii I CurrentbetweenBusiandj I In this section we present some theorems which are developed and proved by the Energy Systems Network Lower limit of variable (*) Laboratory (ESNL). The adaptation of the development of these theorems is to build up the basis for the further sound Uppa limit of variable (*) development of optimal power flow methodologies, and to search for the mathematical formulation of the OPF problem. These theorems are only stated; the proofs in detail are available in [ 91. Proposition 1: The matrices H 2 ~ - 1have exactly 2n-4 TABLE 7: Constraint Parameter Description zeros, two positive and two negative eigenvalues. In other words, they are of rank four unless the bus K is isolated from other buses. As such, the quadratic forms of Pk and Qk can be Variables convex or concave for arbitrary vector X. H" Theorem 2: If X is a solution, for the nonlinear programming problem, then it is necessary that Lx(x, h) = 0 and that the following Extended Kuhn-Tucker(EKT) conditions H must be satisfied for all i > p. I (a) hi=OwhenCis fi(X)lDi (b) hi 2 0 when fi (x) = Di (c) hi 2 0 when fi (x) = Ci H 2k-1 where Lx(X,h) is differentiation of the Lagrange equation: and h are the Lagrange multipliers. Lemma 3: The quadratic optimization has a minimum if and only if it is impossible to find a X to make f(X) + M = 0 as M approaches infinity. Theorem 4: The quadratic optimization has a minimum if there is a set of real numbers a i , ap,... am such that M is __ ~ _ _ positive definite where M is given as Linear term coefficiency,depend on the system See objective function parameter d d p t i o n M=R+ .x aiHi m See objective function parameter description 1= 1 (7) Theorem 5: If there exists a scalar number such that the square matrix Sp + PFxT Fx is positive definite, then the X 263
  4. 4. obtained from the EKT condition is a minimum Substituting equations (4) and (5), L = [ F&x) + W X )I (10) mwhere S=R + z hi ~i (8) L, = [ FOx(x) + hFx(x) I = 0 (11) i= 1and FTx = [HI X +bl, H ~ +b2, x ..., Hmx + bm] (9) Expanding equation (11) yields m Proposition 1 deals with the nonconvexity and =+a+ .Z h i ( H i X+ b i ) I= 1 =o (12)nonconcavity of power systems. Theorems 2 and 4 give theresults of necessary and existence condition of nonlinear solve forprogramming problem in quadratic form, and Theorem 5 isthe sufficiency condition for the problem. x = - s-lw (13) IV. DEVELOPMENT OF GENERALIZED OPTIMUM where m POWER FLOW ALGORITHM S=R+ .x %Hi 1=1 A generalized sensitivity algorithm is developed to solve mthe OPF problem. It satisfies the feasibility, necessary and W=a + , E hibi (15)sufficiency conditions for quadratic model power systems. 1=1Basically, the algorithm consists of three parts. The firstphase is the input data preparation with the capability to ifeasily interface with additional data. The input data consistsof generator limits, voltage limits, load limits, transformertaps and network conditions. In addition, the optimization F[ X ( h )I = k (16)constants include the range of tolerance and criteria forachieving convergence. then X is determined using Newton Raphson method iteratively as : The second phase of the algorithm includes the following:1. Selection of constraints for: a) cases where the number of variables, n, equals the number of constraints, m; X and h are determined from equations (13) and (17), and are used in satisfying feasibility and necessary conditions. b) cases of constraints that are binding or non- binding and The optimal adjustment scheme used for preventing oscillations and overshoot are obtained using one of the c) cases where all candidate constraints are included. a) For constraints with the number of variables equal to2. Satisfaction of the EKT conditions and criteria for the number of constraints, AK is simply calculated optimal adjustment. Two approaches are used in from the optimal adjustment scheme, namely: AK = LE (1 8) a) sensitivity technique and b) least square estimation of eigenvalues and where E is a predetermined number which can be eigen-vectors. specified by users.3. Criteria for convergence evaluation. b ) For the case where all candidate constraints are involved, in the adjustment scheme, AK is obtained as: The third phase of the algorithm involves: AK = A(-Ah) (19) a) criteria for evaluation of termination ana convergence critieria for optimal solution and where A is a normal transformation in m spaces, it is computed from b) display of control variables objective functions and other pertinent data. 264
  5. 5. where pi and its transpose piT are associated eigenvectorswith the corresponding eigenvalues, 81, 82, ..., pm. READ IN DATA 1 I m F’REPROCESS OF THE DATA So that AK= .Z dipi I= 1 L SELECTION OF OBJECTIVEFUNCTION where di = P.c~ Because ti = 0 for i > q, we have AK= Md For cases of binding and nonbinding constraints, newvalues of F(x, k)=k are obtained using equations (13) and(17). The function values are set to either the upper or Fig. 1 Flow Chart o f the OPF Packagelower values by using the sign of the difference between thenew F(x, k) and the given constraint limits. Investigation ofthe direction of h allows us to separate the constraints intobinding and nonbinding sets. The separation of constraints VI. SCENARIOS FOR SELECTING MULTIPLEmust satisfy the EKT conditions thus feasibility and necessity OBJECTIVE FUNCTIONSconditions can be guaranteed. The selection scenarios of objectives and constraints for total economic operation of power systems are given as follows: V. IMPLEMENTATION PROCEDURE 1. Voltage Deviation Objective A program package has been developed for achievingtotal economic and reliable operation of power systems. It is This criterion minimizes the deviation of overvoltagecapable of selecting different objectives and their associated and undervoltage conditions for a given power system. Theconstraints to meet the desired economic objective. For the optimal adjustments of generators, transformer taps, loads,case of optimal power flow problem with the number of etc. alleviate voltage deviation problems. The capability tovariables equal to the number of constraints. the package uses minimize voltage deviation will prevent power systemthe sensitivity method (SM) which is a special case of the instability and improve economic systems operation.generalized SM discussed in the paper. 2. Cost Objective For the general case n # m, two possible approaches areemployed. Firstly, the simplex-like method is used for This criterion, in general, minimizes the productiondetermining which constraints are binding or nonbinding. costs of generating plants. The associated constraints includeThe second approach includes all candidate constraints in the controllable devices such as generators, loads, transformeralgorithm and uses the least square estimate to obtain the taps, etc. It has the additional capability to reduce cost ofoptimal adjustments during the optimization procedure. operation when coupled with the voltage deviation objectives.The program module for the multiple objective optimizationis shown in Figure 1. It consists of network data needed for 3. Loss Objectiveoptimization, the description of selectable objective functionsand the various approximations of optimization techniques, While minimization of voltage objective preventsand finally, the output of optimized objectives and status of system instability and possibly decreases the chance ofconstraint limits. system voltage collapse, the loss minimization increases the optimal power while guaranteeing minimum cost of operation. The associated constraints are given in Table 3 and are adjustable to obtain the desired optimal conditions. 265
  6. 6. 4. Flow Objective Table 8: Sample of Constraints Status for 39 bus system This objective represents the determination ofmaximum mwer transfer capability of a given network. Thisobjective is needed in networks, such as the WSCC system inthe United States, where there exists excess power generation .owbound lalue of Cons upbwabut with limited networks for power transfer. The 1.01 1.05 1.11maximization of the power transfer objective is obtained by 0.918 0.961 1.01satisfying the limits imposed on generator outputs, 5.07 5.59 6.19transformer taps, load demands and power flow, etc. -0.656 -0.656 -0.537 4.57 4.57 5.59 1.43 1.58 1.75 The impact of these objectives, when coupled with 25.2 57.1 75.5appropriate weighting functions, guarantees optimum 1.01 1.06 1.11maintenance costs and increased system reliability. The 0.918 0.957 1.01proposed OPF package includes these objectives and 5.07 6.05 6.19constraints in the hierarchical form discussed earlier in the -0.656 -0.558 -0.537paper. 4.57 5.54 5.59 1.43 1.54 1.75 0.724 1.22 1.36 VII. TEST CASES AND DISCUSSION 4.50 5.06 5.50 A variety of sim,ulations using several medium-size andpractical systems have been performed using the generalized 1.01 1.06 1.11OPF algorithm. The optimum solutions are obtained by 0.918 0.958 1.01utilizing the algorithm to guarantee the feasibility, necessary 5.07 6.08 6.1 9conditions and convergence criteria. Moreover, employing -0.656 -0.551 -0.537optimal adjustments and optimization constants discussed in 4.57 5.22 5.59the paper, the algorithm is prevented from wandering, 1.43 1.47 1.75oscillation and overshoot. 25.2 56.7 75.5 A 39-bus example is presented to demonstrate thedetailed behavior of the algorithm for selected objectivefunctions. Other higher order systems were evaluated toextend the capability of the algorithm. The operatingconstraints limits for different control variables and their Table 9: Results for 39-bus Systemstatus at optimal objective function values are presented inTable 8. Specifically, the tables include the corresponding valuesof voltage, power and loads for different objectives asdetermined by different optimization schemes discussed in thepaper. The computational tractability of the methods are alsoevaluated. The CPU time and number of iterations required toachieve convergence are displayed in Table 9. The special forms of the generalized algorithm based onsimplex-like method (SLM), distinguishes between bindingand nonbinding constraints. The SLM reduces thecomputational burden of the optimization process. Objective Function: CostAccordingly, the S reduces the CPU time for possible cases M Method: SLMwhere the number of constraints equals the number ofvariables. The convergence criteria for various toleranceswere determined. The cost objective using SLM andgeneralized S are displayed in Figures 2 and 3. M Other objective functions are evaluated using thegeneralized algorithm. The potential of these objectives tominimize cost of operation and improve reliability has beenverified. Adaptability of the method to power utility needs isbeing undertaken. 1 2 3 4 Iteration Fig. 2 Convergence of SLM for 39-bus System 266
  7. 7. [ 2,] Happ, H. H., Vierath, D.R., "The OPF for Operation Objective Function: Voltage Planning and for Use On-line", P Q Z B E- of the Snd . . Method: GSD e on Power -S a d Durham, England, pp. 290-295, July n , 1986. 1 [ 3 .I Stott, B., Alsac, O., Marinho, J.L., "The Optimal Power Flow Problem", Power P r o w : t h e MathematicalChallenae, 1986.pp. 327-351, SIAM, 73 [ 4 .] Stott, B., Marinho, L., Alsac, O., "Review of Linear Programming Applied to Power System Rescheduling", the 1 1 th PICA Conference,Cleveland, Ohio pp.142 - 154, May 1979. Fig.3 Convergence of GSD for 39 bus System [5.] Sun, D.I., Ashley, B., Brewer, B., Hughes, A., Tinney, W.F., "Optimal Power Flow by Newton Approach", JFFF Transactkm on Power - , Vol. PAS 103, pp. 2864-2880, 1984. VIII. CONCLUSION [6.] Burchett, R.C., Happ, H.H., Vierath, D.R., A general algorithm extending the basic Kuhn-Tucker "Quadratically Convergent Optimal Power Flow",conditions and employing a quadratic model in formulating ns on P,- oe wr Vol.OPF problem has been developed. The algorithm incorporates PAS-103, pp. 3267-3276, 1984.the general multiple objective functions such as voltagedeviation, losses, production costs, flow and their [7.] Chieh, Hua T., Hsieh, W.C., Optimization Theory withcombinations. The associated constraints of these objectives w a t i v e P r o w, Mon Min Co., 1981.are organized in a hierarchical structure. The proofs of thefeasibility, necessary and sufficiency conditions provide [8.] Gill, P., Murray and Wright, M., . . "minsight in the development of the mathematical basis for the QDtlmlzatian" Academic Press, 1981.algorithm. Special cases of the algorithm to reducecomputation time have been tested on several practical power L9.1 Momoh, J. "Corrective Control of Power Systemsystems. During An Emergency" NMI Ph.D Dissertation Information Service, 1983. Computational memory and execution time required havebeen significantly reduced by exploiting the properties of [lo.] Aoki, K., and Satosh, T. "Economic Dispatch withquadratic forms in the optimization process. The sign of Network Security Constraints Using ParametricLagragian multipliers have been used to convert two-sided Quadratic Programming", Paper No. 82, SM 426-5,constraints into one. Nonbinding constraints using the SLM Presented at the PES Summer Power Meeting, Sanfor certain classes of problems are easily circumvented Francisco, California, July 1982.which reduces the CPU time for the OPF problem. [ 11 .I Sun, D.E., et. al., "Optimal Power Flow Solution By Results of tests performed on other system sizes Newtonian Approach, JFFF T- rn on PAS, Vol.demonstated the same characteristics using the algorithm. In PAS-103, No. 10, October 1984, pp. 2864-2880.general, the algorithm converges in few iterations.Presently, the algorithm is developed on a VAX 11/780.Similar results would be achieved on smaller computers. The [12.] Ponrajah, R.A., and Galiana, F.D., "The Minimum Costpotential of the algorithm to enhance flexibility of the OPF Optimal Power Flow Problem Solved Via the Restartpackage and be suited to different utility needs is being Homotopy Continuation Method", Proceedinps ofinvestigated. JEFF/PFS 1988 Wn i- , New York, New York, 1988. IX. ACKNOWLEDGEMENTS [13.] Aoki, K., Fan, M., and Nishikor, A., "Optimal Var The research is supported by the U.S. Department of Planning by Approximation Method For RecursiveEnergy and National Science Foundation ECS 8657559 PYI. Mixed-Integer Linear Programming" procee-Special gratitude also goes to Dr. Yi Zhang and Dr. ArunsiChuku at the Energy Systems Network Laboratory, Howard Meetiag. New York, New York,University, who provided valuable suggestions. 1988. ( 1 4 -1 Momoh, James A. and B.Wolienberg, "Quadratic Module for Voltage Reduction During an REFERENCES Emergency",- IFFF[1 .I Carpentier, J., "Towards a Secure and Optimal v, 1984. Automatic Operation of Power Systems", ProceedinQs nf the 15th PICA Conference, Montreal, Canada, pp. 2-37, May 1987. 26 7