A new benders decomposition approach to solve power ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 2, MAY 2001

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A New Benders Decomposition Approach to Solve Power Transmission Network Design Problems Silvio Binato, Mário Veiga F. Pereira, and Sérgio Granville

Abstract—In this paper we describe a new Benders decomposition approach to solve power transmission network expansion planning problems. This new approach is characterized by using a linear (0–1) disjuntictive model which ensures the optimality of the solution found and by using additional constraints, iteratively evaluated, besides the traditional Benders cuts. The results obtained, considering a real world power transmission network expansion planning study with the southeastern Brazilian system, show the efficiency of this approach. Index Terms—Benders decomposition, combinatorial optimization, power network design problems.

I. INTRODUCTION

T

HE LONG term power transmission network expansion planning problem consist in choosing, from a predefined set of candidate circuits, those that should be built in order to minimize the investment and operational costs, and to supply the forecasted demand along the planning horizon. This problem is dynamic because it comprises determine the stage-by-stage transmission expansion plan. A subproblem of the dynamic version is the static problem, which aims to determine where and what type of new transmission facilities should be installed. This paper addresses the static version of the transmission design problem. Because of the combinatorial nature of the problem solving it is a very hard task. Among all combinatorial optimization techniques used, Benders decomposition [1] have been used with success since its first application to this problem, see [2]. In the Benders approach, the original network design problem is broken into two subproblems, the master subproblem, which models only investment variables and proposes network expansion plans; and the slave subproblem, which implements the expansion plans suggested by master subproblem and checks its network feasibility. The iteration between both subproblems is characterized by Benders cuts, which are evaluated from the slave’s solution and added to the master subproblem. Usually, a mixed, nonlinear, nonconvex formulation is used to formulate the power transmission network design problems. Neglecting the combinatorial nature of the problem, handling with the nonconvexities of feasible region have been the main drawback for using this formulation in practical studies [3], [4]. Manuscript received September 12, 2000. S. Binato is with the Systems Research Area, Electric Energy Research Center (CEPEL), P.O. Box 68007, Rio de Janeiro 21944-970 Brazil (e-mail: [email protected]; [email protected]). M. V. F. Pereira and S. Granville are with Power System Research Inc., 250, Alberto de Campos St. #101, Ipanema, Rio de Janeiro RJ 22471-020 Brazil (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0885-8950(01)03788-9.

To handle with this nonconvexities difficulties, a Benders hierarchical decomposition approach was proposed in [5], in which the power network constraints were represented by a chain of three models. The two first models relax all the nonconvexities constraints, which results in the optimal solution. Then, the nonconvexities are introduced (third model). Practical experiences showed very good results of this approach with a real world case study [6]. Following this work, in [7] was proposed a two phase Benders hierarchical model but using the Geoffrion heuristic approach [8] within a Branch-and-Bound algorithm to solve the master subproblem. Nevertheless, even the good results, it is not possible to guarantee the optimality of the solution found because the non convexities still exist in the mathematical model used. In this paper we present a new Benders decomposition approach to solve power transmission network design problems. This approach uses a mixed linear (0–1) formulation—the linear (0–1) disjunctive model—which was independently proposed in [9]–[11]. Numerical difficulties introduced by a parameter (“big number”) of this formulation have limited its use. Then, based on analysis we derive minimum values for this parameter. Further, we specialize our Benders approach to allow using small values (less than the minimum) for this parameter. During the run, both the parameter and all Benders cuts previously evaluated must be adjusted. Other important topic we are proposing is the use of Gomory cuts [12]. Balas et al. [13] showed that Gomory cuts could be made valid overall the Branch-and-Bound search tree by a simple lift procedure for mixed (0–1) programs. They also illustrated good computational results of the resulted Branch-and-Cut code for a wide range of mixed (0–1) problems. In this approach, we are also using Gomory cuts but besides Benders cuts in a Benders decomposition approach. This paper is organized as follows: initially we present the mathematical formulation for the power transmission network expansion planning problem; next we show how to apply Benders decomposition with our problem followed by the discussion of how to consider Gomory cuts within the decomposition approach. Then, we illustrate some results of our approach with a real world power transmission expansion planning case study, with the reduced southern Brazilian system. Next, the main conclusions are summarized and, finally, in the appendices, we present a brief review of Benders decomposition and Gomory cuts. II. MATHEMATICAL MODEL FOR THE POWER TRANSMISSION NETWORK DESIGN PROBLEM as

Denoting by the set of all busses (size of ), by the set of existing circuits and by

0885–8950/01$10.00 © 2001 IEEE

is written the set of

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candidate circuits, the power transmission network design problem can be formulated by a mixed (0–1) programming problem as follows: minimize

In fact, as we want to use minimal values for , we might evaluate the minimal existing limit, which can be determined solving the following shortest path problem minimize

(1a) s.t.:

s.t.: if if otherwise.

(1b) (1c) (1d) (1e)

(2) Shortest path problems with non negative objective costs can be efficiently solved via a Dijkstra algorithm. Once the problem can be set up by (2) is solved, the parameter

(1f) (3) (1g) (1h) is the investment cost to build candidate circuit . where and represent the set of all existing and candidate circuits are refdirectly connected with bus . Superscripts indices erences for existing (candidate) network variables, respectively. is the power flow in the branch , Using this notation, is the active generation at node , states the circuit suscepis the -node voltage angle, and tance for branch , state, respectively, -branch power flow and -node generation is the disjunctive parameter used in power limit. Finally, flow equations for the candidate branches . The objective function of problem (1) corresponds to the minimization of all investment costs in building new transmission facilities. The constraints (1b)–(1d) are the linearized power flow equations for the existing and candidate network. Other constraints states operational limits and integrality conditions. is built , its Remark that if the candidate circuit power flow equation (1d) is equal to the power flow equation of an existing circuit (1c). In the opposite case, i.e., if the candi, the parameter must date circuit is not built be large enough to do not impose an implicit limit over voltage angle differences between busses and . Smalls values for create implicit limits over voltage angles differences and require additional investments. In [14] it was illustrated that if the network is all connected and disjunctive parameter is set as no implicit bounds over voltage angles differences will exist and the solution obtained would be optimal. However, this value is large and introduces numerical instabilities in practical applications. In [15] Binato derived minimum values for disjunctive parameter associated with each candidate circuit . These values also ensure the optimality of the solution found and are significantly less than the former. Their computation are based on already existing network restrictions over voltage angle differences, i.e., if there is an existing circuit (or a set of circuits) connecting busses and , there is an existing constraint which limiting the voltage angle difference between these busses, and can be adjusted as a function of these limits. the parameter

For an initial disconnected bus, or if the and belong to different disconnected systems, it is necessary to compute the longest path problem between busses and , but considering the existing and the candidate networks. The reason regards in the unknowability of what candidate circuits will be used to connect the disconnected bus (or system) to the main network. Nevertheless, solving longest path problems is a non polynomial task, but as the number of different paths that could be used to link a disconnected bus is usually small, enumeration can be used. Once the longest path is evaluated, the disjunctive parameter could be set up by (3), but using the longest value instead . of the shortest III. APPLYING BENDERS DECOMPOSITION The resulting problem obtained when the investment variables of problem (1) are fixed corresponds to the slave subproblem, which can be represented as minimize (4a) s.t.: (4b) (4c) (4d) (4e) (4f) (4g) (4h) (4i) , is the solution of the master subproblem where and , are respectively a penalty at th iteration, factor and the amount of network infeasibility (load shedding). The value for must be large enough to ensure no load shedding at final solution.

BINATO et al.: A NEW BENDERS DECOMPOSITION APPROACH TO SOLVE POWER TRANSMISSION NETWORK DESIGN PROBLEMS

Once the slave subproblem is solved, we can evaluate Ben, , , and ders cuts using its Dual solution. Let, , be the vectors of dual variables of slave subproblem associated with constraints (4b), (4d)–(4h) at th iteration, the master subproblem can be stated as: minimize

where

, , and

,

,

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Nevertheless, note that the numeric values of the disjunctive parameters are also used to compute the coefs. of the Benders cuts. Therefore, when they change we need to make an adjust in the coefs. of Benders cuts. In order to illustrated this correction, suppose we have already a set of Benders cut, evaluated with a disjunctive parameters vector . When the disjunctive the coefs. of Benders cuts and parameter changes to must be adjusted according right hand side

,

Other coefs. of Benders cuts remain constant for modifications in the disjunctive parameter. In the classical Benders approach just one cut is evaluated each iteration of the decomposition model. However, it would be desirable to compute a set of cuts in order to improve the , see Appendix I. A simple representation of the function but efficient approach is to compute additional Benders cuts for relaxations of the slave subproblem, e.g., in problem (4) we can relax constraints (4c) and (4d), to obtain the transportation model, or to relax only constraints (4d) and get the hybrid model, see [3]. The solution of the linear programming problem (4), considering the two relaxed models, can be performed efficiently by a Dual Simplex approach. In order to establish a bench mark we will consider the Benders decomposition model presented so far, i.e., using three Benders cuts by iteration and minimum fixed values for the disjunctive parameter, as the reference model. However, reference model does not produce good computational results, as will be illustrated in Section V, because the Benders cuts are bad influenced by the still large value used for disjunctive parameter. One way we developed to handle with this drawback is to start the Benders algorithm using small values for the disjunctive parameters, and adjust these values during the decomposition iterations to the minimal ones (3). This strategy is interesting if we can use all previously generated Benders cuts when the value of disjunctive parameters changes. In order to illustrate how to adjust all Benders cuts for changes , we will use generic concepts of Benders decomposiin tion, presented in Appendix I. We know that Benders cuts are evaluated from the extreme points (vertices) of the feasible region of the Dual of the slave subproblem. We also know that this region is independent of the master solution . However, note that this region is also independent of the value used for the disjunctive parameter, which is considered in the matrix and vector , see problem (9). As a consequence, modifications will not change the feasible region of Dual subproblem in (only its optimal solution will change) and we can conclude that, has changed) are as all vertices already evaluated (before still vertices of the Dual feasible region, they are still good to evaluate Benders cuts.

IV. GOMORY CUTS AND BENDERS DECOMPOSITION The main motivation to consider Gomory cuts within Benders decomposition is to improve the practical convergence to the optimal solution of the Benders approach. Remark that, the master subproblem, that should be solved each iteration, is a mixed linear (0–1) program. A cutting plane approach, using Gomory cuts, allows the resolution of combinatorial problems by a sequence of linear programming problems corresponding to linear relaxations of the original discrete problem. In fact, we could use the Gomory approach to solve the master subproblem, but Branch-and-Bound methods have been showed much more efficient than cutting planes approaches. However, in [13], Balas et al. illustrated, using a simple lifted procedure, that the Gomory cuts can be shared across different branches of the search tree for mixed (0–1) programs. They reported good results by combining the Gomory cuts within a Branch-and-Cut approach. The idea here is lightly different. Note that the master subproblem of th iteration is a relaxation of the master subproblem th iteration. Then, we could evaluate Gomory cuts from of the solution of the former subproblem and use them to help the solution of the later, which has its feasible region enriched with combinatorial aspects of the former subproblem, which allows an improving in the practical convergence when solving it. In order to illustrate how to compute Gomory cuts within the , be the soluBenders decomposition approach, let tion obtained by the master subproblem in a given iterations and be a subset of containing the indices of variables (selected at random) that will remain fixed on the following linear problem minimize

(5)

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Once the problem (5) is solved, the coefficients of Gomory —could be evaluated by equation (11), cuts— illustrated in Appendix II. Note that we do not include in the Gomory cut because is a free variable and so, is a basic variable on the optimal solution in (5). The Gomory cuts should be include in the next master subproblem, which must be redefined to include them.

TABLE I RESULTS OF REF. BENDERS DECOMPOSITION APPROACH

V. COMPUTATIONAL RESULTS All results illustrated in this section was gotten from a SUNUltra 2 workstation with 124 Mbytes of memory running Solaris. The operational system routine etime() was used to compute the CPU time and the computational program with the Benders decomposition approach was implemented in language C, compiled and linked using GNU gcc 2.8 with full optimization. Finally, all mathematical programs, master, slave etc. were built and solved using the CPLEX package version 3.0 [16]. The case study of transmission expansion planning considers a power system with the reduced Southern Brazilian System composed by 46 busses and 66 existing circuits. This case study have been used to illustrate new computational methods to the power transmission network design problems, since [17]. The total number of candidate circuito to be considered in the expansion proccess is equal to 237 (141 are in parallel with any existing circuit—triplication is allowed and 32 are new “right-ofways”). All relevant data of this case study can be found in [17]. The maximum value for the disjunctive parameter , evaluated in accordance with the longest path to connect all 8 initially disconnected busses, is equal to 10 000. Remark that this value is significantly less than the value proposed in [14] and equal by . is used only as a final value in a proceNote that dure that uses for a predefined number of iterations, small values for this parameter. Initially, the disjunctive parameter was set up is incremented by . to 500 and each iteration Thus, at iteration 20 the disjunctive parameter, as all Benders cuts, will be adjusted to the minimal values for the disjunctive is different parameter. It is also important to remark that is for different candidate circuits. The final value used only for the candidate circuits that connect the initial disconnect busses. For other candidate circuits, the minimal value computed by equation (3), is used and in the first iterations the is computed by a linear interpolation. This form of value of parameter was arbitrarily chosen. Numerical expeupdating riences in order to study better empirical rules to update this parameter should be performed in order to improve the efficiency of the approach. Other parameters that should be set up before running the decomposition approach include: the tolerance for convergence and the penalty for load shedding. These parameters was chosen % and US$/MW, . acounding: In order to verify the efficiency of our Benders decomposition approach the transmission expansion case study will be solved first by a reference Benders decomposition model and later by our specialized approach that considers adjusts in the disjunctive parameters and Gomory cuts for master subproblem. The results

TABLE II RESULTS OF OUR BENDERS DECOMPOSITION APPROACH

are compared by CPU time and number of iterations to reach the optimal solution. A. Remarks The optimal solution found, in both cases, has investment cost equal to US $154.4 millions. This solution was first published in [5] but using a non convex approach and consists of the addition of 16 candidate circuits. When using the reference Benders approach, in which the is fixed in 10 000, 135 iterations was perparameter formed to find the optimal solution. The total CPU time used was about 4.6 hours. The Table I illustrates parts of the convergence report, where the first column displays the iteration number, followed by lower and upper bounds, gap for convergence and the average solution time (by iteration) in solving the slave and master subproblems. At all, 319 Benders cuts were computed, 63 using the transportation model and 114 for the hybrid model. On the other hand, when minimimal values for the param, are used, when these values are iteratively eter adjusted as described in this section and when Gomory cuts for the master subproblem are considered, convergence to the optimal solution takes 74 iterations. As a consequence of the smaller number of iterations, the total CPU time consumed was about 0.8 hour. Table II shows parts of the convergence report. At all, 218 Gomory cuts and 186 Benders cuts were computed (46 for the transportation model and 66 for the hibrid model). Based on the results obtained we can conclude that the use of small values in the first iterations for the disjunctive paramand eter, adjusts in the Benders cuts required by changes in mainly the use of Gomory cuts within a Benders decomposition

BINATO et al.: A NEW BENDERS DECOMPOSITION APPROACH TO SOLVE POWER TRANSMISSION NETWORK DESIGN PROBLEMS

scheme fulfilled all its objectives, reducing both CPU time and number of iterations to reach the optimal solution. It is also important to emphasize the use of minimal values for , , equation (3). Using the disjunctive parameter as advocates in [14], i.e., fixed in 203 000, and not consideting Gomory cuts, will require 214 iterations and more than 28 hours of CPU time. Finally, it is also important to remark that compute Benders cuts from relaxations of the slave subproblem is crucial for the success of Benders decompositions. For example, solving the reduced southern brazilian case study by the mixed (0–1) disjunctive formulation and do not considering the relaxed Benders cuts will fail after 350 iterations and more that 4 days of CPU time.

where would be the investment variables while tional ones. If we define

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the opera(7)

then the problem (6) can be written as minimize s.t.: (8) , . Assuming that where is contained in dom , can be obtained from the Dual of problem (7), stated as follows: maximize

VI. CONCLUSION This paper presented a new Benders decomposition approach to solve real world power transmission network design problems. The method is characterized by: a mixed (0–1) model which ensures the optimality of solution found; the use of min; the use of Gomory imum values for disjunctive parameter cuts iteratively evaluated from master subproblem; and the use of Benders cuts evaluated from relaxed versions of the slave subproblem. Based on the results obtained we can conclude that: • The use of minimal values for the disjunctive parameter improves the numerical condition of the problem. Further, we showed that adjusts in the disjunctive parameters and, consequently in the Benders cuts are possible and also helps the convergence of the decomposition approach. • Regarding to the Gomory cuts, we could verify their efficiency in helping the convergence of Benders decomposition. Large CPU time savings was obtained in the case study reported. However, the use of Gomory cuts or within Bender decomposition or within Branch-and-Cut approaches are a very new research topic that needs much more work to be explored at all. • Finally, our model evaluates a set of Benders cuts by each iteration, based on relaxations of the slave subproblem. This strategy was cricial to the success of our Benders decomposition approach. APPENDIX I BENDERS DECOMPOSITION HIGHLIGHTS Considers the following optimization problem

s.t.: (9) Note that the feasible region of problem (9) is a polyhedron, , where that is characterized by its extreme points ( , represents the finite set of all vertices of ), is independent on the first-stage variables . As the optimal solution of problem (9) is a vertex of this polyhedron, it can be solve by enu, meration, i.e., evaluating . As Primal and Dual optimal solution values are where , identical, we can conclude that the constraints are the functions in problem (8). The Benders decomposition is a technique that represent the by a reduced subset of constraints function , . Then, the master subproblem corresponds to a relaxed version of problem (6) and, the value of its solution, , is a lower bound, for the optimal solution of the original problem. , where is the On the other hand, note that the pair solution of problem (7), is a feasible for the original problem but not necessarily the optimal one. Thus, we can see that,

is a upper bound to the optimal solution of the original problem. Denoting by the lower bound, we can check the optimality of the solution computing,

where is a pre-defined tolerance. If the optimality test is the optimal solution. is achieved, the pair Otherwise, a new Benders cut should be evaluated, solving the . This new Benders cut must be slave subproblem (7) for added to the set of constraints of the master subproblem and so on.

minimize s.t.:

(6) , are the variables of the problem. Rewhere mark that we can represent the design problem (1) by this model,

APPENDIX II GOMORY CUTS REVIEW Gomory [12] proposed cutting planes to solve integer problems. The basic idea of cutting planes methods is to solve the continuous linear relaxation of the integer programming

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problem. If the optimal solution of the linear relaxed problem is integer, then the problem is solved. Otherwise, we can cut off the non integer solution (adding new constraints to the problem) but without eliminating any integer solution. After having added the new constraints, the continuous linear relaxation augmented (by the new constraints) must be solved again, and so on. In the Gomory cut approach, the new constraints are evaluated from the Simplex optimal tableau of the continuous linear relaxation. Considers the following mixed (0–1) programming problem

[6] [7] [8] [9] [10] [11]

minimize [12]

s.t.: [13]

(10) is the number of (0–1) variables. where Denoting by and to be the set of basic and nonbasic variables at the optimal solution of problem (10), the coefs. Gomory , can be evaluated by cuts, generically expressed as

[14]

[15]

[16]

;

[17]

, “A zero–one implicit enumeration method for optimizing investments in transmission expansion planning,” IEEE Trans. on Power Systems, vol. 9, no. 3, Aug. 1994. G. C. Oliveira, A. P. Costa, and S. Binato, “Large scale transmission network planning using optimization and heuristic techniques,” IEEE Trans. on Power Systems, vol. 10, no. 4, pp. 1828–1834, Nov. 1995. A. M. Geoffrion and G. W. Graves, “Multicommodity distribution system design by Benders decomposition,” Management Science, vol. 20, no. 5, pp. 822–844, Jan. 1971. S. Granville and M. V. F. Pereira, “Analisys of the linearized power flow model in Benders decomposition,” System Optimization Lab, Dept. of Operations Research, Stanford University, Tech. Rep. SOL 85-04, 1985. A. Sharifnia and M. H. Aashtiani, “Transmission network planning: A method for synthesis of minimum cost secure networks,” IEEE Trans. on PAS, vol. PAS-104, no. 8, Aug. 1985. R. Villasana, “Transmission network planning using linear and linear mixed integer programming,” Ph.D. dissertation, Ressenlaer Poly-technic Institute, 1984. R. E. Gomory, “An algorithms for integer solutions to inner programs,” in Recent Advance in Mathematical Programming, R. Graves and P. Wolfe, Eds: McGraw-Hill, 1963, pp. 269–302. E. Balas, S. Ceria, G. Cornuéjols, and N. Natraj, “Gomory cuts revisited,” Operations Research Letters, 1995. P. Tsamasphyrou, A. Renaud, and P. Carpentier, “Transmission network planning: An efficient Benders decomposition scheme,” in Proceedings of the 13th Power System Computing Conference, June/July 1999, pp. 487–494. S. Binato, “Optimal power transmission expansion planning by Benders decomposition and cutting planes techniques,” D.Sc. dissertation (in portuguese), COPPE-Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil, Apr. 2000. CPLEX Optimization, Inc., “Using the CPLEX Callable Library,”, version 3.0 ed., 1989–1994. M. V. F. Pereira, L. M. V. G. Pinto, G. C. Oliveira, and S. H. F. Cunha, “Composite generation-transmission expansion planning,” EPRI, Tech. Rep. EPRI Report RP 2473-9, 1987.

; ; is the optimal Simplex tableau coef. of where . the fractional part of

(11) and is

ACKNOWLEDGMENT The authors would like to thank the suggestions G. C. Oliveira from Power System Research Inc. and Catholic University of Rio de Janeiro, N. Maculan and P. R. Oliveira both from COPPE, Federal University of Rio de Janeiro and A. J. Monticelli from Campinas University. REFERENCES [1] J. F. Benders, “Partitioning methods for solving mixed variables programming problems,” Numerische Mathematik, vol. 4, pp. 238–252, 1962. [2] M. V. F. Pereira, “Application of sensitivity analysis on generation-transmission system expansion planning,” D.Sc. dissertation (in Portuguese), COPPE-Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil, 1985. [3] S. Granville, M. V. F. Pereira, G. B. Dantzig, B. Avi-Itzhak, M. Avriel, A. Monticelli, and L. M. V. G. Pinto, “Mathematical decomposition techniques for power system expansion planning-decomposition methods and uses,” EPRI, Tech. Rep. RP 2473-6, 1988. [4] , “Mathematical decomposition techniques for power system expansion planning-analysis of the linearized power flow model using the Benders decomposition technique,” EPRI, Tech. Rep. RP 2473-6, 1988. [5] R. Romero and A. Monticelli, “A hierarchical decomposition approach for transmission expansion planning,” IEEE Trans. on Power Systems, vol. 9, no. 1, pp. 373–380, Feb. 1994.

Silvio Binato received the B.Sc. degree in EE (power systems) from Federal University of Santa Maria in 1988, the M.Sc. degree also in electrical engineering from Catholic University of Rio (PUC-Rio) in 1992, and the D.Sc. degree in system and computing engineering in 2000. Since 1989, he has been working at CEPEL—the Brazilian Electric Power Research Center—where he works with power transmission network expansion planning, metaheuristics to solve combinatorial problems and stochastic optimization. In 1996, he was on leave for two months at Bell-Labs (AT&T) where he worked on the Dual Affine Scale method for linear programming problems.

Mário Veiga F. Pereira received the B.Sc. degree in EE (power systems, optimization and computer science) and the M.Sc. and D.Sc. degrees in systems engineering (optimization). From 1975 to 1986, he worked at Cepel, where he coordinated the development of methodology and software in power system planning and operation (PSOP). From 1983 to 1985, Dr. Pereira was a Project Manager at EPRIs PSOP program. In 1987, he co-founded Power Systems Research, Inc. (PSRI), where he has developed computational tools and methodologies for planning and operation for several utilities in Latin America, USA, Europe, New Zealand and China, plus multi-lateral institutions such as the World Bank and IDB. Dr. Pereira has also been Advisor on privatization and power sector restructuring in Brazil, Colombia, Venezuela and South of China.

Sérgio Granville received the B.Sc. degree in mathematics in 1971, the M.Sc. degree in applied mathematics in 1973, both from PUC/RJ, and the Ph.D. degree in operations research in 1978 from Stanford University. From 1979 to 1985, he has been involved in teaching and research activities in optimization techniques for economic planning. From 1986 to 2000, he has been a Senior Researcher at CEPEL and at the moment he is at Power Systems Research, Inc. (PSRI). Since 1986, he has been involved in the development of methodologies and computational tool for transmission planning, reactive power planning, optimal power flow, and more recently, for financial optimization in electricity markets.