IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST

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1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST A Bilevel Approach to Transmission Expansion Planning Within a Market Environment Lina P. Garcés, Student Member, IEEE, Antonio J. Conejo, Fellow, IEEE, Raquel García-Bertrand, Member, IEEE, and Rubén Romero, Senior Member, IEEE Abstract We present a bilevel model for transmission expansion planning within a market environment, where producers and consumers trade freely electric energy through a pool. The target of the transmission planner, modeled through the upper-level problem, is to minimize network investment cost while facilitating energy trading. This upper-level problem is constrained by a collection of lower-level market clearing problems representing pool trading, and whose individual objective functions correspond to social welfare. Using the duality theory the proposed bilevel model is recast as a mixed-integer linear programming problem, which is solvable using branch-and-cut solvers. Detailed results from an illustrative example and a case study are presented and discussed. Finally, some relevant conclusions are drawn. Index Terms Bilevel model, duality theory, electricity market, mixed-integer linear programming, transmission expansion planning. NOTATION T HE main notation used throughout this paper is stated below for quick reference. Other symbols are defined as needed throughout the paper. Constants: Susceptance of line. Load-shedding cost for consumer ( /MWh). Investment cost of building line ( ). Budget for investment in transmission expansion ( ). Maximum power consumed by the th demand in scenario (MW). Minimum power consumed by the th demand in scenario (MW). Size of the th block of the th demand in scenario (MW). Capacity of line (MW). Manuscript received September 17, 2008; revised January 16, First published May 19, 2009; current version published July 22, The work of L. P. Garcés was supported in part by the Programme Alβan, the European Union Programme of High Level Scholarships for Latin America, scholarship no. E07D400725BR. The work of A. J. Conejo and R. García-Bertrand was supported in part by the Government of Castilla-La Mancha, Project PCI , and in part by the Ministry of Education and Science of Spain, CICYT Project DPI Paper no. TPWRS L. P. Garcés and R. Romero are with the Paulista State University, Ilha Solteira, Brazil ( linaneg@aluno.feis.unesp.br; ruben@dee.feis.unesp. br). A. J. Conejo and R. García-Bertrand are with the University of Castilla-La Mancha, Ciudad Real, Spain ( Antonio.Conejo@uclm.es; Raquel.Garcia@uclm.es). Digital Object Identifier /TPWRS Size of the th block of the th generating unit in scenario (MW). Sending-end bus of line. Receiving-end bus of line. Upper limit of the continuous variable. Weight of scenario. Price bid by the th block of the th demand ( /MWh). Price offered by the th block of the th generating unit ( /MWh). Weighting factor to make annual investment cost and social welfare comparable. Variables: Power consumed by the th block of the th demand in scenario (MW). Power flow through line in scenario (MW). Power produced by the th block of the th generating unit in scenario (MW). Load shed by the th demand in scenario (MW). Binary variable that is equal to 1 if line is built and 0 otherwise. Voltage angle at bus in scenario (radians). Auxiliary continuous variable pertaining to line and scenario and used for linearization. Indices and Sets: Bus index where the th generating unit is located. Bus index where the th demand is located. Set of indices of the demands located at bus. Set of indices of the generating units located at bus. Set of indices of the blocks of the th generating unit. Set of indices of the blocks of the th demand. Set of indices of the demands. Set of indices of the generating units. Set of all transmission lines, prospective and existing. Set of all prospective transmission lines. Set of all networks buses. Set of all scenarios /$ IEEE

2 1514 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009 I. INTRODUCTION A. Background and Aim We consider a pool-based electricity market including producers, consumers and retailers that freely trade electric energy. This pool is cleared by a market operator using a network constrained market clearing procedure that seeks maximizing social welfare. A transmission system operator (TSO) is in charge of planning and operating the transmission networks a natural monopoly with the ultimate target of facilitating the trading of electric energy, thus contributing to maximizing the social welfare. Needless to say, this TSO operates within a limited investment budget. This is a common arrangement in the electricity markets across western Europe, e.g., in the electricity market of the Iberian Peninsula [1], [2]. This paper provides a procedure for optimal expansion planning of the transmission network to be derived and implemented by the TSO. The approach is static in the sense that it considers a single target year in the planning analysis. This procedure relies on a bilevel formulation [3]. The upper-level problem represents the objective of the TSO, i.e., minimizing investment cost while facilitating electricity trading. The numerical proxy considered for facilitating electricity trading is maximizing the average social welfare over all considered scenarios. The lower-level problems represent a number of market clearing scenarios. The objective of each of these lower-level problems is to maximize the social welfare pertaining to its corresponding scenario. The scenarios considered include cases with network contingencies, and cases in which the highest demand is located at different buses. Investment decisions made at the upper-level problem condition the network-constrained market clearings at the lowerlevel problems, while the social welfare values computed at the lower-level problems affect the investment decision at the upper-level problem. The proposed bilevel formulation allows considering simultaneously the interactions among the upperlevel problem and the lower-level problems, thus achieving a joint optimal solution. B. Approach Since each lower-level problem representing the market clearing of a market scenario is continuous and convex (it is linear), it can be represented through the constraints of the primal problem, the constraints of the dual problem and the strong duality condition. If the primal and dual constraints and the strong duality condition of all lower-level problems (one per scenario) are incorporated into the upper-level problem representing optimal investment decisions, the resulting problem is a mixed-integer nonlinear problem. This problem is nonlinear since it includes products of variables. However, using results from the duality theory [4] and binary arithmetic, it can be recast as a mixed-integer linear programming problem, which can be solved using commercially available branch-and-cut solvers [5]. C. Literature Review The techniques used to approach the transmission expansion planning problem are diverse but may be classified as constructive heuristic methods [6] [9], classic optimization [10] [15], and intelligent systems [16] [20]. Reference [6] proposes a method for linear flow estimation as an effective guide in the development of preliminary network designs, which are used for expansion planning. Reference [7] considers a dc model to obtain the optimal expansion plan by using sensitivity indices and the so-called least-effort criterion. Reference [8] relies on a modified dc model including fictitious generating units and, as in [7], uses a sensitivity method to find the best expansion plan while minimizing the system load curtailment. In [9], the authors use the method in [6] for selecting network reinforcements. References [10] and [11] present mixed-integer linear approaches to the solution of the transmission expansion planning problem. Reference [10] considers a centralized framework while [11] considers a market one. Reference [12] proposes a branch and cut method for selecting the best expansion plan using a probabilistic reliability criterion. Benders decomposition is combined with heuristic techniques in [13] and [14], while a rigorous decomposition scheme is presented in [15]. Other approaches include simulated annealing [16], genetic algorithms [17], specialized genetic algorithms [18], tabu search [19], and other heuristics [20]. Within the framework above, this paper proposes a novel transmission expansion planning approach within a market environment using a bilevel programming approach. Bilevel programming background is provided, for instance, in [3]. References using bilevel programming in different contexts include [21], which provides a model to find an optimal expansion plan that mitigates the impact of deliberated network attacks, and [22], which addresses the generation capacity expansion planning problem. D. Contributions The contributions of this paper are threefold: 1) to provide a transmission expansion planning algorithm that represents efficaciously market competition through a bilevel programming problem; 2) to use duality results to convert the bilevel formulation of 1) into a mixed-integer linear programming problem solvable via branch-and-cut solvers; 3) to analyze and discuss a number of illustrative examples and case studies that show the interest of the proposed approach. Additionally, the proposed technique is compared with a classical cost-minimization approach. E. Paper Organization The remainder of this paper is organized as follows. Section II presents the proposed bilevel model, the corresponding mixedinteger nonlinear problem, its equivalent linear form, and the linearization procedure carried out. Section III provides results from an illustrative example; these results are discussed and analyzed in detail. Section IV gives some results and provides some discussions for a realistic case study. Section V provides some relevant conclusions.

3 GARCÉS et al.: BILEVEL APPROACH TO TRANSMISSION EXPANSION PLANNING 1515 II. MODEL A. Bilevel Model The decision making problem pertaining to a transmission planner that jointly minimizes network investment and maximizes average social welfare (a proxy for facilitating trade ) can be formulated as a bilevel programming model [3], [23]. The upper-level problem represents the decisions to be made by the planner with the target of deciding transmission investments while both maximizing average social welfare (over the considered scenarios) and minimizing investment cost. The lowerlevel problems represent a market clearing for each scenario and consider investment decisions known. The transmission expansion planning problem can be formulated as the bilevel model below. Note that dual variables are provided after the corresponding equalities or inequalities separated by a colon: subject to where subject to (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) Model (1) (19) comprises the upper-level problem (1) (4) and a collection of lower-level problems (5) (19), one for each scenario. The upper-level problem allows making investment decisions while power produced/consumed by generating units/consumers, and load-shedding result from the solution of all lower-level problems. Note that, are the upper-level decision variables, while the lower-level decision variables are ; ; ; and. The upper-level objective function to be maximized (1) is the average (over the considered scenarios) social welfare [lines 1 and 2 of (1)] minus the transmission investment cost [line 3 of (1)]. The average social welfare is multiplied by the weighting factor to make the annual investment cost and the social welfare comparable quantities. Constraint (2) enforces an upper bound on the investment cost, constraints (3) states that existing lines have already been built and (4) are binary variables declaration. The objective function (5) of each lower level problem is the declared social welfare for the corresponding scenario. This objective function to be maximized is expressed as the summation of the demand energy bids times their corresponding demand price bids minus the summation of the generation energy offers times their corresponding generation price offers minus the cost of load-shedding. Constraints (6) enforce the power balance at every bus. Constraints (7) represent the power flow through each line. Note that each of these constraints is multiplied by a binary variable, thus, if the corresponding line is not physically connected to the network the power flow through it is zero. Constraints (8) (9) enforce the line flow limits. Constraints (10) and (11) establish the sizes of the blocks of the generating units and the demands, respectively. Constraints (12) impose upper bounds on load-shedding. Constraints (13) impose minimum demand consumptions. Constraints (14) (15) impose limits on the voltage angles at every bus and scenario, while constraints (16) fix the voltage angle of the reference bus for each scenario. Finally, constraints (17) (19) are variables declaration. B. Alternative Objectives It is relevant to note that bilevel problem (1) (19) reduce to problem (1) (19) without (5), i.e., a single-level problem, if the objective function of the upper-level problem and the objective functions of all lower-level problems point to the same direction, i.e., all these objective functions align pursuing their respective targets. This may be or may be not the case for the formulation above.

4 1516 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009 An alternative objective function for the upper-level problem that ensures generally that objective functions do not align is (20) that represents the minus average load-shedding cost (multiplied by parameter ) minus the transmission investment cost. C. Mixed-Integer Nonlinear Programming Reformulation Each lower-level problem (5) (19) represents the market clearing of a single scenario. Since each of these lower-level problems is continuous and convex, it can be represented by its constraints, the constraints of its dual problem and the strong duality condition [24]. The bilevel problem (1) (19) can be transformed into an equivalent one-level mixed-integer nonlinear programming problem by incorporating into the upper-level problem the primal and dual constraints and the strong duality condition of each lower level problem. The dual problem corresponding to the lower-level problem (5) (19) for scenario is (33) (34) The optimization variables of problem (21) (34) are, ;, ;, ;, ;,, ;, ;, ;, ;, and,. The strong duality theorem states that a feasible solution of the primal problem and a feasible solution of the dual problem are primal and dual optimal, respectively, if and only if (35) subject to (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) The single-level mixed-integer nonlinear programming problem equivalent to problem (1) (19) is obtained by maximizing the objective function of the upper-level problem considering the constraints of the upper-level problem, the primal and dual constraints of each lower-level problem and the strong duality condition of each lower level problem. The resulting problem is subject to (36) (37) (38) (39) Variables of problem (36) (39) are, ;,,, ;,,, ;,, ;,, ;,, ;,, ;,, ;,,, ;,,, ;,, ;,, ;,, ;,,, and,. D. Equivalent Linear Formulation The single-level mixed-integer nonlinear problem (36) (39) can be transformed into a mixed-integer linear problem by linearizing nonlinear terms. Specifically, the nonlinear terms appear in (7), (26), and (27). These constraints can be replaced by linear ones using wellknown linearization schemes [10], [23], [25]. Nonlinear constraints (7) and (8) (9) are replaced by the equivalent linear ones that follow: (40)

5 GARCÉS et al.: BILEVEL APPROACH TO TRANSMISSION EXPANSION PLANNING 1517 (41) where is the disjunctive parameter that should be equal to a sufficiently large positive constant, providing enough freedom for the voltage angle difference between every unconnected bus of the system to take feasible values. Reference [15] provides insights to tune up the value of. Equations (40) and (41) work as follows. If line is built ( ), then the power flow in this line is limited by its maximum capacity enforced by (40) and its value is given by (41) because the corresponding inequality constraints become equality ones. On the other hand, if line is not built ( ), then (40) sets to zero and constraints (41) are deactivated provided that the constant is appropriately selected. Additionally, dual nonlinear constraints (26) and (27) are replaced by the following linear ones: Fig. 1. Garver s six-bus test system. (42) (43) (44) (45) Equations (42) (45) work as follows. If the positive constant is large enough, and if line is built ( ), from (45) and (44) are not binding. Otherwise, if line is not built ( ), then, it can be concluded from (44) that the difference ( ) is zero and therefore its contributions to the summation in (42) and (43) are zero, too. E. Uncertainty Modeling Relevant uncertainties affecting the transmission expansion planning problem within a market environment include: 1) demand growth; 2) spatial distribution of the demand growth; 3) producers offers; 4) consumers bids; 5) availability of transmission facilities; and 6) availability of the generation facilities. These uncertainties can be adequately taken into account by generating a set of scenarios that considers all possible realizations of the involved uncertain parameters. The use of an appropriate scenario reduction technique is also advisable because the initial number of scenarios may result in an untractable optimization problem. The selection of the final number of scenarios entitles resolving a non trivial tradeoff between accuracy and computational burden: a higher number of scenarios implies higher accuracy but also higher computational burden. Thus, the final number of scenarios should be selected properly balancing accuracy and tractability. However, the generation of an appropriate set of scenarios and its subsequent reduction keeping as much as possible the stochastic information contained in the original set is beyond the scope of this paper; the interested reader can find appropriated information in [26]. For the sake of simplicity only demand growth and line availability scenarios are considered in this paper. III. ILLUSTRATIVE EXAMPLE A. Data The proposed model is analyzed using the classical Garver s six-bus test system [6], depicted in Fig. 1, and comprising six buses, six existing lines, three generating units, and five demands. Note that bus 6, which includes generation, is not initially connected to the network, but connecting lines can be built. Line data are provided in Table I. The first two columns provide origin and destination buses, the third and the fourth columns provide reactances and capacities (obtained from [6]), respectively, and the fifth column gives the investment costs, computed using a building cost of /km and the line lengths obtained from [6]. Note that the maximum number of lines (prospective plus existing) per corridor is limited to three. Table II provides the blocks of power offered and the corresponding offer prices for the generating units; and the blocks of power bid and the bid prices for demands. For the sake of simplicity and without loss of generality, we consider generating offers and demand bids independent of demand scenarios. The load-shedding cost is considered to be one hundred times the bid price of the first block of each demand. Initially, three different scenarios are considered to describe future demand situations. Table III characterizes these demand scenarios. The second column provides the weights of the scenarios, while the third column gives demand factors for the loads at the buses indicated in parenthesis. These factors affect all demand bids, thus representing load growth. For each scenario, the

6 1518 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009 TABLE I LINE DATA FOR GARVER S SIX-BUS EXAMPLE TABLE IV SOLUTION FOR GARVER S SIX-BUS EXAMPLE TABLE II GENERATOR AND DEMAND DATA FOR GARVER S SIX-BUS EXAMPLE TABLE III SCENARIO DATA FOR GARVER S SIX-BUS EXAMPLE minimum power consumption of each demand is considered to be 90% of its total demand bid. The capital recovery factor can be calculated as, where is the interest rate and is the number of considered years. We assume that a line built today is operative for at least 25 years; thus, a 25-year investment return period is considered, as well as a 10% interest rate. With these values, a capital recovery factor value of approximately 10% is obtained, which means that, for the next 25 years, the investment cost in new lines is yearly repaid at a rate of approximately 10% of the total initial investment. Since we considered on one hand the annualized cost, and, on the other hand the social welfare for one hour, the weighting factor needs to be equal to the total number of hours for a year, i.e., Finally, note that the investment budget is limited to 30 million. Numerical simulations allow concluding that an appropriate value for both and (constants needed to linearize nonlinear terms) is B. Results Table IV provides the transmission expansion plan for the bilevel market-oriented approach presented in this paper as well as for a cost-minimization approach. The considered cost-minimization model consists in an optimization problem where the investment cost and the cost of load-shedding are minimized subject to (3) (4) and (6) (19). Only one demand scenario is considered at a time in this cost-minimization approach. The second column of Table IV provides optimal investment plans (if more than one line is built in a corridor, this is indicated in parenthesis). The total investment cost is provided in the third column of Table IV. Note that the annualized investment cost, corresponding to 10% of total investment cost, is represented in the objective function of the upper-level problem. The social welfare corresponding to each demand scenario can be computed as the difference between the summation of the accepted demand blocks times their corresponding bid prices and the summation of the accepted generation blocks times their corresponding offer prices minus the cost of load-shedding. These results are provided in the fourth column of Table IV. Note that the proposed bilevel approach solves the transmission expansion planning problem considering simultaneously several demand scenarios. To make comparable the social welfare for the bilevel and the cost-minimization approaches, an average social welfare is computed for the cost-minimization problem. To do so, problem (36) (39) is solved considering the three demand scenarios and the investment decisions (variables, )fixed to the values obtained from the solution of the corresponding cost-minimization problem. The accepted demand and generation blocks provided by this problem are used to obtain the average social welfare, and this value is given in the fifth column of Table IV. Note that the social welfare values correspond to one year. Finally, the sixth column provides the computing times. The investment plan obtained from the bilevel model results in building lines connecting buses 3 and 6 with the rest of the system. Note that the highest generation capacities are located in buses 3 and 6.

7 GARCÉS et al.: BILEVEL APPROACH TO TRANSMISSION EXPANSION PLANNING 1519 TABLE V SOLUTION FOR GARVER S SIX-BUS EXAMPLE WITH LINE CONTINGENCIES Solutions for the cost-minimization problems have been obtained for the three demand scenarios. Scenario 1 results in a comparatively low total investment cost and a low social welfare since this scenario corresponds to the lowest demand profile. The average social welfare is negative since the demand cannot be fully supplied for some consumers considering the investment plan obtained for scenario 1, which results in an expensive load-shedding. Note that both the total investment cost and the social welfare values are lower that those obtained from the bilevel model. For scenario 2, the demand is comparatively high which results in a higher total investment cost and a higher social welfare than in the case of scenario 1. Note that the solution obtained solving the cost-minimization problem for this scenario is identical to the solution of the bilevel problem. Finally, for scenario 3 the investment cost obtained solving the cost-minimization problem is moderate but the average social welfare for this case is negative since load-shedding affects some consumers. Note that both the total investment cost and the social welfare values are lower that those obtained from the bilevel model. To complement the previous analysis, a new scenario is added to the existing ones. This additional scenario considers the contingency of a single line. The weight of this new scenario is 0.10 and the demand factors are all 1. To keep scenario-weight consistency, the weight of scenario 1 is reduced to The purpose of this analysis is just to make it clear that contingencies matter. Table V provides the investment plans obtained solving the bilevel problem as many times as existing lines. For each run four scenarios are considered, the three original ones and one representing the outage of one of the existing lines. The first column of this table indicates the unavailable lines, the second column provides the optimal investment plans (if more than one line is built in a corridor, this is indicated in parenthesis), the Fig. 2. IEEE 24-bus system. third column gives total investment costs, the fourth column provides the yearly average social welfare value and, finally, the fifth column provides the computing times. For line 1 2, 1 4, 2 3 or 2 4 unavailable, investment plans are identical to that without the contingency scenario. Small discrepancies in annual social welfare are observed because the demand supplied slightly changes depending on the case. If line 1 5 is unavailable, a new line is built linking buses 1 and 5. Similarly, if line 3 5 is unavailable, a new line is built linking buses 3 and 5. These two cases result in the highest investment cost, being the social welfare similar to that of other cases. All problems have been solved using CPLEX 11.0 [5] under GAMS [27] on a Linux-based server with one processor clocking at 2.6 GHz and 32 GB of RAM. IV. CASE STUDY The case study described in this section is based on the IEEE 24-bus Reliability Test System (RTS), [28], depicted in Fig. 2. The transmission network comprises 24 buses, 34 existing corridors, and seven new ones, totaling 41 rights of way. We consider that all lines in the same corridor are identical and that the maximum number of lines per corridor is three. Lines lengths for existing corridors are obtained from [28]. Line lengths, resistances, reactances and capacities of lines for new corridors are obtained from [29, Table I]. Investment costs are /km for 230-kV lines and /km for 132-kV lines. The IEEE RTS includes ten generating units and 16 loads. Table VI provides the location of generators and demands in the network, the blocks of power offered and the corresponding offer prices for generating units, and the blocks of power bid and

8 1520 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009 TABLE VI GENERATOR AND DEMAND DATA FOR IEEE RTS CASE STUDY TABLE VIII SOLUTION FOR THE IEEE RTS CASE STUDY WITH NO CONTINGENCIES TABLE VII CHARACTERISTICS OF THE SCENARIOS FOR THE IEEE RTS CASE STUDY WITH NO CONTINGENCIES model results in building lines 6 10, 10 11, and 14 23, which further interconnect the two areas of the system, one with excess of demand (buses 1 10) and other with excess of generation (buses 13 23). This investment plan results in higher social welfare and investment cost than these pertaining to a cost-minimization approach because a higher amount of comparatively cheap energy can be supplied to the area with excess of demand. Note that the average social welfare using the cost-minimization approach for scenario 1 and 3 are negative since load-shedding affects some consumers. From Table VIII it can be concluded that the proposed formulation exhibit a computational burden similar to the one presented by a cost-minimization approach. However, the proposed formulation is more flexible than a cost-minimization one and allows reproducing actual market functioning, including the uncertainties affecting the market environment. the bid prices for demands. Again, for the sake of simplicity and without loss of generality, offers and bids are considered independent of the demand scenarios. Note that the generation levels and the loads are three times their original values provided in [28]. All problems have been solved using CPLEX 11.0 [5] under GAMS [27] on a Linux-based server with four processors clocking at 2.6 GHz and 32 GB of RAM. A. No Contingency Case For this no contingency case, three different scenarios are considered to model future demand behavior, which are characterized in Table VII. The second column of this table provides scenario weights while the third one gives the demand factors affecting the load buses indicated in parenthesis. For each scenario, we consider that the minimum consumption of each load is 90% of its total bid. Note that the investment cost in new lines is yearly repaid at a rate of 10% of the total initial investment and the maximum total investment should be below 23 million. For this no contingency case, Table VIII provides investment plans (if more than one line is built in a corridor, this is indicated in parenthesis) for both the bilevel and the cost-minimization approaches. The investment plan obtained from the bilevel B. Multicontingency Case A multicontingency case is considered below. The system is geographically divided in four zones to model load growth: northwest (15, 16, 18), northeast (13, 14, 19, 20), southwest (1, 3, 4, 5, 9) and southeast (2, 6, 7, 8, 10). Sixteen different load scenarios are considered to model future demand behavior. Additionally, six scenarios including the contingencies of lines 11 13, 11 14, 12 13, 12 23, and 15 24, respectively, are considered. These lines are important for an appropriate north-south interconnection of the system. Table IX provides detailed information on the considered scenarios using a format similar to that of Table VII. The last column of Table IX provides information on line contingencies. For each scenario, we consider that the minimum consumption of each load is 90% of its total bid. Note finally that the investment budget is limited to 30 million. Table X provides the investment plan obtained solving the the bilevel problem. Similarly to the investment plan obtained in the no contingency case, lines 6 10, and are built. However, line is not built, but reinforcements to the interconnection between the northern and the southern parts of the system are included in this new investment plan (lines 3 9, 7 8 and 13 14). Note that the required investment cost is higher than the one for the no contingency case. However, the

9 GARCÉS et al.: BILEVEL APPROACH TO TRANSMISSION EXPANSION PLANNING 1521 TABLE IX CHARACTERISTICS OF THE SCENARIOS FOR THE IEEE RTS MULTICONTINGENCY CASE STUDY TABLE X SOLUTION FOR THE IEEE RTS MULTICONTINGENCY CASE STUDY V. CONCLUSIONS This paper provides a transmission expansion planning methodology that specifically recognizes and properly models the functioning of a market environment. The transmission planner is a regulated entity that minimizes investment cost while promoting free trade among producers and consumers. The above is accomplished using a bilevel model that materializes in a mixed-integer linear programming problem of tractable size. The proposed model generates solutions with higher social welfare but also higher investment cost than the solutions provided by a classical cost-minimization approach. Computational simulations shows the tractability of the proposed model for realistic problems and its interest in deriving solution involving high social welfare. average social welfare is lower as a result of the aforementioned line contingencies. Although the computing time required to solve this multicontingency case (22 scenarios) is high (5.43 h), this time is still reasonable to address a planning problem of the type considered in this paper. Note that mixed-integer linear programming provides an appropriate tradeoff between modeling accuracy and solution efficiency, which may advantageously compare with other techniques such as decomposition, heuristics, etc. REFERENCES [1] Iberian Electricity Pool, OMEL, Spain and Portugal, [Online]. Available: [2] R. de Dios, F. Soto, and A. J. Conejo, Planning to expand?, IEEE Power Energy Mag., vol. 5, no. 5, pp , Sep.-Oct [3] B. Colson, P. Marcotte, and G. Savard, An overview of bilevel optimization, Ann. Oper. Res., vol. 153, no. 1, pp , Sep [4] E. Castillo, A. J. Conejo, P. Pedregal, R. García, and N. Alguacil, Building and Solving Mathematical Programming Models in Engineering and Science. New York: Wiley, [5] The ILOG CPLEX, [Online]. Available: products/cplex/. [6] L. L. Garver, Transmission network estimation using linear programming, IEEE Trans. Power App. Syst., vol. PAS-89, no. 7, pp , Sep [7] A. Monticelli, A. Santos, M. V. F. Pereira, S. H. Cunha, B. J. Parker, and J. C. G. Praca, Interactive transmission network planning using a least-effort criterion, IEEE Trans. Power App. Syst., vol. PAS-101, no. 10, pp , Oct [8] M. V. F. Pereira and L. M. V. G. Pinto, Application of sensitivity analysis of load supplying capability to interactive transmission expansion planning, IEEE Trans. Power App. Syst., vol. PAS-104, no. 2, pp , Feb [9] R. Villasana, L. L. Garver, and S. J. Salon, Transmission network planning using linear programming, IEEE Trans. Power App. Syst., vol. PAS-104, no. 2, pp , Feb [10] N. Alguacil, A. L. Motto, and A. J. Conejo, Transmission expansion planning: A mixed-integer LP approach, IEEE Trans. Power Syst., vol. 18, no. 3, pp , Aug [11] S. de la Torre, A. J. Conejo, and J. Contreras, Transmission expansion planning in electricity markets, IEEE Trans. Power Syst., vol. 23, no. 1, pp , Feb [12] J. Choi, T. Tran, A. A. El-Keib, R. Thomas, H. Oh, and R. Billinton, A method for transmission system expansion planning considering probabilistic reliability criteria, IEEE Trans. Power Syst., vol. 20, no. 3, pp , Aug [13] R. Romero and A. Monticelli, A hierarchical decomposition approach for transmission network expansion planning, IEEE Trans. Power Syst., vol. 9, no. 1, pp , Feb [14] R. Romero and A. Monticelli, A zero-one implicit enumeration method for optimizing investments in transmission expansion planning, IEEE Trans. Power Syst., vol. 9, no. 3, pp , Aug

10 1522 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009 [15] S. Binato, M. V. F. Pereira, and S. Granville, A new Benders decomposition approach to solve power transmission network design problems, IEEE Trans. Power Syst., vol. 16, no. 2, pp , May [16] R. Romero, R. A. Gallego, and A. Monticelli, Transmission system expansion planning by simulated annealing, IEEE Trans. Power Syst., vol. 11, no. 1, pp , Feb [17] R. A. Gallego, A. Monticelli, and R. Romero, Transmission system expansion planning by an extended genetic algorithm, Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 145, no. 3, pp , May [18] I. J. Silva, M. J. Rider, R. Romero, and C. A. F. Murari, Transmission network expansion planning considering uncertainty in demand, IEEE Trans. Power Syst., vol. 21, no. 4, pp , Nov [19] R. A. Gallego, R. Romero, and A. J. Monticelli, Tabu search algorithm for network synthesis, IEEE Trans. Power Syst., vol. 15, no. 2, pp , May [20] S. Binato, G. C. de Oliveira, and J. L. de Araújo, A greedy randomized adaptive search procedure for transmission expansion planning, IEEE Trans. Power Syst., vol. 6, no. 2, pp , May [21] M. Carrión, J. M. Arroyo, and N. Alguacil, Vulnerability-constrained transmission expansion planning: A stochastic programming approach, IEEE Trans. Power Syst., vol. 22, no. 4, pp , Nov [22] R. García-Bertrand, D. Kirschen, and A. J. Conejo, Optimal investments in generation capacity under uncertainty, in Proc. 16th Power Systems Computation Conf. (PSCC), Glasgow, U.K., Jul [23] J. Fortuny-Amat and B. McCarl, A representation and economic interpretation of a two-level programming problem, J. Oper. Res. Soc., vol. 32, no. 9, pp , Sep [24] A. L. Motto, J. M. Arroyo, and F. D. Galiana, A mixed-integer LP procedure for the analysis of electric grid security under disruptive threat, IEEE Trans. Power Syst., vol. 20, no. 3, pp , Aug [25] G. C. Oliveira, S. Binato, and M. V. F. Pereira, Value-based transmission expansion planning of hydrothermal systems under uncertainty, IEEE Trans. Power Syst., vol. 22, no. 4, pp , Nov [26] J. M. Morales, S. Pineda, A. J. Conejo, and M. Carrión, Scenario reduction for futures market trading in electricity markets, IEEE Trans. Power Syst., to be published. [27] R. E. Rosenthal, GAMS, A User s Guide. Washington, DC: GAMS Development Corporation, [28] Reliability Test System Task Force, The IEEE Reliability Test System 1996, IEEE Trans. Power Syst., vol. 14, no. 3, pp , Aug [29] R. Fang and D. J. Hill, A new strategy for transmission expansion in competitive electricity markets, IEEE Trans. Power Syst., vol. 18, no. 1, pp , Feb Antonio J. Conejo (F 04) received the M.S. degree from the Massachusetts Institute of Technology, Cambridge, in 1987 and the Ph.D. degree from the Royal Institute of Technology, Stockholm, Sweden, in He is currently a full Professor at the Universidad de Castilla La Mancha, Ciudad Real, Spain. His research interests include control, operations, planning and economics of electric energy systems, as well as statistics and optimization theory and its applications. Raquel García-Bertrand (M 06) received the Ingeniera Industrial degree and the Ph.D. degree from the Universidad de Castilla La Mancha, Ciudad Real, Spain, in 2001 and 2005, respectively. She is currently an Associate Professor of electrical engineering at the Universidad de Castilla La Mancha. Her research interests include control, operations, planning and economics of electric energy systems, as well as optimization and decomposition techniques. Rubén Romero (SM 08) received the B.Sc. and P.E. degrees from the National University of Engineering, Lima, Perú, in 1978 and 1984, respectively, and the M.Sc. and Ph.D. degrees from the Universidade Estadual de Campinas, Campinas, Brazil, in 1990 and 1993, respectively. Currently, he is a Professor of electrical engineering at the Universidade Estadual Paulista Julio de Mesquita Filho, Ilha Solteira, Brazil. His research interests include methodologies for the optimization, planning and control of electrical power systems, applications of artificial intelligence in power system, as well as operations research. Lina P. Garcés (S 07) received the B.Sc.EE. and M.Sc.EE. degrees from the Universidad Tecnológica de Pereira, Pereira, Colombia, in 2003 and 2005, respectively. She is currently pursuing the Ph.D. degree at the Universidade Estadual Paulista Julio de Mesquita Filho, Ilha Solteira, Brazil. Her research interests include transmission expansion planning and methodologies for reliability assessment in power composite system.

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