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A Three-Level Static MILP Model for Generation and Transmission Expansion Planning David Pozo, Student Member, IEEE, Enzo E. Sauma, Member, IEEE, and Javier Contreras, Senior Member, IEEE

Abstract—We present a three-level equilibrium model for the expansion of an electric network. The lower-level model represents the equilibrium of a pool-based market; the intermediate level represents the Nash equilibrium in generation capacity expansion, taking into account the outcomes on the spot market; and the upper-level model represents the anticipation of transmission expansion planning to the investment in generation capacity and the pool-based market equilibrium. The demand has been considered as exogenous and locational marginal prices are obtained as endogenous variables of the model. The three-level model is formulated as a mixed integer linear programming (MILP) problem. The model is applied to a realistic power system in Chile to illustrate the methodology and proper conclusions are reached. Index Terms—Equilibrium problem subject to equilibrium constraints (EPEC), mathematical program subject to equilibrium constraints (MPEC), Nash equilibrium, power systems economics, power transmission planning.

Set of all generators that can invest in capacity. Set of all generators that cannot invest in capacity expansion. Note that . Set of all generators belonging to firm that can invest in capacity expansion. This set is given by . Set of all generators belonging to firm that cannot invest in capacity expansion. This set is given by . Set of all generators that do not belong to firm that can invest in capacity expansion. This set is given by . Set of all transmission lines. Set of all candidate transmission lines for investment.

NOMENCLATURE The mathematical symbols used throughout this paper are classified as follows: Indexes:

Set of all generation companies. Set of all discrete investment strategies of generation company . Set of all discrete investment strategies for all companies. This set is given by .

Index of nodes. Index of generation companies.

Set of all the annual demand profiles.

Index of lines. Discrete investment strategy of generator company (GENCO) . Index of the parameter used for the discretization of the generation capacity expansion . Index of the annual demand profile.

Constants: Generation capacity available at node before level 2. Parameters of the generation cost function at node . Inelastic demand at node and demand profile .

Sets:

Number of hours per year of each profile of the demand.

Set of all generators. Set of all generators belonging to company

.

Manuscript received July 27, 2011; revised October 29, 2011, January 11, 2012, and March 29, 2012; accepted June 05, 2012. Date of publication July 19, 2012; date of current version January 17, 2013. This work was supported in part by the CONICYT, FONDECYT/Regular 1100434 grant, by the Fulbright NEXUS Program, by the Junta de Comunidades de Castilla—La Mancha Formación del Personal Investigador (FPI) grant 402/09, and by the Ministry of Science and Innovation of Spain grant ENE2009-09541. Paper no. TPWRS00699-2011. D. Pozo and J. Contreras are with the Escuela Técnica Superior de Ingenieros Industriales, Universidad de Castilla—La Mancha, 13071 Ciudad Real, Spain (e-mail: [email protected]; [email protected]). E. E. Sauma is with the Industrial and Systems Engineering Department, Pontificia Universidad Católica de Chile, Santiago, Chile (e-mail: [email protected]. cl). Digital Object Identifier 10.1109/TPWRS.2012.2204073

Power transfer distribution factor (PTDF) associated to line with respect to a unit injection/withdrawal at node . Annual unit cost of investment in capacity for a generation unit at node . Annual unit cost of investment in capacity for line . Size of the step used in the discretization of the generation capacity at node . Parameter used for the discretization of the generation capacity expansion associated to the number of binary variables. Note that the total number of binary variables is .

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Thermal capacity limit of line already installed before level 1. Maximum capacity allowed for line in level 1. Big-M parameters used in the linearization process.

Variables: Power generated at node with demand profile . Import/export power from/to node with demand profile . Generation capacity available at node after the decisions of level 2 are made. Thermal capacity limit of line after the decisions at level 1 are made. It is a constant for levels 2 and 3. Shadow price/dual variable of the production capacity constraint at node with demand profile . Dual variable associated to the constraint.

balance

Shadow price/dual variable of the thermal capacity constraints of line with demand profile . Locational marginal price (LMP)/dual variable of the energy balance at node with demand profile . Dual variable of the non-negativity constraint of for node with demand profile . Binary variables from the Fortuny-Amat linearization in level 3.

Binary variable that is equal to 1 if the th step of the discretization of is considered, and 0 otherwise. Product of

by

.

Product of

by

.

Superscripts For Variables: Variable in the equilibrium. Variable related to the th generation company choosing the th capacity investment strategy. I. INTRODUCTION A. Literature Review

W

ITH few exceptions, the primary drivers for transmission upgrades and expansions are reliability considerations and interconnection of new generation facilities. However, because the operating and investment decisions by generation companies are market driven, the valuation of transmission expansions must also anticipate the impact of such investments on the market outcomes. Such economic assessments must be

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carefully scrutinized since market outcomes are influenced by a variety of factors including the network topology and uncertainties in the connection-to-the-grid time of generation facilities, among others. Transmission systems are costly infrastructures, implying that their planning must be assertive in technical and economic terms. Accordingly, there are many studies that propose reaching an “optimal” grid planning. They include the use of techniques such as linear programming [1], mixed integer linear programming [2], [3] or Benders decomposition [4]. Other models make use of heuristics, in particular genetic algorithms [5] and simulated annealing [6]. Game theory models have been also applied [7]–[11]. Other models integrate transmission expansion planning within a pool-based market [12] making use of mixed-integer linear programming. In the same vein, [13] formulates a bilevel model where the transmission planner minimizes transmission investment costs in the upper level and the lower level is the market clearing of the pool. The bilevel model is reformulated as a mixed-integer linear problem using duality theory. Additionally, multi-period models have been proposed to characterize investments in the electricity market: [14] proposes a two-stage model of investments in generation capacity where generation investment decisions are made in a first stage while spot market operations occur in the second stage. Accordingly, the first-stage equilibrium problem is solved subject to equilibrium constraints. However, this model does not take into consideration the transmission constraints generally present in network planning problems. Among the aforementioned methods, [9] and [10] are the only ones that assess the economic impact of transmission investment while anticipating the strategic response of oligopolistic generation companies in generation investment and in the subsequent spot market behavior. In both [9] and[10], the authors formulate a model for studying how the exercise of market power by generation firms affects the equilibrium between generation and transmission investments and the valuation of different transmission expansion projects. However, this methodology, based on an iterative process to find the equilibrium, does not solve the optimal transmission planning, but only evaluates the social welfare impact of some predetermined transmission expansion projects. In[15], the authors made a first attempt to formulate the model proposed in [9] and [10] as a mixed integer linear programming (MILP) problem. In this paper, we add to the model in [15] the consideration of uncertainty in demand and the use of profiles and limits in the production of wind and hydro power in order to better represent real power plant conditions. As well, differently than in [15], in this paper we apply the MILP model to a real power system. B. Aims and Contributions In this paper, we formulate a MILP optimization model of transmission planning that extends and transforms the three-level model proposed by Sauma and Oren [9]. Our model integrates transmission planning, generation investment, and market operation decisions and solves the optimal transmission expansion anticipating both the equilibria of generation investments made by firms acting in a decentralized market and the market clearing equilibria. The three-level model is transformed into a compact formulation using the equivalent Karush-Kuhn-Tucker (KKT) conditions for the third level and a discrete approach for the

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Fig. 1. Hierarchical structure of the three-level model.

Nash equilibrium at the second level. We use a Fortuny-Amat reformulation [16] and a binary expansion approach as in [17] in order to convert the nonlinear and non-convex nature of the problem into a MILP, so we can convexify the previous formulations. Accordingly, the contributions of this paper are: 1) Formulation of a MILP model that is able to solve for the optimal transmission expansion anticipating generation investment and market clearing while considering both uncertainty in demand and the use of profiles and limits in the production of wind and hydro power in order to better represent real power plant conditions. 2) Characterization of the equilibria of generation investments (which correspond to the solution of an equilibrium problem subject to equilibrium constraints (EPEC) problem in which the equilibrium constraints come from a perfectly competitive equilibrium) as a set of linear inequalities. 3) Application of the model to a real system: the Sistema Interconectado Central (SIC) in mainland Chile. C. Paper Organization

the GENCOs are able to submit their energy bids. We also assume that the electricity market is perfectly competitive. The model considers transmission network constraints through a lossless DC approximation of Kirchhoff’s laws, assuming perfectly-competitive generators and inelastic demands. Nodal prices (i.e., locational marginal prices, LMP) are given by the Lagrange multipliers of the energy balance constraints at every node. We assume that all nodes can have both demand and generation and that all generation capacity at a node is owned by a single company (although companies can own generation capacity at multiple nodes). We consider a set of different annual demand profiles by selecting a set of equivalent scenarios for each demand profile denoted by . The constant represents the number of hours in a year for each demand profile. Note that the demand profiles can be interpreted as independent scenarios whose probabilities are obtained by normalizing . The demand is assumed inelastic and known for different demand profiles. Marginal generation costs are constant and linearly decreasing with the new installed capacity (1): (1) The ISO problem (2)–(8) consists of the minimization of the total cost subject to generation and network constraints:

(2) subject to: (3) (4) (5)

The remainder of the paper is organized as follows. In Section II we formulate the individual optimization model for each level; the market clearing process, generation investment equilibria and transmission investment planning from the third to first levels, respectively. Section III presents a case study for a stylized version of the SIC in Chile. The main conclusions are summarized in Section IV. The final complete model is shown in the Appendix. II. MODEL The overall hierarchical structure and the optimization problems associated with the proposed model are shown in Fig. 1. The proposed three-level model is simplified to a one-level formulation, using Fortuny-Amat reformulation [16] and binary expansion discretization [17] in order to convexify it. Next, we present a MILP formulation of each of the levels of our model.

(6) (7) (8) The model is split into: 1) the generating units candidates for expansion, , and 2) the units that are not able to expand . The LMPs are obtained from the their capacities, dual variables of the energy balance (7). Other dual variables are represented on the right hand side of the equations. In addition, the problem of a perfectly-competitive GENCO selling in a pool-based market at marginal cost (1), or GENCO problem (9)–(12), assumes a set of given LMPs resulting from the market clearing in order to maximize profit:

A. Third Level: Market Clearing In the third level we obtain the equilibrium that occurs when the ISO clears the market for given generation and transmission capacities. We assume that there is a spot market in which

(9)

POZO et al.: A THREE-LEVEL STATIC MILP MODEL FOR GENERATION AND TRANSMISSION EXPANSION PLANNING

subject to: (10) (11) (12) The optimization models for the ISO and the individual GENCOs are linear and the Karush-Kuhn-Tucker conditions are enough to get the global optimal solution. Note that the perfectly-competitive generators profit maximizing conditions are consistent with the ISO economic dispatch conditions and, thus, (13)–(26) are an exact equivalent formulation of both problems (2)–(12). The Fortuny-Amat linearization [16] is used to transform the slackness conditions into linear constraints (see [17] for more details): (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) Note that (13) and (23) are defined for each node , , (15)–(17), (14) and (24) are defined for the nodes , and (18)–(21) are (22) and (26) are defined for all nodes . All equations are also defined for defined for each line each demand profile . B. Second Level: Generation Investment Equilibria At the second level, each GENCO decides its generation investment in order to increase profits due to the linear decrease in the generation marginal costs, as shown in (1). Note that the return from the investment made at level 2 takes place at level 3. The generation expansion of GENCO is given by the following profit-maximization expression:

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subject to constraints (13)–(26) representing the optimality of the third level. The first term of the objective function consists of the profits obtained selling in the spot market and the second term is the cost of expanding the generation capacity. Taking into account the strong duality condition of a GENCO at the third level [15], we can rewrite the objective function (27) as (28):

(28) We have discretized variable (29) by binary expansion [17] into a linear in order to transform the nonlinear product expression (30) adding constraints (31)–(32). Both a binary expansion [17] and a Fortuny-Amat linearization [16] are used to obtain the MILP model: (29) (30) (31) (32) This formulation represents an MPEC (Mathematical Program with Equilibrium Constraints) model, where the equilibrium constraints are obtained at the third level. From now on, we call MPEC-MILP problem to the objective function in (28), subject to constraints (13)–(26) and (31)–(32) where the term in (28) is replaced by (30). Each GENCO can make a decision to expand its capacity solving the MPEC-MILP model described above. But, since these expansion decisions are dependent on the decisions of the other GENCOs, we use the Nash equilibrium concept to determine the expansion on generation equilibrium for the whole system. In this case, a Nash equilibrium is composed of all GENCOs expansion equilibrium strategies, , where each renders more profit that any other strategy , assuming that . the other GENCOs are fixed in their equilibrium strategies, Equation (33) presents the Nash equilibrium condition, and (34) is an equivalent expression that extends the inequality in (33) for each GENCO and each strategy, , as described in (29). This procedure can be found in detail in [18]. Note that the Nash equilibrium is only for the generation expansion level and assumes perfectly competitive offers into the market (level 3), so this is a restricted Nash equilibrium:

(33) (27) (34)

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The utility functions in both sides of the set of inequalities (34) are computed from (28). They are linearized within the feasible region defined in (13)–(26) and (31)–(32) using (29) or for each expansion decision, On the left-hand side of the set of inequalities we have to replace the decision variables of the MPEC-MILP model for their values in the equilibrium (superscript ). See equations (A6) and (A8)–(A25) in the Appendix. On the right-hand side, the utility function and the feasible (among region are evaluated for each available strategy a discrete set of strategies) and for each GENCO, where the strategies of the competitors are fixed in the equilibrium for all . The rest of the decision variables are depennodes of each GENCO. In other words, the dent on the strategy constraints (13)–(26) and (31)–(32) have to be feasible for any and GENCO , assuming the competitors’ given strategy expansion capacities in the equilibrium are fixed. have to be split Then, the equations given for all , where the expansion capacity into two parts: one for is fixed in the equilibrium, and another one for , where . the expansion capacity is known for this node and GENCO Likewise, the respective summation terms have to be split into similar two terms. See equations (A7) and (A26)–(A45) in the Appendix. The equilibrium problem of the second level is an EPEC problem where all the GENCOs solve simultaneously their MPEC-MILP problems (in which the equilibrium constraints come from a perfectly competitive equilibrium). From now on, we call it the EPEC-MILP problem of the second level for a given for the set of equalities and inequalities. This is represented by equations (A5)–(A45) in the Appendix. Note that the solution of this formulation provides a Nash equilibrium. However, we cannot guarantee that the equilibrium is unique, since there may be more than one or even none. See [19] for details to find all the pure Nash equilibria in a discrete game. C. First Level: Transmission Investment Planning Transmission planning expands the network for a set of candi, and anticipates the second and third levels. The date lines, candidate lines can be either reinforcements of existing lines or new lines. The transmission planner minimizes (35) composed of the operational costs, the capital costs to invest on candidate lines, , and the generation-capacity expansion costs resulting from the equilibrium of the second level:

(38) (39) (40) , is defined as a Note that the decision variable . continuous one whose limits are Naturally, the construction of a new line modifies the PTDF matrix since line impedances change. In this paper, the PTDF matrix is modeled as a function of line capacities, following the approach developed in [15]. In particular, in this paper, we have assumed that: 1) if a line is already built, then it has constant impedance for any capacity expansion; and 2) if a line is a candidate for new construction, then we solve the optimization problem twice (with different PTDF matrices): 1) without the possibility of building the new line, where the PTDF matrix and it is built based on initial impedance values; has size and 2) including the new line, recalculating the line impedances (and, thus, the PTDF matrix). In this last case, the PTDF matrix . has size D. Computational Issues Table I summarizes the computational complexity for the represents the number of EPEC-MILP problem, where binary variables used to discretize the expansion on capacity for generating unit . We assume that the number of discrete . The binary variables is the same for all generation units, symbols are used as the cardinal of the corresponding set that they represent. The second column shows the corresponding size in the MILP model and the third column shows the order of complexity for large systems (where the lines and nodes can be , in the order of hundreds and under the hypothesis: , where and are small). The order of complexity of the problem grows proportionally to the number of nodes, , the number of lines, , the number of time periods, , and the total number of strategies, , considered. However, the total number of strategies grows exponentially with the number of discrete values of generation , the number of candidate generation units to exexpansion, , and the total number of pand belonging to each GENCO, GENCOs, . Fortunately, only a few generation nodes are suitable for expansion. Note that the number of candidate lines for expansion does not affect the size of the problem for large systems because does not appear in the last column of Table I.

(35)

III. CASE STUDY

(36) (37)

We illustrate the proposed model using a stylized representation of the main Chilean power network, i.e., the Sistema Interconectado Central or SIC, as shown in Fig. 2. The SIC is a system composed of both generation plants and transmission lines that operate to meet most of the Chilean electricity demand. The SIC extends over a distance of 1740 km covering a territory of km , equivalent to 43% of the country, where 93% of the population lives. At the end of 2010, the SIC had 12 147 MW of installed power capacity, 54.5%

subject to

EPEC -MILP solution

expansion and the Fortuny-Amat formulation. Thus, we can reusing (38)–(40): place this product by

The above objective function is nonlinear due to the product . The nonlinear term can be decomposed using a binary

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TABLE I COMPUTATIONAL COMPLEXITY

TABLE II LINE EXPANSION DATA

A. Data Main data are presented in Tables II and III.1 Table II provides the data for the candidate lines included in the expansion planning. The third column shows the annualized transmission

investment cost per MW installed. We have not considered economies of scale. The fourth and fifth columns show the current capacity and the maximum line expansion, respectively. Table III provides data for the nodes candidates to expand. The second and third columns show the name of the owner company and the technology installed in the corresponding node. The installed capacity is shown in the fourth column and the maximum expansion in the fifth column. The generation expansion is discretized by binary expansion in four equally-spaced levels between the initial capacity and the maximum expansion. For example, the available expansion values for node 3 are MW. The sixth and seventh columns provide information about the cost parameters. Note that the non-dispatchable (wind) generation has a small marginal cost and there is no possibility of decreasing the marginal cost, i.e., .2 On the other hand, for the dispatchable generation, we have selected the same slope in the marginal cost function based on the historical data of the investments in the SIC system [20]. The last column of Table III shows the annual investment cost per MW installed in each node. To make the model more realistic, we have limited the annual production to 30% of the installed capacity for wind farms and 75% for hydro power plants. Furthermore, based on historical data of wind production [21], we have limited the production of wind farms to a different capacity plant factor in each scenario (or demand profile), as 33.4%, 23.4%, 60.1%, and 23.4% of the installed capacity. Hourly demand forecasts are obtained from [20] for a oneyear period (2010) and simplified into four demand profiles: Summer-off-peak, Summer-peak, Winter-off-peak, and Winterpeak.

1It is worth to mention that Chile does not use LMPs, but a type of “regulated LMPs”. The Chilean National Energy Commission estimates every six months the projected average LMPs for the next 48 months, using a stochastic dual dynamic program, and fixes them until the next revision as “regulated nodal prices” [21].

2The non-dispatchable (wind) generation is modeled similarly to the dispatchable (conventional) generation, but incorporating two main differences: 1) the non-dispatchable generation has a production profile; and 2) investing in nondispatchable generation capacity does not decrease the marginal cost of production (mainly because wind power is considered relatively modular).

Fig. 2. Stylized representation of the Chilean SIC network.

thermal and 44.1% hydroelectric, while the annual gross generation of energy was around 41 062 GWh [20]. Although the decision framework spans a lifetime of 25 years, we have considered a one-year horizon in our model, with annualized investment costs. Nevertheless, our model can be run for investment decisions in a year-by-year fashion. As shown in Fig. 2, the SIC has 34 buses and 38 transmission lines. Four existing lines are candidates for capacity augmentation and two new lines are candidates for construction (represented by the dashed lines in Fig. 2). There are four generation companies (corresponding to the three major generation firms in Chile and the rest are grouped into a fourth firm), each owning generation capacity at multiple locations. The electric characteristics (i.e., resistance, reactance, and thermal capacity rating) of the transmission lines of the network are obtained from [20].

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TABLE III NODE EXPANSION DATA

TABLE IV COMPARISON WITH DIFFERENT CASE STUDIES

B. Results The model has been solved for different case studies: Case 1 is the benchmark case, without considering the expansion on capacity lines and generation; case 2 assumes the new hydro power plant in node 34 (see Table III) is not able to be built and only candidate lines 1 to 4 are possible (see Table II); case 3 considers the line expansion is limited to lines 1 to 4; and case 4 considers all candidates for expansions in Tables II and III are allowed. The comparison of the results for these case studies is summarized in Table IV. The average system LMP shown in the third column of Table IV is calculated as in (41), based on the LMP of the equilibrium from level 2: Average system LMP

TABLE V GENERATION CAPACITY EXPANSION

TABLE VI LINE CAPACITY EXPANSION

(41)

We observe that the total cost (i.e., the sum of the operational and the investment costs) is reduced when the capacity of the lines is increased. This can be explained as a result of a better interconnection of the transmission network, which means less transmission congestion. On the other hand, the average system LMP would likely decrease when the network interconnection improves, but we observe that it does not follow this pattern in case 4 because the GENCOs exercise market power through their investment decisions. Tables V and VI show the results of case 4 in detail. The capacity investment in each node is shown in the second column of Table V. The marginal cost resulting after implementing all capacity investments (third column) is compared with the annual average LMP in each node (fourth column). The annual average LMP is calculated by averaging the LMPs of the four demand patterns of the year at a specific node. Table VI shows the solutions for the line capacity investment and the annualized investment cost per line in the second and

third columns, respectively. Note that the three-level model is formulated with a continuous variable for line capacity investments. Therefore, the optimal values correspond to the minimum investments to comply with the thermal constraints of the candidate lines. Line investment decisions are continuous variables in the model. However, an implementable project would require discrete levels of expansion. For example, an increase of 20 MW in the line connecting nodes 22 and 23 could be simply achieved by changing the material of the conductors. On the other hand, an expansion of 523 MW in the line connecting nodes 11 and 34 is big enough to consider the construction of a line of 600 MW.

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TABLE VII CPU TIMES AND COMPUTATIONAL COMPLEXITY

(A7) Left-Hand Side Constraints

(A8) (A9) (A10) The model has been formulated in the General Algebraic Modeling System (GAMS) [22] and solved with CPLEX 11 solver in a Dell PowerEdge R910x64 computer with 4 processors at 1.87 GHz and 32 GB of RAM. Table VII shows the computation time and computational complexity required for each case study.

(A11) (A12) (A13) (A14)

IV. CONCLUSIONS This paper presents a compact formulation of a transmission investment planning problem that anticipates both generation expansions and market clearing. The model is formulated as a three-level model and rewritten as a compact MILP model, suitable for application in large-scale systems. An illustrative example in the main Chilean power system shows the results of the methodology proposed.

(A15)

APPENDIX COMPACT MILP MODEL We present the complete model of transmission planning formulated as a MILP subject to the EPEC-MILP and market equilibrium constraints:

(A18)

(A16)

(A17)

(A19)

(A20) (A21)

(A22)

(A1) subject to

(A23) (A2) (A3) (A4)

(A24) (A25)

Equilibrium and Profit Definition (A5)

Right-Hand Side Constraints (A26) (A27)

(A6)

(A28)

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(A29) (A30) (A31) (A32) (A33) (A34)

(A35)

(A36) (A37)

(A38)

(A39) (A40) (A41) (A42)

(A43) (A44)

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David Pozo (S’09) received the B.S. degree in electrical engineering from the University of Castilla—La Mancha, Ciudad Real, Spain, in 2006, where he is currently pursuing the Ph.D. degree in electrical engineering. His research interests include power systems economics and electricity markets. Enzo E. Sauma (M’06) received the B.Sc. and M.Sc. degrees in electrical engineering from Pontificia Universidad Católica de Chile (PUC), Santiago, Chile, and the Ph.D. and M.Sc. degrees in industrial engineering and operations research from the University of California, Berkeley. He is an Associate Professor of the Industrial and Systems Engineering Department at PUC. His research focuses on market-based transmission investment in restructured electricity systems. Javier Contreras (SM’05) received the B.S. degree in electrical engineering from the University of Zaragoza, Zaragoza, Spain, the M.Sc. degree from the University of Southern California, Los Angeles, and the Ph.D. degree from the University of California, Berkeley. He is a Professor at the Universidad de Castilla—La Mancha, Ciudad Real, Spain. His research interests include power systems planning and economics and electricity markets.