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14th PSCC, Sevilla, 24-28 June 2002 Session 43, Paper 4, Page 1 EXPANSION PLANNING IN ELECTRICITY MARKETS. TWO DIFFERENT APPROACHES Mariano Ventosa Rafael Denis Carlos Redondo INSTITUTO DE INVESTIGACIÓN TECNOLÓGICA Universidad Pontificia Comillas Alberto Aguilera 23 28015 Madrid, SPAIN [email protected] Abstract – This paper presents two different approaches to model expansion planning in electricity markets under imperfect competitive conditions. Both approaches consider a market in which firms compete in quantity as in the Nash-Cournot game. However, the two models differ from each other in how each firm makes its expansion-planning decisions. In the first model, the key assumption that permits its formulation as a Mixed Linear Complementarity Problem (LCP) is that firms not only decide their output in a Cournot manner but also their new generating capacity. In contrast, in the second model, there is a leader firm that anticipates the reaction of the follower firms as in the Stackelberg game. The latter model is formulated as a Mathematical Program with Equilibrium Constraints (MPEC), wherein the leader firm decides its optimal new capacity subject to a set of market equilibrium constraints. Both models have been implemented in GAMS. A simple example is also presented to illustrate their comparison and application. Keywords: Complementarity Problem, Nash-Cournot, Mathematical Programming with Equilibrium Constraints, Stackelberg. 1 INTRODUCTION In deregulated power markets, electricity generation becomes a liberalized activity in which both expansion planning and operation scheduling no longer depend on administrative and centralized procedures but rather on decentralized decisions of the generation companies whose goals are to maximize their own profit. Given this competitive framework, electricity firms assume much more risk as they become more responsible for their own decisions. Therefore, generation firms need suitable decision-support models that not only consider technical operation constraints but also the interaction of all market participants. The appearance of wholesale electricity markets in many countries has greatly stimulated the efforts of the research community to develop models that consider imperfect competition [1] since in many cases it is only few firms that compete. Scott and Read [2] developed a Cournot-based model with emphasis on hydro operation using dual dynamic programming. An analytical statement of this problem was presented in [3]. More recently, Hobbs utilized the Linear Complementarity Problem (LCP) to model spatial Cournot competition among electricity producers [4] while Wei and Smeers [5] addressed the same challenge using the Variational Inequality (VI) approach. Although there is a large number of papers on the subject of oligopoly pricing and operation of electricity markets, there are few publications regarding investment in a deregulated context. In [6] and [7] two Cournot-based expansion models are described. The first model computes the market equilibrium using an iterative search procedure. The second one calculates the equilibrium with a oneshoot procedure taking advantage of its Variational Inequality (VI) formulation. In addition, another approach can be found in the second article, labeled as close loop Cournot model −equilibrium subject to equilibrium constraints−, which is very difficult to handle numerically. This paper presents two different approaches to model expansion planning1 in electricity markets under imperfect competitive conditions. Both approaches consider a market in which firms compete in quantity as in the Nash-Cournot game. However, the two models differ from each other in how firms make their expansion-planning decisions. In the first approach named in this paper as Cournotbased model, the key assumption that permits its formulation as a Mixed Linear Complementarity Problem (LCP) is that each firm not only decides its output in a Cournot manner but also its new generating capacity. Consequently, in this approach, the NashCournot equilibrium [8] defines a set of prices, quantities and rated capacity of power plants that simultaneously satisfies the first order optimality conditions of all firms. The resulting LCP model can be directly solved taking advantage of its complementarity structure [9], whose particularities allow the use of special resolution methodologies incorporated nowadays in commercial solvers. In the second approach named in this paper as Stackelberg-based model, there is a leader firm that anticipates the reaction of the follower firms as in the Stackelberg game [8]. This model is formulated as a Mathematical Program with Equilibrium Constraints (MPEC), wherein the leader firm decides its capacity expansion subject to a complementarity problem that defines the market equilibrium in prices and quantities. 1 This paper focuses on how firms behave when making investment decisions. For this reason and for the sake of clarity, some important topics of expansion planning have been ignored such as uncertainty, system reliability or the binary nature of investing in a new power plant. 14th PSCC, Sevilla, 24-28 June 2002 It is well known that MPEC models are difficult to solve since there are no powerful methodologies to deal with this kind of problems. Consequently, this model has to be solved by general-purpose non-linear solvers. This paper is organized as follows. Section 2 presents an overview of the modeling assumptions made. Section 3, discusses the mathematical structure of both the models. Section 4 has to do with their implementation. Section 5 describes a case study and Section 6 provides the conclusions drawn from the study. Finally, in order to make the paper easier to read, the detailed mathematical formulation has been placed in appendixes. Appendix A outlines the notation used for the mathematical expressions. Appendix B states the investment optimization program of each firm, and Appendixes C and D show in detail the mathematical formulation of both the proposed models. 2 MODELING ASSUMPTIONS 2.1 Demand and Generation System Representation The expansion model considers a hiperanual scope divided into different time segments: periods, subperiods and load levels (group of hours with a similar demand level). The periods coincide with years, each subperiod groups several months while the grouping of the peak, plateau, and off-peak hours makes up the load levels. In a market environment the modeling of the demand must consider the reaction of the quantity demanded to changes in the price. It is assumed in this paper that the total demand at each load level is a linear function of the price. The model considers the particular characteristics of each type of generation plant: thermal, hydro and pumped plants. Thermal generators are modeled considering their rated power output and their quadratic fuel consumption. Some relevant short-term limitations of thermal units, such as ramp rates and minimum stable output, are neglected in this long-term model. This makes the model more tractable since these limitations have low impact on expansion plans. When modeling hydro plants, their rated power output and reservoirs limited capacity is considered. For each plant, hydro inflows and its initial and final reservoir levels limit the hydraulic energy available. Finally, the pumped-storage plants are modeled considering their rated power output when generating, their rated power consumption when pumping, their maximum energy pumped capacity and their performance in the pumping and generation cycle. 2.2 Short Term Market Assumptions This subsection describes how firms make their operational decisions −output of each firm−. Since market mechanisms in a deregulated context decide the actual operation of the generation units, market equilibrium must be considered in order to Session 43, Paper 4, Page 2 properly model the optimal energy scheduling of each company. As in [6] and [7], in both the models presented in this paper, the well-known Cournot equilibrium has been adopted for representing the short-term market behavior since there typically are only few competitors in electricity markets. In this oligopolistic model, firms compete in quantity, i.e. each firm chooses an output quantity to maximize its profit, and the price is derived from the demand function. The Nash-Cournot market equilibrium defines a set of outputs such that no firm, taking its competitors’ output as given, wishes to change its own output unilaterally. A relevant assumption of this model is that firms’ operation decision-making occur simultaneously. Therefore, modeling this type of market equilibrium requires the simultaneous consideration of each firm’s profit maximization problem. These optimization problems are linked together through the market price resulting from the interaction of all of them. This scheme is shown in Figure 1, where z represents the market profit −market revenues minus operating costs− of each company e∈[1,...,E], x represents the short-term decision variables −output of each unit− and the set of constraints h and g represents the generation limits. The electricity market, which is the link among all optimization problems, is modeled by the demand function that relates the supplied demand −total output of all producers− to the electricity price. Optimization Program of Firm 1 Optimization Program of Firm e Optimization Program of Firm E maximize : z 1 ( x1 ) max imize : z e ( x e ) max imize : z E ( x E ) subject to : h1j = 0 g k1 ≤ 0 subject to : hej = 0 g ke ≤ 0 subject to : h Ej = 0 g kE ≤ 0 Price-m(x)=0 Electricity Market Figure 1 Market equilibrium. 2.3 Two Different Investment Assumptions In this paper, two different hypothesis regarding how firms make investment decisions −new capacity installed− in an imperfect competitive context are considered: Cournot game and Stackelberg game. Cournot-based model: the first model assumes that firms make their capacity-expansion decisions as in the Cournot model (assumption also employed in the shortterm market model described in 2.2). Thereby, each firm chooses its new maximum capacity so that its own profit is maximized. Formally, the Cournot investment market equilibrium defines a set of capacities such that no firm, taking its competitors’ capacities as given, wishes to change its own capacity unilaterally. Unlike the Stackelberg game, no firm has first mover advantage since Cournot assumption implies that firms’ investment decision-making occur simultaneously. 14th PSCC, Sevilla, 24-28 June 2002 Session 43, Paper 4, Page 3 Stackelberg-based model: the second model assumes that firms compete in capacity as in the Stackelberg model. There is a leader firm that first decides its optimal capacity. The follower firms then make their optimal decisions knowing the capacity of the leader firm. This model can be seen as a two-stage game. In the first stage, the leader firm decides its optimal capacity anticipating the future reaction of the follower firms. In the second stage, the leader firm and the follower firms compete in the short-term market in a Cournot manner (as it is described in 2.2). This model is similar to the close loop Cournot model presented in [7]. They are both different types of two-stage model. The Stackelberg-based model seems to be more realistic than Cournot-based one, since it reflects sequential decision-making −first investment and then operation−. Nevertheless, from a theoretical point of view, the main difference between both assumptions is that in the Stackelberg game the leader firm has first mover advantage. That is, the leader firm is allowed to make a new-capacity decision and the followers have to adapt to the new situation. This theoretical gain is illustrated in the case study (Section 5), wherein the leader company (firm C) builds more capacity in the Stackelberg-based model and obtains a slightly higher profit than in the Cournot-based model. these equations make the equilibrium solution technically feasible. The third one is formed by the complementary slackness conditions associated to the inequality constraints g. In fact, the Cournot-based model has the structure of an MCP problem because the complementary slackness conditions have the structure of a Complementarity Problem. This set of equations consists of the inequality constraints multiplied by their corresponding dual variables µ equal to zero; next, the inequality constraints themselves; and finally, the explicit statement of dual variables µ as negative ones. Grouping together all firms’ optimality conditions −gradient of the Lagrangian function with respect to the decision variables x, equality constraints h and the complementary slackness conditions associated to g− leads to a Mixed Complementarity Problem (MCP) formulation [9]. 3 MODELS STATEMENT In this section, the mathematical structure of both the models considered in this paper is stated. Electricity Market 3.1 Cournot-based Model When firms compete in capacity and quantity in a Cournot manner, the investment and operation market equilibrium problem (Figure 1) can be stated in terms of an MCP scheme. This MCP structure is obtained by means of setting the Karush-Kuhn-Tucker’s first order optimality conditions associated to the maximization programs −investment and operation− of each firm (see Figure 2). Therefore, the solution of this non-linear system of equations provides the prices, outputs and new capacity installed that solve the investment and operation Cournot market equilibrium. In Figure 2, for each firm e, L represents the Lagrangian function of the corresponding optimization problem and λ and µ represent the dual variables associated to the set of h and g constraints respectively. The optimality conditions can be grouped together into three sets of equations. The first one cancels the gradient of the Lagrangian function with respect to the decision −investment and operation− variables x. These equations explain how firms must use their resources, i.e. make their decisions, to achieve their maximum profit. The second set, the gradient of the Lagrangian function with respect to the dual variables λ, coincides with the equality constraints themselves h. Therefore, Optimality Conditions of Firm 1 Optimality Conditions of Firm e Optimality Conditions of Firm E ∂L1 =0 ∂x1 ∂L1 1 ∇λ L ( x,λ, µ ) = 1 = h1j = 0 ∂λ j µk1 ⋅ gk1 = 0 gk1 ≤ 0 µk1 ≤ 0 ∂Le =0 ∂xe e ∂L ∇λ Le ( x,λ, µ) = e = hej = 0 ∂λ j µke ⋅ gke = 0 gke ≤ 0 µke ≤ 0 ∂LE =0 ∂x E ∂LE E ∇λ L (x,λ, µ) = E = h Ej = 0 ∂λ j µkE ⋅ gkE = 0 gkE ≤ 0 µkE ≤ 0 ∇xL1 ( x,λ, µ ) = ∇x Le (x,λ, µ ) = ∇x LE ( x,λ, µ) = Price-m(x)=0 Figure 2 Cournot-based model as an MCP. As a result of the modeling assumptions described in Section 2, the Cournot-based model is a Linear Complementarity Problem (LCP), which assures solution existence and uniqueness in case that thermal units’ marginal costs are strictly monotone increasing [5]. Formally, an LCP is an MCP composed only by linear equations, excluding the complementarity slackness conditions [10]. Formulating the Cournot-based model as an LCP problem makes it easy to handle numerically as there exist specific algorithms devoted to Complementarity Problems. 3.2 Stackelberg-based Model In this two-stage game, the leader firm chooses its optimal capacity in a first stage, as in the Stackelberg game, and then in a second stage all the firms compete in quantity as in the Cournot game. Formally, this twostage game can be stated as a bilevel optimization program, i.e. an optimization problem subject to a group of equilibrium constraints. Therefore, the Stackelbergbased model turns out to have a Mathematical Program with Equilibrium Constraints (MPEC) structure [11] (see Figure 3). The upper-level is the investment optimization problem of the leader firm, whose objective function is to maximize its profit z leader . The investment decision variables of the leader firm α are decided in this upper- 14th PSCC, Sevilla, 24-28 June 2002 level taking into account not only the constraints i(α) but also the equilibrium constraints. Notice that i(α) only regard investment decisions and operational variables x −the outputs and prices− are settled by the equilibrium constraints. z 1eader ( x,α ) i (α ) ≤ 0 ← Investment Constraints ∂Le e ∀e ∇ x L ( x,λ , µ ) = e = 0 ∂x ∂Le Equilibrium → ∇λ Le ( x,λ , µ ) = e = hej = 0 ∀e Constraints ∂λ j µke ⋅ gke = 0 gke ≤ 0 µke ≤ 0 ∀e maximize : subject to : Figure 3 Stackelberg-based model as an MPEC. The lower level is stated by the first order optimality conditions of each firm and represents the Cournot short-term market equilibrium. As in the previous model, these constraints have the structure of an MCP due to the so-called complementary slackness condition. However, unlike the Cournot-based model, now x only represents the operation decision variables and h and g only limit such operation variables. In general, MPEC models are non-convex and difficult to solve. As a result of this non-convexity, it is very complicated to derive the conditions that assure solution existence or uniqueness. In addition, there are no powerful solvers to deal with this type of models. 4 IMPLEMENTATION Both models presented in this article were implemented in GAMS version 2.50. In the case of the Cournot-based model, two specific commercial solvers are available for such complementarity problems, MILES and PATH [12]. Both solvers are based on a generalization of the classic Newton method, in which each subproblem is solved as a linear complementarity problem using an extension of the Lemke´s algorithm. When solving our model, PATH showed a better performance as a result of its algorithmic enhancements. In the case of the Stackelberg-based model, the optimization problem has an MPEC structure. Unlike MCP, there is no specific commercial solver for dealing with MPEC. Consequently, these models have to be solved by means of general-purpose Non-Linear Programming (NLP) solvers such as MINOS or CONOPT2. Solving our model, CONOPT2 is more efficient2 than MINOS due to its non-linear equilibrium constraints. Even though the solution of the MCP model 2 Usually, CONOPT2 is better suited for models with very nonlinear constraints than MINOS. On the other hand, if the model has few nonlinearities outside the objective function then MINOS would probably be more convenient. Session 43, Paper 4, Page 4 is used as starting point of the MPEC model in order to accelerate and ensure the convergence of CONOPT2, the MPEC solving time is much longer than the MCP one. 5 CASE STUDY Complementing the theoretical comparison established in previous sections, this case study allows a practical and computational comparison between both the models. Focusing on the results, there are considerable similarities between both approaches; however, from a computational point of view, the MCP approach is significantly easier to handle numerically. 5.1 Test System Description The scope of this sample case has been split into 11 periods (years) with 4 subperiods each (3 months) and 3 load levels (peak, plateau, and off-peak hours) with the same duration. Table 1 describes the initial situation in which the three firms considered are. In this case study, firm C is the only one allowed to build new capacity. Type Nuclear Thermal Hydro Total Units Power (MW) 4 4000 8 3500 3 2600 15 10100 Firm A 25% 24.3% 77% 38.1% Firm B 50% 50% 11.5% 40.1% Firm C 25% 25.7% 11.5% 21.8% Table 1 Generation capacities by firm The slope of the demand curve is 4.82 €/MWh/GW. The system met a maximum peak load of 8322 MW and a yearly energy demand of 55488 GWh in the first year of the study. It is assumed that demand increases at a 4% rate and so the maximum peak of demand in the entire scope of the study is 10087 MW. The hydro energy available every year is 1446 GWh. The only technology considered convenient when building new plants is CCGT. The cost of building new capacity is 482 €/kW. The discount rate utilized for the calculation of the present value is 5%. Depreciation time is ten years since building in the first period is not allowed. 5.2 Expansion Results Some of the results obtained with both models are shown in Table 2. These results are consistent with theoretical predictions, since firm C builds more capacity and makes a higher profit when behaving as the leader firm of the Stackelberg model (MPEC). Variable MCP MPEC New capacity in the first year (GW), firm C 1.208 ↓ 1.375 ↑ Average prices in the first year (€/MWh) 33.55 ↑ 33.37 ↓ Profit Firm A (k€) 5065.8 ↑ 5040.7 ↓ Profit Firm B (k€) 8683.1 ↑ 8637.2 ↓ Profit Firm C (k€) 6028.8 ↓ 6032.2 ↑ Market Share Firm A (%) 24.7 ↑ 24.5 ↓ Market Share Firm B (%) 41.6 ↑ 41.2 ↓ Market Share Firm C (%) 33.7 ↓ 34.3 ↑ Table 2 Main results of the case study. 14th PSCC, Sevilla, 24-28 June 2002 It is seen that firm C builds more capacity in the case of the Stackelberg model. Since there is more capacity in the system, average prices diminish slightly. However, even though firm C has to deal with the cost of higher investments and the slight reduction in prices, its increase in market share provides it bigger profits, as predicted through theory. Its competitors’ decrease in share and prices leads them to lower profits. Still, differences between models are minor ones. Regarding the operation, the new plants run fully in peaks. The rest of the time becomes the technology that sets the marginal price. 5.3 Computational Issues The size of the MCP −Cournot-based model− in this case study is 3023 equations and 3057 continuous variables. The size of the MPEC −Stackelberg-based model− is 7634 equations and 7404 variables, and GAMS identifies up to 10659 non-linear entries. A 1.5 GHz PC using the PATH solver spends 0.9 seconds in the first case and 186 seconds for the MPEC when using CONOPT2. Therefore, in this case study MCP is much more tractable than MPEC from a computational viewpoint. 6 CONCLUSION This paper presents two different approaches for addressing the expansion-planning problem in deregulated electricity markets. Both approaches consider the Cournot model for the short-term competition although they differ in how firms decide their optimal capacity. The first approach, where firms compete in capacity in a Cournot manner, has an MCP structure. The second one, where a leader firm decides its capacity as in the Stackelberg model, has an MPEC structure. Both models have been developed and implemented in GAMS. The application to a sample case clearly shows that there are minor differences in most market figures. However, the study also reveals that, from a computational point of view, MCP solvers are far more powerful than the methods available for computing MPEC models. Therefore, the MCP approach should be the considered when conducting real systems studies. Finally, in order to improve both expansion models, further work must be done so as to take account of uncertainties such as demand growth, regulatory changes and so forth. 7 ACKNOWLEDGEMENTS This work has been developed under a research project funded by Iberdrola, S.A. and with the support of a grant from the Comisión Interministerial de Ciencia y Tecnología. Session 43, Paper 4, Page 5 REFERENCES [1] E. Kahn, “Numerical techniques for analyzing market power in electricity”. The Electricity Journal. pp. 34-43. July 1998. [2] T. J. Scott, and E.G. Read, “Modelling Hydro Reservoir Operation in a Deregulated Electricity Market”. International Transactions in Operational Research, Vol. 3 pp 243-253, 1996. [3] J. Bushnell, “Water and Power: Hydroelectric Resources in the Era of Competition in the Western US”, POWER Conference on Electricity Restructuring, University of California, Energy Institute. 1998. [4] B. F. Hobbs. “LCP Models of Nash-Cournot Competition in Bilateral and POOLCO-Based Power Markets”. Proceedings, IEEE Winter Power Meeting, NY City, Feb. 1999. [5] W. Jing-Yuan, Y. Smeers. “Spatial Oligopolistic Electricity Models with Cournot Generators and Regulated Transmission Prices”. Operations Research, Vol. 47, No. 1, January-February, pp. 102-11, 1999. [6] Chuang, A.S., Wu, F., Varaiya, P., “A Game-Theoretic Model for Generation Expansion Planning: Problem Formulation and Numerical Comparisons”. Transactions on Power Systems, Vol. 16, NO 4, pp. 885-891, Nov. 2001. [7] Murphy, F., Smeers, Y., “Capacity expansion in imperfectly competitive restructured electricity market”. Submitted to Econometrica, 2001. [8] X. Vives, “Oligopoly Pricing: old ideas and new tools”. MIT Press. 1999. [9] M. Rivier, M. Ventosa, A. Ramos, “A Generation Operation Planning Model in Deregulated Electricity Markets based on the Complementarity Problem”, M. Ferris, O. Mangasarian, J. Pang, Chapter 13 of Applications and algorithms of complementarity. Kluwer Academic Publishers, 2001. [10] R.W. Cottle, J.S. Pang, and R.E. Stone. “The Linear Complementarity Problem”. Academic Press, Boston, 1992. [11] Luo, Z., Pang, J. and Ralph, D. “Mathematical Programs with Equilibrium Constraints”. Cambridge University Press, 1996. [12] S.C. Billups, S. P. Dirkse, and M. C. Ferris. “A comparison of large scale mixed complementarity problem solvers”. Computational Optimization and Applications, 7:3-25, 1997. APPENDIX A: NOTATION In this appendix all symbols used in this paper are identified and classified, according to their use, into indices, parameters, variables and dual variables. Index Description Index Description b Pumped-storage plants p Periods e Firms s Subperiods e* Leader firm h Hydro units n Load levels Table 3 Indices. t Thermal plants t* New thermal plants 14th PSCC, Sevilla, 24-28 June 2002 Session 43, Paper 4, Page 6 Parameter Description Ah , p , s bb , bb Variable Associated constraint Hydro inflows for hydro plant h in subperiod s of period p [TWh] µ bR, p , s Scheduling of pumped-storage plant b in subperiod Maximum and minimum capacity of pumped-storage plant b when pumping [GW] µ bR, p , s Maximum hydro energy reserve of pumped-storage s of period p [k€/TWh] plant b in subperiod s of period p [k€/TWh] µ tTA*, p Increasing maximum rated capacity [k€/GW] cit * Capacity investment cost [k€/GW] d ' p, s,n d p0 , s , n Dp, s,n Constant slope of the demand function in load level n in subperiod s of period p [(k€/TWh)/GW] Duration of load level n in subperiod s of period p [kh] hh , hh hb , hb Maximum and minimum capacity of pumped-storage plant b [GW] Heat rate (linear [kTcal/TWh] and quadratic [kTcal/(GW2·kh)] terms) of thermal plant t o 't * , o ''t * Heat rate (linear [kTcal/TWh] and quadratic [kTcal/(GW2·kh)] terms) of thermal plant t* pt , pt Maximum and minimum rated capacity of thermal plant t [GW] p t * , p t * Maximum and minimum rated capacity of thermal plant t* [GW] Rb Rh , Rh tp vt , vt * Maximum hydro energy reserve of pumped-storage plant b [TWh] Maximum and minimum hydro energy reserve of hydro plant h [TWh] Discount rate [p.u.] Fuel cost of thermal plant t and t* [k€/kTcal] ρb Performance of pumped-storage plant b [p.u.] Table 4 Parameters. Variable Description bb , p , s , n Power consumption by pumped-storage plant b in load level n in subperiod s of period p [GW] hb , p , s , n Power generation by pumped-storage plant b in l load level n in subperiod s of period p [GW] hh , p , s , n Power generation by pumped-hydro plant h in load level n in subperiod s of period p [GW] I t *, p New capacity installed in central t* in period p [GW] pt , p , s , n Power generation by thermal plant t in load level n in subperiod s of period p [GW] pt *, p , s , n Rh , p , s Power generation by thermal plant t*in load level n in subperiod s of period p [GW] Hydro energy reserve of hydro plant h at the beginning of subperiod s of period p [TWh] Table 5 Decision variables. Variable Description g e, p , s , n Total power generation sold in the market of firm e in load level n, subperiod s, of period p [GW] π p, s,n System marginal price in load level n, subperiod s, of period p [k€/TWh] Table 6 Auxiliary variables. Variable Associated constraint µ µ hR, p , s ; µ hR, p , s Maximum and minimum hydro energy reserve of Electricity demand at price zero in load level n in subperiod s of period p [(k€/TWh)/GW] Maximum and minimum capacity of pumped-hydro plant h [GW] o 't , o ''t µ tI*, p , s , n Maximum generation of new plants [k€/GW] R h, p, s Hydro scheduling of hydro plant h in subperiod s of period p [k€/TWh] hydro plant h in subperiod s of period p [k€/TWh] µ p t , p, s,n ;µ p t , p, s,n Maximum and minimum rated capacity of thermal plant t in subperiod s of period p [k€/GW] p µ t *, p , s , n Minimum rated capacity of new thermal plant t* in subperiod s of period p [k€/GW] µ I t *, p Minimum installed capacity of new thermal plant t* in period p [k€/GW] µ hh, p , s , n ; µ hh, p , s , n Maximum and minimum capacity of hydro plant h in subperiod s of period p [k€/GW] µ h b, p, s,n ;µ h b, p, s, n Maximum and minimum capacity of pumpedstorage plant b in subperiod s of period p [k€/GW] Table 7 Dual variables. APPENDIX B: INVESTMENT OPTIMIZATION PROBLEM This appendix states in detail the optimization problem that defines the generation companies’ behavior (Figure 1). The goal of each firm is to maximize its own profit subject to the set of constraints that limits its long-term operation decisions. B.1 Objective Function The objective of each generation company is to maximize its profit −market revenues minus operating costs and investment costs− for the entire scope. Maximize : π p , s , n ⋅ g e, p , s , n 2 (1) D t v o p o '' p ⋅ − ⋅ ⋅ + ⋅ ( ) ∑∑∑ p, s,n p t t t , p, s,n t , p,n t , p,s,n ∑ p s n t ∈e 2 − ∑ t p ⋅ vt * ( ocn ⋅ pt *, p , s , n + o ''t *, p , n ⋅ pt *, p , s , n ) t *∈e − ∑∑ cit * ⋅ t p ⋅ ( I p − I p −1 ) ∀e t *∈e p >1 B.2 Investment Constraints The first set of built-capacity constraints (2) states that total capacity in any period is to be greater or equal than capacity in the previous. The second set of constraints (3) limits the power generated by the new plants in a given period. It is to be less than their actual rated capacity. This constraint is analogous to the upper bound in generation output, which affects conventional thermal plants. However, in the case of new plants this output cap is a decision variable. I p −1 − I p ≤ 0 ⊥ µtTA *, p pt *, p , s , n − I p ≤ 0 ⊥ µtI*, p , s , n ∀p, t* ∈ e,e (2) ∀p, s, n, t* ∈ e, e (3) 14th PSCC, Sevilla, 24-28 June 2002 Session 43, Paper 4, Page 7 B.3 Operating Constraints The operating constraints are those that regard resources planning across multiple load levels such a hydro energy scheduling, operation of pumped-hydro units as well as those that represent upper and lower limits of each generation unit. Hydro scheduling. The energy available during each subperiod is limited by the hydro inflows and the initial and final reservoir levels of the subperiod. The initial level of the first subperiod and the final level of the last subperiod are fixed data for the optimization problem. ∑D p, s,n ⋅ hh , p , s , n − Rh , p , s + Rh , p , s +1 − Ah , p , s ≤ 0 ⊥ µ hR, p , s n ∀p, s, h ∈ e, e (4) Pumped-storage constraints. The first constraint establishes a balance between pumped and generated energy (5) while the second limits the maximum pumped energy within each subperiod (6). ∑D n p, s,n ⋅ ( hb , p , s , n − ρb ⋅ bb , p , s , n ) ≤ 0 ⊥ µ R b, p, s C.1 Optimality Conditions Building the Lagrangian function of each firm’s optimization problem3 Le and canceling its gradient with respect to the decision variables leads to the following set of equations: ∂Le = D p , s , n ⋅ t p ⋅ vt ⋅ ( o 't + 2 ⋅ pt , p , s , n ⋅ ot '') ∂pt , p , s , n + D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n ) ( ∀p, s, b ∈ e, e (6) ∂Le = D p , s , n ⋅ t p ⋅ vt * ⋅ ( o 't * + 2 ⋅ pt *, p , s , n ⋅ ot * '') ∂pt *, p , s , n + D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n ) n Variable bounds. The vast majority of the variables involved in the previous formulation are subject to the following bounds: R h ≤ Rh , p , s ≤ R h ⊥ µ hR, p , s ; µ hR, p , s p ( − µ I t *, p , s , n −µ p t *, p , s , n )=0 ∂Le = D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n ) ∂hh , p , s , n (7) p t ≤ pt , p , s , n ≤ p t ⊥ µ t , p , s , n ; µ tp, p , s , n ∀ p, s, n, t ∈ e, e (8) h h ≤ hh , p , s , n ≤ h h ⊥ µ hh, p , s , n ; µ hh, p , s , n ∀ p , s , n , h ∈ e, e (9) bb ≤ bb , p , s , n ≤ bb ⊥ µ bb, p , s , n ; µ bb, p , s , n ∀ p , s , n, b ∈ e , e (10) hb ≤ hb , p , s , n ≤ hb ⊥µ h b, p, s, n ∀ p , s , n, b ∈ e , e (11) ∂Le = D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n ) ∂hb , p , s , n p t * ≤ pt *, p , s , n ⊥µ p t *, p , s , n ∀ p, s, n, t* ∈ e, e (12) − Dp, s, n ⋅ µ I t * ≤ I t *, p ⊥ µ tI*, p ∀ p, t* ∈ e, e (13) ;µ h b, p , s , n B.4 Auxiliary constraints For the sake of the clarity, the following equations have been stated. However, in the implementation, the total power generation and the price have been substituted by these expressions: Total power generation of each firm. The total power generation of each firm represents the effective output that is sold on the market. g e, p , s , n = ∑ pc , p , s , n + ∑ pt *, p , s , n + ∑ hh , p , s , n + ∑ ( hb , p , s , n − bb , p , s , n ) c∈e t *∈e h∈e b∈e ∀e, p, s, n (14) Price equation. The price is represented as a linear function of the total purchased power in the market. π p , s , n = t p ⋅ d ' p , s , n ⋅ d p0 , s , n − ∑ ge , p , s , n ∀ p, s, n e (15) APPENDIX C: COURNOT-BASED MODEL In this appendix, the first order KKT optimality conditions of each firm’s investment problem ( ) − D p , s , n ⋅ µ hR, p , s − µ hh, p , s , n − µ hh, p , s , n = 0 ( R b, p, s +µ R b, p, s ) − (µ h b, p, s, n −µ h b, p,s ,n )=0 ∂Le = D p , s , n ⋅ (π p , s , n − t p ⋅ d ' p , s , n ⋅ g e , p , s , n ) ∂bb , p , s , n ( ) + D p , s , n ⋅ ρb ⋅ µ bR, p , s − µbb, p , s , n − µ bb, p , s , n = 0 ( ) ∂Le = µ hR, p , s − µ hR, p , s −1 − µ hR, p , s − µ hR, p , s = 0 ∂Rh, p , s ∂Le = ( t p − t p +1 ) ⋅ cit * + ∑ µ tI*, p , s , n ∂I t *, p s,n TA I + µ tTA *, p − µ t *, p +1 + µ t *, p = 0 ∀p, n, s, t* ∈ e, e (17) ∀ p, s, h ∈ e, e p ∀p, n, s, t ∈ e, e (16) ) − µ tp, p , s , n − µ t , p , s , n = 0 (5) ∀p, s, b ∈ e, e ∑ Dp, s, n ⋅ hb, p, s , n ≤ Rb ⊥ µbR, p, s formulated in Appendix B is derived. These equations make up the Cournot-based model which has the structure of a Mixed Complementarity Problem (Figure 2). Since there are no equality constraints, the MCP consists of two sets of equations. The equations that set the gradient with respect to the decision variables equal to zero are stated in C.1 and the equations that define slackness conditions are listed in C.2. ∀p, n, s, h ∈ e, e (18) ∀p, n, s, b ∈ e, e (19) ∀p, n, s, b ∈ e, e (20) ∀p, s > 1, h ∈ e, e (21) ∀p, t* ∈ e, e (22) C.2 Complementary Slackness Conditions As previously established in Section 2, in order to complete the set of non-linear equations that define the optimization problem of each company, the following three sets of equations must be added to (16÷22). First 3 In this paper has been considered the equivalent minimization problem of each firm. 14th PSCC, Sevilla, 24-28 June 2002 Session 43, Paper 4, Page 8 of all, the inequality constraints (2÷13) multiplied by their corresponding dual variables µ; next, explicit statement of dual variables µ as negative ones; and finally, the inequality constraints themselves. C.3 Meaning of the Optimality Conditions The optimality conditions provide useful information about the role of each type of power plant not only in firms’ optimal energy scheduling policy [9] but also in firms’ optimal investment policy. Focusing on firms’ optimal investment policy, equation (17) says that when an enlarged-capacity plant is generating at its maximum capacity, the dual variable µ tI*, p , s , n is equal to firm’s marginal revenue minus plant’s marginal cost. This is to say, the profit marginal increase of having an additional MW available. This dual variable could be named as capacity-availability marginal profit. − µ tI*, p , s , n = D p , s , n ⋅ (π p , s , n − t p ⋅ d ' p , s , n ⋅ g e, p , s , n ) − D p , s , n ⋅ t p ⋅ vt * ⋅ ( o 't * + 2 ⋅ pt *, p , s , n ⋅ ot * '') (23) In addition, it can be seen in equation (22) that building new capacity is strongly subject to significant time-coupling since its depreciation will certainly involve the following periods. Therefore, it is easier to comprehend the meaning of those optimality equations avoiding this coupling. Adding together equation (22) of a certain period p’ with the ones of the following periods: ∂Le = 0 → t p ' ⋅ cit * = − ∑ µ tI*, p , s , n p ≥ p ', s , n ∂I t *, p p ≥ p ', s , n ∑ (24) Equation (24) reflects that firm’s maximum-profit is achieved when investment cost is equal to the sum of the capacity-availability marginal profit from p’ to last p. That is to say, each firm should build new capacity up to the point where capacity investment cost −left hand side− reaches the cumulative future profit of such investment −right hand side−. APPENDIX D: STACKELBERG-BASED MODEL This appendix states the mathematical formulation that defines the Stackelberg-based model described in 3.2. In this two-stage model, the equivalent optimization program has the structure of an MPEC problem (Figure 3): profit maximization of the leader firm subject to investment constraints and to the market equilibrium constraints. D.1 First stage. Investment Problem of the Leader Firm In the first stage, the leader firm e* decides its capacity considering that prices and productions are provided by the market equilibrium, which is the second stage. Objective function of leader firm. The leader firm seeks to maximize its profit −market revenues minus operating costs and investment costs− for the entire scope of the model. Maximize : π p , s , n ⋅ g e, p , s , n (25) D p , s , n ⋅ − ∑ t p ⋅ vt ( ot ⋅ pt , p , s , n + o ''t , p , n ⋅ pt2, p , s , n ) ∑∑∑ t∈e* p s n 2 '' t v o p o p − ⋅ ⋅ + ⋅ ∑ p t * ( cn t *, p , s , n t *, p , n t *, p , s , n ) t *∈e* − ∑ ∑ cit * ⋅ t p ⋅ ( I p − I p −1 ) t *∈e* p >1 Investment constraints of the leader firm. In the Stackelberg model, the investment constraints are those that only regard the investment variables of the leader firm. In particular, non-decreasing capacity of new plants (26) and lower bound of new capacity (27) are considered. I p −1 − I p ≤ 0 ⊥ µtTA *, p ∀p, t* ∈ e * (26) I t * ≤ I t *, p ⊥ µ tI*, p ∀p, t* ∈ e * (27) Note that now, equation (3) is not an investment constraint of the leader firm because it regards the power generation of a new thermal plant, and power output is to be considered as a second-stage variable in this model. D.2 Second stage. Market Equilibrium Constraints As explained in Section 3, the short-term market equilibrium −within the Stackelberg-based model− can be stated as a MCP similar to the Cournot-based model. However, in this equilibrium, which is the second decision stage, the investment decision variables of the leader firm are assumed fixed. Like the Cournot-based model, there are no equality constraints. Therefore, this MCP consists of two sets of equations: optimality conditions and complementarity slackness conditions. Optimality conditions. This set of equations is obtained by canceling the gradient of the Lagrangian function of each firm’s optimization problem with respect to the operational decision variables. Therefore, these equations are a subset of the Cournot-based model optimality conditions (see C.1): (16), (17), (18), (19), (20) and (21). Needless to say that the gradient with respect to the new capacity (22) is not to be considered since the investment decisions of the leader firm are variables of the first stage; therefore, they are assumed fixed in these market equilibrium constraints. Complementary slackness conditions. This group of equations consists of three sets of equations: the inequality constraints regarding operational decisions (3÷12) multiplied by their corresponding dual variables µ; next, explicit statement of dual variables µ as negative ones; and finally, the inequality constraints themselves. Obviously, the investment constraints of the leader firm (26) and (27) are not to be considered since they are already included in the first stage of the Stackelberg-based model.