Energy Economics 36 (2013) 135–146

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If you build it, he will come: Anticipative power transmission planning☆ David Pozo a, Javier Contreras a, Enzo Sauma b,⁎ a b

University of Castilla—La Mancha, E.T.S. de Ingenieros Industriales; 13071 Ciudad Real, Spain Pontificia Universidad Católica de Chile, Industrial and Systems Engineering Department; Avenue Vicuña Mackenna # 4860, Raúl Deves Hall, 3rd floor, Macul, Santiago, Chile

a r t i c l e

i n f o

Article history: Received 6 September 2010 Received in revised form 11 December 2012 Accepted 16 December 2012 Available online 22 December 2012 JEL classification: C6 D41 L11 L52 L94 Keywords: Mathematical Program subject to Equilibrium Constraints (MPEC) Equilibrium Problem subject to Equilibrium Constraints (EPEC) Nash equilibrium Power transmission planning Power systems economics Anticipative network planning

a b s t r a c t Like in the film Field of Dreams, the sentence “if you build it, he will come” also applies in power systems. In this sense, if a transmission planner suggests building some lines in anticipation of generation capacity investments, then it can induce generation companies to invest in a more socially efficient manner. In this paper, we solve for the optimal way of doing this anticipative power transmission planning. Inspired in the proactive transmission planning model proposed by Sauma and Oren (2006) we formulate a mixed integer linear programming optimization model that integrates transmission planning, generation investment, and market operation decisions and propose a methodology to solve for the optimal transmission expansion. Contrary to the proactive methodology proposed by Sauma and Oren (2006), our model solves the optimal transmission expansion problem anticipating both generation investment and market clearing. We use the marginalist theory with production cost functions inversely related to the installed capacity in a perfectly competitive electricity market and we find all possible generation expansion pure Nash equilibria. We illustrate our results using 3-node and 4-node examples. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Like in the film Field of Dreams, directed by Phil Alden Robinson, in power systems we can also say that “if you build it, he will come”. In the film, an Iowa corn farmer hears voices commanding to build a baseball diamond in his fields, and the Chicago Black Sox come after building it. In power systems, it has been shown that if a transmission planner suggests building some lines in anticipation of generation capacity investments, then it can induce generation companies to invest in a more socially efficient manner (Sauma and Oren, 2006)— from now onwards SO2006. In this paper, we solve for the optimal way of doing this anticipative power transmission planning.

☆ The work reported in this paper was partially supported by the University of Castilla— La Mancha through the Program of Stays of Latin American Professors, by the Pontificia Universidad Católica de Chile through the Short-Visits Program, by Fulbright through the NEXUS Program, by the CONICYT, FONDECYT/Regular 1100434 grant, and by the Junta de Comunidades de Castilla – La Mancha Formación del Personal Investigador (FPI) grant 402/09. ⁎ Corresponding author. Tel.: +56 2 354 4272; fax: +56 2 552 1608. E-mail addresses: [email protected] (D. Pozo), [email protected] (J. Contreras), [email protected] (E. Sauma). 0140-9883/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.eneco.2012.12.007

Several techniques have been applied to investigate power systems transmission planning. They include mathematical optimization methods such as linear programming (Garver, 1970; Villasana et al., 1985), mixed integer linear programming (Alguacil et al., 2003; Romero and Monticelli, 1994), Benders decomposition (Binato et al., 2001), and dynamic programming (Dusonchet and El-Abiad, 1973). Other models rely on heuristics, in particular intelligent systems applying genetic algorithms (Gallego et al., 1998), simulated annealing (Romero et al., 1996), and agent-based systems (Motamedi et al., 2010). Game theory models have been also successfully applied (Contreras and Wu, 1999, 2000; Sauma and Oren, 2006, 2007). Models that integrate transmission expansion planning within a pool-based market are analyzed in de la Torre et al. (2008) and Garcés et al. (2009) using mixed-integer linear programming and bilevel programming approaches, respectively. One of these works, by Sauma and Oren (SO2006), introduces a methodology for assessing 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 SO2006, the authors formulate a three-period model for studying how the exercise of local market power by generation firms affects the equilibrium between generation and transmission investments and the valuation of different

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Nomenclature The mathematical symbols used throughout this paper are classified below as. Indexes i Node index G Index of generation companies l Index of lines sG Index associated to the discrete investment strategies of generation company G n Index of all pure Nash equilibria found in stage 2 τ Index of states or configurations of the system (according to the discretization of the equivalent impedance of transmission lines) Sets N L Linv Ψ NG SG Τ

Set of all nodes Set of all transmission lines Set of all candidates transmission lines for investment Set of all generation companies Set of generation nodes owned by generation company G Set of all discrete investment strategies of the generation company G Set of all states or configurations of the system (according to the discretization of the equivalent impedance of transmission lines)

Constants gi0 Generation capacity available at node i before stage 2 ai, bi Parameters of the production (generation) cost function at node i di Demand at node i (inelastic demand) φl,i Power transfer distribution factor (PTDF) associated to line l with respect to a unit injection/withdrawal at node i, when the network properties (network structure and electric characteristics of all lines) are known τ φl,i Power transfer distribution factor (PTDF) associated to line l with respect to a unit injection/withdrawal at node i, when the network properties are given by state τ Ki Unit cost of investment in capacity for a generation power plant at node i Kl Unit cost of investment in thermal capacity for line l Δgi Size of the step used in the discretization of the generation capacity gi at node i Λi Parameter used for the discretization of the generation capacity gi, associated to the number of binary variables. Note that the total number of binary variables is Λi + 1. fl0 Thermal capacity limit of line l already installed before stage 1 flmax Maximum thermal capacity allowed for line l in stage 1 flmin,τ Minimum thermal capacity allowed for line l in stage 1 and state τ flmax,τ Maximum thermal capacity allowed for line l in stage 1 and state τ Vector (of dimension n) of pure Nash equilibrium soyki⁎(n) lutions of stage 2 Small positive value used to find all pure Nash equilibria in stage 2 − þ BMγi ; BMλl ; BMλl ; BMξi Big–M parameters used in the linearization process of stage 3 BMyki Big–M parameter used in the linearization process of stage 2

þ

−

BMq ; BMri ; BMz ; BMz Big–M parameters used in the linearization process of stage 1 Variables Power generated at node i qi ri Import/export power from/to node i gi Generation capacity available at node i after the decisions of stage 2 are made fl Thermal capacity limit of line l after decisions of stage 1 (a constant for stages 2 and 3) ξi Shadow price/dual variable of the production capacity constraint at node i α Dual variable associated to the balance constraint of ri λl− Shadow price/dual variable of lower bound of the thermal capacity constraint of line l λl+ Shadow price/dual variable of the upper bound of the thermal capacity constraint of line l βi Locational marginal price (LMP)/dual variable of the energy balance at node i γi Dual variable of the non-negativity constraint of qi for node i − þ ηγi ; ηξi ; ηλl ; ηλl Binary variables from the Fortuny-Amat linearization in stage 3 yki Binary variable that is equal to 1 if the k-th step of the discretization of gi is considered, and 0 otherwise y^ ki Product of ξi and yki Product of qi and yki y~ ki uτ Binary variable that is equal to 1 for state τ, and 0 for the rest of states wiτ Product of u τ and ri + − zlτ , zlτ Product of u τ and λl+ and of u τ and λl−, respectively

Superscripts for variables e Variable in the equilibrium sG Variable associated to the G-th generation company choosing the sG-th capacity investment strategy

Functions ci(gi,gi0) Marginal production cost function at node i CIG(gi,gi0) Investment cost in generation capacity at node i to increase generation capacity from gi0 to gi CIL(fl,fl0) Investment cost in line l to increase transmission capacity from fl0 to fl e UG(gi,g− ) i Profit for generation company G when having generation capacities gi (i ∈ NG) and when the competitors are at the equilibrium e g− i

transmission expansion projects. Their model is named “proactive network planning” since the network planner may influence generation investment and the subsequent spot market behavior. Comparisons of this proactive model with an ideal integrated resource network planning model and a reactive network planning model are analyzed in SO2006. Although the findings in SO2006 are important, the methodology used by the authors, based on an iterative process to find the generation expansion equilibrium, does not solve the optimal transmission planning, but only evaluates the social-welfare impact of some predetermined transmission expansion projects. To avoid the problem of computing the equilibrium of generationcapacity investments subject to the equilibrium of the market operations presented in SO2006, Motamedi et al. (2010) use an agent-based

D. Pozo et al. / Energy Economics 36 (2013) 135–146

system and search-based optimization techniques to solve a similar problem. In Motamedi et al. (2010), the authors model each generation company (GENCO) as a Q-learning agent and use a heuristic to solve a three-stage four-level optimization problem. In their problem, the four levels considered are: (i) GENCOs' bidding strategy, (ii) market clearing, (iii) GENCOs' generation investments, and (iv) transmission expansion. In this paper, we formulate a mixed integer linear programming optimization model of transmission planning that is inspired in the model proposed in SO2006, which allows us to solve the optimal transmission expansion problem. The proposed model integrates transmission planning, generation investment, and market operation decisions. Contrary to SO2006's proactive methodology, it 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. As in SO2006, our model accounts for the transmission network constraints through a lossless DC approximation of Kirchhoff's laws. However, differently than in SO2006, we assume that the electricity market is perfectly competitive in order to guarantee that the linear transformation of the three-period problem is convex. Within this framework we are able to solve the three-period problem and find the optimal transmission expansion. One of the main contributions of this paper is the idea of characterizing the equilibria of generation investments made by the decentralized firms (which correspond to the solution of an EPEC – Equilibrium Problem subject to Equilibrium Constraints – problem) as a set of linear inequalities. This idea allows considering the generation investment EPEC equilibria as a set of linear constraints that the network planner can impose in its transmission planning optimization problem, making possible to obtain an optimal transmission plan that anticipates both generation investments and market operation equilibria. Although our work has been inspired by SO2006, we have improved it in different ways. Firstly, the model proposed in this paper is formulated as a single MILP optimization problem, allowing us to find global solutions. However, the model proposed by SO2006 uses a diagonalization methodology for solving the second and third stages; therefore, the solution found using SO2006 may be a global solution, a local solution or a saddle point, without any guarantee of finding a global solution. Secondly, we compute and characterize all pure Nash equilibria of generation investments at the stage 2, unlike SO2006. Thirdly, the model proposed here is based on the marginalist theory (spot market prices come from Lagrange multipliers). However, SO2006 sets prices from elastic demand functions. This fact makes both models not directly comparable. Fourthly, SO2006 solves the optimal transmission planning at stage 1 by evaluating a finite number of line investment projects subject to the results of the second stage, the generation expansion equilibrium, and the third stage, the spot market equilibrium. However, in this work, we have posed a single optimization model that integrates the three stages in a single optimization problem without limiting the transmission investment options to a discrete set of expansion projects, as in SO2006. Finally, an approximation of the line impedance values as functions of the installed transmission capacities allows incorporating the changes in the operation due to the new topology resulting from line investments. Other authors have proposed multi-period models to characterize investments in the electricity market. Murphy and Smeers (2005), for instance, propose a two-stage model of investments in generation capacity in restructured electricity systems. In this two-stage game, 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. Additionally, Garcés et al. (2009) propose a bilevel model where the transmission planner minimizes transmission investment

137

costs in the upper level and the lower level represents the market clearing representing pool trading. The bilevel model is reformulated as a mixed-integer linear problem using duality theory. In CAISO (2004) and Sheffrin (2005), the authors describe the “Transmission Economic Assessment Methodology (TEAM)”, developed by the California Independent System Operator (CAISO) and based on the “gain from trade” economic principle. The TEAM's model considers that transmission planning anticipates the equilibrium of a perfectly-competitive energy market, but it ignores the potential strategic response by generation investments to the transmission upgrades. That is, the TEAM's model assesses the economic impact of transmission upgrades, given the current estimation of the generation capacity. On the other hand, Borenstein et al. (2000) present an analysis of the relationship between transmission capacity and generation competition in the context of a two-node network. They argue that relatively small transmission investments may yield large payoffs in terms of increased competition. The main contributions of this article are: • We formulate a mixed integer linear programming optimization model that integrates transmission planning, generation investment, and market operation decisions; • We propose and apply a methodology to solve for the optimal transmission expansion (anticipating both generation investment and market clearing); • We characterize the equilibria of generation investments made by the decentralized firms (which correspond to the solution of an EPEC – Equilibrium Problem subject to Equilibrium Constraints – problem) as a set of linear inequalities; and • We compute all possible pure Nash equilibria on generation investment (EPEC) problem by creating holes in the equilibrium solution space. The rest of this paper is organized as follows. Section 2 describes the proposed three-stage transmission planning model, as well as the linearization process we follow to characterize the market equilibria as a set of linear constraints. Section 3 presents a methodology in order to find all possible generation investment pure Nash equilibria. In Section 4, we introduce some changes to the proposed mixed integer linear programming model to capture the complexity of modeling the power transfer distribution factors under a changing network. Section 5 shows the computational complexity of the proposed model. Section 6 illustrates our model using 3-bus and 4-bus examples. Conclusions are presented in Section 7. 2. The three-stage transmission planning model We assume that the transmission planning model consists of three stages that are described in reversed order, since the first stage represents the final decision by the transmission planner. Our model is of the Stackelberg type, where the transmission planner (first level) anticipates generation expansions (second level), and the clearing of the market (third level). This (three-stage) hierarchy is motivated by the fact that transmission planners should consider the expansions in generation that may take place, as well as the clearing of the market related to generation expansion, in order to make their decisions. As shown in SO2006, ignoring the interrelationship between the transmission and the generation investments (and, thus, optimizing the social-welfare impact of transmission expansions based only on the changes they induce in the spot market equilibrium) may lead to suboptimal network plans. Fig. 1 shows a graphical depiction of the overall model. In the market-operation layer, we assume that demand is exogenous and spot prices are determined by the independent system operator (ISO) on the basis of a usual dispatch model. In the generation-investment layer, investments in generation capacities

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D. Pozo et al. / Energy Economics 36 (2013) 135–146

Fig. 1. Three-stage transmission planning model.

satisfy a Nash equilibrium taking into account the spot prices determined by the ISO. Finally, transmission investment planning is modeled as an optimization problem for accommodating these generation capacities so that total cost is minimized. Each of the three stages is described in detail in the following subsections. Since the three-stage problem is not convex, we use both the Fortuny-Amat linearization and the binary expansion in order to convexify it, with the purpose of guaranteeing a global solution of the overall problem.

power at marginal costs in our setting. In contrast, the problem of a price-taker generation company (GENCO) is to maximize its profit given the nodal prices resulting from the market clearing. The models for the ISO and individual GENCOs follow. 2.1.1. ISO problem formulation The ISO problem is modeled as a cost minimization one (dual variables are presented to the right of each constraint):  
 
  0 0 min ∑ ci g i ; g i qi ¼ min ∑ ai qi −bi g i −g i qi

2.1. Third stage: Market clearing

qi ;r i

The third stage models the energy market operation. At this stage we compute the equilibrium that occurs when the ISO clears the perfectly competitive market for given generation and transmission capacities. Our model accounts for transmission network constraints through a lossless DC approximation of Kirchhoff's laws. Moreover, we assume price-taker generators and inelastic demand, where nodal prices are given by the Lagrange multipliers of the energy balance constraint at every node. As in SO2006, we assume that all nodes are both demand and generation nodes and that all generation capacity at a node is owned by a single firm (although firms can own generation capacity at multiple nodes). In addition, marginal generation costs are constant in the power generated and inversely related to the installed capacity, as shown in Fig. 2; specifically, they can be expressed as: ci(gi,gi0) = ai − bi(gi − gi0). The ISO problem generally consists of the maximization of social welfare subject to network constraints. Since we assume both a perfectly competitive market and inelastic demands, this is equivalent to perform cost minimization. Additionally, generators offer their

ci

qi ;ri

i

subject to: qi ≤g i

: ξi

∑ ri ¼ 0

i∈N

∀i∈ N :α

−f l ≤ ∑ φl;i r i ≤f l i∈N

−qi −r i ¼ −di qi ≥0

: γi

ð2Þ ð3Þ −

þ

: λl ; λl : βi

∀l∈ L

ð6Þ

The objective function (1) minimizes the total cost of generation, constraint (2) establishes the maximum power that the GENCOs can produce, constraint (3) represents an energy balance of the net injections/withdrawals for the whole electrical network (given that network losses are assumed negligible), constraint (4) expresses the maximum flow through lines as a function of the power transfer distribution factors (PTDFs), constraint (5) meets demand at every node as a function of the injection in the node and the flow coming through the lines connected to this node, and constraint (6) forces power generation to be nonnegative at every node. The Karush–Kuhn–Tucker (KKT) conditions equivalent to Eqs. (1)–(6) are given by:

l∈L

0≤γ i ⊥qi ≥0 ∀ i∈ N

gi

gi0 Fig. 2. Marginal generation cost functions.

ð5Þ

∀i ∈N:


 þ − α þ ∑ λl −λl φl;i −βi ¼ 0

bi

ð4Þ

∀i∈ N


 0 ai −bi g i −g i −γi −βi þ ξi ¼ 0

ai

ð1Þ

i

0≤ξi ⊥g i −qi ≥0 −

i∈N

: ri

∀i ∈N ∀i∈ N

ð7Þ ð8Þ ð9Þ

∀i∈ N

0≤λl ⊥f l þ ∑ φl;i r i ≥0

: qi

ð10Þ ∀l∈L

ð11Þ

D. Pozo et al. / Energy Economics 36 (2013) 135–146 þ

0≤λl ⊥f l − ∑ φl;i ri ≥0 i∈N

∀l∈L

ð12Þ

plus Eqs. (3) and (5), and considering that the symbol ⊥ represents the complementarity relationship (i.e., x ⊥ y means x Ty = 0). Note that, if nodal prices are equal to marginal costs, a GENCO has no profit. Thus, a GENCO does not have incentives to invest in generation capacity unless producing at its maximum limit, which takes place only when qi = gi, yielding to yi = 0 and ξi > 0. Since complementary constraints (9)–(12) are nonlinear, they can be replaced by an equivalent set of linear constraints using the Fortuny-Amat linearization formula (Fortuny-Amat and McCarl, 1981) 1 in order to obtain a mixed integer linear optimization model. This yields to Eqs. (17) to (24), shown later in this section. 2.1.2. GENCO problem formulation Each individual GENCO maximizes its profit considering the income from sales at nodal market prices provided by the ISO cost minimization. Hence, a GENCO solves: n h
 io 0 βi qi − ai qi −bi g i −g i qi

max ∑ qi

i∈N G

ð13Þ

γ

λþ

qi ≥0

ξ

2.2. Second stage: Generation investment equilibria In the second stage, each GENCO determines the investments in generation capacity in order to increase its profits due to the linear decrease in the generation marginal costs, as seen in Fig. 2. Since the investments in new generation capacity reduce the marginal cost of production, the return from the investments made in stage 2 occurs in stage 3. Accordingly, there are no spot market decisions in stage 2. In this stage, the spot market decisions are given as parameterized equilibrium constraints (3), (5), (7), (8), and (17)–(24), which are anticipated by the generation expansion investments. Moreover, a GENCO considers its capacity expansion against the capacity expansion of its competitors. Therefore, the generation expansion problem for each GENCO results: n

o 0 0 βi qi −ci g i ; g i qi −CIG g i ; g i i∈N G n h
 i
o 0 0 ¼ ∑ βi qi − ai qi −bi g i −gi qi –K i gi −g i

max U G ¼ ∑ gi

: ξi

∀i ∈NG

ð25Þ

ð14Þ

∀i∈ NG :

ð15Þ

Let's call primal problem to Eqs. (13)–(15). Thus, from the duality theorem (Luenberger and Ye, 2008), we know that if either the primal or the associated dual problem has an optimal solution, then the other one has the same optimal solution. Since both primal and dual problems are linear in this case, the problem is convex and we can also apply the strong duality theorem (Luenberger and Ye, 2008). Thus, we get Eq. (16) from applying the strong duality theorem (which we will use later in this section): n h
 io 0 βi qi − ai qi −bi g i −gi qi ¼ ∑ g i ξi i∈NG

∀G:

ð16Þ

Using Eq. (16), it is easy to see that, extending the problem defined in Eqs. (13)–(15) to all GENCOS, we obtain a set of KKT conditions that is equivalent to constraints (7), (9) and (10). Therefore, using the Fortuny-Amat linearization formula, we have that the set of constraints (3), (5), (7), (8), and (17)–(24) fully represents the stage 3 of our model: 0≤γ i ≤BM 0≤qi ≤BM

γi  γ  ηi

γi 

∀ i∈ N γ

1−ηi

−

0≤λl ≤BM

λ− l

ð18Þ

∀l∈ L

ð19Þ

λ− l

i∈N

þ

0≤λl ≤BM

λþ l


þ λ ηl

0≤f l − ∑ φl;i r i ≤BM i∈N


 ξ 0≤ξi ≤BM ηi ξi

ð17Þ

∀i ∈N


− λ ηl

0≤f l þ ∑ φl;i ri ≤BM


 λ− 1−ηl

∀l∈ L

∀ l∈L λþ l


 λþ 1−ηl

∀i ∈N

ð20Þ ð21Þ

∀l∈L

ð22Þ ð23Þ

s.t. Eqs. (3), (5), (7), (8), and (17)–(24). The problem formulated in Eq. (25) can be stated as a Mathematical Program subject to Equilibrium Constraints (MPEC), where the equilibrium constraints are defined by the linearized-equivalent KKT conditions of the third-stage problem; i.e. Eqs. (3), (5), (7), (8), and (17)–(24). Using Eq. (16), we can rewrite the problem as: max U G ¼ ∑ gi

i∈N G

n
o 0 g i ξi –K i g i −g i

Note that the Fortuny-Amat linearization is an exact approximation, disregarding any discretization of the variables.

ð26Þ

s.t. Eqs. (3), (5), (7), (8), and (17)–(24). The only non-linear term in Eq. (26) is giξi. Since the gi variables are controlled by the generation firms and we assume that generation expansion is made in discrete steps, then we use a binary expansion (Pereira et al., 2005) to discretize gi: 0

Λi

k

g i ¼ g i þ Δg i ∑ 2 yki k¼0

i ∈N G :

ð27Þ

Accordingly, the non-linear product giξi can be replaced by the expression: Λi

0 k g i ξi ¼ g i ξi þ Δg i ∑ 2 y^ ki k¼0

i ∈NG

ð28Þ

where we define y^ ki by constraints (29) and (30), using the Fortuny-Amat linearization formula: 0≤ξi −y^ ki ≤BMð1−yki Þ

∀i∈ NG ; k ¼ f0; 1 ; …;Λi g

ð29Þ

0≤y^ ki ≤BMðyki Þ ∀i ∈N G ; k ¼ f0; 1 ; …;Λi g:

ð30Þ

Thus, the generation expansion problem of each GENCO can be set as a linear MPEC, as stated next: (

1

ð24Þ

i∈N G

qi ≤g i

i∈N G

λ−

∀i∈ N

where ηi ; ηl ; ηl ; ηi ∈f0; 1g and βi, α are free variables.

s.t.

∑

139


 ξ ξ 0≤g i −qi ≤BM i 1−ηi

max U G ¼ ∑ gi

i∈N G

Λi

!

Λi

0 k k g i ξi þ Δg i ∑ 2 y^ ki –K i Δg i ∑ 2 yki k¼0

k¼0

!) ð31Þ

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D. Pozo et al. / Energy Economics 36 (2013) 135–146

s.t. Λi

! k

ai −bi Δg i ∑ 2 yki −γi −βi þ ξi ¼ 0 k¼0

: qi

∀i ∈N

ð32Þ

Eqs. (3), (5), (8), and (17)–(23) 0

Λi

k

0≤g i þ Δg i ∑ 2 yki −qi ≤BM k¼0

ξi


 ξ 1−ηi

∀i ∈N

ð33Þ

Eqs. (29)–(30). Accordingly, the stage 2 problem can be formulated as an Equilibrium Problem with Equilibrium Constraints (EPEC), in which each firm faces (given the other firms' commitments and the system operator's import/export decisions) a mixed integer linear programming (MILP) MPEC problem. This EPEC represents the equilibrium when all the GENCOs expand their capacity simultaneously subject to the market equilibrium of stage 3. 2 In SO2006, this EPEC is solved using a heuristic approach that sequentially solves each MPEC taken as given the decisions of the other firms. Specifically, they sequentially solve each firm's profitmaximization problem using as data the optimal values from previously solved problems. Thus, starting from a feasible solution, they solve for g1 using g−1 as data in the first firm's optimization problem (where g−1 means all firms' generation capacities except for firm 1's), then solve for g2 using g−2 as data, and so on. One problem of this heuristic approach is that there is no guarantee of convergence to an equilibrium. Another (and more practical) problem of this approach is that it does not allow characterizing the EPEC as a set of constraints to be imposed by the network planner at stage 1. To avoid these difficulties, in this paper we enumerate the GENCOs' investment strategies and express the Nash equilibria conditions as a finite set of inequalities, which can be used to find all the Nash equilibria. Since the GENCOs' strategies match the MPEC problem decision variables, the same discretization used in the MPEC problem can be used to enumerate the strategies of each firm. This idea of characterizing the equilibria of the GENCOs' generation capacity investments (which correspond to the solution of the EPEC problem) as a set of linear inequalities allows formulating the transmission planning (stage 1) problem as a mixed integer linear programming optimization problem, for which we can compute an optimal transmission plan that anticipates both generation investments and market operations equilibria. 3 Accordingly, in order to characterize the EPEC equilibria as a set of linear inequalities, we discretize all generation investment strategic variables of the problem. The general expression for the Nash equilibrium is given by:  e e U G g i ; ∀i ∈N ≥



 e max U G g i ; g −i ; ∀i∈ NG ; ∀−i∉N G gi

∀G∈ Ψ

ð34Þ

where, for all GENCOs, UGe(gie) is the profit function of each GENCO G given its strategic decision variable gie in the Nash equilibrium, which is always better than any other profit resulting from a different 2

The reader should be aware that the EPEC presented here represents an oligopolistic generation investment problem subject to the outcomes of a short-run market with a single-period demand. Thus, as Kreps and Scheinkman (1983) have shown, this game gives the same outcome whether the last period is Cournot or perfectly competitive. However, this is no longer true when characterizing the EPEC equilibria as a set of linear inequalities using the discretization of generation investment strategic variables (as in our final model—as described later on this section) and/or when considering different states of the system (as in our final model—as described later in Section 4). 3 SO2006's approach is not able to compute an optimal transmission plan because, contrary to the model proposed here, their model does not allow characterizing the EPEC equilibria as a set of constraints to be imposed in the first-stage problem. This explains why, in SO2006, the authors do not solve the stage 1 problem, but only evaluate the social welfare impact of some pre-determinate transmission expansion projects.

strategy, assuming that the other GENCOs use their Nash equilibrium e strategies, g− i. Hence, the Nash equilibrium in Eq. (34) is solved by approximating its solution using discrete strategies. In doing that, we replace expression (34) by a set of inequalities, where the strategic variables are discretized for each GENCO. Thus, the Nash equilibria of the GENCOs' capacity investment decisions are given by the following set of inequalities:   e e s  s e U G g i ; ∀i ∈N ≥U GG g i G ; g −i ; ∀i ∈NG ; ∀−i∉NG

∀G∈Ψ; ∀sG ∈S G ð35Þ

where we have to distinguish between the left hand side (LHS) and the right hand side (RHS) of Eq. (35). The LHS in Eq. (35) is the profit function of each GENCO given its strategic decision variable in the Nash equilibrium. That is, the definition of the profit function for GENCO G in the equilibrium is given by: ( e UG

0 e g i ξi

¼ ∑

i∈NG

þ

!

Λi

k Δg i ∑ 2 y^ eki k¼0

!)

Λi

k e Δg i ∑ 2 yki k¼0

–K i

∀G ∈Ψ

ð36Þ

subject to the linearized constraints of stage 3 in the equilibrium, which correspond to constraints (3), (5), (8), (17)–(23), (29)–(30) and γ − λþ λ− ξ (32)–(33), replacing yki ; y^ ki ; qi ; r i ; γi ; βi ; ξi ; α; λþ l ; λl ; ηi ; ηl ; ηl and ηi þe

−e

e e e e e e e e þe −e γe λ λ ξe by yki ; y^ ki ; qi ; r i ; γi ; βi ; ξi ; α ; λl ; λl ; ηi ; ηl ; ηl and ηi ; respectively, and considering Eqs. (29) and (30) for all i ∈ N. The RHS in Eq. (35) is the profit function of each GENCO given a particular value of the strategic decision variable. That is, considering firm G chooses strategy sG (which involves investing in generation capacity at node i up to the capacity g si G , with i ∈ NG, ∀ G ∈ Ψ), the definition of the profit function for GENCO G is given by: 4 s

U GG ¼ ∑

i∈NG

n
o s s s 0 g i G ξi G –K i g i G −g i

∀G∈Ψ; ∀sG ∈S G

ð37Þ

subject to the corresponding constraints of stage 3, which correspond to constraints (3), (5), (8), (17)–(23), (29)–(30) and (32)–(33), but considering them ∀G∈Ψ; ∀sG ∈S G , replacing sG − γ λþ λ− ξ ^ sG sG sG yki ; y^ ki ; qi ; r i ; γ i ; βi ; ξi ; α; λþ l ; λl ; ηi ; ηl ; ηl ; and ηi by yki ; y ki ; qi ; r i ; s

s

s

þs

s

−s

λþsG

γs

λ−sG

ξs

respectively, γi G ; βi G ; ξi G ; α G ; λl G ; λl G ; ηi G ; ηl ; ηl ; and ηi G ; and replacing Eq. (32) by Eqs. (38) and (39), Eq. (33) by Eqs. (40) and (41), Eq. (29) by Eq. (42), and Eq. (30) by Eq. (43):
 s 0 s s s ai −bi g i G −g i −γi G −βi G þ ξi G ¼ 0 Λi

∀i∈NG ; ∀G∈Ψ; ∀sG ∈S G

ð38Þ

! k e

s

s

s

ai −bi Δg i ∑ 2 yki −γ i G −βi G þ ξi G

ð39Þ

k¼0

¼0

∀i∉NG ; ∀G∈Ψ; ∀sG ∈S G s

s

0≤g i G −qi G ≤BM 0

Λi

ξi


 ξs ∀i∈N G ; ∀G∈Ψ; ∀sG ∈S G 1−ηi G

k e

s

0≤g i þ Δg i ∑ 2 yki −qi G ≤BM

ξi




k¼0

ξ sG

1−ηi



ð40Þ

∀i∉NG ; ∀G∈Ψ; ∀sG ∈S G ð41Þ

 s s e  0≤ξi G −y^ kiG ≤BM 1−yki k ¼ f0; 1 ; …;Λi g; ∀i∉NG ; ∀G∈Ψ;

ð42Þ ∀sG ∈S G

 e s k ¼ f0; 1 ; …;Λi g; ∀i∉N G ; ∀G∈Ψ; 0≤y^ kiG ≤BM yki

∀sG ∈S G :

ð43Þ

4 Note that, since g si G is known, it is possible to directly replace its value in Eqs. (3), (5), (8), (17)–(23), (29)–(30) and (32)–(33), without having the non-linear term that motivates the binary expansion used before.

D. Pozo et al. / Energy Economics 36 (2013) 135–146

141

With all these definitions, Eq. (35) represents the EPEC of the GENCOs' capacity investment decisions (stage 2 equilibrium).

found, as described in Eq. (48) for each stage-2 Nash equilibrium found (indexed by n):

2.3. First stage: Transmission investment plan

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ∑ yki ðnÞ−yeki 2 ≥

∀n:

i;k

In stage 1, the network planner – which we model as a Stackelberg leader in our 3-stage game – maximizes the social welfare subject to the transmission constraints while anticipating the solutions from stages 2 and 3. Since we have considered inelastic demands, this problem is equivalent to minimize the total cost: sum of generation dispatch costs and transmission investment costs. Thus, the objective function of the transmission planner in stage 1 is: (

)

 0 0 min ∑ ci g i ; g i qi þ ∑ CIL f l ; f l f l ;l∈Linv i l∈Linv ( )

 0 0 ¼ min ∑ ai qi −bi g i −g i qi þ ∑ K l f l −f l : f l ;l∈Linv

i

Each one of the quadratic terms in Eq. (48) is expanded as: 



e

yki ðnÞ−yki

2

  2  e 2  e ¼ yki ðnÞ þ yki −2 yki ðnÞyki

e are binary numbers, the term and, using the fact that yki⁎(n) and yki in Eq. (49) is equivalent to:





e

e

ð44Þ

The problem of minimizing Eq. (44) subject to the transmission constraints and the constraints representing the solutions from stages 2 and 3 is non linear. Moreover, the variables that represent the solution to the EPEC are equilibrium results, thus, having qie instead of qi, and gie instead of gi. The non-linear term in the objective function, gieqie, can be decomposed using the binary expansion applied to gie and linearized using the Fortuny-Amat formulation. This yields: 0 e

Λi

k¼0

k e

∀i∈NG

ð45Þ

e e q e  ∀i∈NG ; k ¼ f0; 1 ; …;Λi g 0≤qi −˜yki ≤BM 1−yki

ð46Þ

e q e  ∀i∈NG ; k ¼ f0; 1 ; …;Λi g 0≤˜yki ≤BM yki

ð47Þ

where y˜ eki is a continuous variable taking optimally the values of either cero or qie. In Eq. (44), we have considered that there is a set of transmission lines that are candidates for investment (Linv). That means that the previously-constant maximum active flows (fl) are now variables of the problem in stage 1. Note that, contrary to the assumptions in SO2006, the network planner now solves stage 1 for the optimal transmission expansion capacities in both new and existing lines within the set of candidate locations. Therefore, we can formulate stage 1 problem as a mixed integer linear programming optimization program subject to EPEC and other-equilibrium constraints. Thus, the final model is given by Eqs. (63)–(103), shown in Appendix A. 3. Methodology to obtain all feasible generation investment pure Nash equilibria in stage 2 The EPEC for the stage 2 problem may have multiple equilibria. The model described in the previous section finds only one EPEC equilibrium, but we could be interested in detecting more than one equilibrium, or even all of them. In this section, we modify the previous section model of the stage-2 problem in order to find all pure strategy EPEC equilibria. To do that, we generate “holes” in the feasible region for each solution found within the set of dise crete strategies: yki . That is, after we found a solution (Nash equilibrium) for the stage-2 problem (characterized by yki⁎ for all k and i), e we impose a new constraint avoiding that the optimal value of yki is close to the previously found solution (within a distance of ). We repeat this procedure with any new solution found. Thus, given a solution vector yki⁎(n) of the EPEC problem of stage 2, we include a set of new constraints to generate holes in the solutions already

ð50Þ

which is a linear expression. Thus, Eq. (48) becomes:   e  e  2 ∑ yki ðnÞ þ yki −2 yki ðnÞyki ≥ i;k

e e

ð49Þ

yki ðnÞ þ yki −2 yki ðnÞyki

l∈Linv

g i qi ¼ g i qi þ Δg i ∑ 2 y˜ ki

ð48Þ

∀n:

ð51Þ

To account for all pure strategy EPEC Nash equilibria, we need to add Eq. (51) to the set of constraints defined in Eq. (35). Note that this methodology to obtain all pure strategy Nash equilibria applies only to the stage 2 problem of the model presented in the previous section. Since the stage 1 problem is not an equilibrium, but an optimization problem, the application of the methodology of creating holes in the feasible region to the stage 1 problem has not a clear intuition. 4. Methodology to account for the variation of the line impedance as a function of the installed transmission capacity in stage 1 The network planning model described in Section 2, Eqs. (63)–(103), allows the network planner to solve the optimal transmission expansion capacities within the set of candidate locations, considering that the investments in transmission capacity can be done without changing the impedances of the links. This assumption may be not realistic and we now propose an approximation of the line impedance values as a function of the installed transmission capacity and incorporate this modification to our model. Assume that an existing line has an impedance whose value is xl. If another line with the same impedance is placed in parallel, the total link impedance is xeq ¼ x2l . In general, if there are n lines in parallel, l the equivalent impedance is xeq ¼ xnl . If fl0 is the initial capacity of the l link, we can express the change in the link impedance as a function of the transmission capacity in a continuous fashion, as shown in Fig. 3. This approximation neglects line resistance. In order to apply this modification to the model proposed for stage 1, and keep the stage-1 formulation as a mixed integer linear programming optimization program, continuously changing the power transfer distribution factors (PTDFs) seems unviable due to the nonlinearities involved. Instead, we consider a discretization of the equivalent impedance in the potentially-expanded lines and calculate the associated PTDFs. Thus, if the initial transmission capacity of link l is

Impedance

2

0

2

0

Line capacity

Fig. 3. Link impedance as a function of transmission capacity.

142

D. Pozo et al. / Energy Economics 36 (2013) 135–146

Eqs. (94), (95), (97) and (98), the modifications are similar, but replacing the product uτ rsi G by wsiτG . In Eqs. (71) and (88), there are terms containing the PTDFs as a result of the KKT conditions of stage 3, which involve the PTDFs multiplied by some dual variables. Since only one state occurs from all configurations, considering the variable that accounts for the possible states, Eq. (71) becomes:

Equivalent impedance 0 1 2 3


 e þe −e τ τ e α þ ∑ ∑ λl −λl u φl;i −βi ¼ 0 ∀i∈N: 0

1

2

3

=

Fig. 4. Discretization of the equivalent impedance as a function of installed transmission capacity.

fl0 and investments can be done up to a capacity whose value is flmax, we can approximate the equivalent impedance by performing a discrete approximation, between fl0 and flmax, as illustrated in Fig. 4. Fig. 4 shows a 3-interval discretization where the limits of each interval are represented by a superindex. Within this discretization, we use the average impedance per interval. Note that a good discretization should consider many intervals for small investments and few intervals for large investments, in agreement with Fig. 4. Assume that there exists a set of lines, Linv, who could be either expanded or constructed. Let NLinv be the number of possible investments and J be the number of intervals in which we divide the expansion. Then, we have J + 1 independent investment options for the rest of the lines (including the no-investment option). Therefore, we have ðJ þ 1ÞNLinv possible configurations, or states, of the system. Denoting each state by super-index τ and associating a binary variable, u τ, to each state, it must happen that only one configuration occurs. That is: τ

∑ u ¼ 1:

ð52Þ

τ

In the mixed integer linear programming optimization model of stage 1, Eqs. (63)–(103), each of the states has an associated PTDF τ matrix, φl,i . As well, each transmission capacity limit after investment depends on the resulting state or configuration (i.e., flmax,τ). Consequently, Eq. (64) becomes: τ min;τ

∑u fl τ

τ max;τ

≤f l ≤ ∑ u f l τ

∀l∈Linv :

ð53Þ

In addition, all constraints where the PTDFs appear must be modified. In the model, i.e. Eqs. (63)–(103), the constraints that must be modified are Eqs. (71), (77), (78), (80), (81), (88), (94), (95), (97), and (98). In Eqs. (77), (78), (80), (81), (94), (95), (97) and (98) the PTDFs are multiplied by the variable ri in order to compute the transmission flow limits. Due to the similarity in the procedure, we will only explain the modifications needed in Eq. (77). Now, Eq. (77) results: τ

0

τ e

0≤f l þ ∑ ∑ u φl;i ri ≤BM

λ− l

τ i∈N


 λ− e 1−ηl

∀l∉Linv :

ð54Þ

e Replacing the product u τrie by wiτ , and using the Fortuny-Amat linearization formula, we obtain:

−BM −BM

ri

ri

 

τ e e r  τ 1−u ≤r i −wiτ ≤BM i 1−u ∀i∈NG ; ∀τ∈Τ τ e r  τ u ≤wiτ ≤BM i u

∀i∈NG ; ∀τ∈Τ:

τ

Final capacity

Now, there are two non-linear products of a binary variable by a +e continuous variable. Thus, replacing the product u τλl+ e by zlτ and τ −e −e the product u λl by zlτ , and, using the Fortuny-Amat linearization formula, we can replace Eq. (71) by Eqs. (58)–(62):
 e þe −e τ e α þ ∑ ∑ zlτ −zlτ φl;i −βi ¼ 0 τ

l∈L

þe

þe

þe

z

0≤λl −zlτ ≤BM 0≤zlτ ≤BM

þ

−e

−e

−e

z



zþ

τ

0≤zlτ ≤BM

−



z

−

τ

u

τ

1−u

Therefore, we must change Eq. (77) by Eq. (54), but replacing the e , and adding Eqs. (55) and (56) as constraints. product u τrie by wiτ Similar changes must be done to Eqs. (78), (80), and (81). For

∀i∈N

∀l∈L; ∀τ∈Τ

∀l∈L; ∀τ∈Τ

u

0≤λl −zlτ ≤BM





τ

1−u

∀l∈L; ∀τ∈Τ

∀l∈L; ∀τ∈Τ:

ð58Þ

ð59Þ ð60Þ ð61Þ ð62Þ

A similar change must be done with Eq. (88), but replacing the sG sG G G product uτ λþs by zþs , and the product uτ λ− by z− l lτ . l lτ With all these changes, the final mixed integer linear programming formulation of the stage-1 problem is the one described by Eqs. (63)–(103) and Eq. (52), but replacing Eq. (64) by Eq. (53); e Eq. (71) by Eqs. (58)–(62); Eq. (77) by Eqs. (54)–(56) with wiτ =uτrie; Eqs. (78), (80), (81), (94), (95), (97), and (98) by equations similar e to Eqs. (54)–(56) with wiτ =uτrie and wiτsG ¼ uτ r i sG ; and Eq. (88) by G G equations similar to Eqs. (58)–(62) with zþs ¼ uτ λþs and lτ l sG τ − sG z− ¼ u λ . l lτ 5. Computational issues Multi-stage models are generally difficult to solve and the proposed model is not an exception. Although the many advantages of the proposed model (and the fact that transmission planning is an off-line process), the model has the potential shortcoming of having an exponential number of constraints, which may challenge the computational solution of large systems. Table 1 summarizes the computational complexity for our MILP model, when considering that every node can expand generation using the same number of discrete steps (intervals), represented by Λ binary variables. As well, in Table 1 we assume the same number of expansion strategies (S) for every generating company. The rest of the symbols used in Table 1 represent the cardinal of the corresponding set. As it is evident from Table 1, the number of impedance discretization levels considered for each line (i.e., the number of configurations of the Table 1 Computational complexity.

ð55Þ ð56Þ

ð57Þ

l∈L

Dimension # of binary variables # of positive variables # of free continuous variables # of inequality constraints # of equality constraints

(2N + 2L)(1 + Ψ·S) + N(1 + Λ) + T (2N + 2L)(1 + Ψ·S) + N(1 + 2Λ) + (1 + S)Ψ + N + Linv (N·T + 2L·T + N)(1 + Ψ·S) + (1 + N)·Ψ·S (5N + 6L + 16N·T)(1 + Ψ·S) + 6N·Λ + 3N·Λ(Ψ − 1) Ψ·S + Ψ·S (3N + 8L·T + N·T)(1 + Ψ·S) + 2Ψ·S + Ψ + 2

D. Pozo et al. / Energy Economics 36 (2013) 135–146

network) and the number of the discrete strategies considered for every GENCO make the set of constraints grow fast (in an exponential manner). Fortunately, in real-world applications, only few lines and nodes are suitable for expansion. Moreover, most markets are oligopolies with few GENCOs participating in them. In applying the proposed model to large systems, we recommend the following simplifications: (i) consider a limited number of GENCOs that are able to expand generation capacity, (ii) consider a limited number of generation investment strategies for each GENCO, (iii) consider a limited number of configurations of the network, and (iv) assume that PTDFs of large-capacity existing transmission lines do not change when increasing their thermal capacity. 6. Case studies We illustrate the methodology proposed with two examples: a 3-node example and a 4-node example. The first system consists of 3 nodes connected by 3 lines, as shown in Fig. 5. The nodal data are shown in Table 2. Inelastic demand load is provided in the second column, initial production capacity is in the third column, the parameters of the production cost functions are shown in the fourth and fifth columns, and the hourly-equivalent unit cost of investment on capacity for each generation plant is shown in the sixth column. Note that some parameters are transformed into their equivalent annual hourly values (i.e., the cost of investment in capacity represents the actual value on $/MW of the annual cost divided by 8760 hours). The three nodes are initially connected with 3 lines, having the same electric characteristics. The initial thermal capacity for each line is 70 MW. The annualized present values of the investment costs are: $306,600/MW for lines and $131,400/MW for generating units. They are transformed into their equivalent hourly values, Kl = $35/MW (for transmission lines) and Ki = $15/MW (for generating units). We solve the stage-1 problem using the methodology presented in Section 4 and considering 4 possibilities for transmission investment: investment in line 1, investment in line 2, investment in line 3, and investment affecting lines 1, 2, and 3 simultaneously. In the first case, if we invest in line 1 only, the link flow limit according to our discretization process is 140 MW and there are four states for the expansion line capacity: no investment, line flow bound between 70 and 84 MW, line flow bound between 84 and 105 MW, and line flow bound between 105 and 140 MW. See Fig. 6 for the relationship between the line expansion factor (from 1 – equivalent to 70 MW – to 2 —equivalent to 140 MW) and the impedance of line 1. In addition, the PTDFs for the corresponding states are shown in Table 3. Similar calculations were made for the other 3 transmission investment possibilities. For the second stage of our model, we assume the three GENCOs can invest in generation capacity from 300 MW up to 540 MW at intervals of 16 MW. Solving the problem of stage 1 formulated in Sections 2 and 4, for the case of investing in line 1 only, we obtain that the optimal value is to invest up to 138 MW of capacity for line 1. We provide the optimization solution in Table 4. Note that the transmission planner has

143

Table 2 Three-node case study data. Node i 1 2 3

Demand

Generation units: production cost parameters

Unit cost of investment

di [MW]

gi0 [MW]

ai [$/MWh]

bi [$/(MW·MWh)]

Ki [$/MW]

300 250 200

300 300 300

25 24 24

0.06 0.06 0.06

15 15 15

already anticipated the equilibrium solution for stage 2 in a proactive manner. The GENCO in node 1 invests 208 MW in generation capacity, meaning that its total production becomes 508 MW in stage 3. The GENCO in node 1 becomes the most economic unit, whose marginal cost is $12.25/MWh, and the production of this GENCO is partially consumed at node 1 (300 MWh) and partially sent through lines 1 and 2. This yields the same LMPs for all the nodes and the minimum cost of dispatch. It is remarkable that, although we assume that the electricity market is perfectly competitive and GENCOs bid their true marginal costs (i.e., assuming a cost minimization framework), the GENCO in node 1 exercises market power through its generation capacity investment decision, obtaining a profit. This implies another potential benefit to society of anticipating generation capacity investment decisions through a proactive transmission plan that mitigates GENCOs' market power. The transmission planner problem for stage 1 is also solved for the other cases: investing in line 2, investing in line 3, and investing in lines 1, 2 and 3, jointly. The solutions are summarized in Table 5. Note that the discretization applied is the same for all cases (i.e. four states of investment for each line). From Table 5, we observe that there is a manifold of solutions with identical total costs, in the case of possible investment in lines 1, 2, and 3, jointly. All these solutions incentivize the GENCO at node 1 to expand its capacity. Thus, in terms of the title of this article, we may say that “if you build it” (either adding capacity in line 1 or in line 2), “he will come” (i.e., the GENCO in node 1 will come to invest in generation capacity). It is interesting to note that, if we fix the investment in line 1 and we solve the EPEC (stage 2) problem by applying the methodology to find all pure Nash equilibria, we obtain more equilibria. Each equilibrium has a different cost of dispatch, but all of them are perfectly valid. In the optimization process of stage 1, the transmission planner attempts to anticipate the EPEC equilibrium by choosing the best possible solution for stage 2 (minimum cost of dispatch). However, this cannot be guaranteed. This implies that, while the methodology presented in Section 3 is useful for finding all pure strategy Nash equilibria of the stage 2 problem, this methodology is not useful for solving for all possible instances of the stage 1 problem. Hence, we 0.32

Link impedance (p.u.)

0.3

2

3

1

0.28 0.26 0.24 0.22 0.2 0.18 0.16

1

3 2 Fig. 5. Three-node case study.

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Capacity investment factor Fig. 6. Link impedance as a function of the capacity in line 1.

1.9

2

144

D. Pozo et al. / Energy Economics 36 (2013) 135–146

Table 3 PTDFs for the four considered states in the 3-node network, when investing in line 1 onlya. Case of investment in interval [84 105] MW

Case of investment in interval [105 140] MW

0.333 0 0.686 0.343 0 0.727 0.363 0.667 0 0.314 0.657 0 0.273 0.636 0.333 0 −0.314 0.343 0 −0.273 0.363

0 0.774 0.387 0 0.226 0.613 0 −0.226 0.387

Case of investment in interval [70 84] MW

Case of no link investment 0 0 0

0.667 0.333 −0.333

a For each case, the matrix represents each element of the PTDF matrix, where the rows correspond to lines and columns correspond to injection nodes.

Table 4 Optimal market clearing values given the solutions of stages 1 and 2 in the 3-node network. Node

Profits [$]

Available capacity for each GENCO [MW]

LMP [$/MWh]

Production [MWh]

1 2 3

2295.84 0 0

508 300 300

24 24 24

508 101.5 140.5

Table 5 Optimal values of the problem for stage 1 of the 3-node network. Case

Cost for the transmission planner [$]

Line capacity [MW]

Available capacity [MW] g1 g2 g3

‘1 ‘2 ‘3 ‘1 ; ‘2 &‘3

14,548.16 14,548.16 15,140.96 14,264

138 (‘1 ) 138 (‘2 ) 70 (‘3 ) 100 (‘1 ), 140 (‘1 ), 105 (‘1 ), 135 (‘1 ),

508 508 428 540 540 540 540

140 100 135 105

(‘2 ) (‘2 ) (‘2 ) (‘2 )

300 300 300 300 300 300 300

300 300 300 300 300 300 300

solve the stage 1 problem using what we call an optimistic solution for the transmission planner, which considers that the transmission planner anticipates the best (from the social-welfare viewpoint) EPEC equilibria. The reader should note that there is also a pessimistic solution for the transmission planner, which considers the worst EPEC equilibria.

Fig. 7 depicts the whole range of equilibria seen by the transmission planner (in the case of investing in line 1 only), where there exist optimistic (lower part of Fig. 7) and pessimistic (upper part of Fig. 7) equilibria, according to the energy dispatch plus the investment costs incurred by the transmission planner. The values in this figure are obtained by solving the EPEC problem for the stage 2, where the line-1 investment is settled up and discretized using 1000 values, ranging from 70 MW of capacity to 140 MW. We observe that the optimal optimistic solution consists of investing 68 MW, with a total cost of $14,548, and the optimal pessimistic solution is to invest 6 MW, with a total cost of $16,156, corresponding to the EPEC with the highest cost of dispatch. Based on the 3-node network case study, we add a new node and a new line. Fig. 8 shows the 4-node system. Lines 1 to 3 have already been built (although they can still be expanded) and line 4 can be built by the transmission planner. Table 6 provides the data for the 4 nodes and Table 7 shows the data for the lines and the investment options. We solve the stage-1 problem considering potential expansions of lines ‘1 ; ‘2 and ‘3 and/or investment in the new line ‘4 . Solutions are shown in Table 8. In this case, we use a 4-step discretization for the equivalent impedance of the existing lines ( ‘1 ; ‘2 ; and ‘3 ) and a 6-step discretization for the equivalent impedance of the potential new line (‘4 ). From Table 8, we show six optimistic optimal solutions, but there are infinite solutions with identical total costs for this problem (for instance, investing 100 MW in line 4; or 70 MW in line 1 and 30 MW in line 2; or 35 MW in line 1, 15 MW in line 2 and 50 MW in line 4). There are infinite solutions for transmission investments because unit line investment costs are identical, there is network symmetry and the production cost functions of all the generation units are identical. For all of these infinite solutions, the second stage has the same generation capacity expansion equilibrium. Note that the transmission planner anticipates the second stage equilibrium by stimulating the expansion of the GENCO in node 1 (which is the more socially-efficient generation expansion possibility). This is done by expanding 100 MW of transmission capacity among lines 1, 2 and 4, using any combination. All case studies have been solved using CPLEX 11 under GAMS (Brooke et al., 2003). We have used an Intel Core Duo E7500 computer at 2.93 GHz and 4 GB of RAM. Table 9 shows the running times and computational complexity required for solving the problems. The second to fifth columns show the 3-node network CPU times and

Dispatch plus line investment cost [Thousands of dollars]

17

16.5 16.16 16

15.5

15

14.55

14 70

76

80

90

100

110

120

130

Capacity of line 1 Fig. 7. Optimistic and pessimistic stage-1 solutions for the case of investing only in line 1.

138

D. Pozo et al. / Energy Economics 36 (2013) 135–146 Table 8 Optimal values of the problem for stage 1 of the 4-node network.

2

3

1

1

4

7. Conclusions We propose a 3-stage mixed integer linear program model where a transmission planner decides on the first stage upon the best line investments given the optimistic pure Nash equilibria in generation investment, second stage, and market clearing, third stage. The transmission expansion model anticipates the decisions on generation investment à la Stackelberg, where we set the equilibrium on generation investment using an EPEC framework. This anticipative transmission plan is important because (i) it may induce more sociallyefficient (and/or environmentally-convenient) generation capacity investments and (ii) it may help to mitigate the market power exercised by GENCOs through their generation capacity investment decisions. We also propose an approximation of the line impedance values as a function of the installed transmission capacity. This assumption allows us to incorporate the changes in the operation due to the new topology resulting from line investments within the transmission planning stage. Several case studies, including investment in existing and new lines, illustrate the methodology proposed. Finally, although the proposed methodology has several advantages, it is important to recall its limitations. First, the model used here is static. This fact does not represent the dynamic nature of investments, but this assumption is made due to tractability issues. Secondly, we assume perfect competition of the electricity market and inelastic demand (in order to deal with convex problems), but the reader should be aware of the possibility that GENCOs exercise Table 6 4-node example data.

1 2 3 4

‘1

‘2

‘3

‘4

g1

Available capacity [MW] g2

g3

g4

20,264

70 140 100 70 100 105

70 100 140 100 70 85

70 70 70 70 70 70

100 0 0 70 70 50

540

300

300

300

Table 9 CPU times and computational complexity of the 3- and 4-node networks.

computational complexity for the cases shown in Table 5. The sixth column shows the CPU time and the computational complexity to solve the case in Table 8.

i

Line capacity [MW]

2 Fig. 8. 4-node case study.

Node

Cost for the transmission planner [$]

3

4

145

Demand

Generation units: production cost parameters

Unit cost of investment

di [MW]

gi0 [MW]

ai [$/MWh]

bi [$/(MW·MWh)]

Ki [$/MW]

300 250 200 250

300 300 300 300

25 24 24 24

0.06 0.06 0.06 0.06

15 15 15 15

Table 7 4-node example line data. Line

Initial thermal limit capacity

Unit transmission investment cost

Maximum thermal limit capacity

‘

fl0 [MW]

Kl [$/MW]

flmax [MW]

1 2 3 4

70 70 70 0

35 35 35 35

140 140 140 140

3-node network

CPU time # of binary variables # of positive variables # of free continuous variables # of inequality constraints # of equality constraints

4-node network

‘1

‘2

‘3

‘1 ; ‘2 &‘3

3.84 s 607 670 2103

4.17 s 607 670 2103

5.31 s 607 670 2103

140.13 s 36 min 42 s 667 1316 672 1152 28,563 200,260

14,603 14,603 14,603 155,727 5834 5834 5834 85,214

‘1 ; ‘2 ; ‘3 &‘4

1,077,204 599,954

market power in the market operation as well. And thirdly, our model considers that transmission capacity investments are continuous variables, although we recognize that they are lumpy due to scale economies. In this sense, the numerical results of our model should be taken as approximations of the transmission capacities to be added to the network in order to induce the desired response by generation capacity investors. Appendix A. Supplementary data Supplementary data to this article can be found online at http:// dx.doi.org/10.1016/j.eneco.2012.12.007. References Alguacil, N., Motto, A., Conejo, A.J., 2003. Transmission expansion planning: a mixedinteger LP approach. IEEE Trans. Power Syst. 18 (3), 1070–1077. Binato, S., Pereira, M., Granville, S., 2001. A new benders decomposition approach to solve power transmission network design problems. IEEE Trans. Power Syst. 16 (2), 235–240. Borenstein, S., Bushnell, J., Stoft, S., 2000. The competitive effects of transmission capacity in a deregulated electricity industry. RAND J. Econ. 31 (2), 294–325. Brooke, A., Kendrick, D., Meeraus, A., Raman, R., 2003. GAMS/CPLEX 9.0. User Notes. GAMS Development Corp, Washington, DC. California Independent System Operator (CAISO), 2004. Transmission Economic Assessment Methodology. Available at www.caiso.com/docs/2003/03/18/ 2003031815303519270.html. Contreras, J., Wu, F., 1999. Coalition formation in transmission expansion planning. IEEE Trans. Power Syst. 14 (3), 1144–1152. Contreras, J., Wu, F., 2000. A kernel-oriented algorithm for transmission expansion planning. IEEE Trans. Power Syst. 15 (4), 1434–1440. de la Torre, S., Conejo, A.J., Contreras, J., 2008. Transmission expansion planning in electricity markets. IEEE Trans. Power Syst. 23 (1), 238–248. Dusonchet, Y., El-Abiad, A., 1973. Transmission planning using discrete dynamic optimization. IEEE Trans. Power App. Syst. PAS-92 (5), 1358–1371. Fortuny-Amat, J., McCarl, B., 1981. A representation and economic interpretation of a two-level programming problem. J. Oper. Res. Soc. 32, 783–792. Gallego, R., Monticelli, A., Romero, R., 1998. Transmission system expansion planning by an extended genetic algorithm. IEE Proc. Gener. Transm. Distrib. 145 (3), 329–335. Garcés, L.P., Conejo, A.J., García-Bertrand, R., Romero, R., 2009. A bilevel approach to transmission expansion planning within a market environment. IEEE Trans. Power Syst. 24 (3), 1513–1522. Garver, L., 1970. Transmission network estimation using linear programming. IEEE Trans. Power App. Syst. PAS-89 (7), 1688–1697. Kreps, D., Scheinkman, J., 1983. Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell J. Econ. 14 (2), 326–337. Luenberger, D.G., Ye, Y., 2008. Linear and Nonlinear Programming, 3rd ed. Springer, New York.

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