Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions

European Journal of Operational Research 174 (2006) 457–479 www.elsevier.com/locate/ejor

O.R. Applications

Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions Raquel Garcı´a-Bertrand a, Antonio J. Conejo b

a,*

, Steven Gabriel

b

a Department of Electrical Engineering, Universidad de Castilla—La Mancha, Ciudad Real, 13071 Spain Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742, USA

Received 14 October 2003; accepted 21 March 2005 Available online 23 June 2005

Abstract This paper provides a tool to determine the near-equilibrium of an electric energy market. This market works under locational marginal pricing, i.e., generating units and demand loads are paid and pay, respectively, the locational marginal prices corresponding to the nodes they are connected to. The near-equilibrium is defined as the energy transaction levels for which generating companies maximize their respective profits and consumption companies maximize their respective utilities. An independent system operator clears the market maximizing the social welfare. Conditions that ensure minimum profit for generating units can be included. However, these conditions may render a generating unit uncompetitive and expel it from the market. Demands are taken to be non-constant and values are determined as part of the solution. The near-equilibrium is obtained through the solution of a mixed-integer quadratic problem equivalent to a mixed linear complementarity problem that includes the minimum profit conditions. It is important to note that the near-equilibrium concept presented in this paper does not solve a market equilibrium when indivisibilities such as start up costs or the like are present. Lastly, we validate the proposed model on a case study using data from the IEEE Reliability Test System.  2005 Elsevier B.V. All rights reserved. Keywords: Quadratic programming; Market equilibrium; Pool-based electric energy market; Linear complementary problem; Minimum profit conditions

1. Introduction This paper analyzes a competitive market near-equilibrium for an electric energy market in a single period of time, typically an hour. The market includes generating companies (GENCOs), energy service *

Corresponding author. Tel.: +34 926 295 433; fax: +34 926 295 361. E-mail address: [email protected] (A.J. Conejo).

0377-2217/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.03.037

458

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

companies (ESCOs) that include retailers and consumption companies, as well as an independent system operator (ISO) [1–4]. The market has a pool format in which each GENCO bids a set of energy production blocks and their corresponding minimum selling prices, and each ESCO bids a set of consumption energy blocks and their corresponding maximum selling prices. In turn, the ISO clears the market seeking maximum social welfare. The type of market under consideration uses the concept of locational marginal pricing (LMP) [5]. In this scheme, a generating unit injecting power at a given node is paid the locational marginal price corresponding to that node; and conversely, a demand receiving power from a given node pays the locational marginal price corresponding to that node. Mature electricity markets such as the New England ISO [6] or PJM [7] in the US use this locational marginal pricing scheme. We consider a detailed network representation of the electric power system, consisting of a linearized power flow model that includes a precise representation of losses [8]. This linear model permits one to compute locational marginal prices while taking into account the effects of line congestion and transmission losses in an accurate and efficient manner. Contrary to what may happen if a Lagrangian decomposition approach is used [9,10], the proposed model includes constraints that force any generating unit that operates to meet a pre-specified minimum profit. These minimum profit requirements are used in some markets to promote generation capacity investments [11]. However, these conditions may render a generating unit uncompetitive and expel it from the market. In the time frame of the model that we consider, the market near-equilibrium is defined as the level of energy transactions that results in: • maximum profit for every individual GENCO, • maximum utility for every individual ESCO, • maximum net social welfare for the ISO. As such, the GENCOs and ESCOs are modeled as solving appropriate optimization problems with locational marginal prices as inputs. The LMPs are then determined as the dual prices to balance constraints as part of an optimization problem faced by the ISO. Using realistic simplifications concerning lines losses and power flows, each of these three sets of optimization problems results in a linear program (LP). Thus, the Karush–Kuhn–Tucker (KKT) optimality conditions [12] are both necessary and sufficient for describing optimal points. Consideration of the KKT conditions for these three sets of LPs in conjunction with LMPs computed as the appropriate dual variables in the ISO problem, results in a mixed linear complementarity problem (LCP) to be solved to determine the market near-equilibrium. We obtain the solution of this mixed LCP as the optimal solution of an equivalent quadratic problem [13,14]. Minimum profit conditions for generating units that declare such conditions are linearized using SchurÕs decomposition [15] and binary variables [16] and included as constraints of the quadratic problem, turning the problem into a mixed-integer quadratic problem. We describe numerical results on a test network. As compared to the optimal power flow (OPF) model [17–19] in which total system costs are minimized subject to various technical constraints such as power balance for each node, the LCP approach allows modeling the activities of the various market participants and what is more relevant, allows including minimum profit conditions. Current research seeks to identify and compute multi-period near-equilibrium through the use of decomposition techniques. The electricity market near-equilibrium as defined in this paper is computed once the bidding stacks of every unit of each producer and the bidding stacks of every consumer are known. Note also that price bids by producers as reflected in the ISO problem may not coincide with the actual cost values represented in the problem of each producer. In the case study price bids and costs are not differentiated for simplicity. A similar consideration applies to consumers.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

459

Quadratic cost/bidding curves can be straightforwardly incorporated in the model. However, most markets require piece-wise linear bidding curves and this is why this paper uses such piece-wise linear curves. This paper can be seen as an extension to the previous works by Hobbs [20] and Boucher and Smeers [21] in three respects. First, the model in the current paper is more general in that it includes step demand functions and can also include the continuous demand function considered in [20,21]. Second, the current paper includes a detailed representation of the transmission network including the effects of both congestion and losses; losses were not considered in [20,21]. Third, the current paper includes minimum profit conditions that ensure a minimum profit for any generating unit that declares such a requirement and is not expelled from the market. By contrast, these conditions are not included in either [20] or [21] and including these constraints is a relevant modeling advance not seen before due to its computational challenges. It is important to discuss these minimum profit conditions in somewhat more depth since they are computationally challenging yet realistic aspects of electricity markets. For clarity we describe the market clearing procedure of the pool-based electric energy market of mainland Spain which uses these minimum profit conditions. Hours are considered one at a time and a simple auction mechanism is used to identify the clearing price for each hour as the intersection of the stepwise increasing production stack and the stepwise decreasing demand stack. Inter-temporal coupling due to ramping constraints is taken into account using a myopic mechanism that conditions for each unit the power available in 1 hour to the power output in the previous or the next hour. Once the 24 hours have been processed, minimum profit constraints imposed by generating units are checked. If one or some of these constraints are violated, the unit with the higher violation is expelled from the market (not to be readmitted) and the whole procedure rerun. The clearing concludes once all units declaring minimum profit constraints meet these constraints. The model in this paper is formulated to achieve market clearing subject to units not meeting profit constraints being expelled. There may be multiple equilibria of this type, and if this is the case, the solution of this model may or may not correspond to the solution of the Spanish market heuristic just described. From this point of view, these minimum profit conditions can be considered ‘‘outside’’ of any of the generator, consumer, or ISO optimization problems discussed below. This is the motivation for adding them later, after the market equilibrium is presented, first as a linear complementarity problem and then as an equivalent quadratic program. Moreover, even if they could be included for example, as part of the producer or ISO problem, the optimality conditions for this problem would not apply due to the binary nature of these constraints. This is a more general issue which researchers have struggled with over the years, namely, solving a market near-equilibrium problem for which there are integer or more generally non-convex constraints. One must take great care to show that such an near-equilibrium actually makes sense in this setting. In dealing with problems with integer constraints, one approach adopted by OÕNeill et al. [22] is to solve the related optimization problem to optimality, then add constraints forcing the integer variables to be at the optimal levels. The result is a linear (or convex) program which has a defendable interpretation for an equilibrium. The LCP formulation presented in this paper is a useful tool to various market participants. In particular, the market regulator benefits by identifying market equilibria as part of its market monitoring. Additionally, this competitive market model can assist generating and consuming companies in their planning process. The rest of this paper is organized as follows. Section 2 provides the notation used throughout the paper. In Section 3, the problems of the market participants are modeled. Section 4 provides the formulation of the market near-equilibrium problem. In Section 5, the effect of imposing minimum profit conditions in the market equilibrium is discussed. In Section 6, a realistic case study is analyzed and relevant results are reported. Section 7 provides several conclusions of interest. Finally, the linearization of the minimum profit condition can be found in the Appendix A.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

460

2. Notation The main mathematical symbols used throughout this paper are classified below for quick reference. Variables PDjk power block k that demand j is willing to buy at price kDjk ð2Þ P Djk power block k that demand j is willing to buy at price kDjk. This variable is equal to PDjk and it is used in the problem of the ISO PGib power block b that generating unit i is willing to sell at price kGib ð2Þ P Gib power block b that generating unit i is willing to sell at price kGib. This variable is equal to PGib and it is used in the problem of the ISO P loss power loss in line n  m nm xi on-line status for generating unit i (1 if generating unit i is on and 0 otherwise) dþ nm ð‘Þ positive part of the voltage angle difference block ‘ between nodes n and m d nm ð‘Þ negative part of the voltage angle difference block ‘ between nodes n and m qn(i) locational marginal price corresponding to generating unit or demand i that is located at node n Dual variables cnm dual variable Cþ dual variable nm C dual variable nm rj dual variable /ib dual variable ujk dual variable

associated associated associated associated associated associated

with the transmission capacity constraint of line n  m with the upper bound of the positive part of voltage angle block, dþ nm ð‘Þ with the upper bound of the negative part of voltage angle block, d nm ð‘Þ with demand j minimum with the maximum capacity limit for the block b of the generating unit i with the maximum capacity limit for the block k of the demand j

Constants Bnm susceptance of line n  m Gnm conductance of line n  m Ki positive constant that represents the minimum profit for generating unit i P min minimum power supplied to demand j Dj P max maximum power consumed in block k of demand j Djk P max maximum power output of generating unit i Gi max P Gib maximum power output of block b of generating unit i P max transmission capacity limit of line n  m nm kBDjk price bid by demand j to buy power block k U kDjk utility ($/MWh) associated to power block k of demand i B kGib price bid by generating unit i to sell power block b kC production cost of power block b of generating unit i Gib Dd upper bound on the piece-wise angle blocks Sets and numbers n(i) indicates that generating unit or demand i is at node n N total number of nodes ND number of demands NDj number of blocks requested by demand j NG number of generating units

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

NGi Xn hn #n

461

number of blocks bid by generating unit i set of indexes of nodes connected to node n set of indexes of generating units at node n set of indexes of demands at node n

3. Modeling of market participants 3.1. The problem of a generating company We first consider a generating company owning NG power units. We assume that its market activity is to maximize profit and the following linear program problem is considered: maximize P Gib

NG X N Gi X i¼1

ðqnðiÞ  kCGib ÞP Gib ;

ð1Þ

b¼1

subject to: P max Gib P P Gib : /ib P Gib P 0

8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi ;

8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi .

ð2Þ ð3Þ

We note that the decision variables of this problem are the amounts of power to be generated by generating unit i in block b, i.e., PGib and that the prices qn(i) are fixed values for the generating unit but variables in the larger near-equilibrium problem. For convenience, the notation used is stated in Section 2. The objective function (1) represents the total profit of all units of the generating company which is to be maximized subject to a maximum capacity limit (2) for each block of each power unit as well as nonnegative levels of power to be generated by unit i in block b (3). The dual variables of the constraints in (2) are given by /ib.1 The optimality conditions for this problem are to find generation power blocks levels PGib and dual variables /ib such that 0 6 kCGib  qnðiÞ þ /ib þ Mð1  xi Þ ? P Gib P 0 8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi ; 06

P max Gib

 P Gib ? /ib P 0

8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi .

ð4Þ ð5Þ

The meaning of the term M(1  xi) in Eq. (4) is explained in Section 3.4. Note that M is a large enough positive constant. The symbol ? indicates that the product of the variable used to derive the corresponding optimality condition and the associated equation must be zero, i.e., x ? y is the same as xTy = 0. We note that these optimality conditions are independent for each unit. 3.2. The problem of a consuming company Next, we consider a consuming company in which ND demands for electricity are made. We assume that such a company can be modeled as maximizing its economic utility resulting in the following linear programming formulation: 1

The dual variables to constraints appear to the right of these constraints in the problem formulations.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

462

maximize P Djk

N Dj ND X X j¼1

ðkU Djk  qnðjÞ ÞP Djk ;

ð6Þ

k¼1

subject to: N Dj X

P Djk P P min Dj : rj

k¼1 P max Djk

ð7Þ

8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj ;

P P Djk : ujk

P Djk P 0

8j ¼ 1; . . . ; N D ;

8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj .

ð8Þ ð9Þ

We note that unlike in the generating companyÕs problem where there are machine constraints on the maximum electricity that can be generated, for the consuming company no such explicit upper bounds on the demands are needed. However, as shown below via the power balance constraints (14), it can be shown that there is an implied upper bound on the demands. The objective function (6) represents the sum of the economic utilities for each individual demand of the consuming company. Eq. (7) represents the minimum demands that must be supplied with the dual variable of this equation being rj. Eq. (8) represents the maximum power consumed in each block of each demand and its dual variable is ujk. Eq. (9) imposes that the power to be consumed by demand j in block k is nonnegative. The optimality conditions for this optimization problem are independent for each demand and can be formulated as finding the demand power blocks PDjk and dual variables rj, ujk such that 0 6 qnðjÞ  kU Djk  rj þ ujk ? P Djk P 0 06 06

N Dj X

P Djk  P min Dj ? rj P 0

k¼1 P max Djk

 P Djk ? ujk P 0

8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj ;

8j ¼ 1; . . . ; N D ;

ð10Þ ð11Þ

8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj .

ð12Þ

3.3. The problem of the ISO Lastly, the ISO clears the market seeking maximum social welfare. The problem of the ISO is formulated as the following linear program: maximize

ð2Þ ð2Þ  P Gib ;P Djk ;dþ nm ð‘Þ;dnm ð‘Þ

N Dj ND X X j¼1

ð2Þ

kBDjk P Djk 

k¼1

NG X N Gi X i¼1

ð2Þ

kBGib P Gib ;

ð13Þ

b¼1

subject to: N Gi XX

ð2Þ

P Gib 

i2hn b¼1

N Dj XX j2#n k¼1

ð2Þ

P Djk 

X m2Xn

Bnm

L X

 ðdþ nm ð‘Þ  dnm ð‘ÞÞ

‘¼1

" # L X 1 X þ  Gnm Dd ð2‘  1Þðdnm ð‘Þ þ dnm ð‘ÞÞ ¼ 0 : qnðiÞ  2 m2Xn ‘¼1 8n ¼ 1; . . . ; N ;

X m2Xn

Bnm

L X ‘¼1

 max ðdþ nm ð‘Þ  dnm ð‘ÞÞ 6 P nm : cnm

ð14Þ 8n ¼ 1; . . . ; N ; 8m 2 Xn ;

ð15Þ

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479 ð2Þ

P Gib  P Gib ¼ 0 : lGib

8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi ;

463

ð16Þ

ð2Þ P Djk  P Djk ¼ 0 : mDjk 8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj ; þ 8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L; Dd P dþ nm ð‘Þ : Cnm   Dd P dnm ð‘Þ : Cnm 8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L; dþ 8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L; nm ð‘Þ P 0 d ð‘Þ P 0 8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L. nm

ð17Þ ð18Þ ð19Þ ð20Þ ð21Þ

The objective function (13) is the social welfare [23] and it is subject to enforcing power balance at every node (14), line capacity limits (15), that the power generated and demanded in the problem of the ISO are equal to the power generated and demanded in the problems for each generating company and each consuming company, (16) and (17), respectively, and upper and lower bounds on the piece-wise angle blocks (18)–(21). Note that power generation and demand variables are replicated to make the problems of producers and consumers compatible with the problem of the ISO. Note also that (16) and (17) can be used to eliminate ð2Þ ð2Þ variables P Gib and P Djk from (13) and (14), resulting in a constant value of the objective function. As a result, the ISOÕs problem reduces to a load flow problem whose purpose is find a feasible set of dþ nm ð‘Þ and d ð‘Þ so that flow from generators to loads satisfies all transmission constraints. nm For every node, Eq. (14) includes four terms: generation injected in the node, demand extracted from the node, power reaching the node from adjacent nodes and losses. The linearized version of losses included in Eq. (14) is similar to the one presented inP[8], where the convexity and good computational behavior of this  linearization is proved. The term Gnm Dd L‘¼1 ð2‘  1Þðdþ nm ð‘Þ þ dnm ð‘ÞÞ represents losses in line n  m. Note þ  that this expression is linear and that variables dnm ð‘Þ and dnm ð‘Þ are used to represent the absolute value of dnm(‘), and that Dd(2‘  1)Prepresents the value of the slope of the ‘th block of voltage angle difference L (anm(‘)); that is, P loss nm ¼ Gnm ‘¼1 anm ð‘Þjdnm ð‘Þj. The last two expressions are linear approximations of losses 2 loss in line n  m expressed as P nm ¼ Gnm ðdnm Þ . Moreover, losses in line n  m can be interpreted as additional demands in nodes n and m, which are divided equally between node n and m. For every line, Eq. (15) states that the power flowing through any line (in either of the two directions) should be below a security bound. In the context of an optimal power flow, the pioneering reference [5] proposes a formulation similar to (13)–(21), although nonlinear. In an equilibrium modeling context, the loss-affected piece-wise formulation (13)–(21) is similar to the one used in [24]. ð2Þ The optimality conditions for problem (13)–(21) are to find the generation power blocks levels P Gib , the ð2Þ þ demand power blocks levels P Djk , the positive part of the voltage angle difference blocks dnm ð‘Þ, the negative þ  part of the voltage angle difference blocks d nm ð‘Þ and dual variables qn(i), cnm, lGib, mDjk, Cnm and Cnm such that 0 ¼ kBGib  qnðiÞ þ lGib

8i ¼ 1; . . . ; N G ; 8b ¼ 1; . . . ; N Gi ;

ð22Þ

kBDjk

þ mDjk 8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj ; 0 ¼ qnðjÞ    1 þ 0 6 qnðiÞ Bnm þ Gnm Ddð2‘  1Þ þ cnm Bnm þ Cþ 8n ¼ 1; . . . ; N ; 8m 2 Xn ; nm ? dnm ð‘Þ P 0 2

ð23Þ

8‘ ¼ 1; . . . ; L;   1  0 6 qnðiÞ Bnm þ Gnm Ddð2‘  1Þ  cnm Bnm þ C nm ? dnm ð‘Þ P 0 2

ð24Þ

8‘ ¼ 1; . . . ; L;

8n ¼ 1; . . . ; N ; 8m 2 Xn ; ð25Þ

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

464 N Gi XX

ð2Þ

P Gib 

i2hn b¼1

N Dj XX

ð2Þ

P Djk 

X m2Xn

j2#n k¼1

"

Bnm

L X  ðdþ nm ð‘Þ  dnm ð‘ÞÞ ‘¼1

# L X 1 X þ  Gnm Dd ð2‘  1Þðdnm ð‘Þ þ dnm ð‘ÞÞ ¼ 0  2 m2Xn ‘¼1

0 6 P max nm 

X

Bnm

m2Xn ð2Þ

P Gib  P Gib ¼ 0 P Djk 

ð2Þ P Djk

0 6 Dd  0 6 Dd 

¼0

dþ nm ð‘Þ d nm ð‘Þ

L X

8n ¼ 1; . . . ; N ;

 ðdþ 8n ¼ 1; . . . ; N ; 8m 2 Xn ; nm ð‘Þ  dnm ð‘ÞÞ ? cnm P 0

ð26Þ ð27Þ

‘¼1

8i ¼ 1; . . . ; N G ;

8b ¼ 1; . . . ; N Gi ;

ð28Þ

8j ¼ 1; . . . ; N D ; 8k ¼ 1; . . . ; N Dj ;

ð29Þ

? ?

Cþ nm C nm

P0

8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L;

ð30Þ

P0

8n ¼ 1; . . . ; N ; 8m 2 Xn ; 8‘ ¼ 1; . . . ; L.

ð31Þ

Free dual variables, qn(i), lGib and mDjk, are associated to Eqs. (26), (28) and (29), respectively. 3.4. Minimum profit condition Next, we consider the minimum profit condition below for each generating unit that declares such condition. This minimum profit condition can be used to internalize fixed and other costs that do not directly appear in the bidding stack. N Gi X

ðqnðiÞ  kCGib ÞP Gib P K i xi  Mð1  xi Þ 8i ¼ 1; . . . ; N G

ð32Þ

b¼1

such that PN Gi xi P

b¼1 P Gib ; P max Gi

xi 2 f0; 1g 8i ¼ 1; . . . ; N G ;

ð33Þ

where Ki is a positive constant that represents minimum profit for generating unit i, xi is the on-line status for generating unit i and M is a large enough positive constant. Note that conditions (32) and (33) either enforce minimum profit (xi = 1) or expel the generating unit from the market (xi = 0). If a unit is expelled from the market (PGib = 0), constraints (5) are satisfied with /ib equal to zero; and constraint (4) is satisfied as a result of including the additional term M(1  xi). It should be noted that Eq. (32) is nonlinear because the left-hand side is the sum of bilinear terms. However, this condition can be closely approximated in a linear fashion through SchurÕs decomposition of the Hessian of the quadratic terms in the left-hand side of (33) and the use of binary variables as explained in Appendix A. Minimum profit constraints such as (32) and (33) are used in some actual markets [11] to ensure peaker profitability and to promote generation capacity investment.

4. Formulation of the market near-equilibrium problem The mixed-linear complementarity problem defined by the optimality conditions for the problem of any generating company, Eqs. (4) and (5), for the problem of any consuming company, Eqs. (10)–(12), and for the problem of the ISO, Eqs. (22)–(31), that determine the market equilibrium can be written in compact form as

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

465

qð1Þ þ M ð11Þ zð1Þ þ M ð12Þ zð2Þ P 0; ð2Þ

þM

ð1Þ

P 0;

q z

ð1Þ T

ðz Þ ðq

ð21Þ ð1Þ

ð1Þ

z

þM

ð22Þ ð2Þ

z

ð34Þ

¼ 0;

ð35Þ ð36Þ

þM

ð11Þ ð1Þ

þM

z

ð12Þ ð2Þ

z Þ ¼ 0;

ð37Þ

where M(11) and M(22) are real square matrices of size n and m, respectively, M ð12Þ 2 R , M ð21Þ 2 Rmxn , qð1Þ 2 Rn , qð2Þ 2 Rm and the variables are zð1Þ 2 Rn and zð2Þ 2 Rm . Note that the variable vector z(2) is not restricted to be nonnegative and corresponds to locational marginal prices, qn(i), and dual variables lGib and mDjk. The variable vector z(1) is nonnegative and corresponds to the rest of variables of the problem. The conditions shown above assume that the particular generators have been selected a priori by some mechanism. In the sequel, we provide minimum profit constraints which expel certain generators from the market if the minimum profit is not achieved. Thus, the near-equilibrium that results is in fact a restricted market equilibrium in which these minimum profit conditions (32) and (33) determine which generators participate in the market. The mixed LCP (34)–(37) can be solved using an equivalent quadratic problem [13,14], which is nxm

minimize zð1Þ ;zð2Þ

T

ðzð1Þ Þ ðqð1Þ þ M ð11Þ zð1Þ þ M ð12Þ zð2Þ Þ;

ð38Þ

qð1Þ þ M ð11Þ zð1Þ þ M ð12Þ zð2Þ P 0;

ð39Þ

subject to :

q

ð2Þ

þM

ð21Þ ð1Þ

z

þM

ð22Þ ð2Þ

z

¼ 0;

zð1Þ P 0.

ð40Þ ð41Þ

Note that piecewise linear losses (Section 3.3) are used to obtain a mixed-integer quadratic programming problem, which requires all the constraints to be linear. If no minimum profit condition is imposed, the solution of problem (38)–(41) provides a market equilibrium. If, on the other hand, minimum profit conditions are considered, problem (38)–(41) should be extended to include linear constraints representing the minimum profit conditions. These constraints, which include binary variables are explained in Appendix A and shown as (49)–(61). The solution of the mixedinteger quadratic programming problem (38)–(41), (33) and (49)–(61) provide the market near-equilibrium under minimum profit conditions. Note that the objective function value at the optimal solution should be zero corresponding to an actual LCP solution. In any case, profit for each generating unit, utility for each demand and social welfare are jointly maximized. We define near-equilibrium as the optimal solution of problem (38)–(41) with constrains (33), (49)–(61) that results in an optimal objective function value slightly different from zero, which implies that one or some of the optimality conditions of the equilibrium problem (34)–(37) are slightly not satisfied.

5. On uniqueness, degeneracy and infeasibility In the following, we discuss the effect of imposing constraints (33), (49)–(61) on problem (38)–(41). There are two cases for problem (38)–(41) that are considered below. Case 1. Problem (38)–(41) is degenerate and has multiple prices (dual solutions). This case is shown in Fig. 1(a). Note that there is a range of prices at which the power supplied equal the power demanded. In this situation, adding constraints (33), (49)–(61) to problem (38)–(41) may results in a feasible problem whose optimal solution meets the minimum profit conditions.

466

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479 Demand

Demand

Δ price

Δ power Δ power

Supply

Supply

Price

Price

Power

a. Degeneracy

Power

b. Near-degeneracy

Fig. 1. Degeneracy and near-degeneracy for problem (38)–(41).

Case 2. Problem (38)–(41) has a unique solution in prices, generations, demands and flows. By adding constraints (33), (49)–(61) to problem (38)–(41), we simply create infeasibilities, assuming that the minimum profit condition was not attained for this unique solution beforehand. However, it should be noted that for practical applications, these infeasibilities are generally negligible. The reason for that is as follows. Power supply curves tend to be ‘‘hockey-stick’’ shaped around the market clearing price in most practical markets, while demand curves are rather inelastic around the market clearing price. The supply curve has a hockey-stick shape because many bids are made at zero price to ensure acceptance. The demand curve is rather inelastic due to the nature of electricity consumption. Moreover, the number of steps in the supply curve is usually large (72 units times 25 steps each results in 1800 steps for the day-ahead market of mainland Spain). The above can be observed in the demand and supply curves shown in Fig. 2 that correspond to the electric energy market of mainland Spain (January 14, 2004, hour 9). The conclusion is that market clearing is often in the vicinity of these ‘‘near-degeneracy’’ regions in which there is a unique equilibrium albeit with steep supply and demand curves as illustrated in Fig. 1(b). In this figure, it can be observed that small increments in power (which create small infeasibilities) result in significant price differences (which allow meeting minimum profit constraints). Small infeasibilities cause an optimal objective function value slightly different from zero in problem (38)–(41) plus constraints (33), (49)–(61). This can be seen in Fig. 5 for the case study presented. These small infeasibilities are related to prices because power balance is enforced in every bus. The cost incurred due to price infeasibilities is allocated pro rata among market participants.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

467

20 Demand curve Supply curve

18

Price [ /kWh]

16 14 12 10 8 6 4 2 0

0

0.5

1

1.5

2

2.5

Power [MW]

3

3.5

4

4.5 4 x 10

Fig. 2. Demand and supply curves in the electric energy market of mainland Spain: January 14, 2004, hour 9.

Thus, in general, the optimal solution of problem (38)–(41) plus constraints (33), (49)–(61) represents a near-equilibrium that may include small complementary infeasibilities of negligible practical significance. An appropriate metric of the importance of such infeasibilities is the optimal value of the objective function (38). The closer to zero is the optimal value of (38), the closer to complementary feasibility is problem (38)– (41) with constraints (33), (49)–(61). Nevertheless, the significance of infeasibilities due to minimum profit conditions will be investigated for a range of realistic cases in future work.

6. Case study A case study based on the IEEE Reliability Test System (IEEE RTS), depicted in Fig. 3, is presented in this section. Topology, line and generating unit data can be found in [25] (Fig. 1 and Tables 12 and 9, respectively, of that reference). Transmission capacity limits of the lines are also given in [25] (Table 12 of that reference). The size and price of each block of each generating unit that the producer is willing to sell are shown in Table 1. The sum of all blocks is considered to be the maximum power output. Table 2 shows the location of the generating units throughout the network as well as their respective capacities. It is assumed that every consuming company uses three blocks. The minimum power requirement of each demand and the size and price of each block of each demand are shown in Table 3. Table 4 provides equilibrium results concerning generator output, revenue, and profit for the pool-based electric power market if no minimum profit conditions are imposed for generating units. These results are obtained by directly solving problem (34)–(37). Table 5 shows the power consumed and the corresponding demand payments. The total losses are the difference between the total generator power output and the total power consumed, and are equal to 67.51 MW.

468

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

The equilibrium of the pool-based electric market provides locational marginal prices for all nodes, which are shown in Table 6. It should be noted that price differences throughout the network are small as no significant congestion occurs in this network. The next set of results is obtained considering that generating units 12–14, which are located at node 13, each imposes a minimum profit requirement of 100 €/h. This requires the solution of the problem (38)–(41), (33) and (49)–(61). Table 7 provides results concerning generator output, revenue and profit under these conditions. Note that the power output by generator matches that shown in Table 4, only the revenues and profits differ. Note also that only those generating units at nodes whose prices increase experience increments in revenues and profits. For instance, generating unit 3, located in node 1 (see Table 2), produces 76 MW in both cases (without and with minimum profit constraints on generating units 12–14), as state in Tables 4 and 7, respectively. However, the profit of unit 3 is higher in the second case (588.90 €) than in the first case (581.08 €). This is due to a change in the LMP of node 1 that increases from 20.89 €/MWh to 20.99 €/ MWh. However, if larger minimum profit constraints are imposed, both generation and demand may change with respect to the case of no minimum profit constraints and any of the generating units imposing minimum profit may be expelled from the market.

22

18 21 17

23

16

19

20

14 15

13

24

12

11

3

10

9

6 4 5

1

2

Fig. 3. IEEE reliability test system.

8

7

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

469

Table 1 Generating unit data Unit

1,2 3,4 5,6 7,8 9,10,11 12,13,14 15,16,17,18,19 20,21 22,23 24,25,26,27,28,29 30,31 32

Block 1

Block 2

Block 3

Block 4

Size (MW)

Price (€/MWh)

Size (MW)

Price (€/MWh)

Size (MW)

Price (€/MWh)

Size (MW)

Price (€/MWh)

15.8 15.2 15.8 15.2 25.0 68.95 2.4 54.25 100.0 50.0 54.25 140

29.58 11.46 29.58 11.46 18.60 19.20 23.41 9.92 5.31 0 9.92 10.08

0.2 22.8 0.2 22.8 25.0 49.25 3.6 38.75 100.0

30.42 11.96 30.42 11.96 20.03 20.32 23.78 10.25 5.38

3.8 22.8 3.8 22.8 30.0 39.4 3.6 31.0 120.0

42.82 13.89 42.82 13.89 21.67 21.22 26.84 10.68 5.53

0.2 15.2 0.2 15.2 20.0 39.4 2.4 31.0 80.0

43.28 15.97 43.28 15.97 22.72 22.13 30.40 11.26 5.66

38.75 87.5

10.25 10.66

31.0 52.5

10.68 11.09

31.0 70.0

11.26 11.72

Table 2 Generating unit locations and capacities Node

Generating unit number (unit size [MW])

1 2 7 13 15 16 18 21 22 23

1(20) 5(20) 9(100) 12(197) 15(12) 21(155) 22(400) 23(400) 24(50) 30(155)

2(20) 6(20) 10(100) 13(197) 16(12)

3(76) 7(76) 11(100) 14(197) 17(12)

25(50) 31(155)

26(50) 32(350)

4(76) 8(76)

18(12)

19(12)

20(155)

27(50)

28(50)

29(50)

Table 8 shows the power consumed and the corresponding demand payments. The losses are the difference between the total generator power output and the total power consumed and equals to 67.51 MW. The near-equilibrium of the pool-based electric market provides locational marginal prices for all nodes, which are shown in Table 9. It should be noted that these prices have increased to satisfy the minimum profit condition for generating units 12–14. Fig. 4 shows locational marginal prices throughout the network for the cases with and without minimum profit constraints. Note that the changes in locational marginal prices due to minimum profit constraints are larger in nodes where generating units imposing minimum profit constraints are located. However, these constraints have an influence throughout the system as reflected in Fig. 4. Minimum profit conditions (32) and (33) generate small complementarity infeasibilities since the objective function optimal value of problem (38)–(41) with constraints (33) and (49)–(61) is slightly above zero (1.26 · 103 per unit), as can be observed in Fig. 5. A single small infeasibility that concerns expression (35) occurs. This infeasibility is related to the optimality condition (10) for the block 2 of the consuming company 14. The infeasibility mismatch is obtained as the product of the optimality condition (10) and the corresponding dual variable:

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

470 Table 3 Demand data Node

Minimum power output (MW)

Block 1

Block 2

Block 3

Size (MW)

Price (€/MWh)

Size (MW)

Price (€/MWh)

Size (MW)

Price (€/MWh)

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

97.20 87.22 162.00 66.61 63.91 122.42 112.51 153.90 157.52 175.50 238.52 174.61 285.31 90.02 299.70 162.92 115.21

104.40 93.68 174.00 71.54 68.64 131.48 120.84 165.30 169.18 188.50 256.18 187.54 306.44 96.68 321.90 174.98 123.74

22.80 22.81 22.56 23.33 23.20 23.30 23.83 24.15 22.84 22.94 22.42 22.45 21.69 21.74 21.24 21.83 21.74

7.20 6.46 12.00 4.93 4.73 9.06 8.33 11.40 11.66 13.00 17.66 12.93 21.13 6.66 22.20 12.06 8.53

20.73 20.74 20.51 21.21 21.09 21.18 21.67 21.96 20.76 20.85 20.38 20.41 19.72 19.77 19.31 19.85 19.76

7.20 6.46 12.00 4.93 4.73 9.06 8.33 11.40 11.66 13.00 17.66 12.93 21.13 6.66 22.20 12.06 8.53

18.65 18.66 18.46 19.08 18.98 19.06 19.50 19.76 18.68 18.76 18.34 18.36 17.74 17.79 17.37 17.86 17.79

Table 4 Results for generating units; no minimum profit constraints Generating unit 1,2 3,4 5,6 7,8 9 10,11 12,13,14 15,16,17,18,19 20 21 22 23 24,25,26,27,28,29 30,31 32 Total

Power output (MW)

Revenues (€/h)

Profits (€/h)

0.00 76.00 0.00 76.00 73.60 50.00 118.20 0.00 155.00 155.00 400.00 400.00 50.00 155.00 350.00

0.00 1587.39 0.00 1582.32 1594.85 1083.50 2412.73 0.00 3020.88 3063.08 7690.02 7583.90 934.41 3020.41 6820.28

0.00 581.08 0.00 576.00 117.75 117.75 88.13 0.00 1405.39 1447.59 5504.62 5398.50 934.41 1404.92 3073.70

2902.21

57 164.90

28 177.90

0 6 q14  kU D14;2  r14 þ u14;2 ? P D14;2 P 0; 0 6 20.51  20.41  0 þ 0 ? 0.0126 P 0. The incurred cost due to this infeasibility, C, is: C ¼ ðq14  kU D14;2 ÞP D14;2 ¼ ð20.51  20.41Þ1.26 ¼ 0.126 €=h. This cost is allocated pro rata among all market participants as it is shown below:

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

471

Table 5 Results for demands; no minimum profit constraints Load

Power consumed (MW)

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20 Total

Load payments (€/h)

104.40 93.68 186.00 71.54 68.64 131.48 129.17 165.30 169.18 188.50 256.18 188.80 327.57 103.34 344.10 174.98 131.84

2180.58 1950.41 3776.89 1532.12 1474.59 2815.81 2799.11 3684.69 3522.48 3937.06 5229.21 3853.31 6384.18 2042.18 6615.34 3507.24 2605.69

2834.70

57 910.89

Table 6 Locational marginal prices; no minimum profit constraints Node

Locational marginal price (€/MWh)

Node

Locational marginal price (€/MWh)

1 2 3 4 5 6 7 8 9 10 11 12

20.89 20.82 20.31 21.42 21.48 21.42 21.67 22.29 20.82 20.88 20.70 20.70

13 14 15 16 17 18 19 20 21 22 23 24

20.41 20.41 19.49 19.76 19.49 19.23 20.04 19.76 18.96 18.69 19.49 20.12

PN Gi C P Gib PN G b¼1 PN Gi 2 i¼1 b¼1 P Gib PN Dj C P Djk C j ¼ PN D k¼1 PN Dk 2 j¼1 k¼1 P Djk

Ci ¼

8i ¼ 1; . . . ; N G ;

ð42Þ

8j ¼ 1; . . . ; N D ;

ð43Þ

where Eq. (42) represents the cost corresponding to each unit due to infeasibility, which is to be subtracted to the corresponding profit in Table 7; and Eq. (43) represents the amount paid by each demand due to infeasibility, which must be added to load payments in Table 8. Note that the infeasibility cost, C, should be subtracted from demand 14Õs payment in Table 8.

472

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

Table 7 Results for generating units; minimum profit constraints Generating unit

Power output (MW)

Revenues (€/h)

Profits (€/h)

1,2 3,4 5,6 7,8 9 10,11 12–14 15–19 20 21 22 23 24–29 30,31 32

0.00 76.00 0.00 76.00 73.60 50.00 118.20 0.00 155.00 155.00 400.00 400.00 50.00 155.00 350.00

0.00 1595.21 0.00 1590.11 1594.85 1083.50 2424.61 0.00 3020.88 3063.08 7690.02 7583.90 934.41 3020.41 6820.28

0.00 588.90 0.00 583.79 117.75 117.75 100.00 0.00 1405.39 1447.59 5504.62 5398.50 934.41 1404.92 3073.70

Total

2902.21

57 231.78

28 244.79

Table 8 Results for demands; minimum profit constraints Load 1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20 Total

Power consumed (MW)

Load payments (€/h)

104.40 93.68 186.00 71.54 68.64 131.48 129.17 165.30 169.18 188.50 256.18 188.80 327.57 103.34 344.10 174.98 131.84

2191.32 1960.02 3795.49 1539.67 1481.86 2829.68 2799.11 3684.69 3539.83 3956.45 5254.97 3872.29 6384.18 2042.18 6615.34 3507.24 2605.69

2834.70

58 060.01

To illustrate the effect of the minimum profit conditions, Table 10 shows the evolution of the locational marginal prices (for nodes exhibiting higher changes) as minimum profit conditions impose higher and higher profits for the corresponding generating units. Note that price differences throughout the network increase as higher profits are imposed by generating units. For case 1, generating units 12–14 impose a minimum profit requirement of 100 €/h, for case 2, 150 €/h, for case 3, 200 €/h, and for case 4, 300 €/h for generating units 12 and 13, and 350 €/h for generating unit 14. It should be noted that generating unit 14 is expelled from the market in Case 4 due to the overly restrictive minimum profit constraints (32) and

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

473

Table 9 Locational marginal prices; minimum profit constraints Node

Locational marginal price (€/MWh)

Price variation (%) vs. Table 6

Node

Locational marginal price (€/MWh)

Price variation (%) vs. Table 6

1 2 3 4 5 6 7 8 9 10 11 12

20.99 20.92 20.41 21.52 21.59 21.52 21.67 22.29 20.92 20.99 20.80 20.80

0.4787 0.4803 0.4924 0.4669 0.5121 0.4669 0.0000 0.0000 0.4803 0.5268 0.4831 0.4831

13 14 15 16 17 18 19 20 21 22 23 24

20.51 20.51 19.49 19.76 19.49 19.23 20.04 19.76 18.96 18.69 19.49 20.22

0.4900 0.4900 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4970

23 22.5

without minimum profit constraints with minimum profit constraints

LMP [ /MWh]

22 21.5 21 20.5 20 19.5 19 18.5 18 17.5

0

5

15

10

20

25

Node Fig. 4. Locational marginal prices throughout the network.

(33) for this case. Note that the minimum profit imposed for unit 14 is about 300% higher than its initial profit, 88.13 €/h. To illustrate the numerical behavior of the linearization of the minimum profit conditions, the optimal objective function value of problem (38)–(41), (33) and (49)–(61) considering a minimum profit requirement of 100 €/h for generating units 12–14 is plotted in Fig. 5 as a function of the number of blocks used for the linearization. Observe that the objective function optimal value of quadratic programming problem (38)– (41), (33) and (49)–(61) approaches zero more and more accurately as the number of blocks of the linearization increases. Note that the sequencing of the linear blocks for loss computation has been checked in all simulations and it is correct. Table 11 provides the CPU time required to solve problem (38)–(41), (33) and (49)–(61) considering a minimum profit requirement of 100 €/h for generating units 12–14 as the number of blocks of the linearization of the minimum profit constraints increases. It also includes, in the first row of the table, the time

474

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479 x 10

5

-3

Optimal objective function value

4.5 4 3.5 3 2.5 2 1.5 1

20

15

10

25

30

Number of blocks

Fig. 5. Evolution of optimal objective function value (per unit on a 100 MW base) of the quadratic problem with the numbers of blocks.

needed to solve problem (34)–(37), which considers no minimum profit constraint (No MPC). The computer used is a Dell PowerEdge 6600 with 2 processors at 1.60 GHz and 2 Gb of RAM memory. Problems are solved using SBB under GAMS [26].

Table 10 Evolution of locational marginal prices Node

Case 1 (€/MWh)

Case 2 (€/MWh)

Case 3 (€/MWh)

Case 4 (€/MWh)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

20.99 20.92 20.41 21.52 21.59 21.52 21.67 22.29 20.92 20.99 20.80 20.80 20.51 20.51 19.49 19.76 19.49 19.23 20.04 19.76 18.96 18.69 19.49 20.22

21.35 21.35 20.76 21.96 21.96 21.96 21.67 22.29 21.35 21.42 21.23 21.23 20.93 20.93 19.79 20.07 19.79 19.52 20.35 20.07 19.25 18.98 19.79 20.57

21.78 21.78 21.18 22.41 22.41 22.41 21.78 22.41 21.78 21.85 21.66 21.66 21.36 21.35 20.19 20.47 20.19 19.92 20.77 20.48 19.64 19.36 20.19 20.99

22.50 22.50 21.88 23.15 23.15 23.15 22.50 23.15 22.50 22.57 22.37 22.37 22.06 22.06 20.86 21.15 20.86 20.58 21.45 21.16 20.29 20.00 20.86 21.68

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

475

Table 11 CPU time Number of blocks

CPU time (seconds)

No MPC 10 15 20 25 30

0.43 46.55 40.10 81.49 128.54 128.65

7. Conclusions This paper analyzes the single-period near-equilibrium of a pool-based electric energy market working under locational marginal pricing that may include minimum profit constraints for on-line generating units. The market near-equilibrium results in maximum profit for every generating company and maximum utility for every consuming company. The market near-equilibrium tool can be seen as an efficient market-clearing mechanism in electric energy markets based on hourly auctions. A tool to identify the near-equilibrium of an electricity market is of interest for the market regulator that may use it for market monitoring. It is also of interest for generating and consuming companies to foresee their respective most appropriate strategies. Unlike optimal power flow based tools, this approach allows incorporating minimum profit constraints for on-line generating units which are relevant in actual markets and represents an important modeling advantage. Current research seeks to identify and compute multi-period near-equilibrium through the use of decomposition techniques.

Acknowledgments The Ministry of Science and Technology of Spain provided partial financial support for this work through grant CICYT DPI2003-01362. We thank the reviewers for insightful comments and suggestions.

Appendix A. Linearization of minimum profit condition Consider the minimum profit condition (44) N Gi X ðqnðiÞ  kCGib ÞP Gib P K i xi  Mð1  xi Þ 8i ¼ 1; . . . ; N G . b¼1

ð44Þ

PN Gi PN Gi ðkCGib ÞP Gib , and bilinear terms, b¼1 ðqnðiÞ ÞP Gib . The left-hand side of (44) is the sum of linear terms, b¼1 The bilinear terms can be expressed more compactly as 12 rTi Hri , where rTi ¼ ð qnðiÞ P Gi;1 . . . P Gi;N  Gi Þ and 0 eT where H is the Hessian matrix of the sum of quadratic terms and has the form H ¼ , where e 0 eT ¼ ð 1 . . . 1 Þ. Since H is a real symmetric matrix, SchurÕs decomposition [15] shows that there is an orthogonal matrix Q such that H = QDQT, where D is a diagonal pffiffiffiffiffiffiffiffi pmatrix ffiffiffiffiffiffiffiffi whose eigenvalues match those of H. It can be shown that these eigenvalues are f0; . . . 0; N Gi ;  N Gi g so that

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

476

1 T 1 1 1 pffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffi r Hri ¼ rTi QDQT ri ¼ vTi Dvi ¼ N Gi ðvi;N Gi 1 Þ2  N Gi ðvi;N Gi Þ2 2 i 2 2 2 2 with the linear constraint QTri = vi where vTi ¼ ð vi;0 . . . vi;N Gi Þ and 0 1 pffiffiffi 0 0 0 B 0 2 B B B 1 1 1 1 1 B pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi B 2 2N Gi 6 p þ p2 B ðN Gi  1Þ þ ðN Gi  1Þ2 B B B 1 1 1 1 1 B pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi B 2 2 2 2N 6 p þ p Gi B ðN Gi  1Þ þ ðN Gi  1Þ B B B .. 2 1 1 B 0 p ffiffiffi    .    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi B B 2 2N Gi 6 ðN Gi  1Þ þ ðN Gi  1Þ B B B .. B 1 1 1 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 0 . pffiffiffiffiffiffiffiffiffiffiffiffiffi Q¼B 0 2 2N Gi pþp B ðN Gi  1Þ þ ðN Gi  1Þ2 B B B p 1 1 B 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 0 B 2 2N Gi p þ p2 B ðN Gi  1Þ þ ðN Gi  1Þ B B B .. 1 B 0 pffiffiffiffiffiffiffiffiffiffi 0 0 . B 2N B Gi B B . . . . 1 B .. .. .. .. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  B 2 B ðN Gi  1Þ þ ðN Gi  1Þ B B B ðN Gi  1Þ 1 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 0 0 @ 0 2 2N Gi ðN Gi  1Þ þ ðN Gi  1Þ

1 1  pffiffiffi C 2 C C 1 C pffiffiffiffiffiffiffiffiffiffi C 2N Gi C C C C 1 C pffiffiffiffiffiffiffiffiffiffi C 2N Gi C C C C 1 C pffiffiffiffiffiffiffiffiffiffi C C 2N Gi C C C C 1 C pffiffiffiffiffiffiffiffiffiffi C. 2N Gi C C C C 1 C pffiffiffiffiffiffiffiffiffiffi C 2N Gi C C C C 1 C pffiffiffiffiffiffiffiffiffiffi C C 2N Gi C C .. C . C C C C C 1 C C pffiffiffiffiffiffiffiffiffiffi A 2N Gi

Consequently, (44) is equivalent to N Gi X 1 pffiffiffiffiffiffiffiffi 2 2 N Gi ½ðvi;N Gi 1 Þ  ðvi;N Gi Þ   kCGib P Gib P K i xi  Mð1  xi Þ 2 b¼1

8i ¼ 1; . . . ; N G

ð45Þ

with the additional linear constraints Q T r i ¼ vi

8i ¼ 1; . . . ; N G .

ð46Þ

In light of the fact that there are no constraints on ð vi;1 N Gi X 1 1 pffiffiffi qnðiÞ þ pffiffiffiffiffiffiffiffiffiffi P Gib ¼ vi;N Gi 1 2N Gi 2 b¼1 Gi X 1 1 pffiffiffiffiffiffiffiffiffiffi P Gib ¼ vi;N Gi  pffiffiffi qnðiÞ þ 2N Gi 2 b¼1

...

vi;N Gi 2 Þ these linear restrictions reduce to

8i ¼ 1; . . . ; N G ;

ð47Þ

8i ¼ 1; . . . ; N G .

ð48Þ

N

Therefore, Eq. (44) can be replaced by Eqs. (45), (47) and (48). Then, the quadratic terms of Eq. (45) can be approximated by piece-wise linear functions with integer constraints [16]. This approximation is explained below.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

477

Consider the break points (points where the slope of the piece-wise linear function changes) ðiÞ ðiÞ ðiÞ 2 2 b1 ; b2 ; . . . ; bK for the functions ðvi;N Gi 1 Þ and ðvi;N Gi Þ . Then, vi;N Gi 1 can be written as vi;N Gi 1 ¼

K X

ðiÞ ðiÞ

8i ¼ 1; . . . ; N G ;

uk bk

k¼1

where K X

ðiÞ

uk ¼ 1 8i ¼ 1; . . . ; N G ;

k¼1 ðiÞ

0 6 uk 6 1 8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K. P ðiÞ ðiÞ The quadratic term ðvi;N Gi 1 Þ2 is replaced by the equation ðvi;N Gi 1 Þ2 ¼ Kk¼1 uk ðbk Þ2 . For the approximating function to yield a good approximation we must add the following adjacency assumption. For ðiÞ ðiÞ k = 1, . . . , K, at most two uk can be positive. If for a given k, two uk are positive, they must be adjacent. The adjacency constraints for vi;N Gi 1 are as follows: ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

u1 6 z1 ; u2 6 z1 þ z2 ; . . . ; uK1 6 zK2 þ zK1 ; 8i ¼ 1; . . . ; N G ;

uK 6 zK1 K X

ðiÞ

zk ¼ 1 8i ¼ 1; . . . ; N G ;

k¼1 ðiÞ

zk 2 f0; 1g

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K.

In the similar manner, vi;N Gi can be written as vi;N Gi ¼

K X

ðiÞ ðiÞ

y k bk

8i ¼ 1; . . . ; N G ;

k¼1

where K X

ðiÞ

8i ¼ 1; . . . ; N G ;

ðiÞ

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K.

yk ¼ 1

k¼1

0 6 yk 6 1

P ðiÞ ðiÞ The quadratic term ðvi;N Gi Þ2 is replaced by the equation  Kk¼1 y k ðbk Þ2 . And the adjacency constraints for vi;N Gi are as follows: ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

y 1 6 w1 ; y 2 6 w1 þ w2 ; . . . ; y K1 6 wK2 þ wK1 ; y K 6 wK1 K X

ðiÞ

wk ¼ 1

8i ¼ 1; . . . ; N G ; 8i ¼ 1; . . . ; N G ;

k¼1 ðiÞ

wk 2 f0; 1g

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K.

To sum up, the following Eqs. (49)–(61) are the linearization of the minimum profit condition (44).

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

478

K K X 1 pffiffiffiffiffiffiffiffi X ðiÞ ðiÞ 2 ðiÞ ðiÞ 2 N Gi uk ðbk Þ  y k ðbk Þ 2 k¼1 k¼1

! 

N Gi X

kCGib P Gib P K i xi  Mð1  xi Þ

8i ¼ 1; . . . ; N G ;

ð49Þ

b¼1

N Gi K X X 1 1 ðiÞ ðiÞ pffiffiffi qnðiÞ þ pffiffiffiffiffiffiffiffiffiffi P Gib ¼ uk bk 2N 2 Gi b¼1 k¼1

8i ¼ 1; . . . ; N G ;

K Gi X X 1 1 ðiÞ ðiÞ pffiffiffiffiffiffiffiffiffiffi P Gib ¼  pffiffiffi qnðiÞ þ y k bk 2N 2 Gi b¼1 k¼1

ð50Þ

N

K X

8i ¼ 1; . . . ; N G ;

ð51Þ

ðiÞ

8i ¼ 1; . . . ; N G ;

ð52Þ

ðiÞ

8i ¼ 1; . . . ; N G ;

ð53Þ

uk ¼ 1

k¼1 K X

yk ¼ 1

k¼1 ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ w1

ðiÞ ðiÞ w2 ; . . . ; y K1

ðiÞ

ðiÞ

ðiÞ

ðiÞ

u1 6 z1 ; u2 6 z1 þ z2 ; . . . ; uK1 6 zK2 þ zK1 ; uK 6 zK1 ðiÞ y1

6

K X

ðiÞ ðiÞ w1 ; y 2

ðiÞ

zk ¼ 1

6

þ

6

ðiÞ wK2

þ

ðiÞ ðiÞ wK1 ; y K

6

8i ¼ 1; . . . ; N G ;

ðiÞ wK1

8i ¼ 1; . . . ; N G ;

ð54Þ ð55Þ

8i ¼ 1; . . . ; N G ;

ð56Þ

ðiÞ

8i ¼ 1; . . . ; N G ;

ð57Þ

0 6 uk 6 1

ðiÞ

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K;

ð58Þ

ðiÞ yk

61

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K;

ð59Þ

2 f0; 1g

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K;

ð60Þ

wk 2 f0; 1g

8i ¼ 1; . . . ; N G ; 8k ¼ 1; . . . ; K.

ð61Þ

k¼1 K X

wk ¼ 1

k¼1

06 ðiÞ zk ðiÞ

References [1] G.B. Sheble´, Computational Auction Mechanisms for Restructured Power Industry Operation, Kluwer Academic Publishers, Boston, 1999. [2] M. Shahidehpour, M. Marwali, Maintenance Scheduling in Restructured power systems, Kluwer Academic Publishers, Norwell, Massachusetts, 2000. [3] M. Shahidehpour, H. Yamin, Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling and Risk Management, Wiley, New York, 2002. [4] M. Ilic, F. Galiana, L. Fink, Power Systems Restructuring: Engineering and Economics, Kluwer Academic Publishers, Boston, 1998. [5] F. Schweppe, M. Caramanis, R. Tabors, R. Bohn, Spot pricing of electricity, Kluwer Academic Publishers, Boston, 1988. [6] ISO New England. Available from http://www.iso-ne.com. [7] The PJM Interconnection. Available from http://www.pjm.com. [8] A.L. Motto, F.D. Galiana, A.J. Conejo, J.M. Arroyo, Network-constrained multi-period auction for a pool-based electricity market, IEEE Transactions on Power Systems 17 (2002) 646–653. [9] A.L. Motto, F.D. Galiana, A.J. Conejo, M. Huneault, On Walrasian equilibrium for pool-based electricity markets, IEEE Transactions on Power Systems 17 (2002) 774–781. [10] A.L. Motto, F.D. Galiana, Equilibrium of auction markets with unit commitment: The need for augmented pricing, IEEE Transactions on Power Systems 17 (2002) 798–805.

R. Garcı´a-Bertrand et al. / European Journal of Operational Research 174 (2006) 457–479

479

[11] OMEL: Market Operator of the electric energy market of mainland Spain. Market rules. Available from http://www.omel.es/en/ pdfs/EMRules.pdf. [12] M.S. Bazaraa, H.D. Sheraly, C.M. Shetty, Nonlinear Programming, Theory and Algorithms, John Wiley and Sons, New York, 1993. [13] R.W. Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992. [14] J.J. Ju´dice, A.M. Faustino, I.M. Ribeiro, On the solution of NP-hard linear complementarity problems, TOP-Sociedad Espan˜ola de Estadı´stica e Investigacio´n Operativa 10 (1) (2002) 125–145. [15] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [16] W.L. Winston, Operations Research: Applications and Algorithms, Duxbury Press, Belmont, California, 1994. [17] M. Huneault, F.D. Galiana, A survey of the optimal power flow literature, IEEE Transactions on Power Systems 6 (1991) 762– 770. [18] J.L. Carpentier, Optimal power flows: Uses, methods and developments, Proceedings of IFAC Conference, 1985. [19] A.J. Wood, B.F. Wollenberg, Power Generation Operation and Control, second ed., John Wiley and Sons, New York Inc., 1996. [20] B.F. Hobbs, Linear complementarity models of Nash-Cournot competition in bilateral and poolco power markets, IEEE Transactions on Power Systems 16 (2001) 194–202. [21] J. Boucher, Y. Smeers, Alternative models of restructured electricity systems. Part 1: No market power, Operations Research 49 (2001) 821–838. [22] R.P. OÕNeill, P.M. Sotkiewicz, B.F. Hobbs, M.H. Rothkopf, W.R. Stewart, Efficient market-clearing prices in markets with nonconvexities, European Journal of Operational Research 164 (2005) 269–285. [23] T. Takayama, G.G. Judge, Spatial and Temporal Price and Allocation Models, North-Holland Publishing Company, Amsterdam, 1971. [24] Drayton Analytics, PLEXOS for Power Systems, 2000. http://www.plexos.info/kb5/. [25] Reliability test system task force, The IEEE reliability test system 1996, IEEE Transactions on Power Systems 14 (1999) 1010– 1020. [26] A. Brooke, D. Kendrick, A. Meeraus, R. Raman, GAMS: An userÕs guide, GAMS Development Corporation, Washington, 1998.