Centro de Investigación Operativa

I-2005-01

Strategic bidding in continuous electricity auctions: an application to the Spanish electric market

Juan Aparicio, J.C. Ferrando, Ana Meca

and Julia Sancho

January 2005

ISSN 1576-7264 Depósito legal A-646-2000

Centro de Investigación Operativa Universidad Miguel Hernández de Elche Avda. de la Universidad s/n 03202 Elche (Alicante) cio@umh.es

Strategic bidding in continuous electricityauctions: an application to the Spanish

electricity market∗

Juan Aparicio, J.C. Ferrando, Ana Meca y Julia Sancho†

Operations Research CenterUniversidad Miguel Hernández de Elche, Spain

December 10, 2004

Abstract

In this paper we introduce an asymmetric model of continuous elec-tricity auctions with limited production capacity and bounded rate bidsupply functions. For this model, we study the strategic bidding throughthe electricity market game. We prove that for every electricity marketgame with continuous production cost functions, there always exists amixed-strategy Nash equilibrium. In particular, we focus on the behav-ioral way of producers in the Spanish electricity market. We consider avery simple form for the Spanish electricity market: an oligopoly just withindependent hydro-electric power production units in a single wet period.We show that there always exists a pure-strategy Nash equilibrium forthe Spanish electricity market game. In addition, a procedure based onoptimization tools is given to calculate a pure-strategy equilibrium for anumerical example. A sensitive analysis of the procedure on the resultsfor the above example is provided as well.

Key words: continuous electricity auctions, Spanish electricity mar-ket, electricity market game, Nash equilibrium.

1 IntroductionSince the beginning of the nineties electricity markets have been significantlyrestructured through the world. Researchers have been engaged on developingmethodologies which tackle the new electricity industry situation. This dereg-ulation on the sector was produced in Spain in 1998. It led to consumers andproducers were able to get benefits from a more competitive marketplace with∗The research of the authors is partially supported by Spanish Ministry of Education and

Science and Generalitat Valenciana grant numbers: SEC2002-00112 and GRUPOS04/79. Theauthors thank Juan Parra for helpful comments.

†Corresponding author.

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a greater choice in buying and selling electricity. Since then, producers and themarket operator started to be interested on modelling and analyzing electric-ity markets. Accordingly, several papers studying the competition in electricityproduction and trading have been published.When studying problems concerning electricity markets, two clear and well-

known points of view can be followed. The first one is the optimization problemfor individual power producers. The second one concerns the strategy optimiza-tion for them. This work is addressed to the last problem.There exists a lot of theoretical and empirical analysis on describing the

strategic behavior for firms competing on electricity markets. All of them areperformed by auction-like trading subjected to several technical restrictions.One of the main research lines consists of debating over discriminatory ver-sus uniform-price auctions. Even though there exists a lot of papers tacklingthis problem, it is well-known amongst auction theorists that none of them isgenerally superior to the other one (see [4] and [9]). In multi-unit auctionsthe comparison between these two forms is particularly complex. There is noclear evidence which tells us discriminatory-price auctions perform better thanuniform-price auctions in markets such as those for electricity. In this work,following the Spanish electricity market setting, we take the uniform-price pro-cedure. Note in passing that electricity wholesale markets differ in several di-mensions, but until recently all of them have been organized as uniform-priceauctions.On the other hand, nowadays, it is a hot topic to come up for discussion if

electricity markets are either discrete multi-unit auctions or continuous auctions.[12] claim that all electricity auctions are discrete multi-unit auctions rather thancontinuous “share auctions” or auctions for perfectly divisible goods. However,[5] propose that continuous auctions is the best way to describe the strategicbehavior for firms competing on electricity markets. When having a look togeneral electricity markets literature, we find out more interest on modellingelectricity markets as continuous auctions than discrete ones.In this work, we adopt the continuous electricity auctions point of view.

We consider a generic electricity market, with n producers, which works asfollows: each electricity producer submits, independently, for an hour of thenext day a bid supply function for power which he is willing to sell at leastat the corresponding prices. The bid supply functions of all firms are thenaggregate by the market operator to form the market supply for this hour. Themarket marginal price (spot price henceforth) for the current hour equals tothe one for which the market supply coincides with the market demand forthis hour, which is supposed to be known and constant. As for uniform-priceauction, all producers obtain the spot price. Hence, the production of eachfirm is determined. This electricity market situation will be called continuouselectricity auction.The analysis of supply function equilibrium models is not new. Thus, one can

find in the literature of competition market models approached from this point ofview. Klemperer and Meyer in [19] study an oligopoly facing uncertain demandin which each firm chooses as its strategy a supply function relating its quantity

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to its price. In the absence of uncertainty, there exists an enormous multiplicityof Nash equilibria in supply functions (SFE henceforth) [see also [16] and [17]],but uncertainty dramatically reduces the set of SFE. For this context, they provethe existence of a SFE for a symmetric oligopoly producing a homogeneousgood with no limited capacity, and give sufficient conditions for uniqueness.Recently, [11] propose a version of the above model, without uncertainty butsymmetric, where the strategies of the firms are restricted to non-decreasingsupply functions. It turns out that restricting the strategy spaces significantlyreduces the set SFE: the set of prices that can be sustained by SFE is theinterval between the competitive price and the Cournot price. Nevertheless,the set SFE remains large. Then, they introduce the notion of coalition-proofsupply function equilibrium and prove that if the number of firms is above athreshold, the Cournot outcome is the unique coalition-proof supply functionequilibrium outcome.On the other hand, in recent papers, supply function equilibrium models

have been applied to analysis of electricity markets [[1], [2], [3], [5], [6], [7], [8],[10], [14], [15], [18], [23], [25]]. Unfortunately, without restrictive assumptionson the nature of the costs and capacity constrains, on the number of firms, oron the form of the allowed bid functions, it has proven difficult to find equilibriain supply functions. For example, in [15] the supply functions are restrictedto being linear (in the sense that the intercept of the supply function is zero).In [[8],[18],[25]], the supply functions are affine (in the sense that there is aconstant slope and a (possible zero) intercept) but either the intercepts or theslopes are assumed constant. In [[14],[15],[23]], the marginal cost functions allhave zero (or all have the same) intercept. Finally, in [5], to obtain a convenientcharacterization of the equilibrium, the authors assume that each firm mustsubmit either an affine supply function, or a piece-wise affine supply functionwhere the number of pieces is relatively small.Our model of continuous electricity auctions describes many electricity mar-

kets more realistically that the above ones. There are three main differences be-tween our model and the above ones: (1) each producer has a limited productioncapacity, (2) different production cost functions for each producer (asymmetricmarket), (3) less restrictive assumptions on the form of the allowed bid supplyfunctions.We start by introducing definitions and notations in section 2. In section 3,

we give a complete description of the continuous electricity auctions and studythe strategic bidding in it. On doing that, we define the corresponding nonco-operative game which will be called electricity market game. The main resultin this section states that for every electricity market game with continuousproduction cost functions, there always exists a mixed-strategy Nash equilib-rium. Section 4 presents an application to the Spanish electricity market. It isinspired on the behavioral way of producers in the Spanish electricity market,following the data published by the market operator OMEL. We prove thatthere always exists a pure-strategy Nash equilibrium for the Spanish electricitymarket game. In addition, a procedure based on optimization tools is givento calculate a pure-strategy equilibrium for a numerical example. A sensitive

3

analysis of the procedure on the results for the above example is provided aswell. Section 5 completes the paper.

2 PreliminariesThe theory of noncooperative games studies the behavior of agents in any sit-uation where each agent’s optimal choice may depend on his forecast of thechoices of his opponents. For example, the electricity market where the produc-ers (players) have competitive interests in market shares, but a common interestin high prices. The word “noncooperative” means that the players’ choices arebased only on their perceived self-interest, it does not mean that the players donot get along, or that they always refuse to cooperate.Game theory has been applied to many fields, being one of the most useful

the study of economic problems. The game-theoretic point of view is more usefulin settings with a small number of players, for then each player’s choice is morelikely to matter to his opponents. For example, when the number of producersin a market is small, each producer’s output is likely to have large impact onthe market price; thus, it is not reasonable for each producer to take the marketprice as given.Formally, a game in strategic (or normal) form has three elements: the set

of players i ∈ N , which we take to be the finite set {1, 2, ..., n}, the pure-strategy space Xi for each player i, and payoff functions Hi that give playeri’s von Neumann-Morgenstern utility Hi(x) for each profile x = (x1, ..., xn) ofstrategies; i.e. Hi : X → R where X :=

Yi∈N

Xi. We will frequently refer to all

players other than some given player i as “player i’s opponents” and denote themby “−i”. To avoid misunderstanding, let us emphasize that this terminologydoes not mean that the other players are trying to “beat” player i. Rather, eachplayer’s objective is to maximize his own payoff function, and this may involve“helping” or “hurting” the other players. For economists, the most familiarinterpretation of strategies may be as choices of prices or output levels, whichcorrespond to Bertrand and Cournot competition, respectively. By the otherhand, depending on the pure-strategy spaces for the game, we could distinguishbetween finite or infinite games. Finite games are those for which X is finite;finiteness should be assumed wherever we do not explicitly note otherwise.A mixed strategy σ is a probability distribution over pure strategies (see

[13] for the motivation for mixed strategies). Each player’s randomization isstatistically independent of those of his opponents, and the payoff to a profileof mixed strategies are the expected values of the corresponding pure-strategypayoffs.A Nash equilibrium is a profile of strategies such that each player’s strategy

is an optimal response to the other player’s strategies. Formally, a mixed-strategy profile σ∗ is a Nash equilibrium if, for all players i,Hi(σ∗i ,σ

∗−i) ≥

Hi(xi,σ∗−i) for all xi ∈ Xi. A pure-strategy Nash equilibrium is a pure-strategy

profile that satisfies the same conditions. Since expected utilities are “linear in

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the probabilities”, if a player uses a nondegenerate mixed strategy in a Nashequilibrium (one that puts positive weight on more than one pure strategy)he must be indifferent between all pure strategies to which he assigns positiveprobability (this linearity is why, in equation above, it suffices to check that noplayer has a profitable pure-strategy deviation).Not all games have pure-strategy Nash equilibrium. Two examples of games

whose only Nash equilibrium is in (nondegenerate) mixed strategies are “match-ing pennies” and “inspection game”. Many games have several Nash equilib-ria, for example “the battle of the sexes”. Nash in [22] proves that every fi-nite strategic-form game has a mixed-strategy equilibrium. However, since theeconomists often use models of games with an uncountable number of actions,some might argue that prices or quantities are “really” infinitely divisible, theexistence of Nash equilibrium in infinite games with continuous payoff has beenalso studied. Hence, two classical results on it can be found in [13] (Theorems1.2 and 1.3).Finally notice that for section 3 we use some topology tools which can be

found in [20]. Analogously, for Theorem 11, in section 4, some results relatedto the cuasi-concavity (see [21]) have been employed.

3 Electricity market gamesTo start with, the basic form of the continuous electricity auctions is now de-scribed. We consider a competitive electricity market where n producers, whoare risk neutral, participate every day. Denote by N = {1, 2, ..., n} the setof producers. Demand just for an hour of the next day, D > 0, is constantand known to all of them with certainty. All producers have not access to thesame technology and have capacity constrains, so each one has a different costfunction, being Ci : [0, ki] → R the cost function for producer i, with ki themaximum production capacity. In this setting, each producer i bids, indepen-dently, a schedule of price-quantities for power which he is willing to sell at leastat these prices; i.e. a bid supply function for this hour, gi : [0, p]→ [0, ki], where_p is the maximum market marginal price. According to the market rules, eachgi is non-decreasing on

£0,_p¤. The reader may notice that for producer i, Ci (q)

is the cost of producing a quantity q ∈ [0, ki] and gi (p) is the power quantityoffered at price p ∈ £0, _p¤ . Then, the market operator arranges the supply func-tions of all firms in the aggregate supply function and forms the market supplyfor this hour G :

£0,_p¤→ [0, k], defined by G (p) =

Pni=1 gi (p) for all p ∈

£0,_p¤.

Note that G (p) is the total power quantity produced in the market at price pand k =

Pni=1 ki. The spot price for the current hour bp equals to the one for

which the market supply coincides with the market demand for this hour; i.e.the solution to equation G(bp) = D. As for uniform-price auction, all producersobtain the spot price bp, hence the production of each firm i is gi(bp).A sufficient condition to guaranty the existence and uniqueness of the spot

price for continuous electricity auctions is that D belongs to rank of G and Gbe invertible in a neighborhood of D. In order to get it, we have to impose some

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properties to the supply functions gi.

Definition 1 Let f : [a, b]→ R be a non-decreasing function on [a, b] . We saythat f is a bounded rate function on [a, b] if there exists 0 < ² ≤ L such that

² · |s− t| ≤ |f(s)− f(t)| ≤ L · |s− t| ,

for all s, t ∈ [a, b] .

We denote by B ([a, b]) the set of all bounded rate functions on [a, b], fromnow on BR functions on [a, b] , and by B (², L, [a, b]) the set of BR functions on[a, b] with constant parameters ² and L.

Remark 2 The reader may notice that every BR function on [a, b] is strictlyincreasing and then it is invertible along its rank. Moreover it is Lipschitz’s, sothat it is continuous on [a, b] . Hence B ([a, b]) ⊂ C ([a, b]) .

Next Lemma shows that if the demand belongs to the rank of the aggregatesupply function, we are able to guaranty the existence and uniqueness of thespot price for every continuous electricity auction with BR bid supply functionson [0, p]; i.e. if D ∈ G([0, p]) then, there exists a unique bp such that G(bp) = D.Hence, the production of each firm i is given by gi(bp).Lemma 3 Consider a continuous electricity auction with gi BR bid supplyfunctions on [0, p] , for all i ∈ N. Then the aggregate supply function G isinvertible in its rank G([0, p]).

Proof. Take gi a BR bid supply functions on [0, p] with parameters ²i and Li.Then, for each s > t

²i · (s− t) ≤ gi(s)− gi(t) ≤ Li · (s− t),

with 1 ≤ i ≤ n holds, and so(nXi=1

²i

)· (s− t) ≤ G(s)−G(t) ≤

(nXi=1

Li

)· (s− t).

Hence, G is injective in£0,_p¤and then, it is invertible in G([0, p]).

The reader may notice that since gi(0) < gi(p) for all i ∈ N,G([0, p]) is nota simple interval reduced just to a point.Next we are able to define the electricity market game corresponding to a

continuous electricity auction with BR bid supply functions on [0, p] and con-stant parameters, as an infinite noncooperative game (i.e. finite number ofplayers but the pure-strategy space for each one of them is infinite).

Definition 4 The electricity market game is defined by the 2-tupla < N, {Xi,Hi}i∈N >, where

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• Xi :=©gi ∈ B

¡²i, Li,

£0,_p¤¢ ±

0 ≤ gi(0) ≤ 1nD ≤ ²i ·

_pªis the pure-strategy

space for producer i ∈ N , and• Hi : X → R is such that for all g = (g1, ..., gn) ∈ X

Hi(g) := G−1 (D) · gi

¡G−1 (D)

¢− Ci £gi ¡G−1 (D)¢¤ ,that is, Hi is the payoff function for producer i ∈ N.

The reader may notice that for each player i ∈ N,

(1) the strategies gi are uniformly bounded functions by 1nD + Li ·

_p.

Proof is easy. Taking into account that ²i(s− t) ≤ gi(s)−gi(t) ≤ Li(s− t)for each s > t, we can take s = p, t = 0 and then, gi(p)−gi(0) ≤ Li ·p. Thisimplies that gi(p) ≤ gi(0)+Li·p ≤ 1

nD+Li·p, and hence gi(p) ≤ 1nD+Li·p

for all p ∈ [0, p].Moreover, note that the strategies are BR functions with starting point lo-cated between the origin and the n-part of the demand, i.e. gi(0) ∈ [0, 1nD],and the aggregate supply function generated by them has an enough bigdomain to contain the demand, i.e. 1

nD ≤ ²i ·_p.

(2) the payoff function is well defined because of the demand belongs to rankfor all profile of strategies, i.e. D ∈ G([0, p]), for all g ∈ X.Proof is also easy. Since ²i · p ≤ gi(p) − gi(0) ≤ Li · p, then gi(p) ≥gi(0)+²i ·p ≥ ²i ·p. Taking into account that gi

¡£0,_p¤¢ ⊇ [gi(0), ²i · p] , we

get G¡£0,_p¤¢ ⊇ [Pn

i=1 gi(0), (Pn

i=1 ²i) · p] , and then G¡£0,_p¤¢ ⊇ [D,D] ;

hence D ∈ G([0, p]) for every G ∈ Pni=1Xi, which is equivalent, for all

g ∈ X.

It is well-known that we cannot guaranty, in general, the existence of Nashequilibria for infinite games. The main result in this section states that therealways exist Nash equilibria in mixed strategies for the electricity market gamewith continuous cost functions.Three technical Lemmas are needed previously.

Lemma 5 Let {Gm} be a sequence in B¡², L,

£0,_p¤¢. If Gn → G pointwise on£

0,_p¤and G(0) < D < G (p), then there exist m0 ∈ N and a non empty closed

interval J ⊆ [0, k], where k = G (p), such that D ∈ J ⊆ ∩m≥m0Gm¡[0,

_p]¢.

Proof. Take bp ∈ £0, _p¤ such that G(bp) = D. Since the sequence {Gm} isequicontinuous and bp ∈ ¡0, _p¢ there exists a closed interval [a, b] and m0 ∈ Nsuch that

(i) bp ∈ (a, b),(ii) 0 < a < b < p,

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(iii) Gm ([a, b]) ⊆ G¡¡0,_p¢¢for each m ≥ m0.

Set mm := Gm (0) and Mm := Gm¡_p¢. Obviously mm ≤ Gm (a) <

Gm (b) ≤ Mm for each m ≥ m0, so that limmm ≤ G (a) and G (b) ≤ limMm

for each m ≥ m0. Then Let ² > 0 be such that G (a) + ² < D < G (b) − ².The reader may notice that since a < bp < b, then G(a) < G(bp) < G(b) andso G(a) < D < G(b). Moreover, since limmm = infk supn≥kmm, there ex-ists k1 ≥ m0 such that supm≥k1mm ≤ G (a) + ². And, by the other hand,since limMm = supk infm≥kMm, there exists k2 ≥ m0 such that G (b) − ² ≤infm≥k2Mm. Hence, if m0 = max{k1, k2} then

supm≥m0

mm ≤ G (a) + ² < G (b)− ² ≤ infm≥m0

Mm,

which implies[G (a) + ²,G (b)− ²] ⊆ ∩m≥m0 [mm,Mm].

Hence taking J := [G (a) + ²,G (b)− ²] we conclude that J ⊆ Gm¡[0,

_p]¢for

each m ≥ m0.

Remark 6 Given a sequence {Gm} in B¡², L,

£0,_p¤¢such that Gm → G point-

wise on£0,_p¤, there exist the inverse functions of Gm, for all m ≥ m0, and

G, being consider as functions from J to [0,_p], which will be denoted by G−1m |J

and G−1 |J , respectively.

Lemma 7 Let {Gm} be a sequence in B¡², L,

£0,_p¤¢. If Gm → G pointwise

on£0,_p¤and D ∈ G ((0, p)) , then G−1m |J → G−1 |J uniformly on J.

Proof. If {Gm} is a sequence in B¡², L,

£0,_p¤¢, then

² · |s− t| ≤ |Gm(s)−Gm(t)| ≤ L · |s− t| ,

for all s, t ∈ £0, _p¤, and so1

L· |x− y| ≤ ¯̄G−1m (x)−G−1m (y)¯̄ ≤ 1² · |x− y| ,

for every x, y ∈ J . Suppose without lost of generality that m0 = 1 in Lemma5. Then, since

¯̄G−1m (x)−G−1m (y)

¯̄ ≤ 1² · |x− y| for all x, y ∈ J, it turns out that

the sequence©G−1m |J ª is equicontinuous. Moreover, since supy∈J ¯̄G−1m (y)¯̄ ≤ _

p

then, by Ascoli’s theorem (see [20]),©G−1m |J ª is a relatively compact set in

C(J).This implies that for each subsequence

©G−1mk

|J ª in ©G−1m |J ª we can obtaina subsequence

nG−1mkl

|Jowhich converges uniformly to a continuous function,

say H. Since uniform convergence implies pointwise convergence, it is obviousthat

1

L· |x− y| ≤ |H(x)−H(y)| ≤ 1

²· |x− y| ,

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for all x, y ∈ J. Hence H is injective in J and there exists H−1 in its rank.Let us see that H = G−1 |J . Then, for each subsequence in ©G−1m |J ª we

may get a subsequence converging to G−1 |J , which implies that ©G−1m |J ªitself converges to G−1 |J . For the sake of simplicity, we may suppose thatG−1mk

|J → H uniformly. We have just to prove that H = G−1 on J.

1. Since G−1mk|J → H pointwise on J , given δ > 0 and t ∈ £0, _p¤ such that

G(t) ∈ J, there exists k0 ∈ N (depending on δ and t) such that¯̄G−1mk

(G(t))−H(G(t))¯̄ < δ

for each k ≥ k0. Hence¯̄Gmk

¡G−1mk

(G(t))¢−Gmk

(H(G(t)))¯̄ ≤ L ¯̄G−1mk

(G(t))−H(G(t))¯̄ < L δfor each k ≥ k0, which implies that

|G(t)−Gmk (H(G(t)))| < L δ

for each k ≥ k0. Hence limkGmk(H(G(t))) = G(t) and, consequently,

G (H(G(t))) = G(t). Thus H(G(t)) = t for all t ∈ G−1(J). So we concludethat H ◦G = I1 where I1 is the identity function on G−1(J).

2. Since Gmk→ G pointwise on [0, p], given η > 0 and y ∈ J there exists

k1 ∈ N such that|Gmk

(H(y))−G(H(y))| < η

for each k ≥ k1. Hence¯̄G−1mk

(Gmk(H(y)))−G−1mk

(G(H(y)))¯̄ ≤ 1

²|Gmk

(H(y))−G(H(y))| < η

²,

for each k ≥ k1, which implies that¯̄H(y)−G−1mk

(G(H(y)))¯̄<

η

²

for all k ≥ k1. Hence limkG−1mk(G(H(y))) = H(y) and, consequently,

H (G(H(y))) = H(y). This ensures that G(H(y)) = y for all y ∈ J . So weconclude that G ◦H = I2 where I2 is the identity function on J .

So, considering G as a function from G−1(J) onto J , we conclude that His its inverse, i.e. H = G−1 |J .

Lemma 8 If D ∈ ∩G∈Pni=1Xi

G ((0, p)), the operator g 7−→ G−1(D) from Xinto R is continuous.

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Proof. It is obvious that the operator g 7−→ G from X tonXi=1

Xi ⊆ C¡£0,_p¤¢

is continuous on C¡£0,_p¤¢. Let us prove that the operator G 7−→ G−1(D) from

nXi=1

Xi to R is also continuous. In this way the operator g 7−→ G−1(D) from X

in R will be continuous as well.Let {Gm} be a sequence of functions in

Xi∈N

Xi such that Gm → G uniformly

on£0,_p¤. According to Lemmas 5 and 7 there exist J ⊆ [0, k] such that D ∈ J

and G−1m |J → G−1 |J uniformly on J. Hence, G−1m (D)→ G−1(D) in R and theoperator G 7−→ G−1(D) from X to R is continuous.

Theorem 9 Let < N, {Xi,Hi}i∈N > be the electricity market game corre-sponding to a continuous electricity auction with Ci continuous functions on[0, ki], for all i ∈ N. Then, there always exists a Nash equilibrium in mixedstrategies for it.

Proof. Let < N, {Xi,Hi}i∈N > be the electricity market game correspondingto a continuous electricity auction with Ci continuous functions on [0, ki], forall i ∈ N.Take i ∈ N. If D ∈ ∩G∈Pn

i=1XiG ((0, p)), by Lemma 8 the operator g 7−→

G−1(D) is continuous, and so g 7−→ G−1 (D) · gi¡G−1 (D)

¢−Ci £gi ¡G−1 (D)¢¤is continuous as well. Hence, Hi is continuous on X. Let us see that Xi iscompact in the Banach space C(

£0,_p¤), so that by Glicksberg’s theorem (see

[13]) we can conclude that there exists a Nash equilibrium in mixed strategiesfor the electricity market game above.Since |gi(s)− gi(t)| ≤ Li · |s− t| for all s, t ∈

£0,_p¤, Xi is equicontinuous.

Moreover since functions gi are uniformly bounded, by Ascoli’s theorem Xiis relatively compact. The proof is completed showing that Xi is closed inC(£0,_p¤). Take a sequence of functions {gmi } ⊆ Xi converging uniformly to gi.

Then, since²i · |s− t| ≤ |gmi (s)− gmi (t)| ≤ Li · |s− t| ,gmi (0) ≤ 1

nD ≤ ²i ·_p,

taking limits when m→∞, we get²i · |s− t| ≤ |gi(s)− gi(t)| ≤ Li · |s− t| ,gi(0) ≤ 1

nD ≤ ²i ·_p

Hence, gi ∈ Xi.The following example exhibits the results obtained in this section.

Example 10 Consider the following continuous electricity auction:

• for each i ∈ N, gi :£0,_p¤ → R such that gi(p) := bi + ai · p, for all

p ∈ £0, _p¤ , with bi ∈ £0, 1nD¤ and ai ∈ [²i, Li], 0 < ²i ≤ Li,10

• 1nD ≤ ²i ·

_p,

• Ci(q) = ci · q for all q ∈ [0, ki], ki ≤ 1nD + Li ·

_p.

The reader may notice that for each i ∈ N, gi ∈ B¡²i, Li,

£0,_p¤¢satisfies

that gi(0) ≤ 1nD ≤ ²i ·

_p.

The corresponding electricity market game, which it is called affine electricitymarket game, is given by < N, {Xi,Hi}i∈N > where:

• Xi =£0, 1nD

¤× [²i, Li] for all i ∈ N,• X =

£0, 1nD

¤n × nYi=1

[²i, Li],

• Hi : X → R such that for all (b1, b2, ..., bn, a1, a2, ..., an) ∈ X,

Hi(b1, ..., bn, a1, ..., an) =

"D −Pn

j=1 ajPnj=1 bj

− ci#·"ai + bi ·

ÃD −Pn

j=1 ajPnj=1 bj

!#.

By Theorem 9 we can guaranty the existence of a Nash equilibrium in mixedstrategies for the affine electricity market game.

In this context, the reader may notice that mixed strategies are (Borel)probability measures over pure strategies, which we endow with the topologyof weak convergence (see [20]). In practice, to calculate mixed strategies forinfinite games is not an easy problem. Then, the following question comes im-mediately to our mind: there exits a continuous electricity auction for whichwe can guarantee the existence of pure-strategy Nash equilibria for the corre-sponding electricity market game? If the answer is yes, is possible to provide aprocedure to calculate them?Next section shows that there always exists a pure-strategy Nash equilibrium

for the Spanish electricity market game. In addition, we provide a procedure,based on optimization, to settle it.

4 Spanish electricity marketThis section is devoted to study the strategic bidding of producers in the Span-ish electricity market. A full description and a wide statistical analysis of theSpanish electricity market is developed in [24]. There exist three kinds of pro-duction technologies for the above market: conventional thermal, hydro-electricpower and combined cycle thermal. Since there exists a strong dependence be-tween the spot price and the raw material for the hydro-electric power one, twodifferent scenarios are originated: wet and dry periods. For each one of them,one or several production technologies are brought out as the more competitiveones.

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Following [24] we can conclude that, in general, the technology that setthe spot price for different hours a day for wet periods are the hydro-electricpower production units. Since their costs are lower for rainy periods, the hydro-electric power production units are allowed themselves to behave in a morecompetitive way, which lead these production units to supply all the marketdemand in the top hours. Hence, it can be considered just a market with hydro-electric power production units for the top hours in wet periods; and so, thebid supply functions for the production units could be model in the same way.This restriction, which initially could be too much strong, is well fitted to thereal Spanish electricity market. In addition, the main difference among the bidfunctions for the hydro-electric power production units is related to the interceptfor the function, i.e. the supply to zero price.Taking into account all above, we will consider the simplest form for the

Spanish electricity market: an oligopoly just with independent hydro-electricpower production units in a single wet period.Formally, the Spanish electricity market can be described as the following

continuous electricity auction:

• N = {1, 2, ..., n} is the set of production units,• for each i ∈ N, gi :

£0,_p¤ → R such that gi(p) := bi + ai · m(p), for all

p ∈ £0, _p¤ , with bi ∈ £0, 1nD¤ ,m ∈ B ¡², L, £0, _p¤¢ being convex such thatm(0) = 0,m−1 ∈ C1, and 1

nD ≤ ai · ² ·_p,

• for each i ∈ N,Ci(q) = ci · q for all q ∈ [0, ki], ki ≤ 1nD + ai · L ·

_p, where

0 < ci < p is the marginal production cost, which is well-known by allproduction units in the market and it is never higher to the maximummarket marginal price.

Hence

G (p) =nXj=1

bj +

nXj=1

aj

·m (p) ,and so

G−1 (D) = m−1

D −

nXj=1

bj

nXj=1

aj

.Suppose that for all i ∈ N, ai and m are well-known by all production

units. By definition 4, the corresponding electricity market game, which is calledSpanish electricity market game, is described by the 2-tupla < N, {Xi,Hi}i∈N >where for each i ∈ N,

• Xi =£0, 1nD

¤,

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• Hi : X → R such that, for every (bi, b−i) ∈£0, 1nD

¤× £0, 1nD¤n−1 ,

Hi (bi, b−i) =

m−1D −

nXj=1

bj

nXj=1

aj

− ci ·bi + ai ·

D −

nXj=1

bj

nXj=1

aj

.(1)

The reader may notice that by Theorem 9 we can guaranty the existence ofa Nash equilibrium in mixed strategies for the Spanish electricity market game.However, we take one step further and are able to guarantee the existence of apure-strategy Nash equilibrium for such a game.The main result in this section states that there always exist pure-strategy

Nash equilibria for the Spanish electricity market game.

Theorem 11 There always exists a pure-strategy Nash equilibrium for the Span-ish electricity market game.

Proof. Let < N, {Xi,Hi}i∈N > be the Spanish electricity market game. Takei ∈ N. The reader may notice that Xi is a compact and convex subset in R andmoreover, Hi is a continuous function in X.Let us prove that Hi is cuasi-concave in bi, and then by Theorem 1.2 of

[13] we can conclude that there exists a pure-strategy Nash equilibrium for theSpanish electricity market game.Taking logarithms in both sides of (1) we obtain

lnHi (bi, b−i) = ln

m−1D −

nXj=1

bj

nXj=1

aj

− ci+ ln

bi + ai ·D −

nXj=1

bj

nXj=1

aj

.

In order to prove that lnHi (bi, b−i) is concave in bi, it is just needed to see thatevery term in the above sum is concave in bi.Denote the first term by the following function:

f1 (bi, b−i) := ln

m−1D −

nXj=1

bj

nXj=1

aj

− ci .

This expression can be rewritten as the composition of functions h and s in the

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following way: f1 (bi, b−i) = h (s (bi, b−i)) where

s (bi, b−i) := m−1

D−

nXj=1

bj

nXj=1

aj

− ci,h (y) := ln (y) .

It is well-known that the logarithmic function is concave, increasing and of classC1. Let us prove that s is concave in bi, and then by Theorem 13.6 in [21] wecan conclude that f1 is concave in bi.To do that, we rewrite s as the composition of functions m−1 and t such

that s (bi, b−i) = m−1 (t (bi, b−i))− ci where

t (bi, b−i) :=

D −nXj=1

bj

nXj=1

aj

.

Since the function t is a line with respect to bi, it is concave. By the otherhand, since m is convex, m−1 is a concave and increasing function. Again by

Theorem 13.6 of [21], we know that m−1

D−

nXj=1

bj

nXj=1

aj

is concave in bi. The

above function minus a constant ci is also concave, and so we can conclude thats is concave in bi.In a similar way, we denote the second term by the following function:

f2 (bi, b−i) := ln

bi + ai ·D −

nXj=1

bj

nXj=1

aj

.

Again, we rewrite f2 as the composition of functions h and s in the followingway: f2 (bi, b−i) = h (s (bi, b−i)) , where

s (bi, b−i) := bi + ai ·

D−

nXj=1

bj

nXj=1

aj

,h (y) := ln (y) .

Taking into account that s is a line with respect to bi, by Theorem 13.6 in [21]we can conclude that f2 is concave in bi.

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Since lnHi (bi, b−i) = f1 (bi, b−i) + f2 (bi, b−i) is the sum of concave func-tions in bi, then lnHi (bi, b−i) is a concave function in bi. To prove thatHi (bi, b−i) is cuasi-concave in bi, we just denote f (bi, b−i) := lnHi (bi, b−i) ,then Hi (bi, b−i) = exp {f (bi, b−i)} . By the other hand, since f is concave in bi,then f is cuasi-concave in bi. Finally, by Theorem 13.8 in [21] we conclude thatHi (bi, b−i) is cuasi-concave in bi.

4.1 A numerical example

Once we have proved that there always exist pure-strategy Nash equilibria forthe Spanish electricity market game, we exhibit a natural example of it in whichwe focus on providing a procedure to calculate them.To start with, there exist two main producers for the Spanish electricity

market game: Iberdrola and Endesa, whose production units supply the 70%of all the market demand, approximately. However, taking into account thatwe are considering an electricity market just with independent hydro-electricpower production units, we should include a third producer: Unión Fenosa,which owns several production units which are located at the main Spanish riverbasins. Hence, we focus on 6 independent hydro-electric power production unitswhich belong to the three producers above. Each producer has two productionunits. They correspond to the rivers Duero, Tajo, Sil, Ebro and Miño.First of all, we had just to decide which period would be used to estimate the

function m. In order to do that, we studied the evolution of the dam of thesebasins and the period with more water. It turned out to be the period fromJanuary to May 2001. Once the period had been chosen, we proposed to studythe hour 12th because of it was the one with higher demand. At the same time,we realized that the bid supply functions of the production units fluctuatedbetween working days and non-working days. All of this lead us to focus ouranalysis on the hour 12th for non-working days in May 2001. The technologythat set the spot price for all of these auctions was the hydro-electric power one.The market behavior of the hydro-electric power production units included

in the example could be modeled by a quadratic function. For each productionunit, it was estimated making use of the information provided by the marketoperator, OMEL, about the bid supply functions of the production units, threemonths just before.Formally, the market situation can be described as follows:

• N = {1, 2, ..., 6} is the set of hydro-electric power production units,• the market demand, which is known by all production units, is D = 7000MW ,

• the maximum production capacity for each production unit is given in the

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following table:

Production unit Maximum Production CapacityEndesa 1 12765 MWEndesa 2 6400 MWUnión Fenosa 1 1600 MWUnión Fenosa 2 3200 MWIberdrola 1 13600 MWIberdrola 2 25600 MW

• the bid supply function for each production unit i ∈ N is given bygi (p) := bi + ai · m (p), for all p ∈ [0, 20], with bi ∈ [0, 1166.667], andm ∈ B (15, 20, [0, 20]) being convex such that m(0) = 0,m−1 ∈ C1, andai ≥ 1166.667

300 for all i = 1, ..., 6. The reader may notice that in our examplep = 20 euro cents per KW, and 1

6D = 1166.667.

Taking into account the bid supply functions of the above productionunits, the best function m which satisfies all the properties required bythe model to describe their average behavior is the one given by m(p) =(p+ 0.01)2 − 0.012, for all p ∈ [0, 20].In order to give the specific bid supply function for each one of the 6production units, we have to determine the values for ai, i = 1, ..., 6. Thefollowing table resume the marginal costs ci, ai and the bid supply func-tion for each production unit i = 1, ..., 6. Values for ai are obtained byregression analysis over the data market for the period selected1.

Production unit ci ai Bid Supply FunctionEndesa 1 1.70 29 b1 + 29 · [(p+ 0.01)2 − 0.012]Endesa 2 1.90 16 b2 + 16 · [(p+ 0.01)2 − 0.012]Unión Fenosa 1 1.86 4 b3 + 4 · [(p+ 0.01)2 − 0.012]Unión Fenosa 2 1.95 8 b4 + 8 · [(p+ 0.01)2 − 0.012]Iberdola 1 1.50 34 b5 + 34 · [(p+ 0.01)2 − 0.012]Iberdola 2 1.92 64 b6 + 64 · [(p+ 0.01)2 − 0.012]

Next we are going to define the Spanish electricity market game correspond-ing to this example. It is given by the 2-tupla < N, {Xi,Hi}i∈N > where foreach production unit i ∈ N,

• the pure-strategy space is Xi = [0, 1166.667],1The reader may notice that some parameters used have been specified randomly without

any similarity to the realilty. In particular, the maximum capacities and the marginal costsfor the production units.

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• the payoff function Hi : X → R is given by:

Hi(bi, b−i) =

µ7000−Pj∈N bj155

+ 0.012¶ 1

2

− 0.01− ci×

bi + 45.26 · ai − ai155

Xj∈N

bj

.In order to get a pure-strategy Nash equilibria for the game above, we have

to solve the optimization problem for each i = 1, ..., 6:

maxbi Hi(bi, b∗−i)

s.t. 0 ≤ bi ≤ 1166.667It means we should solve 6 non-linear optimization problems simultaneously.

The reader may notice that the objective function for each one of those problemsis a concave one, which implies that every local optimum for the problem equalsto the global optimum. Hence, our problem consists on solve the followingequations system:

∂

∂biHi(bi, b

∗−i) = 0, i = 1, 2, ..., 6,

which turns out to be a system with 6 equations and 6 variables. Using Matlab6.5, we obtain that it has a unique solution, i.e. there exists a unique pure-strategy Nash equilibrium, (b∗1, ..., b∗6), for the Spanish electricity market gamecorresponding to this example.The following table shows the pure-strategy Nash equilibrium and the cor-

responding bid supply function for each the hydro-electric power productionunits:

Production unit b∗i g∗i (p)Endesa 1 956.5 956.5 + 29 · [(p+ 0.01)2 − 0.012]Endesa 2 1029.3 1029.3 + 16 · [(p+ 0.01)2 − 0.012]Unión Fenosa 1 1166.667 1166.667 + 4 · [(p+ 0.01)2 − 0.012]Unión Fenosa 2 1132.1 1132.1 + 8 · [(p+ 0.01)2 − 0.012]Iberdrola 1 1002.0 1002.0 + 34 · [(p+ 0.01)2 − 0.012]Iberdrola 2 84.4 84.4 + 64 · [(p+ 0.01)2 − 0.012]

The spot price in equilibrium is bp∗ = 3.2319 euro cents per KW, and theamount of energy which every hydro-electric power production unit sells in themarket at that price (equilibrium energy amount, henceforth) is given in the

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following table:

Production unit Equilibrium energy amountEndesa 1 1261.28Endesa 2 1197.45Unión Fenosa 1 1208.70Unión Fenosa 2 1216.17Iberdrola 1 1359.33Iberdrola 2 757.03

Note is passing that the above continuous electricity auction is that one withasymmetric producers and no limited capacity. In this context, we have got aunique pure-strategy Nash equilibrium which is stable in the sense that if weapply a light perturbation on the marginal costs of the hydro-electric powerproduction units, the equilibrium does not change significantly. The followingtable shows the effects produce when increasing lightly the marginal costs of thehydro-electric power production units:

Production unit b∗i (ci) b∗i (ci + 0.001) b∗i (ci + 0.005)Endesa 1 956.5 956.4 (−0.1) 956 (−0.5)Endesa 2 1029.3 1029.2 (−0.1) 1028.8 (−0.5)Unión Fenosa 1 1166.667 1166.667 (0) 1166.667 (0)Unión Fenosa 2 1132.1 1132.0 (−0.1) 1131.7 (−0.4)Iberdrola 1 1002.0 1001.9 (−0.1) 1001.4 (−0.6)Iberdrola 2 84.4 84.1 (−0.3) 82.9 (−1.5)

where b∗i (ci), b∗i (ci + ²) mean the original equilibrium and the one after a light

perturbation ² > 0, respectively, and b∗i (ci + ²)− b∗i (ci) brackets off.The reader may notice that a light increase in the marginal costs produce a

light decrease in the strategies. This fact reflects that the less competitive theproduction units are the less amount of energy supply at the same price. Inaddition, it produces a light increase in the spot price: bp∗(ci + 0.001) = 3.2326and bp∗(ci + 0.005) = 3.2354.By the other hand, a decrease in the marginal costs produce the reverse

effect. The following table shows that the more competitive the productionunits are the more amount of energy supply at the same price,

Production unit b∗i (ci) b∗i (ci − 0.001) b∗i (ci − 0.005)Endesa 1 956.5 956.7 (+0.2) 957.1 (+0.6)Endesa 2 1029.3 1029.4 (+0.1) 1029.8 (+0.5)U. Fenosa 1 1166.667 1166.667 (0) 1166.667 (0)U. Fenosa 2 1132.1 1132.2 (+0.1) 1132.4 (+0.3)Iberdrola 1 1002.0 1002.1 (+0.1) 1002.5 (+0.5)Iberdrola 2 84.4 84.7 (+0.3) 86.0 (+1.6)

where now b∗i (ci − ²) means the original equilibrium after a light perturbation² > 0, and b∗i (ci− ²)− b∗i (ci) brackets off. Hence, it produces a light decrease inthe spot price: bp∗(ci − 0.001) = 3.2311 and bp∗(ci − 0.005) = 3.2284.

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5 Concluding RemarksIn this paper we have presented an asymmetric model of continuous electricityauctions with limited production capacity and bounded rate bid supply func-tions. It describes many electricity markets in a very realistic way. For thismodel, we study the strategic bidding through the corresponding noncooper-ative game: the electricity market game. We prove that for every electricitymarket game with continuous production cost functions, there always exists amixed-strategy Nash equilibrium.In particular, we focus on the behavioral way of producers in the Spanish

electricity market. Taking into account the data published by the market oper-ator OMEL, we consider a very simple form for the Spanish electricity marketas an oligopoly just with independent hydro-electric power production units ina single wet period. We show that there always exists a pure-strategy Nashequilibrium for the Spanish electricity market game. In addition, a procedurebased on optimization tools is given to calculate a pure-strategy equilibrium fora numerical example. A sensitive analysis of the procedure on the results for theabove example is provided as well. We note in passing that the game proposedabove could be used in more general contexts for which the production unitswill employ the same technology.Additional topics for further research on the model of continuous electric-

ity auctions considered in this paper are: (1) existence of pure-strategy Nashequilibria for the electricity market game, (2) uniqueness of pure-strategy Nashequilibrium for the Spanish electricity market game above, (3) to extend theSpanish electricity market to an oligopoly with production units employing dif-ferent technologies.Apart from the model studied in this paper it could be also considered among

others the same model with linear or quadratic cost function being the para-meters random variables or a different model of dynamic continuous electricityauctions (each electricity producer submits for each hour of the next day hisschedule consisting of bid supply functions).

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