Nash equilibria in competitive electric energy markets

Electric Power Systems Research 60 (2002) 153– 160 www.elsevier.com/locate/epsr

Nash equilibria in competitive electric energy markets Diego Moitre * Fac. De Ingeniena, Campus Uni6ersitario, Uni6ersidad Nacional de Rı´o Cuarto, Ruta Nacional 36, Km. 601, 5800 Rio Cuarto, Cordoba, Co´rdoba, Argentina Received 23 February 2001; received in revised form 4 September 2001; accepted 27 September 2001

Abstract Several countries have modified the structure of their Electric Energy Markets (EEM) by the introduction of various levels of competence in the generation, transmission and distribution areas, which allows the generators to sell their production at a short-term market price (or spot price). The fundamental premise of the regulation is that the global efficiency can be improved through a strong competence in a market structure governed by explicit rules. The Game Theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. In this paper, the Game Theory is proposed to analyze the economic behavior of the generators to make their offers to the short-term EEM. The IEEE 9-bus system is used to illustrate the main features of the proposed method. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Electric energy markets; Economic dispatch; Game theory

1. Introduction In the last decade, the Electric Industry in Argentine, has changed from a vertically integrated structure of EEM, to a vertically and horizontally segmented one. The introduction of competence by means of the vertical and horizontal disintegration of the Electric Industry requires different forms of regulation compared with the ones traditionally applied in the sector. This approach puts emphasis both in the design of the EEM structure, and the set of rules that regulates, it where their ‘kindness’ is measured by the way in which they establish incentives for those behaviors that would contribute to the global efficiency [1]. The general spirit of the regulation in Argentine incorporates these innovative elements in orders to control the regulated companies through the verification of the fulfillment of obligations, subject to penalties and incentives. This EEM, includes different commercial and financial agreements, by means of contracts of different type and duration, shared risks, short-term transactions, among others. The nucleus of these agreements is the spot market in which the electric energy is valued and com* Tel./fax: +54-358-4676-252. E-mail address: [email protected] (D. Moitre).

mercialized. This task is carried out by CAMMESA, the Argentine Independent System Operator (ISO) [2]. The programming of the economic dispatch (ED) is carried out using models of optimization and simulation of the operation, where the objective is to minimize the cost of operation plus failure of the generating units. The prize of the energy reflects the cost of the next MW of load to be supplied subject to the restrictions associated to the transport and maintenance of the level of quality of the service and security established. From the results obtained in the daily unit commitment, CAMMESA determines the prevision of prices of the energy for each hour. In this paper, a methodology based on the noncooperative Game Theory [3–5] is used to analyze the economic behavior of the generating companies. The paper is organized as follows: First the application of the game theory on EEMs is reviewed, then the wholesale competitive spot market is described. Finally, the proposed method is presented.

2. Game theory The Game Theory can be defined as the study of mathematical models of conflict and cooperation be-

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tween decision-makers. The Game Theory is a mathematical technique that analyzes situations in which two or more individuals make decisions that influence one another’s welfare. The modem Game Theory was introduced in 1944 by Von Neumann and Morgenstern [6]. Others had anticipated some ideas, Zermelo (1913), Borel (1921) and even Von Neumann (1928). Important developments in this field took place during the period 1950– 1960. The Game Theory gain additional relevance in 1994 when the Nobel Prize of Economy was awarded to John Harsanyi, John Nash and Richard Selten for their contributions to the analysis of the equilibrium in noncooperative games. In theory, a game refers to any social situation, which involves two or more players. Two basic hypotheses exist that are made about the players: they are rational and intelligent. Each of these adjectives is employed in a technical sense. A player is rational if he makes consistent decisions with the achievement of his own objectives. It is supposed that the aim of each player is to maximize the expected value of its own payment, which is measured in some scale of utility. A player is intelligent if he knows everything that is relative to the game and can make inferences concerning the situations, which can take place. A first classification of the games, regarding the movements that each player makes and the information he has, is presented in Table 1. Von Neumann and Morgenstern investigated two different approaches. One is the strategic or noncooperative approach. This requires a very detailed specification of the rules of the game, so that the strategies available to the players could be known in detail. The objective is to find an adequate group of strategies of equilibrium, which will be called the solution of the game. What is best for a player depends on what the other players think to do and this in turn depends on Table 1 Games type Game

Characteristic

Static

The players perform their actions (strategies) simultaneously and receive payoffs that depend on the combination of previously executed actions The players perform their actions sequentially and receive payoffs, which depend on the combination of previously executed actions Each player’s payoff function is common knowledge among all the players At least one player is uncertain about another player’s payoff function In each movement the player that moves knows the complete story of the game In each movement the player who does not know the complete story of the game

Dynamic

Complete Information Incomplete information Perfect information Imperfect information

what they think the first player will do. Von Neumann and Morgenstem solved this problem in the particular case of games with two players whose interests are diametrically opposed. These games are called strictly competitive or of zero-sum because any player’s gain is always exactly balanced with a loss corresponding to the other player. The solution of games of non zero-sum, those in which the gain of a player is not the same as the loss of others, was first formulated by John Nash [7]. Several applications to EEM of noncooperative Game Theory has been suggested: in [8], the competitive behavior of the generators and the eventual coalitions’ that could be formed in order to exploit the situations of imperfect competence is discussed. The analysis is made resorting to the criterion max–min (or characteristic function) to make decisions based on the use of pure strategies. The next step in the analysis of the problem of generator’s information is treated in [9], the competence is modeled like a noncooperative game with incomplete information and is solved calculating the Bayesian Nash equilibrium. In both articles the authors deal with static games and the study cases correspond to situations with two or three players where the computational aspects of the resolution of the game are not relevant. The application of the games with complete information is presented in [10]. Three types of competitive behavior between generators are examined: perfect competence, imperfect competence and monopoly based on the use of pure strategies, analyzing the impact of the network on the market power. Reference [11] examines the competitive behavior between generators using mixed strategies, analyzing the impact of the congestion of the transmission network on the market power. The application of the games with incomplete information is presented in [12], where the unit commitment is modeled with restrictions in the transmission and requirements of spinning reserve. It is supposed that each generator only knows its own payment function and uses a Bayesian approach to deal with the information relative to the other generators. The second approach is the coalitional or cooperative, which adopts a less rigid attitude [13– 18]. It deals with situations in which the players can negotiate on how to develop the game before starting it. Besides, it is supposed that these negotiations can finish by the signing of a bonding agreement, which obliges them. In these conditions the concrete strategies which are available in the game are not too important in front of the structure of preferences of the game since it determines which contracts are feasible. The cooperative approach can be applied to assignation problems and the different solutions proposed can be interpreted as alternative solutions to a problem of assignation or distribution.

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3. Wholesale competitive spot market

dCi fpi ] u, dPGi

3.1. The mathematical model

dCi fpi 5 u, if PGi = P Max Gi M dPGi N % PGi = PL + % PDj

The problem of the active power ED of a thermal electric power system can be set a non-linear optimization problem, subject to the generation-load balance, the technological characteristic of the generating units, and the capacity of the transmission system restrictions [19,20]:

!

Minimise

"

M

N

Subject to: % PGi =PL + % PDj P Gi 5PGi 5 P Gi ; Max

Pk 5 P Max ; k

M



(1)

N

M

j=1

i=1

= % Ci (PGi )+ u % PDj +PL − % PGi i=1

n

Max + % [v Min (P Min (PGi −P Max i Gi − PGi ) + v i Gi )] i=1 TL

+ % [|k (Pk −P Max )] k

(2)

k=1

where u, Lagrange multiplier (generation-load balance); , v Max , Lagrange multipliers (technological characv Min i i teristics of the generating units); |k, Lagrange multipliers (capacity of the transmission system). The marginal costs of the thermal units are affected by the spatial and temporal quantification of the transmission losses, via the penalty factors of the corresponding generation bus, which are defined in the classic ED of active power by: (P fpi = 1− L (PGi

n

(4)



−1

(3)

The necessary conditions of the first order of Karush –Khun –Tucker [21] are: for i= 1,..., M Max if P Min Gi BPGi BP Gi

fpi



n

dCi (Pk + % |k ]u dPGi k  LIN (PGi

or, if PGi = P Max Gi fpi

n

dCi (Pk + % |k =u dPGi k  LIN (PGi



n

dCi (Pk + % | 5 u, dPGi k  LIN k(PGi

M

N

% PGi = PL + % PDj i=1

j=1

Pk B P

Max k

Pk = P

Max k

; kQLIN ; kLIN

|k ] 0; kLIN

M



k= 1,…, TL

or, if PGi = P Min Gi

i =1,…, M

k=1,…, TL

L(PG1,…, PGM )

;

The inclusion of active restrictions (LIN) into Eq. (4) is given by:

fpi

where Ci (PGi ), production cost of the unit i; PGi, active power output of the unit i; PDj, active power load at bus j; M, number of generating units; N, number of system buses; Pk, P Max , active power flow and its limit k Max , P , active power limits of the unit i; on line k; P Min Gi Gi PL, transmission losses; TL, number of transmission line. The Lagrange function is:

dCi fpi =u, dPGi

Pk B P

j=1

Min

j=1 max k

Max if P Min Gi B PGi B P Gi

i=1

i=1

i=1

for i= l,…, M

% Ci (PGi )

M

if PGi = P Min Gi

(5)

If the restrictions on the transmission capacity are not active, in the optimum, all thermal units operate at market price (u of the system) within its technological limits. For the ones which generate their minimum (or maximum) power, they operate, respectively, at i-bus price: zi = fpi

dCi dPGi

(6)

If the restrictions on the transmission capacity are active, in the optimum, all thermal units operate at the market price within its technological limits. For the ones, which generate their minimum (or maximum) power, they operate, respectively, at i-bus price: zi = fpi



dCi (Pk + % | dPGi k  LIN k (PGi

n

(7)

3.2. The competiti6e game The ISO programs the operation with the objective of minimizing the costs of fuel. To do so, a unit commitment is run to determine the cheapest generating units. The decision of the generators to take part in the spot market to supply the load is reflected in the

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Table 3 Thermal unit technological limits

Table 2 Generator cost data Generator

a

B

c

Startup cost ($)

Unit

Pmin (MW)

Pmax (MW)

Qmin (MVAr) Qmax (MVAr)

G1 G2 G3

150 100 335

5 1.2 1.0

0.11 0.085 0.1225

1500 2000 1500

G1 G2 G3

10.0 10.0 10.0

250.00 300.00 270.00

−300.00 −300.00 −300.00

function of prices declared to the ISO. In this work, a noncooperative game is considered, where generating companies represent the players that try to maximize their profit using different pure strategies (the price that they declare for their production). Their benefits are given by: Bk = % {zj ×PGj − [aj +bj PGj +cj PG2j ]

(8)

j  dk

where dk is the set of generating units belonging to the utility k and zj is the market price at bus j. The generating costs are represented by: C(PG) =a+ b PG +c PG2 ($/h)

(9)

The payoff function of the player k is: c

u k : 5 Si “ R i=1

(S,…, Sc )“ Bk

(10)

where c is the number of utilities, however, each generator knows only its own function of payment. The complete information is known by the ISO. Therefore, the economic behavior of the generators in the spot market can be analyzed by the ISO as a Static Game with Complete and Perfect Information (see Appendix A). 4. Simulation of a wholesale competitive spot market In this section, a simple example to simulate the ED of a wholesale competitive spot market is presented, using the IEEE 9 Bus Test Case. This system is conformed by nine transmission lines, three load bus and three generation bus denoted by {G1, G2, G3} with an installed capacity of 820.0 MW. It will be assumed that there are three generation utilities, denoted d1 ={G1}, d2 ={G2}, and d3 = {G3}. The corresponding curves of generation cost ($/h) are shown in Table 2, and the thermal unit technological limits are shown in Table 3. Three pure strategies are defined for each of these players (Table 4). Three load levels of were considered and for each level 33 unit commitment were run using MATPOWER® simulator version 2.0 under MATLAB® version 5.3. The algorithm, which allows it to shut down the expensive units (runuopf ) is based on a simplified version of the decommitment technique proposed in

300.00 300.00 300.00

[22]. The results correspond to the possible combinations of the pure strategy proposed for the schedule load (see tables B1, B2 and B3). A player does not need to know anything about the opponent to decide if a strictly dominated strategy can never be optimal (see definition A2). If we assume that the rational players do not play strictly dominated strategies, an algorithm of iterated elimination of strictly dominated, strategies can be designed [23]. In other words, to justify a number of elimination of strictly dominated strategies, it is necessary to assume that is common knowledge that no player is sufficiently irrational as to play a strictly dominated strategy. The Nash equilibrium (see definition A3) obtained by an algorithm of iterated elimination strictly dominated strategies for several loads are shown in the following tables Tables 5–7. The first two games present multiple Nash equilibria. The assumption that Nash equilibrium is played relies on there being some mechanism or processes that leads all players to expect the same equilibrium. In this case, the mechanism is provided by the ISO, which will dispatch those generators that produce the lowest value of the objective function. For example, for the load of 100 MW, the ISO will dispatch the generators G2 and G3 and consequently the best strategy for the utility d2 is HIGH while for the utility d3 is LOW, without caring which strategy the utility d1 takes. Observe that the remaining two equilibria produce the same value of the objective function, but correspond to the profiles of different strategies. A slightly different situation is presented for a load of 150 MW, in the sense that the three Nash equilibria produce three different values of the objective function. The ISO will dispatch the generators G1 and G2 and consequently the best strategy for the utility d1 is BASE while for the utility d2 the best one is LOW; in this case it is the generator G3, the one that does not result dispatched. Finally, the case corresponding to the 250 MW presents a simple Nash equiTable 4 Strategies of generation utilities Strategies

Prices

LOW BASE HIGH

75% of the marginal cost of the unit Marginal cost of the unit 25% over the marginal cost of the unit

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Table 5 Pure-strategy Nash equilibrium (Load 100 MW)

Table 7 Pure-strategy Nash equilibrium (Load 250 MW)

Nash equilibrium

Generator

Strategy

Benefit ($/h) Objective ($/h)

Nash equilibrium

Generator

Strategy

Benefit ($/h) Objective ($/h)

1

G1 G2 G3 G1 G2 G3 G1 G2 G3

BASE HIGH LOW BASE HIGH BASE HIGH BASE HIGH

0 172.83 18.39 0 780.77 0 0 780.77 0

1

G1 G2 G3

HIGH HIGH HIGH

355.09 907.93 382.42

2

3

990.49

1102.88

1102.88 0

librium corresponding to the profile of strategies (HIGH, HIGH, HIGH) where the three utilities are dispatched. It is concluded that the design of this EEM induces the competence between generators in scenes of a low load, but not for a high load where the generators could agree to offer all HIGH to maximize its benefits. The computational experience of the author indicates that the number of Nash equilibria that takes place in this structure of competitive EEM diminishes with the increase of load of the system, with a tendency to single Nash equilibrium. This equilibrium corresponds to the profile of strategies where all the generator offers HIGH. The point of transition of multiple equilibria to single equilibrium depends on the relation load versus installed capacity, and in this particular example it is of the order of 30%.

5. Conclusion The current structure of the Argentine EEM considers the possibility of the generators offering their production of the spot market from the definition of a price curve. In this paper, the economic behavior of the generators is modeled by the ISO as a Static Game with Complete and Perfect Information. The participants’ bids are modeled as pure strategies and the Nash Table 6 Pure-strategy Nash equilibrium (Load 150 MW) Nash equilibrium

Generator

1

G1 G2 G3 G1 G2 G3 G1 G2 G3

2

3

Strategy

BASE HIGH HIGH BASE LOW BASE LOW BASE BASE

Benefit ($/h) Objective ($/h) 342.99 836.39 0 73.47 358.27 0 34.10 469.25 0

3032.93

1998.42

1499.66

1542.60

equilibrium concept is used. From the results, it is concluded that the number of Nash equilibrium that takes place in the unit commitment diminishes as the load of the system increases. Multiple restrictions such as: spinning reserve, thermal unit (minimum up time, minimum down time, crew constraints), hydro-constraints, fuel constraints, that the unit commitment has in real systems, will affect the value of the point of transition. Nevertheless, it will persist the fact that in the cases in which the multiple Nash equilibria exist, the ISO defines a mechanism that solves the game by programming the ED of the generating units. The design of this EEM induces the competence between generators in scenes of a low load, but not for a high one. In competitive EEM with a high number of generating companies, the possibility of establishing coalitions to increase their benefits is remote, however, it could happen in the case of EEM structures with a reduced number of agents. In this context, the Game Theory provides a methodology to study the way the rules affect the incentives and consequently, the strategies. In some cases, it could be used by individual agents of the EEM to devise strategies that would benefit them, but the most important function of the Game theory is to identify the way the rules must be modified to improve the global efficiency of the EEM.

Appendix A. Static games with complete and perfect information Definition A1: the normal-form representation of an N-player game specifies the players’ strategy space S1,…,SN and their payoff functions u1,…, uN. For k= 1,…, N: N

uk : 5 Si “ R i=1

We denote the game by G= {S1,…,SN ; u1,...,uN }. Definition A2: in the normal-form game G= {S1,…,SN ; u1,…, uN}, let si% and s ¦i by feasible strategies for player i (s %S i i ; S1 Si ). Strategy s ¦ i is strictly dominated by strategy s ¦i if for each feasible combination of the other players’ strategies, i’s payoff from playing s %i is strictly less than i’s payoff from playing s ¦: i

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ui (S1,…, Si − 1, Si, Si + 1,…, SN ) Bui (S1,…, Si − 1, Si, Si + 1,…, SN ) for each: (S1,…, Si − 1, Si + 1,…, SN ) that can be constructed from the other players’ strategy spaces S1,…, Si − 1, Si + 1,…, SN. Definition A3: in the N-player normal-form game G = {S1,…, SN ; u1,…, uN }, the strategies (s *,…, s *) 1 N are a Nash equilibrium if, for each player i, s *i is (at least tied for) player i’s best response to the strategies specified by the N-1 other players, (S *,…, S *i − 1, S *i + 1,…, 1 S *): N ui (S*,…, S*i − 1, S*, 1 i S* i + 1,…, S* N) S*i − 1, Si, S*i + 1,…, S*N ) ] ui (S*,…, 1 for every feasible strategy si in Si ; that is, s *i solves: max ui (s*,…, s*i − 1, si, s*i + 1,…, s*N 1 si  Si

The Nash equilibrium is a strategy profile in which

each player’s part is as good a response to what the others are meant to do as any other strategy available to that player. The relationship between the two former concepts is the following: if iterated elimination of strictly dominated strategies eliminates all but the strategies (s *,…, i then these strategies are the unique Nash equis *), N librium of the game. If we quantify the uncertainty of a player with respect to what the other players will do, it is proved that the concept of Nash equilibrium is stronger than the iterated elimination of strictly dominated strategies. Definition A4: in the normal-form game G= {S1,…,SN ; u1,…,uN }, we assume that each space of strategies is finite. Then a mixed strategy for each player is a discrete probability distribution. Nash (1950) proved that if the mixed strategies are admitted in any finite game (i.e. where the number of players and the spaces of strategies are finite) then there exists at least one Nash equilibrium, possibly involving mixed strategies.

Appendix B. Study case Result of the simulation (Load 100 MW). Table B1. Unit commitment. Number Strategies

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

G1 L L L L L L L L L B B B B B B B B B H H H H H H

G2 L L L B B B H H H L L L B B B H H H L L L B B B

G3 L B H L B H L B H L B H L B H L B H L B H L B H

Generation (MW)

Market price ($/MWh)

Benefit ($/h)

G1 34.48 34.48 34.48 43.53 43.53 43.53 50.64 50.64 50.64 0 0 0 0 34.48 34.48 0 41.60 41.60 0 0 0 0 0 0

G1 9.474 9.474 9.474 10.976 10.976 10.976 12.156 12.156 12.156 8.640 14.350 14.350 10.117 12.586 12.586 11.297 14.152 14.152 8.640 14.350 14.350 10.117 19.164 19.164

G1 −126.51 −126.51 −126.51 −98.30 −98.30 −98.30 −69.71 −69.71 −69.71 0 0 0 0 −19.21 −19.21 0 40.36 40.36 0 0 0 0 0 0

G2 66.21 66.21 66.21 57.08 57.08 57.08 49.93 49.93 49.93 59.09 101.79 101.79 51.29 66.20 66.20 45.08 59.02 59.02 59.09 101.79 101.79 51.29 101.79 101.79

G3 0 0 0 0 0 0 0 0 0 42.02 0 0 49.82 0 0 56.04 0 0 42.02 0 0 49.82 0 0

G2 9.374 9.374 9.374 10.904 10.904 10.904 12.116 12.116 12.116 8.463 13.929 13.929 9.919 12.455 12.455 11.084 14.048 14.048 8.463 13.929 13.929 9.919 18.505 18.505

G3 9.463 9.463 9.463 10.998 10.998 10.998 12.213 12.213 12.213 8.472 14.151 14.151 9.906 12.573 12.573 11.050 14.171 14.171 8.472 14.151 14.151 9.906 18.799 18.799

G2 68.58 68.58 68.58 176.96 176.96 176.96 233.13 233.13 233.13 32.38 314.98 314.98 123.59 272.57 272.57 172.83 362.20 362.20 32.38 314.98 314.98 123.59 780.77 780.77

Objective ($/h) G3 0 0 0 0 0 0 0 0 0 −237.32 0 0 −195.35 0 0 18.39 0 0 −237.32 0 0 −195.35 0 0

755.59 755.59 755.59 878.44 878.44 878.44 980.20 980.20 980.20 796.61 829.75 829.75 901.79 1005.20 1005.20 990.49 1132.21 1132.21 796.61 829.75 829.75 901.79 1102.88 1102.88

D. Moitre / Electric Power Systems Research 60 (2002) 153–160

25 26 27

H H H

H H H

L B H

0 45.08 0 53.06 34.50 66.19

56.04 11.297 11.084 11.050 0 48.04 13.038 12.780 12.771 0 0 15.737 15.572 15.719 89.50

159

172.83 −156.51 990.49 275.13 −52.23 1169.62 478.89 0 1256.72

Result of the simulation (Load 150 MW). Table B2. Unit commitment. Number Strategies

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

G1 L L L L L L L L L B B B B B B B B B H H H H H H H H H

G2 L L L B B B H H H L L L B B B H H H L L L B B B H H H

G3 L B H L B H L B H L B H L B H L B H L B H L B H L B H

Generation (MW)

Market price ($/MWh) Benefit ($/h)

G1 36.06 57.11 57.11 42.39 69.57 69.57 47.00 53.17 79.37 0 45.08 45.08 31.22 36.12 57.14 35.43 41.16 66.95 0 0 36.13 0 0 47.61 0 32.37 36.12

G1 9.736 13.231 13.231 10.787 15.299 15.299 11.552 12.577 16.925 12.820 14.916 14.916 11.868 12.946 17.572 12.795 14.055 19.728 12.820 14.292 16.187 15.010 17.055 19.342 16.764 15.151 16.184

G2 67.41 94.49 94.49 55.27 81.84 81.84 46.45 51.29 71.94 89.66 106.78 106.78 61.20 67.47 94.46 51.94 57.83 84.50 89.66 100.62 115.95 78.30 89.79 104.19 69.26 62.64 67.45

G3 47.84 0 0 53.51 0 0 57.65 46.54 0 63.35 0 0 58.98 47.71 0 63.95 52.20 0 63.35 52.43 0 74.72 63.22 0 83.79 56.37 47.73

G2 9.528 12.994 12.994 10.595 15.112 15.112 11.376 12.404 16.795 12.377 14.568 14.568 1.605 12.670 17.257 12.543 13.795 19.464 12.377 13.780 15.742 14.511 16.465 18.912 16.225 14.818 15.840

G3 9.542 13.165 13.165 10.586 15.288 15.288 11.346 12.401 16.970 12.395 14.780 14.780 11.591 12.689 17.484 12.504 13.790 19.697 12.395 13.845 15.989 14.483 16.490 19.182 16.150 14.810 15.864

G1 G2 −122.26 75.14 −38.70 255.50 −38.70 255.50 −102.35 159.61 34.10 469.25 34.10 469.25 −85.05 189.28 −58.11 251.05 103.53 582.00 0 218.82 73.47 358.27 73.47 358.27 −42.80 185.38 −650 286.32 209.22 658.32 −11.90 259.85 36.35 344.10 342.99 836.39 0 218.82 0 305.23 110.59 443.37 0 421.13 0 585.35 283.48 822.69 0 532.89 63.33 419.51 110.45 500.76

Objective ($/h) G3 −206.71 0 0 −172.81 0 0 −145.68 −69.73 0 −104.75 0 0 −136.48 −56.16 0 −100.30 −1.15 0 −104.75 1.72 0 −11.48 154.67 0 74.37 54.22 95.38

1279.52 1328.81 1328.81 1401.14 1542.60 1542.60 1496.08 1674.93 1716.05 1337.84 1499.66 1499.66 1520.37 1703.56 1766.60 1630.05 1830.46 1988.42 1337.84 1537.69 1632.69 1535.44 1780.97 1944.36 1698.08 1950.54 2129.62

Result of the simulation (Load 250 MW). Table B3. Unit commitment. Number Strategies

1 2 3 4 5 6 7 8

G1 L L L L L L L L

G2 L L L B B B H H

G3 L B H L B H L B

Generation (MW)

Market price ($/MWh) Benefit ($/h)

G1 51.56 57.04 78.66 59.57 66.33 70.99 65.40 73.21

G1 12.310 13.218 16.808 13.639 14.761 15.535 14.607 15.902

G2 87.67 94.77 122.92 72.31 78.98 83.59 61.14 67.33

G3 61.97 49.35 0 69.21 55.72 46.43 74.51 60.43

G2 12.121 13.030 16.634 13.492 14.626 15.410 14.499 15.815

G3 12.140 13.091 16.910 13.472 14.650 15.468 14.444 15.806

G1 −65.52 −39.14 98.20 −25.72 13.48 43.53 7.81 58.57

G2 204.13 257.71 512.85 344.39 430.17 493.89 395.36 498.69

Objective ($/h) G3 −115.09 −36.65 0 −58.59 45.25 72.67 −13.38 112.38

1822.93 2014.17 2073.71 2005.03 2222.38 2398.06 2144.21 2382.59

D. Moitre / Electric Power Systems Research 60 (2002) 153–160

160

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

L B B B B B B B B B H H H H H H H H H

H L L L B B B H H H L L L B B B H H H

H L B H L B H L B H L B H L B H L B H

78.64 38.27 43.18 63.38 45.48 51.66 55.97 50.81 58.03 63.13 0 33.35 36.41 35.43 41.06 45.02 40.28 46.93 51.67

71.65 95.64 103.99 138.42 79.76 87.74 93.32 68.03 75.54 80.86 118.86 110.56 117.00 85.10 94.10 100.46 73.03 81.59 87.71

50.66 67.52 54.20 0 76.07 61.80 51.87 82.39 67.53 57.06 83.70 57.66 48.15 80.97 66.22 55.85 88.11 72.75 61.82

16.804 13.419 14.500 18.944 15.005 16.365 17.313 16.179 17.767 18.889 16.693 15.422 16.262 15.993 17.540 18.630 17.328 19.155 20.459

16.734 13.142 14.211 18617 14.758 16.116 17.064 15.964 17.560 18.692 16.114 15.052 15.876 15.666 17.198 18.277 17.027 18.846 20.147

16.761 13.161 14.279 18.959 14.731 16.140 17.132 15.894 17.545 18.722 16.135 15.127 15.995 15.633 17.223 18.353 16.945 18.825 20.178

References [1] M. Hesse, Regulacio´ n del Sector Ele´ ctrico: objetivos y principios. Presente y Futuro del Sector Ele´ ctrico. Revista del instituto de Estudios Econo´ micos No. 4, Madrid, 1991. [2] H. Rudnick, R. Varela, W. Hogan, Evaluation of alternatives for power system coordination and pooling in a competitive environment, IEEE Transactions on Power Systems 12 (2) (1997) 605 – 613. [3] R. Myerson, Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA, 1991. [4] D. Fudemberg, J. Tirole, Game Theory, The MIT Press, Cambridge, MA, 1991. [5] H. Kuhn (Ed.), Classics in Game Theory, Princeton University Press, Princeton, NJ, 1997. [6] J. Von Neumann, O. Morgenstern, The Theory of Games and Economic Behavior, Princenton University Press, Princeton, NJ, 1944. [7] J.F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences 36 (1950) 48 –49. [8] R.W. Ferrero, S.M. Shahidehpour, V.C. Ramesh, Transaction analysis in deregulated power systems using Game Theory, IEEE Transactions on Power Systems 12 (3) (1997) 1340 –1347. [9] R.W. Ferrero, J.F. Rivera, S.M. Shahidehpour, Application of games with incomplete information for pricing electricity in deregulated power pools, IEEE Transactions on Power Systems 13 (1) (1998) 184 – 189. [10] C. Berry, B. Hobbs, W. Meroney, R. O’Neill, R. Stewart, Analizing Strategic Bidding Behavior in Transmission Networks, IEEE Tutorial on Game Theory Applications in Electric Power Markets, 9TP136-0 (1999) 7 –32. [11] S. Stoft, Using Game Theory to study Market Power in Simple Networks, IEEE Tutorial on Game Theory Applications in Electric Power Markets, 9TP136-0 (1999) 33 – 40. [12] H.Y. Yamin, S.M. Shahidehpour, Risk Management Using

[13]

[14]

[15]

[16]

[17]

[18]

[19] [20]

[21] [22]

[23]

98.0 11.09 55.11 291.90 77.50 143.55 194.57 134.02 220.45 288.42 0 75.23 114.22 101.40 179.44 240.67 168.10 272.03 355.09

576.64 264.64 333.83 682.25 440.65 554.37 640.20 511.01 650.80 758.64 471.82 392.48 453.53 515.49 652.75 757.72 602.51 773.90 907.93

149.06 −72.36 24.86 0 0.65 132.80 172.18 60.57 223.65 27.38 73.60 72.29 103.00 46.71 202.11 252.06 118.90 313.43 382.42

2573.90 1969.84 2180.78 2334.55 2181.31 2426.11 2622.15 2344.04 2616.77 2833.99 2044.10 2305.71 2487.65 2313.67 2579.87 2792.05 2494.75 2794.75 3032.93

Game Theory by Transmission Constrained Unit Commitment in a Deregulated Power Market, IEEE Tutorial on Game Theory Applications in Electric Power Markets, 9TP136-0 (1999) 40 –60. J. Ruusunen, H. Ehtamo, R.P. Hamalainen, Dynamic cooperative electricity exchange in power pool, IEEE Transactions on Systems, Man and Cybernetics 21 (4) (1991) 758 – 766. A. Haurie, R. Loulou, G. Savard, A two-player game model of power cogeneration in New England, IEEE Transactions on Automatic Control 37 (9) (1992) 1451 – 1456. J.W. Lima, M.V. Pereira, J.L. Pereira, An integrated framework for cost allocation in multiowned transmission system, IEEE Transactions on Power Systems 10 (2) (1995) 971 – 977. Y. Tsukainoto, I. Joda, Allocation of fixed transmission cost to Wheeling transactions by Cooperative Game Theory, IEEE Transactions on Power Systems 11 (2) (1996) 620 – 629. X. Bai, S.M. Shahidehpour, V.C. Ramesh, E. You, Transmission analysis by Nash game method, IEEE Transactions on Power Systems 12 (3) (1997) 1046 – 1052. D. Moitre, H. Rudnick, Integration of wholesale competitive electric markets: an application of the Nash bargaining generalised solution, International Journal of Electrical Power and Energy Systems 22 (7) (2000) 507 – 510. A. Wood, B. Wollenberg, Power Generation, Operation, and Control, second ed., Wiley, New York, 1996. A.A. El-Keib, X. Ma, Calculating short-run marginal costs of active and reactive power production, IEEE Transactions on Power Systems 12 (2) (1997) 559 – 565. M. Bazaraa, H. Sherali, C. Shety, Nonlinear Programming, second ed., Wiley, New York, 1993. C. Li, R.B. Johnson, A.J. Svoboda, A new unit commitment method, IEEE Transactions on Power Systems 12 (1) (1997) 113 – 119. R. McKelvey, A. McLennan, Computation of equilibria in finite games, in: Handbook of Computational Economics, vol. 1, Elsevier, Amsterdam, 1996.