Direct approach in computing robust Nash strategies for generating companies in electricity markets

Electrical Power and Energy Systems 54 (2014) 442–453

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Direct approach in computing robust Nash strategies for generating companies in electricity markets Damoun Langary a, Nasser Sadati a,b,⇑, Ali Mohammad Ranjbar b a b

Intelligent Systems Laboratory, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran Center of Excellence in Power System Management and Control (CEPSMC), Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 7 February 2013 Received in revised form 24 July 2013 Accepted 25 July 2013

Keywords: Game theory Supply function equilibrium (SFE) Strategic bidding Nash equilibrium (NE) Electricity market

a b s t r a c t Supply function equilibrium (SFE) is often used to describe the behavior of generating companies in electricity markets. However, comprehensive analytical description of supply function models is rarely available in the literature. In this paper, using some analytical calculations, a novel direct approach is proposed to compute the Nash equilibrium (NE) of the supply function model under uniform marginal pricing mechanism. An explicit mathematical proof for its existence and uniqueness is also presented. The proposed methodology is then generalized to accommodate practical market constraints. In addition, a new concept of robust NE is introduced and calculated based on this approach. Finally, numerical simulations demonstrate the applicability and effectiveness of the proposed solution scheme.  2013 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the trend of electricity industry in many countries has been towards a less regulated and more competitive energy market. Several methods and theories have been introduced and utilized to model different aspects of deregulated pool-based electricity markets. Agent-based modeling [1–3], statistical methods [4,5], artificial intelligence based approaches [6–8], optimization theory [9], and of course game-theoretic approaches [10–14] have been all frequently used in the literature. Among these approaches, game theoretic methods not only have better ways of realistically simulating the oligopolistic competition in electric power markets, but also are perfectly applicable to a wide range of market models such as Bertrand, Cournot, Stackelberg and supply function models [15–17]. Among all aforementioned market models, the Cournot and SFE models seem to fit better and hence are more frequently exploited in the analysis of electricity markets. The Cournot models are simpler in essence and provide more intuition into the actual behavior of the market through analytical results. On the other hand, supply function models grant much more consistency with what really goes on in most electricity markets, and as a result, many of recent publications in the literature have implemented SFE in their

⇑ Corresponding author at: Center of Excellence in Power System Management and Control (CEPSMC), Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran. Tel.: +98 (21)6616 4365; fax: +98 (21)6616 5939. E-mail addresses: [email protected], [email protected] (N. Sadati). 0142-0615/$ - see front matter  2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.07.031

simulations. In spite of certain well-known advantages of the supply function models [18], there are some known limitations in properly applying SFE to analyze strategies of generating companies in power markets [19]. The first and probably most important drawback of supply function models is the uniqueness issue; it is straightforward to show that an infinite number of NEs could exist [20,21]. To tackle this issue, Baldick [21] and others have examined some arbitrary parameterization methods to restrict the results to a unique NE point, which is frequently used in the literature. This simplification process of course leads to some other questions on how to set those parameters. Another downside to supply function models is that the NE is usually calculated through some iterative algorithms that not only do not have any guarantees of convergence, but also provide little insight into the characteristics of the calculated NE. In this paper, we present a new approach to directly calculate the NE of electricity markets using the supply function model. The solution is then generalized to support all aforementioned parameterization methods [21]. This direct computation approach not only guaranties the existence and uniqueness of the NE, but also opens way to analyze the effect of parameterization techniques in the resulted equilibrium point. Given this new perspective into the NE calculation problem, we then proceed to some more advanced ideas to obtain a more eligible assessment of independent power producers’ behavior according to the characteristics of the actual market. As it will be shown, the proposed method is also applicable to computing NE of Cournot models.

D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

The remainder of this paper is structured as follows. Section 2 contains the literature review. The problem formulation is given in Section 3. In Section 4, the proposed solution method is presented and discussed in detail. The method is then generalized in Section 5, to cope with existent constraints of the actual markets. In Section 6, the new concept of robust Nash strategy is introduced and resolved. The simulation studies are presented in Section 7. Finally, concluding remarks are drawn in Section 8. 2. Literature survey The concept of SFE was originally developed by Klemperer and Meyer [20] as a better way of modeling independent players’ competing behaviors in markets under uncertain demand conditions. The SFE approach was later adopted by Green and Newberry [22,23] as a model for strategic bidding in the British and Wales’ electricity spot market, assuming that suppliers must bid the same supply function across multiple pricing periods. Their researches, however, implied some oversimplifying restrictions on the cost functions of the participating suppliers. In addition, they did not explain how to incorporate transmission constraints into their approach. Using similar restricting assumptions of [1], Rudkevich proved that if the players begin the bidding based on their marginal costs and then update their strategies according to a described learning process, in the limit, the supply functions converge to a SFE which provides an explanation of how might the bids of actual players evolve into their Nash strategies. A detailed literature review of bidding strategies in electricity markets are presented in [24]. In a distinguished work in 2002 [21], Baldick compared the main variations of supply function models of bid-based electricity markets in presence and absence of transmission constrains. Based on artificial restrictions that are arbitrarily placed on the strategic parameters of players, he classified the supply function models used in literature into four categories, namely, R – parameterization, c – parameterization, (R / c) – parameterization and (R, c) – parameterization. Whilst the latter parameterization method indicates the true nature of supply function models, the restricted models are more frequently used because of their simplicity. Baldick concluded that parameterization of the SFE has a significant effect on the calculated results, to such an extent that several SFE results demonstrated in the literature are, in fact, artifacts of assumptions about the choices of particular bid parameters. de la Torre et al. [25,26] presented a detailed model of the electricity market consisted of profit-maximizing generating companies (GENCOs), by taking multi-period bidding, price elasticity of demand and network modeling into account. The behavior of market participants and the market itself were characterized through a repetitive simulation procedure. Each firm, using a price-quota curve, is able to solve its profit maximization problem and construct its optimal bidding strategy. The method can be of interest to regulators that study the behavior of oligopolistic markets and also to GENCOs who want to know the best strategy to follow. In [27], the authors extended the proposed idea of [10] to develop a more general method for analyzing the competition among transmission-constrained GENCOs with incomplete information. Each GENCO models its opponents’ unknown information with specific types to transform the game of incomplete information into a complete game with imperfect information. Using SFE to model GENCOs’ bidding strategies, the competition was modeled as a bi-level optimization problem: the upper subproblem represents the GENCOs’ profit maximization and the lower subproblem is the ISO’s market clearing problem, which is solved by minimizing consumers’ total payment as a measure of social welfare. Price elasticity of demand was assumed to be zero in this study. Using GAMS programming language, the same bi-level problem was

443

solved in [28], in which the authors used a similar iterative algorithm to find the NE, and thus, the bidding strategies of GENCOs in a day-ahead energy market. In [29], a similar bi-level optimization problem was solved to find the NE using MPEC approach. In [30], using analytical computations, the authors have provided closed formulas for price, quantities and profits for a shortterm electricity market whose participants rely on Cournot models. Based on those formulas, the case of several identical Cournot GENCOs and the case of one dominant GENCO were compared in a detailed manner. In [31], the authors have proposed a new algorithm for the calculation of Cournot equilibrium in the absence of transmission constraints, and hereby analyzed and investigated the properties of deregulated electricity markets using game theory. The algorithm is based on transforming the game into a three level decision making process with economic signal exchange. They argued that even though SFE models are better representatives of actual biddings in electricity markets, they do not possess the attractive features of the Cournot models as far as ease of modeling and eligible computation of NE are concerned. The main drawback of Cournot models, however, is their high level of sensitivity to the price elasticity of demand curve. In [13], Bompard et al. have provided a comparative analysis of the application of game theoretic models to simulate the oligopolistic competition in network constrained electricity markets, by focusing on strategic behavior of electricity producers. Several models such as SFE, Cournot, Stackelberg and conjectural supply function models were considered and their appropriateness to model the electricity was discussed. Furthermore, the effects of both strategic bidding and network constraints on the efficiency of the market were investigated. 3. Problem formulation: linear SFE Assuming that independent power producers (IPPs) are rational self-interested players, the objective of each player is to maximize its respective profit function. Since each player’s profit not only depends on its own bidding strategy, but also on its rivals’ bidding behavior, the bidding strategies of GENCOs and the NE of the market needs to be determined at once. The market clearing mechanism is assumed to have uniform pricing. Assuming a linear supply function model for GENCOs’ bidding strategies, we have

q ¼ ai Pi þ bi ; i ¼ 1; . . . ; nG

ð1Þ

where q is the market clearing price (MCP) and Pi is the production level of ith player (GENCO). Also ai and bi present the strategic bidding parameters of ith player in the supply function model. Each player, willing to increase its own profit, might be able to do so either by increasing the MCP using higher values of strategic parameters ai and bi (offering higher prices for the same amount of electric energy) or by increasing its output via decreasing its strategic parameters (offering higher amounts of electric energy for the same price). The objective of each GENCO is then to find a profit-maximizing supply function, by tuning its control variables ai and bi via an optimization process. The demand function of the market is assumed to be elastic (price-sensitive), expressed as follows

D ¼ D0  c  q

ð2Þ

where D is the actual demand level at market price q. D0 is an initial demand value and c is a price elasticity coefficient, accounting for a mild level of elasticity in demand of electric power markets. Now, by introducing new control variables xi = 1/ai and hi = bi/ai for all players, Eq. (1) can be rewritten as follows

Pi ¼ xi q  hi ;

i ¼ 1; . . . ; nG

ð3Þ

444

D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

in which xi and hi are the new (substitute) strategic parameters for ith GENCO. At the economic equilibrium point, the aggregate production quantity must be equal to the total demand, that is,

D ¼ D0  c  q ¼ ) qMCP

nG X

nG X

i¼1

i¼1

Pi ¼

P D0 þ i hi P ¼ c þ i xi

ðxi q  hi Þ ð4Þ

P D0 þ j hj P  hi ; c þ j xj

i ¼ 1; . . . ; nG

ð5Þ

Also, the cost function of generating companies is usually approximated by a quadratic function of its production level, i.e.

1 C i ðPi Þ ¼ ai P2i þ bi þ ci 2

ð6Þ

where ai and bi are coefficients of ith GENCO’s marginal cost function. Now by assuming the market pricing mechanism to be uniform, the profit function of ith player can be formulated as follows

ui ðx; hÞ ¼ qPi 

 @ui  ¼0 @hi NE  @ui @ q @ui @Pi   þ  ¼0 ) @ q @hi @Pi @hi NE

  1 2 ai Pi þ bi þ ci 2

ð7Þ

Pi 

1

ð9bÞ

r

   1 þ ðq  ai P i  bi Þ  xi   1 

r

where x and h are vectors consisting of player 1 to nG’s strategic parameters. We remark that the market price q and generation level Pi in the right hand side of Eq. (7) could be replaced with terms consisting of x and h using Eqs. (4) and (5). Now, by assuming GENCOs to be rational self-interested players, each tries to maximize its profit by adjusting its own strategic parameters accordingly. Thus at the NE, in which every player’s strategy is optimal given the other players do not alter their respective strategic parameters, the first-order necessary conditions imply that

ð8aÞ ð8bÞ

i NE

Therefore, there exist 2nG equations to be solved and 2nG strategic parameters to be adjusted. However, it is easy to show that the two equation sets above are not linearly independent. Hence, we are left with nG degrees of freedom, which raises a major question: How to find one of these many solutions? A prevalent way to tackle this issue in the literature is to use one of the parameterization techniques described by Baldick in [21] where, one of the two classic strategic parameters ai or bi or their ratio bi/ai(=hi) are assumed to be constant pre-chosen quantities for all participating GENCOs. Therefore, it reduces the problem of finding NE for the supply function model to a nG-variable nG-equation system. It is easy to prove that any NE of the reduced Supply function model is indeed a NE of the original model. However, such answer for the above question raises yet another question: Why is this particular equilibrium chosen out of infinite number of existing NEs? 4. The proposed solution method: direct computation of NE 4.1. Modeling Because the two sets of Eqs. (8a) and (8b) are not independent, from now on, we will only consider a single set (8a) in our calcula-

¼0

ð10Þ

NE

P where r ¼ c þ j xj . Because strategic parameters aj (hence xj) are usually positive, and also price elasticity coefficient c is nonnegative, r is nonzero (positive). Therefore multiplying the above equation by r yields

    Pi þ q  ai Pi  bi  xi  r ¼ 0

ð11Þ

in which the star indicates the value of the corresponding variables at any NE point. Eq. (11) is a key statement for us; the basic necessary condition we incorporate here to find the NE. Now let’s suppose that strategic parameters xi are being fixed, by adopting some pre-assigned values for all players i = 1, . . . , nG. Using Eqs. (3) and (11) could be rewritten as follows





xi q  hi þ ð1  xi rÞq þ ai hi  bi ðxi  rÞ ¼ 0 ⁄

 @ui  ¼ 0; i ¼ 1; . . . ; nG @ xi NE  @ui  ¼ 0; i ¼ 1; . . . ; nG @h 

ð9aÞ

Now, using Eqs. (4), (5) and (7), by replacing terms with their respective values, we obtain

We remark that unless a constraint is violated, the above equation always holds, no matter what strategy each player may have adopted. Given the MCP, each GENCO’s production level could be calculated using its strategic parameters

P i ¼ xi q  h i ¼ xi

tions. Thus, by taking the partial derivative of ith GENCO’s profit function with respect to parameter hi, we get

ð12Þ

hi

in which, q and are the unknown variables; the other parameters are either arbitrarily chosen or assumed to be known. Now considering Eq. (12) for all players i = 1, . . . , nG, there exists a set of nG linear equations in contrast to (nG + 1) unknown variables to be calculated. Thus an extra equation needs to be added to this system of linear equations using Eq. (4), that is

r  q 

X  hj ¼ D0

ð13Þ

j

Now,   if we define T a new vector of unknown variables x ¼ h1    hnG q , the set of equations defined by Eqs. (12) and (13) could be rewritten as a linear matrix equation of rank (nG + 1), as follows

Fx ¼j

ð14aÞ

in which, 2 6 6 6 6 F¼6 6 6 6 4

a1 ðx1  rÞ  1 0 .. . 0 1

0

 0 .. .. a2 ðx2  rÞ  1 . . .. . 0   0 anG ðxnG  rÞ  1 1  1

x1 þ ðx1  rÞð1  a1 x1 Þ

3

7 x2 þ ðx2  rÞð1  a2 x2 Þ 7 7

7 7 7 7 7 xnG þ ðxnG  rÞð1  anG xnG Þ 5 .. .

r ð14bÞ

3 b1 ðx1  rÞ 7 6 .. 7 6 . 7 j¼6 7 6 4 bnG ðxnG  rÞ 5 D0 2

ð14cÞ

Hence, if the coefficient matrix F(a, c, x) to be nonsingular, in which T a ¼ ½ . . . ai . . .  is consisted of coefficients of the quadratic cost functions and x ¼ ½ . . . xi . . . T is a vector of arbitrarily pre-chosen strategic parameters, then there exists a unique solution as follows for the matrix Eq. (14a).

x ¼ F1  j and consequently we get

ð15Þ

D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

2

3

.. 6 . 7 6 7 h ¼ 6 hi 7 ¼ ½InG j0  F1  j 4 5 .. . 

ð16Þ

where InG is the identity matrix of size nG and 0 is a vector of all zeros. Furthermore, the last element of vector x⁄ gives us q⁄, which is the MCP of supply function model at the resulted NE point. Then, for every player i, the ordered pair ðxi ; hi Þ is a NE strategy for LSF model. As a result, one can easily calculate the corresponding ai and bi of the original supply function model (Eq. (1)) as follows

aNE ¼ 1=xi i bNE ¼ hi =xi i

ð17aÞ ð17bÞ

For the sake of simplicity, we avoid switching back to original parameters in the rest of the paper. We remark that in process of finding NE, all xis (or equivalently ais) are first assigned to prechosen arbitrary values and then each player should adjust its only remained (free) parameter in a way that the resulted economic equilibrium is also a NE, that is, no player can single-handedly increase its profit by readjusting its strategic parameters. This is indeed equivalent to the so-called c – parameterization technique introduced by Baldick [21]. Later, we will develop our solution method to be applicable to other parameterization techniques mentioned in the same reference. 4.2. Existence/uniqueness of the NE

2

a1 ðx1  rÞ  1

6 6 6 6 det F ¼ det 6 6 6 6 4

0 .. . 0 1

0   0 . .. a2 ðx2  rÞ  1 . . . .. . 0   0 anG ðxnG  rÞ  1 1   1

3

0

xnG

7 .. 7 7 . 7 þ ðxnG  rÞð1  anG xnG Þ 7 5 r  x1 þ 1þar1ðrx1x1 Þ

By continuing the procedure with multiplying ith column by bxi + (xi  r)/(1 + ai(r  xi))c and then adding the resultant to the last column for i = 2, . . . , nG, we get 6 6 6 6 6 det F ¼ det 6 6 6 6 6 4

a1 ðx1  rÞ  1 0 .. .

0

0 .. .

  .. a 2 ðx 2  r Þ  1 . .. .

0 .. .

0

0

 

0 anG ðxnG  rÞ  1

1

1

 

1

3

0

0 X X r  xi þ 1þari ðrxixi Þ i

7 7 7 7 7 7 7 7 7 7 5

The above matrix is a lower-triangular matrix, thus its determinant is equal to the product of its diagonal entries. On the other hand, from the definition of r, we have

X

xi ¼ c

i

Hence nG X

!

nG Y r  xi  ð1 þ ai ðr  xi ÞÞ det F ¼ ð1Þ  c þ 1 þ ai ðr  xi Þ i¼1 i¼1 nG

We also remark that,

r  xi ¼ c þ

X

xj

j–i

In this section, the proposed solution is generalized to show that it can be used to find the NE of supply function models, under different parameterization assumptions introduced by Baldick [21], that is (R / c) – parameterization and R – parameterization. A. hi has a fixed pre-chosen value for every player i. In this case, the only free parameter left to any player i would be

xi. The vector x in Eq. (14a) can be rephrased as a function of xis as follows

2

3

h1 .. .

7 6 7 6 7 6 7 6 xðxÞ ¼ 6 h 7 n G 7 6 P 4 D0 þ hi 5 P

ð20Þ

xi

P Considering that r ¼ c þ xi , the only unknown variables in F, x and j are the strategic parameters xi, "i. Now to find the NE, we should solve the following minimization problem

x ¼ arg min kFðxÞ  xðxÞ  jðxÞk x

ð21Þ

B. hi/xi has a fixed pre-chosen value for every player i. In a similar way, we can rewrite the vector x as follows

3 x1 f1 7 6 .. 7 6 . 7 6 7 xðxÞ ¼ 6 6 xnG fnG 7 7 6 P 4 D0 þ x f 5 P ii 2

cþ

ð22Þ

xi

i

ð18bÞ

r

4.3. Generalization of the method (to other Baldick parameterizations)

If the value corresponding to the resulted minimum point is   equal to zero, then the pair xi ; hi is the NE strategy for each player i. This is equivalent to Baldick’s (R / c) – parameterization method.

7 x2 þ ðx2  rÞð1  a2 x2 Þ 7 7

ð18aÞ

2

Thus, in a market with no price sensitivity (c = 0), the positivity of arbitrarily chosen strategic parameters xi (equivalently ai) is a sufficient condition for det F to be nonzero, and as a result, for the NE point to exist. But if the demand function is elastic (c – 0), then for any nonnegative pre-assigned set of xis, there exists a set of corresponding hi s that results in a NE point. The special case of xi = 0, "i, represents the NE of the electricity market using Cournot model.

cþ

In order to examine the non-singularity of matrix F, we try to compute its determinant by applying elementary column operations. First, we compute the first column multiplied by b[x1 + (x1  r)/(1 + a1(r  x1))c. Then we add the resultant to the last column, which yields

445

ð19Þ

in which, fi = hi/xi is the fixed pre-chosen value. The rest of solution is identical to the previous case. Therefore

x ¼ arg min kFðxÞ  xðxÞ  jðxÞk x

ð23Þ

Existence of a NE point, once again, requires that the value of the above term is zero at the minimum point. This is equivalent to Baldick’s R – parameterization method. We should remark that there is no guarantee, in general, for either of Eqs. (21) or (23) to result in a unique NE point. These generalized methods are presented here only to account for aforementioned Baldick parameterization techniques that have already been used in former approaches in the literature. However, in fact, it is easy to show that any of the solutions of those approaches can also be obtained using Eqs. (21) or (23).

446

D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

 xi P i  ð1Þdhi dui ¼ ðqmax  ai Pi  bi Þ 1 

5. NE in presence of constraints

r

In previous sections, the parameters of the electricity market were modeled freely. However, in actual markets, there are many limitations that change the dynamics of the system. In this section, it will be shown that how the proposed solution can be modified to be effectively applicable in modeling different conditions of the real market.

r

qmax < q & Pi > Pi max

) ðq

ð28Þ

 xi P i   xi Pi  < q  ai P i  bi 1   ¼0  ai P i  bi Þ 1 

r

r

r

r

Thus, a small decrease in hi (i.e., increasing Pi) will decrease the profit of the firm i, and therefore, it is a NE point. 5.2. Capacity constraints

5.1. Price caps In case there is an upper-bound limit for the MCP, we first compute the NE of the unconstrained market using the proposed algorithm. If the MCP of the market at the computed NE (q⁄) does not exceed the upper limit qmax, then the solution is also valid for the constrained model. Otherwise, it will be shown that the strategic parameter hi is not optimal, hence, an update is required.

(

q > qmax ) qa def ¼ qactual ¼ qmax a a ) P i ¼ xi q  h i

ð24Þ

Now, considering that qa is not affected by small perturbations of hi, by computing differentials from Eqs. (5) and (7), we get

(

dPi ¼ dhi   dui ¼ qa  ai Pai  bi  dPi

ð25Þ

Because all rational firms offer prices higher than their marginal costs, the term between the parentheses above is positive. Hence, decreasing hi will increase the profit of ith GENCO. As a result, the actual point is not a NE point. To tackle this issue, the idea is to increase the production of GENCOs as much as it increases their profits. To do so, we first calculate both the demand required and the maximum aggregate profitable production at price qmax, that is

D ¼ D0  cqmax X X P ¼ qmax 1=ai  bi =ai

ð26aÞ ð26bÞ

Now if P 6 D, every player increases its production level until its marginal cost equals the price cap qmax. From Eq. (25) it’s clear that no player can improve its strategy anymore, and thus, the following settlement provides a NE point

Pi

qmax  bi

ð27Þ

ai

However, if P > D, the following prototype of the algorithm is implemented

Pi sðiÞ

8i;

8i;

Pi 1

;

IF Pi >

8i

qmax  bi

ai qmax  bi Pi ai sðiÞ 0

P i þ

q bi X X   Pi ai P P P D  cqmax  Pi  Pi qmax ð1=ai Þ  ðbi =ai Þ  Pi 0 sðiÞ¼0 sðiÞ¼1

X 8 e 0 ¼ D0  D Pmax > i > > > violated ones < X r~ ¼ r  xi > > > violated ones > : ~ G ¼ nG  hnumber of violated onesi n

ð29Þ

This will result in a reduced solution method for remaining zGENCOs, i.e. the ones that are producing within their allowed range of production. If the new solution implies some other firms to produce over their upper bound limits, then this process is repeated until no firm is over-producing. 5.3. System contingencies In the case of load reduction because of line outage, the demand parameters in the model are updated accordingly. In the case of a generator outage, the size of the matrices is reduced and the problem is similarly solved for the other players. 5.4. Network congestions If the transmission lines of the power system model have limited capacities, then possible line congestions may cause market separations and price differences in the separated zones. In that case, Eqs. (14a)–(14c) will be derived separately for each zone based on their own (separated) demands and generating companies. The trick is to treat the amount of energy imports/exports into/from each separated zone as a negative/positive load (demand), with an absolute value equal to the capacity of the congested line. As demonstrated in the above subsections, it is fairly straightforward to include market constraints in the proposed solution method. However, this paper is mainly focused on other areas of electricity market, namely the strategic bidding. As a result, further discussion on the effects of market constraints will be avoided in the rest of this paper. 6. Robustness of the resulted NE

IF sðiÞ ¼ 1; THEN

max

Pi

; THEN

To consider GENCO’s production limits in the proposed solution method, the problem is solved first by ignoring the capacity constraints. Then all firms that their production levels at the NE point violate their maximum possible level are excluded from the model, and the solution method is implied on the remaining players.

!

The above algorithm is repeated until no player’s assigned Pi is over its maximum profitable level. At that point, for all players producing under their maximum profitable level, we have

As mentioned in Section 3, there exist an infinite number of NE points for an electricity power market modeled by SFE, from which every player is to adjust its own strategic parameters. Now, from a GENCO’s point of view, the question arises as which NE point is more appropriate or more desirable? This appropriateness could of course be translated into many different characteristics of the NE for players of different vantage points. Surprisingly, the prevalent approach in the context is often to ignore this problem by assuming prefixed values for some of the parameters and simply reducing the number of available strategic parameters. In this paper, however, we take a more considerate approach.

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

Understandably, the first approach that comes to mind for a producer to face this issue, might be trying to find a NE that provides a larger profit for its firm. As the simulation in Fig. 2 suggests, the smaller value of xi is chosen, the larger profit ui() of ith GENCO would be. The problem is that, for very small values of xi, the sensitivity of the players’ profit function to the demand would be very small. This means that a player’s strategy does not adapt well with unexpected changes in demand (see Fig. 6). Therefore, unexpected variations in actual parameters of the market might significantly harm ith GENCO’s profitability, if they adopt a greedy approach and select an inappropriately small value for xi. As a result of many parameters not being directly available to a player, including other players’ cost parameters, strategies and also because of uncertainties over demand parameters, a more reasonable approach for a GENCO participating in a real market seems to be finding a particular NE point, in which the optimality of its chosen strategic parameters has small sensitivity to external parameters of the model (from a producer’s perspective), in particular, the total demand level of the market or the price elasticity of demand. The desirability of such a NE point, for each player, is that it ensures the GENCO still plays its optimal strategies in response to other players’ strategies (hence, it would earn the best possible profit) even if the parameters of the market differ from their expected values. We call such a NE point, a robust Nash equilibrium point in this paper. In the following subsections, a solution method is proposed to compute such robust bidding strategies for all participating GENCOs. 6.1. Uncertainty in D0 Because F in Eq. (14a) is independent of D0, by taking derivatives with respect to D0 we get

F

 0nG 1 @x @j ¼ ¼ @D0 @D0 1

ð30Þ

 0nG 1 @h ) ¼ ½InG j0F1 @D0 1





@h

¼ arg min ½In j0F1 ðxÞ 0nG 1 arg min

x @D0 x G 1

2

x





@h

¼ arg min ½In j0F1 ðxÞ b arg min x @c x G 0 

@F 1  ðxÞF ðxÞjðxÞ

@c

ð31Þ

ð32Þ

  dD0  N 0; r2D

dc  N 0; r2c

ð33Þ

0



 @F  F1 j @c

ð37Þ

Then we have

dhi ¼

@hi @h dD0 þ i dc @D0 @c

ð38Þ

where @hi =@D0 and @hi =@ c are the ith element of vectors @h⁄/@D0 and @h⁄/@ c, respectively, which are calculated using Eqs. (31) and (34). Given the characteristics of the Gaussian distribution, dhi is also a stochastic variable of Normal distribution, that is



dhi  N 0; r2hi

r2hi ¼ r2D 



@hi @D0

ð39Þ

2

þ r2c 

  2 @hi @c

ð40Þ

Now, it is safe to assume that every player i wants to select its decision variable xi in a way that along with decision variables of the others, it results in minimizing rhi . Thus, the game of GENCOs’ bidding strategies is not exactly a game of normal form anymore, but rather a sequential (dynamic) game of imperfect information. Therefore, for an exact solution, a subgame-perfect Nash equilibrium [32] is to be found. For the sake of simplicity, we evade this complexity by assuming that all players are trying to minimize a common aggregate objective function. Hence, instead of every player trying to minimize

h

@h J i ¼ f @Di

ð1  fÞ

0

By taking partial derivative with respect to c, from Eq. (14a) and using the Leibniz rule, we get

b

ð36Þ

Suppose that uncertainties in both D0 and c can be modeled as two independent Gaussian distribution functions, given by

Therefore,



ð35Þ

6.3. Uncertainties in both D0 and c

6.2. Uncertainty in c

@h ) ¼ ½InG j0F1 @c

7 7 0 aj xj  1 7 7 7 .. .. 7 . . 5 0 1

In a similar way to the previous section,

J¼

@x @j @F ¼  x @c @c @c

3

.. .

6 . 0 6 6 0 aj @F ðxÞ ¼ 6 6 . .. @c 6 . . 4 . 0 

Now, by computing x based on Eq. (32) and then calculating h from Eqs. (15) and (16) using this particular set of xis, the specific bidding strategies that result in a robust NE point are obtained for all GENCOs.

F

..

in which,

Now if the vector @h⁄/@D0 has a small norm, it means that the value of hi does not change a lot if the level of demand fluctuates; thus, it is a robust NE in the sense that each player’s adjusted strategic parameters are still close to its optimal strategies even if the player miscalculates the demand level of the market. The ideal case of (@ h⁄/@D0) = 0 would guarantee that the NE exactly would be preserved even if the demand level varies. Using the above intuition, the ith player‘s aim prior to computing its NE bidding strategies using Eq. (16), is to choose xi such that the resulted NE is robust to uncertainties in demand, i.e.,

x

where b is a nG  1 vector of cost function parameters bi and

@hi @c

i

;

f¼

rD rD þ rc

ð41Þ

it is assumed that all players’ objective is to minimize the following cost function

x

qX ffiffiffiffiffiffiffiffiffiffiffiffi J2 i i

h

@h arg min f @D 0 x

ð42Þ i  ð1  fÞ @h @c

F

ð43Þ

where kkF denotes the Frobenius norm of the matrix and @h⁄/@D0 and @h⁄/@ c are computed via Eqs. (31) and (34), respectively. 6.4. Special case; bi = bi, "i

ð34Þ

As indicated in Section 4.1, the first order necessary conditions at a NE point imply that

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453



Pi þ xi  r

 



q  ai Pi  bi ¼ 0

ð44Þ

Now by considering a special case of R – parameterization method, in which the pre-chosen values of bi are all set to bi, we have

hi ¼ bi =ai ¼ bi xi ¼ bi xi

ð45Þ

) Pi ¼ xi q  hi ¼ xi ðq  bi Þ

ð46Þ

Using the above equation, Eq. (44) can be rewritten as follows







xi ðq  bi Þ þ xi  r q  bi  ai xi ðq  bi Þ ¼ 0

ð47Þ

Because q  bi = aiPi, the above equation yields

ai Pi







xi þ xi  r 1  xi ai



¼0

ð48Þ

For all GENCOs that are actually participating in the market, Pi > 0. Thus we have







xi þ xi  r 1  xi ai ¼ 0; i ¼ 1; . . . ; nG P

⁄

ð49Þ

Substituting r with c þ j x in the above equation results in a system of nG nonlinear equations with nG unknown variables xi . The solution of this system of equations yields the NE strategies of the players i = 1, . . . , nG, corresponding to the pre-chosen values of bi. The interesting property of such a solution is that since all above equations are independent of D0, the resulted NE strategies are also independent of D0. Therefore, the resultant NE is perfectly robust to changes of D0. As expected, this outcome is consistent with the results of simulations of the method in Section 6.1.  j

7. Simulation results To further investigate the analytical results illustrated in previous sections, some relevant numerical results from a hypothetical test case, similar to the one used in [30], are provided in this section. Consider an electricity market with 5 GENCOs associated with cost functions given as

1 C 1 ðP 1 Þ ¼ 0:1 þ 0:002P1 þ 0:001P21 2 1 C 2 ðP 2 Þ ¼ 0:1 þ 0:002P2 þ 0:001P22 2 1 C 3 ðP 3 Þ ¼ 0:1 þ 0:0024P3 þ 0:001P23 2 1 C 4 ðP 4 Þ ¼ 0:1 þ 0:002P4 þ 0:0012P24 2 1 C 5 ðP 5 Þ ¼ 0:1 þ 0:0024P5 þ 0:0012P25 2

ð50aÞ

5 X Pi

! ð50bÞ

i¼1

which yields

D ¼ D0  cq;

D0 ¼ 100;

The method derived in Section 4 is applied again to the above oligopoly model, only this time to compute the NE corresponding to the Cournot model. For this purpose, the pre-assigned strategic parameter xi is set to zero, for all participating producers. The resulted strategic parameters of the players, at NE point, and the production quantities and profit values corresponding to all players are presented in Table 2. We should remark that, as expected, the resulted MCP is higher than the one calculated for the NE of supply function model in Table 1 [20,21]. This is because the supply function model gets further away from the perfect competition model (Bertrand) as the value of the strategic parameter xi decreases. In Figs. 1 and 2, the alterations in the characteristics of the resulted NE are displayed, as the pre-assigned strategic parameter x1 varies. Fig. 3 illustrates the characteristics of the NE point, when all pre-chosen strategic parameters xi are being altered together. The alteration coefficient value equal to 1 indicates the same exact quantities used in Section 7.1. 7.3. Capacity limits and transmission constraints To put the effectiveness of the methods introduced in Section 5 into a preliminary test, an Electricity market consisting of 2 distinct zones is considered, assuming that each zone contains 5 GENCOs with cost functions given by Eq. (50a). It is also assumed that the two zones have different initial demand values DZ01 ¼ 100; DZ02 ¼ 200Þ and they are connected with a transmission line of capacity 40. The NE of this market is computed using the proposed solution method of Section 4 and by incorporating the adjustments of Section 5. The simulation results are presented in Table 3. As can be seen from Table 3, the capacity constraints (CCs) of the GENCOs are assumed to be identical over the two zones. We remark that at the NE point, the transmission line is at full capacity. Hence the two zones are virtually separated and each has its own (local) cleared price. However, if the initial demand levels of the two zones changes to DZ01 ¼ 120; DZ02 ¼ 180 , a transmission line congestion will not happen at NE point, and the two zones will have identical MCPs equal to .0352. Because of the increased level of competition, the MCP in this case is smaller compared to the local (zonal) MCPs of the former case, even though the aggregate (initial) demand level has not changed. 7.4. Computation of the robust NE

and also the inverse demand function given by

q ¼ 0:2  0:002

7.2. NE for the Cournot model

c ¼ 500

In the next subsections, the NE of the above competition is calculated based on different assumptions and the results are presented. 7.1. Computation of NE First, it is assumed that each GENCO has chosen its pre-assigned strategic parameter xi equal to 1/ai. The corresponding NE parameters hi are then computed using the proposed method, from which the results are presented in Table 1.

Now consider the market model described by Eqs. (50a) and (50b). Suppose that there are some uncertainties in the initial level of demand D0. A robust NE, that is, one which preserves its optimality for all players best, should the amount of demand vary from its expected value, is computed as follows. First the pre-assigned strategic parameters xi are chosen using Eq. (32). Then the characteristics of NE are calculated via Eq. (15) using the computed xis. The results are illustrated in Table 4. Figs. 4 and 5 demonstrate the variation in the sensitivity of every player’s NE strategy hi to changes of the initial demand value D0, when the pre-chosen strategic parameter of player 1, x1, varies. In Fig. 4, the alterations are applied to the pre-chosen values stated in Section 7.1. In Fig. 5, the values corresponding to the robust NE are used. In each of these figures, the upper curve indicates the value of k@ h⁄/@D0k and the lower curves denote separate values of j@hi =@D0 j for all players. As expected, the upper curve’s value reaches zero for a specific value of x1, for the robust NE case (Fig. 5).

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453 Table 1 Results of the supply function model at the NE point. GENCO i

ai

bi

ci

xi

hi

ai

bi

Pi

ui

1 2 3 4 5

.001 .001 .001 .0012 .0012

.002 .002 .0024 .002 .0024

.1 .1 .1 .1 .1

1000 1000 1000 833.3 833.3

6.4654 6.4654 6.7879 4.7676 5.0472

.001 .001 .001 .0012 .0012

.0065 .0065 .0068 .0057 .0061

18.6056 18.6056 18.2831 16.1249 15.8453

.1562 .1562 .1474 .1160 .1086

MCP = .0251

Table 2 Characteristics of NE for the corresponding Cournot model. GENCO i

xi

hi

ai

bi

Pi

ui

1 2 3 4 5

0 0 0 0 0

15.5699 15.5699 15.4366 14.5968 14.4817

1 1 1 1 1

1 1 1 1 1

15.5699 15.5699 15.4366 14.5968 14.4817

.5061 .5061 .4957 .4540 .4445

MCP = .0487

Total production: 75.6451

7.5. Numerical comparison: the effect of demand uncertainty In this subsection, it is assumed that players 2–5 are willing to play their robust Nash strategies driven by Eq. (32), but player 1 refuses to play its robust Nash strategy and decides to set its pre-chosen strategic parameter x1 to a different value. Hence, it results in a different NE point which is not robust to variations of D0. After the problem is solved and the resulted NE is computed for the expected demand level, another actual demand value is adopted and the players’ actual payoff values are observed and compared. The robust NE for the proposed market model is computed and the results are already presented in Table 4. Assuming that player 1 uses a different pre-chosen value for x1 as given below

x1 ¼ 77:22 The new NE point is then computed and the results are presented in Table 5. Now if the actual demand value is different from its nominal value, e.g., Da0 ¼ 130, the actual MCP and accordingly, all GENCOs’

Total production: 87.4645

production levels and payoff values vary based on Eqs. (4), (5) and (7), respectively. These actual results are presented in Table 6. It is clear that player 1 earns a significantly smaller payoff value compared to player 2 in the new market condition, even though they are assumed to be identical GENCOs. We remark that because player 1 is refusing to play its robust Nash strategy, the resulted economic equilibrium of the cleared market presented in Table 6 is not anymore a NE point. Fig. 6 illustrates the behavior of the cleared market for all players as initial demand value D0 varies from its nominal value. As it can be seen, the resulted payoff value for the new cleared market is smaller for player 1 compared to player 2, as D0 increases, even though they have the same cost function parameters. 7.6. GENCOs with different cost parameters In the market model represented by Eq. (50a), a case of similar GENCOs were considered. Consequently, the parameters of GENCOs’ cost functions were approximately the same. To further examine the proposed solution method, another market model consisted of GENCOs with different cost function parameters is considered in this subsection. For this case, we consider an electricity market of 4 GENCOs with cost functions given as

1 C 1 ðP1 Þ ¼ 0:1 þ 0:002P1 þ 0:001P21 2 1 C 2 ðP2 Þ ¼ 0:5 þ 0:01P2 þ 0:005P22 2 1 C 3 ðP3 Þ ¼ 0:2 þ 0:004P3 þ 0:002P23 2 1 C 4 ðP4 Þ ¼ 0:4 þ 0:007P4 þ 0:004P24 2

Fig. 1. Alterations in NE strategies of all players when player 1 changes its pre-assigned strategic parameter x1.

ð51Þ

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

Fig. 2. Alterations in MCP, production levels and profits of all players at NE when player 1 changes its pre-assigned strategic parameter x1.

Fig. 3. Alterations in MCP, production levels and profits of all players when all players change their pre-chosen strategic parameter xi by a coefficient factor.

Table 3 NE data for a market with capacity/network constraints. GENCO i

1 2 3 4 5

Zone 1

CCs

xi

hi

Pi

ui

1000 1000 1000 833.3 833.3

10.20 8.259 10.50 7.361 7.6277

25.97 25.00 25.66 22.78 22.51

.4500 .4417 .4372 .3669 .3561

LMCP = .0362

Total production = 121.9

It is assumed that the demand parameters are also given as D0 = 150 and c = 500. For this market model, a NE is computed using the method presented in Section 4, equivalent to Baldick’s c – parameteriztion. Also, the robust NE is computed using the approach described in Section 6. The results are presented in Table 7.

40 25 30 40 40

Zone 2

xi

hi

Pi

ui

1000 1000 1000 833.3 833.3

15.65 9.155 11.93 11.14 11.39

29.57 25.00 30.00 26.54 26.29

.7406 .6679 .7344 .6242 .6109

LMCP = .0452

Total production = 137.4

As it can be seen from Table 7, the GENCOs with smaller cost parameters are able to produce higher levels of electric power at the NE point. Therefore, they have considerably higher profit margins. The same applies for the robust NE case. By considering the results of the above two equilibriums, all GENCOs also have a smal-

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453 Table 4 Robust NE point considering uncertainties in D0.

Table 5 NE point when player 1 refuses to play robust.

GENCO i

xi

hi

ai

bi

Pi

ui

GENCO i

xi

hi

ai

bi

Pi

ui

1 2 3 4 5

772.2 772.2 772.2 672.7 672.7

1.544 1.544 1.853 1.345 1.614

1.295e3 1.295e3 1.295e3 1.487e3 1.487e3

2.000e3 2.000e3 2.400e3 2.000e3 2.400e3

18.48 18.48 18.17 16.09 15.83

.1714 .1714 .1624 .1297 .1220

1 2 3 4 5

77.22 772.2 772.2 672.7 672.7

17.0782 2.6069 2.8986 2.1085 2.3653

1.295e2 1.295e3 1.295e3 1.487e3 1.487e3

2.212e1 3.3759e3 3.7537e3 3.1345e3 3.5162e3

19.15 18.08 17.79 15.92 15.66

.1915 .1849 .1758 .1427 .1349

MCP = .0259

Total production: 87.04

MCP = .0268

Total production: 86.60

Table 6 Actual cleared market data.

ler value of xi for the robust NE case compared with the classic NE, that results in a higher MCP value, as expected from the curves of Figs. 2 and 3. In this particular case, it is also interesting to note that robust Nash strategies not only provide more sustainability to variations in demand according to equations of Section 6, but also provide higher nominal profit value for all participating GENCOs.

GENCO i

xi

hi

ai

bi

Pi

ui

1 2 3 4 5

77.22 772.2 772.2 672.7 672.7

17.0782 2.6069 2.8986 2.1085 2.3653

1.295e2 1.295e3 1.295e3 1.487e3 1.487e3

2.212e1 3.3759e3 3.7537e3 3.1345e3 3.5162e3

19.82 24.77 24.48 21.74 21.48

.3665 .4217 .4094 .3436 .3331

MCP = .0354

Total production: 112.28

Fig. 4. Sensitivities of players’ Nash strategies to perturbations in D0 when player 1 changes its pre-chosen strategic parameter x1.

Fig. 5. Sensitivities of players’ Nash strategies to perturbations in D0 when player 1 changes its pre-chosen strategic parameter x1 at the robust NE.

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D. Langary et al. / Electrical Power and Energy Systems 54 (2014) 442–453

Fig. 6. The MCP, production levels and profits of all players when the demand level differs from its expected value.

Table 7 NE and robust NE for a case of GENCOs with different cost parameters. GENCO i

1 2 3 4

NE

Robust NE

xi

hi

Pi

ui

xi

hi

Pi

ui

1000 200 500 250

34.77 3.180 9.989 3.670

47.52 13.28 31.16 16.90

2.586 0.019 1.269 0.301

558.8 178.3 372.0 216.4

1.118 1.783 1.488 1.515

46.61 13.45 30.28 16.96

2.701 0.062 1.348 0.354

MCP = .0823

Total production = 108.8

MCP = .0854

Total production = 107.3

8. Conclusion

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Practical electricity markets are not perfectly competitive because of particular characteristics of power systems and as a result, it is critical for GENCOs to devise an appropriate bidding strategy to maximize their potential revenues. In this paper, under certain simplifying assumptions, closed formulae for market variables at the NE point of a short-term electricity market, whose participants rely on supply function models, has been provided and a mathematical proof for its existence and uniqueness is presented. It is shown that the proposed method could be utilized for Cournot models as well. From these formulations, analytical expressions for price and production quantities were derived, that resulted in insightful understanding of the qualitative behavior of the market. Based on this, the concept of robust NE was introduced and computed to account for uncertainties of market parameters. The numerical results of the test case confirm the efficiency of the proposed method to find NEs, for both the supply function model and the Cournot model. The computed robust NE point showed consistency with theoretical conclusions presented in Section 6.4. Finally, using a numerical example, it was shown that refusing to play a robust NE strategy might harm a player’s profitability in the case of deviations in the level of demand.

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