Generation companies decision-making modeling by linear control theory

Electric Power Systems Research 80 (2010) 815–827

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Generation companies decision-making modeling by linear control theory G. Gutiérrez-Alcaraz a,∗ , Gerald B. Sheblé b a Programa de Graduados e Investigación en Ingeniería Eléctrica. Departamento de Ingeniería Eléctrica y Electrónica, Instituto Tecnológico de Morelia. Av. Tecnológico 1500, Col. Lomas de Santiaguito 58120. Morelia, Mich., México b INESC Porto, Faculdade de Engenharia, Universidade do Porto, Campus da FEUP, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal

a r t i c l e

i n f o

Article history: Received 4 April 2008 Received in revised form 17 December 2009 Accepted 19 December 2009 Available online 18 January 2010

a b s t r a c t This paper proposes four decision-making procedures to be employed by electric generating companies as part of their bidding strategies when competing in an ologopolistic market: naïve, forward, adaptive, and moving average expectations. Decision-making is formulated in a dynamic framework by using linear control theory. The results reveal that interactions among all GENCOs affect market dynamics. Several numerical examples are reported, and conclusions are presented. © 2009 Elsevier B.V. All rights reserved.

Keywords: Oligopolistic competition Cournot model Electricity market

1. Introduction Generation companies (GENCOs) are the competitive players in today’s liberalized electricity markets. The competition and interactions among them are the major sources of market dynamics. Analyzing their interactions and decision-making processes becomes particularly important when electricity markets are ologopolistic and there are few market players. To survive and succeed under such circumstances, GENCOs analyze the market behavior and expected output of their competitors by comparing actual market outcomes with forecasted outcomes based on simulations or static analyses. GENCOs must adaptively adjust market strategies as each new piece of information is gathered and understood to maximize profits, and will utilize several methods to accomplish this learning process, i.e. forward expectation, moving average expectation, and adaptive expectation [1,2]. Expectation plays a very important role in electric market dynamic analysis to understand dynamic interactions between players. The strategies one player anticipates other players may use will determine its decision and action in one of the next periods. Different expectations by market participants lead to different actions and market transitions. Several models that attempt to formally model expectations in an effort to better understand decision-making in a world of uncertainty have been reported in the literature. Various limited information estimation methods applied to models within the

∗ Corresponding author. Tel.: +52 4433171870; fax: +52 4433171870. E-mail address: [email protected] (G. Gutiérrez-Alcaraz). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.12.007

framework of standard econometric estimators are reported in [2]. Game theory, evolutionary game theory, stochastic simulation, and agent-based modeling have been used to model the behavior of players and the impacts of their decision on market dynamics [3–5]. Maiorano et al. [6] and Yu et al. [7] studied dynamics of noncollusive oligopolistic electric markets. In [8] the authors present a theory and method for estimating the conjectural variations (CV) of GENCOs. CV is a game-theoretical concept in which players have a conjecture about the behavior of their opponents [16]. Based on these estimates of CV in an actual electricity market, an empirical methodology is also proposed to analyze the dynamic oligopoly behaviors underlying market power. Recently, there has been considerable interest in oligopoly models with “consistent” CV. A CV is considered consistent if it is equivalent to the optimal response of the other firms at the equilibrium defined by that conjecture. A new unified framework of electricity market analysis based on coevolutionary computation for both the one-shot and the repeated games of oligopolistic electricity markets is presented in [4]. Discrete event system simulation (DESS) has also been applied to model competition to study the effect of some control policies [7]. DESS is very useful when one is including a decision support system (DSS) as part of the trader’s support via front, middle, or back office support software. DSS are often considered a database of rules to be applied to the observed conditions to authorize certain trader actions. DESS facilitates the study of transitions and changes in variables over time. Additional properties, controllability, and observability are also useful for monitoring market performance and for DSS rule analysis. Yang and Sheblé [1] introduced the expectations of GENCOs and electricity consumers and studied market dynamics in a discrete-time setting. Later, the authors modeled the electricity

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G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

market as a DESS and studied various GENCOs’ decisions as specific events unfolded [1]. Stothert has proposed and analyzed a model of an electricity generation bidding system formulated as a control problem [9]. The reasons for approaching generation bidding as a control problem center on the general principle that feedback reduces errors in strategy selection under uncertainty. In [10,11], the repeated bidding process in (hourly-based) real-time electricity markets as a dynamic feedback system is modeled. Recently, several other dynamic models have been developed to study the behavior of players in electricity markets, because their decisions are the most important variable. A dynamic replicator model of the power suppliers’ bids in an oligopolistic electricity market, derived for fixed and variable demand cases, is reported in [17]. The replicator model is presented in a state-space structure. A dynamic model based on invasive weed colonization optimization is reported in [18]. It derives on concepts from game theory and supply function. The method is integrated with a power system simulator to consider all of the constraints of a realistic power system. Simple learning rules have been implemented in dynamic models for strategic bidding in spot electricity markets [21]. This paper introduces more extensive models to simulate GENCOs’ dynamic decision-making process with learning in an oligopolistic market. The paper is organized as follows: the electricity market model is described in Section 2. Section 3 develops different models of decision-making in a dynamic framework with learning effect. Numerical examples are reported in Section 4. Conclusions are presented in Section 5. 2. Electricity market model

k, Q (k) =

qi (k) is the total generation in the market at period

i=1

k, qi (k) is the quantity generated by GENCO i at period k, a and b are positive market demand coefficients, publicly known for every GENCO, and k is the time period index. GENCOs make decisions according to their internal estimates of the aggregated electricity buyer, competitor behavior, and delayed market information. GENCO i must determine its generation output qi (k) to maximize its profit at period k, mathematically represented by: maxi (k) = (a − bQ (k))qi (k) − Ci (qi (k)) n 

(1)

qi (k)

i=1

qmin ≤ qi ≤ qmax i i where qmin and qmax are the lower and upper generation limits. i i Its generation cost is given by Ci (qi (k)) = di + ei qi (k) + fi q2i (k) ∀i = 1, ..., n where di , ei and fi are the coefficients of GENCO i’s production cost function. Treating inequality constraints (the lower and upper generation limits) as if they did not exist, the corresponding optimization problem for GENCO i is represented by:

⎛ ⎜ ⎜ ⎝

⎞

maxi (k) = ⎜a − b qi

n 

j=1 i∈j

⎟ ⎟ ⎠

qj (k)⎟ qi (k) − (di + ei qi (k) + fi q2i (k))

n

(3)

i= / j

Grouping terms in (3), we have: 2(b + fi )qi (k) + b

n 

qj (k) = a − ei

(4)

j=1 i= / j

Similarly, GENCO j’s profit maximization decision is represented as:

⎛

⎞

⎜ ⎝

maxj (k) = ⎜a − b qj

 n

⎟ ⎠

qi (k)⎟ qj (k) − (dj + ej qj (k) + fj q2j (k))

i=1

(5)

j∈i

The first-order condition to maximize profits at period k is: ∂j (k) ∂qj (k)

= a − 2bqj (k) − b

n 

qi (k) − ej − 2fj qj (k) = 0

(6)

i=1 j= / i

Grouping terms in (6), we have: n 

qi (k) + 2(b + fj )qj (k) = a − ej

(7)

i=1 j= / i

For the n-GENCOs, in matrix form we have:

⎡

2(b + f1 )

⎢b ⎢ ⎢ ⎢ .. ⎣.

b



P E (k) = −b

b

...

b

2(b + f2 )

...

b

.. .

..

.. .

b

...

−b

...

.

−b

⎤⎡

⎤

⎡

a − e1

⎤

⎥ ⎢ q2 (k) ⎥ ⎢ a − e2 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥=⎢ ⎥ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎦⎣. ⎦ ⎣. ⎦

2(b + fn )

⎡

q1 (k)

q1 (k)

⎤

qn (k)

a − en

(8)

⎥ ⎢  ⎢ q2 (k) ⎥ ⎥+a ⎢ ⎥ ⎢ .. ⎦ ⎣. qn (k)

qi

S. to Q (k) =

 ∂i (k) = a − 2bqi (k) − b qj (k) − ei − 2fi qi (k) = 0 ∂qi (k) j=1

b

In this paper we consider a spot market operated on an hourly basis. The market model is based on the Cournot oligopoly. Consider a n-GENCOs spot electricity market with an inverse market demand curve P E (k) = a − bQ (k), where P E (k) is the price at period n 

To achieve the maximum profit, its first-order condition for optimality should be satisfied; this is:

(2)

Given that electricity markets operate repeatedly on an hourly basis, GENCOs might learn from available historical market data to forecast or estimate their competitors’ strategic behavior. Similarly, market analysts can learn from such market simulation to determine unintended outcomes, misstated market rules, etc. In the next section four kinds of expectations are introduced to the basic Cournot model to build Cournot–like models with learning to improve strategic bidding performance. A representation of the electricity market used in this paper is shown in Fig. 1. In the “Market Clearing Mechanism” block in Fig. 1, the market operator conducts a market clearing mechanism. Once market equilibrium and price-quantity are discovered, this information is made public. A GENCO observes this new market information and chooses from a finite set of actions. GENCO i does not know its competitors’ decisions at the time of decision-making. Therefore, it needs to estimate the decisions of its rivals, which is the “estimator” block in Fig. 1. The key target of estimating rivals’ decisions is to obtain accurate adjusting factor values. It should be noted that

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

817

Fig. 1. A n-GENCO electricity market.

adjusting factors are usually not a constant during the iterative process of electricity market operation, which makes estimation more difficult. Data mining is a key tool to determine the strategies (rules) that each competitor is apparently using. In addition, each GENCO may assess other information it gathers over time, especially the data most likely to influence its present choice, i.e. fuel price variations and demand uncertainty. Data mining the trade magazines, other markets (such as NYMEX), as well as forms submitted to state and federal commissions are useful for such model building.

more plants or units can be built in a particular area due to environmental considerations), or other combinations. GENCO i’s profit is:

⎛ maxi (k) = qi

⎞⎞

⎛

n ⎜ ⎜ ⎟⎟  ⎜ ⎜ ⎟⎟ qˆ j (k)⎟⎟ qi (k) − (di + ei qi (k) ⎜a − b ⎜qj (k) + ⎝ ⎝ ⎠⎠

j=1 j= / i

+ fi q2i (k)) 3. Decision-making with learning effect When companies are willing to make trade-offs between present and future profits, GENCOs need to incorporate learning strategies in their decision-making. For example, GENCO i may understand so little about its rival’s past actions and the underlying rationales that GENCO i comes to believe (“static assumption”) and accept that the circumstances it observes in the immediate past will repeat themselves. However, repetition can also offer many opportunities for GENCOs to successfully learn to “play the game.” GENCOs also benefit from studying how the output decisions of their competitors change with time, and the effects on market dynamics. An important issue to address is what happens to learners, non-learners, and the market interactions when some GENCOs do not learn, or only one or a few GENCOs learn. It is clear that GENCOs have incentives to learn only when they can receive more benefits (maximize profit) when learning than when not learning. 3.1. GENCOs under naïve expectation This scenario may be caused by financial problems (insufficient fixed asset investment funding source), regulatory limitation (no

(9)

Under naïve expectation, GENCO i believes that GENCO j will not change its output such as: qˆ j (k) = qj (k − 1)

(10)

where qj (k) represents the expectation of GENCO i about the decision of GENCO j at period k. Substituting (10) in (9) and finding the first-order conditions for optimality gives (11):

 ∂i (k) = a − 2bqi (k) − b qj (k − 1) − ei − 2fi qi (k) = 0 ∂qi (k) j=1 n

(11)

i= / j

After manipulating (11) and solving for qi (k) it yields: qi (k) =

b a − ei − 2(b + fi ) 2(b + fi )

n 

j=1 i= / j

qj (k − 1)

(12)

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G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

Eq. (12) can be written as: qi (k + 1) =

Forward expectations can be expressed as:

b a − ei − 2(b + fi ) 2(b + fi )

 n

qj (k)

(13)

j=1

qˆ j (k) = j q∗j + (1 − j )qj (k − 1)

(18)

i= / j

Similarly for GENCO j: qj (k + 1) =

a − ej 2(b + fj )

−

b 2(b + fj )

n 

qi (k)

(14)

i=1 j= / i

The linear system of Eqs. (8) can be turned into a system of difference equations [22]. In the discrete form [12], the system state-space model is: X(k + 1) = AX(k) + BU(k)

(15)

Y (k + 1) = CX(k) + DU(k)

where X is the vector of state variables, Y is the vector of output variables, and A, B, C, D are system matrices function of k. Considering Eqs. (13) and (14), the discrete-time linear system is:

⎡ 0

⎡

⎤ ⎢ q1 (k+1) ⎢ −b 2(b+f2 ) ⎢ q2 (k+1) ⎥=⎢ ⎣ ⎦ ⎢ ⎢. ⎢ .. ⎣ qn (k+1) −b 2(b+fn )

−b 2(b+f1 )

...

0

...

. . .

..

−b 2(b+fn )

⎡q

.

...

1 (k)

P E (k + 1) =



−b

−b . . .

−b

where j is the adjusting coefficient for GENCO j and j ∈ [−1 < j ≤ 1], and q∗j is the possible output equilibrium of GENCO j. Adjusting factors assume a key role in the determination of market equilibrium. The process for estimating the adjustment coefficients is as complex as the one for determining conjectural variations factors [24], and it is beyond the scope of this paper. Now, assuming that all GENCOs adopt forward expectations, the profit maximization decision for GENCO i becomes:

⎤ ⎡ a−e1 −b 2(b+f1 ) ⎡ q (k) ⎤ 2(b+f1 ) ⎥ 1 a−e2 −b ⎥ q2 (k) ⎢ ⎢ ⎥ ⎢ 2(b+f2 ) 2(b+f2 ) ⎥⎢ ⎥⎢ . ⎥+⎢ ⎥⎣ . ⎦ ⎢ .. . . ⎥ . . ⎦ q (k) ⎣ .

0

⎤

n

⎡

⎛

⎢ ⎢ ⎣

⎜ ⎜ ⎝

⎞⎤

i (k) = ⎢a − b ⎜qi (k) +

i= / j

(19)

⎤

a − en 2(b + fn )

⎥ ⎥ ⎥u ⎥ ⎥ ⎦

When satisfying the first-order optimal condition:

 ∂i (k) = a − 2bqi (k) − b qˆ j (k) − ei − 2fi qi (k) = 0 ∂qi (k) j=1 n

(20)

i= / j

⎥ + [a]u ⎦

.

⎟⎥ ⎟⎥ ⎠⎦

qˆ j (k)⎟⎥ qi (k) − (di + ei qi (k) + fi q2i (k))

j=1

 ⎢ q2 (k) ⎥ ⎢. ⎣.

n 

Substituting (18) in (20) we have:

qn (k)

(16) To prove stability, we apply Gershgorin’s theorem to the A matrix [23]. According to the theorem, every eigenvalue of a matrix lies in a circle centered at diagonal elements aii with a radius of Ri =  n j = 1 |aij |. The radius is calculated as:

  ∂i (k) = a − 2bqi (k) − bj q∗j − b(1 − j ) qj (k − 1) ∂qi (k) j=1 j=1

j= / i

Ri =

n 

j=1 j= / i

n     −b  |aij | < 1 =  2(b + f )  < 1

j=1

n

n

i= / j

i= / j

−ei − 2fi qi (k) = 0

(21)

(17)

i

Solving for qi (k):

j= / i

Therefore the eigenvalues lie in a circle centered at aii with a radius less than one. This area is a subset of the unit, circle. Hence the system is stable. If all GENCOs have linear cost function, fi = 0 ∀i = 1, ..., n, the system is always stable (with eigenvalues ±0.5). Otherwise, stability depends on market demand (b), and production cost parameters (fi ). It can be seen that the market under naïve expectation and the traditional Cournot model have the same equilibrium when it exists. The traditional Cournot model can be seen as a special case of n-GENCOs under naïve expectation.

a − ei − bj

n 

q∗j

j=1 qi (k) =

i= / j

2(b + fi )

−

b(1 − j ) 2(b + fi )

n 

qj (k − 1)

(22)

j=1 i= / j

Similarly for GENCO j:

3.2. GENCOs under forward expectation Forward expectation is valid when GENCOs anticipate a possible future equilibrium for competitors. GENCOs believe that their rivals may try to gradually reach specific equilibrium because of certain constraints and will not move away from the equilibrium due to the risk associated with any action significantly deviating from that equilibrium.

a − ej − b

n 

i q∗i

i=1 qj (k) =

j= / i

2(b + fj )

−

b 2(b + fj )

n 

i=1 j= / i

(1 − i )qi (k − 1)

(23)

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

The new dynamic n-GENCOs’ market representation can be written as:

⎡

⎡

0

⎤ ⎢ ⎢ b(1 − 1) ⎢ 2(b + f ) 2 = ⎣ ⎦ ⎢ ⎢. ⎢ .. qn (k + 1) ⎣ q1 (k + 1) ⎢ q2 (k + 1) ⎥

b(1 − 1) 2(b + fn )

⎡

b(2 − 1) 2(b + f1 )

...

0

...

.. . b(2 − 1) 2(b + fn )

..

n 

∗

⎢ a − e1 − b i qi ⎢ i=2 ⎢ ⎢ 2(b + f1 ) ⎢ n  ⎢ ⎢ a − e2 − b i q∗i ⎢ ⎢ i=1 ⎢ +⎢ i= / 2 ⎢ 2(b + f2 ) ⎢ ⎢. ⎢ .. ⎢ n ⎢  ⎢ i q∗i ⎢ a − en − b ⎣ i=n−1

2(b + fn )

⎡



P E (k + 1) = −b

−b

...

.

...

b(n − 1) 2(b + f1 ) b(n − 1) 2(b + f2 ) .. . 0

⎤

⎡ ⎤ ⎥ q1 (k) ⎥ ⎥ ⎢ q2 (k) ⎥ ⎥⎢. ⎥ ⎥⎣. ⎦ ⎥ . ⎦ qn (k)

Under adaptive expectation, GENCOs adjust their output expectation of the current period according to both their internal expectations and the actual output of the last period. In other words, GENCOs learn from past errors. In our n-GENCOs model, adaptive expectation is expressed as: qˆ j (k) − qˆ j (k − 1) = ˇj (qj (k − 1) − qˆ j (k − 1))

⎢ ⎢ ⎣

⎜ ⎜ ⎝

⎞⎤

n 

⎟⎥ ⎟⎥ ⎠⎦

qˆ j (k)⎟⎥ qi (k) − (di + ei qi (k) + fi q2i (k))

j=1

(27) The first-order conditions for optimality gives:

 ∂i (k)  = a − 2bqi (k) − b qj (k) − ei − 2fi qi (k) = 0 ∂qi (k) j=1 n

Substituting (26) into (28), we have:

⎤

 ∂i (k) qˆ j (k − 1) + ˇj [qj (k − 1) − qj (k − 1)] = a − 2bqi (k) − b ∂qi (k) j=1 n

q1 (k)  ⎢ q2 (k) ⎥ ⎥ + [a ]u −b ⎢ . ⎦ ⎣ .. qn (k)

j= / i

−ei − 2fi qi (k) = 0

n

(25)

(29)

Solving for: a − ei − bˇj

n

j = 1 qj (k − 1) j= / i

qi (k) =

−

2b + 2fi

b(1 − ˇj ) 2b + 2fi

j=1

j= / i

When all GENCOs have linear cost function, the system will be always stable, since j ∈ [−1 < j ≤ 1] ∀j = 1, . . . , n. On the other hand, when production costs are not linear, stability depends on market demand (b), and production cost parameters (fi ).

Adaptive expectation is another useful technique by which GENCOs can learn from experience, and has proven to be one of the most effective forecasting tools [13,14].

⎡ ⎤

⎢ q1 (k + 1) ⎢ ˇ1 ⎢ qˆ 1 (k + 1) ⎥ ⎢ bˇ1 ⎢ q2 (k + 1) ⎥ ⎢ − ⎢ ⎥ ⎢ 2(b + f2 ) ⎢ ⎢ qˆ 2 (k + 1) ⎥ ⎢ ⎢ ⎥ = ⎢0 . ⎢. ⎥ ⎢. ⎢. ⎥ ⎢. ⎣ ⎦ ⎢. qn (k + 1) ⎢ bˇ1 ⎣− qˆ n (k + 1)

qj (k − 1) (30)

j=1

Proceeding similarly for GENCO j: a − ej − bˇi

−

bˇ2 2(b + f1 )

1 − ˇ1 0 b(ˇ1 − 1) 0 2(b + f2 ) 0 ˇ2 .. .. . . b(ˇ1 − 1) bˇ2 − 2(b + fn ) 2(b + fn ) 2(b + fn ) 0 0 ⎤0 ⎡ q1 (k)   ⎢ q2 (k) ⎥ ⎥ + [a]u P E (k + 1) = −b −b . . . −b ⎢ . ⎦ ⎣ .. qn (k)

b(ˇ2 − 1) 2(b + f1 ) 0

n 

qi (k − 1)

i=1 qj (k) =

3.3. GENCOs under adaptive expectation

0

n 

j= / i

j= / i

0

(28)

j= / i

      b(j − 1)    aij  < 1 = Ri =  2(b + fi )  < 1

⎡

⎛

i= / j

To prove stability we apply Gershgorin’s theorem to the A matrix. The radius is calculated as:

j=1

⎡

i (k) = ⎢a − b ⎜qi (k) +

(24)

n

(26)

where ˇ is adjusting coefficient and ˇ ∈ [0, 1].

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥u ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

819

i= / j

−

2b + 2fj

b(1 − ˇi ) 2b + 2fj

n 

 qi (k − 1)

(31)

i=1 i= / j

Rearranging the resulting equations in discrete-time linear system to describe the GENCOs’ decision dynamics with adaptive expectation, the market is represented by the following system: ···

−

··· 0

0

···

−

1 − ˇ2 .. . b(ˇ2 − 1) 2(b + fn ) 0

··· .. .

0 .. .

bˇn 2(b + f1 ) bˇn 2(b + f2 )

b(ˇn − 1) 2(b + f1 ) 0 b(ˇn − 1) 2(b + f2 ) 0 .. .

··· 0

0

· · · ˇn

1 − ˇn

⎤

⎡ a−e ⎤ 1 ⎥ ⎡ q1 (k) ⎤ ⎢ 2(b + f1 ) ⎥ ⎥ ⎥ 0 ⎥ ⎢ qˆ 1 (k) ⎥ ⎢ ⎢ a − e2 ⎥ ⎥⎢ ⎢ ⎥ ⎥ ⎢ q2 (k) ⎥ ⎢ 2(b + f2 ) ⎥ ⎥ ⎢ qˆ (k) ⎥ ⎢0 ⎥ ⎥⎢ 2 ⎥ +⎢ ⎥u ⎥⎢ . ⎥ ⎢. ⎥ ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎥ . ⎥⎣ . ⎥ ⎥ qn (k) ⎦ ⎢ ⎢ a − e n ⎥ ⎥ ⎣ ⎦ ⎦ qˆ n (k) 2(b + fn ) 0

(32)

820

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

According to Gershgorin’s theorem:

  aij  < 1

The stability criteria can be obtained in similar fashion as described in the previous case. Until now we have assumed that all n-GENCOs are adopting the same expectations. When GENCOs have different expectations, however, the market can still be modelled as a linear control system. Thus, we next, we develop the market system under different expectation combinations. We restrict our attention to two GENCOs for algebraic manipulation simplicity without loss of generality.

n

Ri =

(33)

j=1 j= / i

The system is always stable when ˇj = 0 ∀j = 1, ..., n. Otherwise, it depends on market demand (b), and production costs parameters (fi ).

3.5. GENCOs under naïve and forward expectation 3.4. GENCOs under moving average expectation Assume that GENCO 1 is under naïve expectation and GENCO 2 is under forward expectation. After manipulating equations the market representation in discrete-time linear form is:

GENCOs can learn from historical experience to improve the quality of their forecasting and decision-making. Under the moving average expectation, collecting and analyzing data from the past few periods can assist in estimating new data. Each “past” data set is assigned a weight to reflect its forecasting ability. For example, assume GENCO i uses GENCO j’s output in the two most recent periods to forecast qj (k). The weights are 0 ≤ w1m ≤ 1, m = 1, 2,  w1m = 1. Therefore, the expected output of GENCO j at period k is: qˆ j (k) =

2

m=1

wjm qj (k − m)

⎡ ⎢ ⎣

q1 (k + 1)

a − ei −





qˆ j (k)

j=1

a − ei = − 2(b + fi )

j= / i

qi (k) =

2(b + fi )

P E (k

2

wjm qj (k − m)

j = 1 m=1

m=1

2(b + fi )

wim qi (k − m)

a − ej −

2 n  

qˆ i (k)

i=1 i= / j

qj (k) =

=

2(b + fj )

a − ej 2(b + fj )

i=1

0

2(b + fj )

⎢ ⎢ ⎢ q (k) ⎥ ⎢1 ⎢ 1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ q2 (k + 1) ⎥ ⎢ − bw11 ⎢ ⎥ ⎢ 2(b + f2 ) ⎢ q (k) ⎥ ⎢ ⎢ 2 ⎥ = ⎢0 ⎢ ⎥ ⎢ ⎢ .. ⎥ ⎢ ⎢. ⎥ ⎢ .. ⎢ ⎥ ⎢. ⎢ ⎥ ⎢ ⎣ qn (k + 1) ⎦ ⎢ ⎢ − bw11 ⎣ 2(b + fn ) qn (k) q1 (k + 1)



+ 1) = −b



−b

0



P (k) = −b

0

−b 0

⎡

0

0

···

bw12 2(b + f2 )

0

0

···

0

1

0

···

.. .

.. .

.. .

..

bw12 2(b + fn )

bw21 − 2(b + fn )

bw22 2(b + fn )

···

0

0

0

··· 1

q1 (k)

⎤

⎥ ⎢ ⎢ q1 (k − 1) ⎥ ⎥ + [a] u ⎢ ⎦ ⎣ q2 (k) q2 (k − 1)

 q1 (k) q2 (k)



+ [a]u

.

⎤

⎡ a−e ⎤ 1 ⎤ ⎥ ⎡ q (k) ⎥ 1 ⎢ 2(b + f1 ) ⎥ 0 0 ⎥ ⎢ q1 (k − 1) ⎥ ⎢ ⎥ ⎥⎢ ⎢0 ⎥ ⎥ ⎥ ⎢ bwn1 bwn2 ⎥ ⎢ a − e2 ⎥ q (k) ⎥⎢ ⎥ 2 − ⎢ ⎥ ⎢ ⎥ 2(b + f2 ) 2(b + f2 ) ⎥ ⎢ 2(b + f2 ) ⎥ ⎥ ⎢ q2 (k − 1) ⎥ ⎢ ⎥ ⎢0 ⎥ ⎥u 0 0 ⎥+⎢ ⎥⎢ ⎥ ⎥ ⎢ .. ⎥⎢ ⎥ ⎥ ⎢. .. .. ⎥⎢ ⎥ . . ⎢ ⎥ ⎥ ⎢ ⎥ . . . ⎥ ⎢ ⎥⎢ ⎦ ⎢ a − en ⎥ ⎥ ⎣ qn (k) ⎥ ⎥ ⎣ ⎦ 0 0 2(b + fn ) ⎦ qn (k − 1)

···

bw21 2(b + f1 )

(39)

(37)

bw22 2(b + f1 )

−

0

q2 (k)

⎤

Assume that GENCO 1 is under forward expectation and GENCO 2 is under adaptive expectation. After formulating each GENCO’s maximizing profit problem and finding their first-order conditions, the market representation in discrete-time linear form is:

wim qi (k − m)

⎡

⎤

⎥ ⎦

⎢ ⎥ ⎣ a − e − b q (k − 1) ⎦ u 2 1 1

For the n-players in a discrete-time linear form the market can be then represented by the following system:

⎡

0

⎤

3.6. GENCOs under forward and adaptive expectation

m=1

i= / j

−

⎥⎢ ⎦⎣

q1 (k)

(36)

and GENCO j’s output is n 

b(1 − 1) 2(b + f2 ) a − e1 2(b + f1 )

⎤⎡

From equation (39) we can see that in comparison with equation (24) from GENCOs under forward expectations, this is a simplification with the consideration that 2 is set to zero all the time. The condition for existence of the equilibrium implies that | −b2 (1 − 1)/4(b + f1 )(b + f2 )| < 1. System stability depends on market demand (b), production costs parameters (fi ), and adjusting factor . On the other hand, when fi = 0 ∀i = 1, 2, system stability depends only on the adjusting factor.

j= / i

Similarly, assume GENCO j includes GENCO i’s output in the two most recent periods in its forecasting. The weights are 0 ≤ w2m ≤  1, m = 1, 2, w2m = 1. Thus, the expected output of GENCO i at period k is:

2

−b 2(b + f1 )

0

2(b + f2 )

(35)

qˆ i (k) =

⎡ +

Under average expectation, GENCO i’s output will be n

⎡

⎥ ⎢ ⎦=⎣

q2 (k + 1)

(34)

n

⎤

−

bwn1 2(b + f1 )

bwn2 2(b + f1 )

0

0

(38)

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

821

Table 1 Market demand and generation unit parameters. Scenario

d1 $

e1 $/MW

f1

1

w11

w12

ˇ1

d2 $

e2 $/MW

f2

2

w21

w22

ˇ2

BASE 1 2 3 4

26 26 26 26 26

3 3 3 3 3

0 0 0.0126 0.0126 0.0126

0 0.5 0.2 1.0 1.0

1.0 0.0 1.0 0.5 0.0

0.0 1.0 0.0 0.5 1.0

1.0 0.5 1.0 0.0 0.3

18 18 18 18 18

3 5 3 5 5

0 0 0.035 0.035 0.045

0 0.3 -0.4 -0.9 1.0

1.0 0.0 0.8 0.5 0.2

0.0 1.0 0.2 0.5 0.8

1.0 0.5 0.0 0.5 0.6

⎡

q1 (k + 1)

⎤

⎡

0

0

⎢ ⎢ ⎥ ⎢ ⎣ qˆ 1 (k + 1) ⎦ = ⎢ ˇ1 ⎣ q2 (k + 1)

−

b(1 − 1) ⎤ ⎡ ⎤ 2(b + f1 ) ⎥ q1 (k) ⎥⎢ ⎥ 0 ⎥ ⎣ qˆ 1 (k) ⎦ ⎦

1 − ˇ1

bˇ1 2(b + f2 )

b(ˇ1 − 1) 2(b + f2 )

q2 (k)

0

⎡ a − e − b q (k − 1) ⎤ 1 2 2 2(b + f1 ) ⎢ ⎥ ⎢ ⎥ +⎢0 ⎥u ⎣ ⎦ (a − e2 ) 2(b + f2 )



P E (k + 1) = −b

−b

   q1 (k) q2 (k)

(40)

Table 2 Market properties: Case I. Scenario

BASE

1

2

3

4

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.667 1183.9 377.1

171.33 84.11 255.44 10.401 872.33 356.98

177.18 64.25 241.43 10.654 934.63 200.80

180.31 53.60 233.91 10.789 968.89 163.04

equilibrium. Dynamic simulation, therefore, can provide decisionmakers with significant, new, and even unusual information with the potential to alter market behavior. The solution reported in Table 2 is the final solution after reaching the stability displayed graphically in Fig. 2. In the base scenario, if the GENCOs’ marginal costs are equal,

+ [a] u

The condition for existence of the equilibrium is:

⎧ ⎨



(b + f1 )((b + f2 )(1 − ˇ1 )) ±



0,

⎩

⎭

2(b + f1 )(b + f2 )

System stability depends on all system parameters and adjusting factors. If both GENCOs have linear cost function, the resulting eigenvalues are: 0,

⎫ ⎬

2

(b + f1 )(b + f2 ) b2 ˇ1 (1 − 1) + (ˇ1 − 1) (b + f1 )(b + f2 )

1 2

!

"

 1 − ˇ1 −

1 − ˇ1 + ˇ12 − ˇ1 1

,

1 2

!

then their outputs are half of the total market quantity. Once the GENCOs’ marginal costs are different, a new market equilibrium is reached which is different with respect to the base scenario, market

"#

 1 − ˇ1 +

1 − ˇ1 + ˇ12 − ˇ1 1

Thus, the system stability depends only on the adjusting factors. There are other combinations of similar scenarios in which the two GENCOs use either a predefined or an adaptive expectation. 4. Numerical examples

(41)

(42)

price increase, and market quantity decrease. GENCO 2’s greater marginal cost means that its production output is now lower. We can also observe from Table 2 that, generally speaking, the aggregation of quadratic production costs increases the market price. The overall impact on GENCO 2 is a decreasing production

In this section, a simple duopoly market and a 5-GENCOs market are used as sample markets. 4.1. Duopoly market We consider one numerical example for each model described above. Five scenarios are simulated within each case. The first two scenarios assume a linear–linear case, a linear demand and linear cost. Next, the effect of quadratic production costs is analyzed, a linear demand and quadratic cost. The two GENCOs’ parameters are shown in Table 1. Demand curve P(k) = 15 − 0.018Q (k) is used and remains constant in all scenarios. All values have been selected arbitrarily. 4.1.1. Case I. GENCOs have naïve expectations Case I assumes that both GENCOs have naïve expectations. The price dynamics for all these scenarios are plotted in Fig. 2. In Fig. 2 it is possible to observe that the market prices have a dynamic transition, that they reach stability after a certain period, and that the parameters do have an impact on the transition processes and the time needed for the price to converge to final

Fig. 2. Electricity price dynamics: Case I.

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G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827 Table 4 Market properties: Case III. Scenario

BASE

1

2

3

4

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.667 1183.9 377.1

196.07 79.91 275.98 10.032 868.44 320.45

168.33 94.33 262.66 10.271 841.09 167.88

180.31 53.60 233.91 10.789 968.89 163.04

adjusting coefficients will fall between monopoly and perfect competition models, and eventually one GENCO will act as leader in the market, and will still reach the standard Nash equilibrium [15].

Fig. 3. Electricity price dynamics: Case II.

output as we proceed by modifying its production costs, which in this instance makes it more costly; therefore GENCO 2’s output decreases. In all scenarios the standard Nash equilibrium is reached. 4.1.2. Case II. GENCOs have forward expectations In Case II we assume that both GENCOs have forward expectations. Price dynamics in all scenarios are shown in Fig. 3. We observe that the market has different transition processes and oscillation is lower in the dynamic solution when both adjusting coefficients are moving in the same direction (in this case when both = 0.5). On the other hand, when both coefficients move in the opposite direction, the solution presents more complex dynamics. Nonetheless, the system is stable in both conditions. The market equilibrium under different forward expectation values from GENCO 1 of GENCO 2 is shown in Table 3. Table 3 shows that Scenario 1 presents the lowest price and the highest market quantity which is the result of adjusting coefficients values. Since the adjusting coefficients, moves in the same direction, both tend to 1, and the market equilibrium moves to perfect competition [15]. On the other hand, as discussed in the case above, the effect of quadratic production cost in market equilibrium will be an increase in price resulting in a decreasing market total quantity. Given that we are now introducing learning through adjusting factors, the market equilibrium will be different as we observe by comparing Tables 2 and 3. The learning effect means that scenario 5 determines the lowest market price while the opposite occurs when GENCOs are naive. In Table 3 we also observe that the market equilibrium moves in both directions as soon as we alter the values of the adjusting coefficients. The extreme values −1 and 1 reach monopoly and perfect competition (Bertrand). The traditional Cournot equilibrium is achieved when both coefficients are 0. Any other combination of

4.1.3. Case III. GENCOs have adaptive expectations Table 4 shows the market equilibrium under different conditions when both GENCOs have adaptive expectations. Similar to Case II, they are between monopoly and perfect competition equilibrium models. In all simulated scenarios the standard Nash equilibrium is reached. Under adaptive expectation, the GENCO makes different decisions from previous simulations as the expectations alter the strategy used. The market has different dynamics even when demand is the same. In Fig. 4 we observe that the system is stable in all scenarios. In cases with quadratic production cost parameters, it is clear that GENCO 1 is cheaper and therefore we expect a higher production level in each scenario. However, given the learning aspect (represented by ˇ), GENCO 1’s production decreases in scenario 3 whereas GENCO 2’s increases substantially. 4.1.4. Case IV. GENCOs have moving average expectations In this case, the model implements moving average expectations for both GENCOs. The resulting price dynamics are shown in Fig. 5. Under moving average expectation, market equilibrium and system stability depend on all system parameters except when both GENCOs have linear cost function where system stability depends only on adjusting the factors’ values. Fig. 5 clearly shows this where all scenarios present a transition process. In Table 5, we observe that the base scenario results once again in the classical Cournot outcome with equal output for both GENCOs, given that both have the same production cost curves and expectations. Once the production costs are different, however, another

Table 3 Market properties: Case II. Scenario

BASE

1

2

3

4

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

258.75 213.08 471.83 6.506 881.45 303.12

158.31 91.69 250.00 10.499 845.54 375.42

143.35 94.33 237.69 10.721 821.99 210.29

196.07 79.36 275.43 10.042 870.38 98.74

Fig. 4. Electricity price dynamics: Case III.

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

823

Table 6 Market properties: Case V. Scenario

BASE

1

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

222.22 222.22 444.44 7.000 862.89 426.45

2 144.16 176.49 320.65 9.228 610.01 −8.972

3

4

148.01 163.39 311.40 9.394 644.46 −234.30

148.01 163.39 311.40 9.394 644.46 −501.26

Table 7 Market properties: Case VI.

Fig. 5. Electricity price dynamics: Case IV.

Case

BASE

1

2

3

4

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

286.19 134.68 420.87 7.424 1240.2 308.5

160.70 85.91 246.61 10.560 863.66 373.24

143.35 94.33 237.68 10.721 821.99 210.29

196.07 51.35 247.42 10.546 969.24 148.15

ing factors as well as marginal costs makes the base scenario more dynamic. It is interesting to note that scenarios 3 and 4 reach the same market equilibrium, and even when they have different production costs, the learning effect impacts their decision behavior as if in scenario 3. The combination of learning models determines the market equilibrium for each scenario. The base scenario is still the Cournot outcome. In scenario 1 the market equilibrium is above the Cournot but below the monopoly level. The effect of learning can be clearly observed in scenarios 2–4 by comparing Table 6 with Table 2. In all of these scenarios, GENCO 2’s outputs increase substantially while the market price decreases with respect to those reported in Table 2. This means more social welfare of the market. 4.1.6. Case VI. GENCOs have forward and adaptive expectations In this case, the GENCOs are also modelled with different expectations. The effect of such expectations determines the market equilibrium in companion with the demand curve and production costs. Additionally, expectations also influence the transition process until stability is reached (Table 7). Comparing Figs. 2 and 7, we can observe that in all cases the final equilibrium is reached, but the price transitions to the equilibrium are quite different. Some paths approaching the equilibrium

Fig. 6. Electricity price dynamics: Case V.

market equilibrium is reached; learning also contributes to the new values for this market equilibrium. However, the learning effect leads the output of the two GENCOs to differ from the quadratic scenario. 4.1.5. Case V. GENCOs have naïve and forward expectations In this case, we model as if GENCO 1 is under naïve expectation and GENCO 2 is under forward expectation. In Fig. 6, we observe that the same market price is reached in two scenarios even when there are different learning parameters. For instance, when both GENCOs have linear production cost, the Cournot outcome is reached. However the effect of different adjustTable 5 Market properties: Case IV. Scenario

BASE

1

2

3

4

q1 (MW) q2 (MW) Q (MW) Price ($/MW) 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.667 1183.9 377.1

171.33 84.11 255.44 10.401 872.33 356.98

177.18 64.25 241.43 10.654 934.63 200.80

180.31 53.60 233.91 10.789 968.89 163.04

Fig. 7. Electricity price dynamics: Case VI.

824

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

Table 8 Market equilibrium and GENCOs’ quantities and profits. Case

BASE

1

2

3

4

I

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.766 1183.9 377.1

171.33 84.11 255.44 10.401 872.33 356.98

177.18 64.25 241.43 10.654 934.63 200.80

180.31 53.60 233.91 10.789 968.89 163.04

II

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

258.75 213.08 471.83 6.506 881.45 303.12

158.31 91.69 250.00 10.499 845.54 375.42

143.35 94.33 237.69 10.721 821.99 210.29

196.07 79.36 275.43 10.042 870.38 98.74

III

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.666 1183.9 377.1

171.33 84.11 255.44 10.401 868.44 320.45

177.18 64.25 241.43 10.654 841.09 167.88

180.31 53.60 233.91 10.789 968.89 163.04

IV

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

259.25 148.14 407.39 7.666 1183.9 377.1

196.07 79.91 275.98 10.032 872.33 356.98

168.33 94.33 262.66 10.271 934.63 200.80

180.31 53.60 233.91 10.789 968.89 163.04

V

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

222.22 222.22 444.44 7.000 862.89 426.45

144.16 176.49 320.65 9.228 610.01 −8.972

148.01 163.39 311.40 9.394 644.46 −234.30

148.01 163.39 311.40 9.394 644.46 −501.26

VI

q1 (MW) q2 (MW) Q (MW) Price 1 ($) 2 ($)

222.22 222.22 444.44 7.000 862.89 870.89

286.19 134.68 420.87 7.424 1240.2 308.5

160.70 85.91 246.61 10.560 863.66 373.24

143.35 94.33 237.68 10.721 821.99 210.29

196.07 51.35 247.42 10.546 969.24 148.15

are more desirable than others. A different choice of parameters influences market outcomes. Market equilibrium depends on all system parameters except the fixed-cost parameters. Adjusting factors becomes a key driver to determine of market equilibrium since each factor modifies the reaction functions. Table 8 summarizes the GENCOs’ outputs, market quantity, and market price for each scenario for each case. We note that the market equilibria differ from case to case as a result of learning, modeled through the different adjusting factors. In Table 8, we observe that the base scenario reaches the classical Cournot equilibrium in all cases. Scenario 1 in Case 2 reaches the maximum market quantity, and therefore the minimum market price with respect to all the cases. This implies that the market equilibrium in such cases tends to a Bertrand outcome, whereas in the rest of the cases, except Case 5, the market equilibria move to monopoly. 4.2. 5-GENCOs market Two numerical examples of the models described above are reported in this section. The demand function for the day-ahead market is P(Q (k)) = 35 − 0.018 Q (k). The production cost data shown in Table 9 is derived from reference [19] and modified. The adjusting coefficients for the different models are shown in Table 10. These values were specifically assigned for this case, to ensure that all models achieve the classical Cournot equilibrium [20], and hence to observe the effect of the costs of production on the dynamic properties of the system.

All the models achieve the classic Cournot outcome as shown numerically in Table 11 and graphically in Fig. 8. Table 11 shows that outputs vary due to different production cost curves. It is clear that GENCO 2’s is the cheapest and therefore its production will be greater with respect to the others. The market clearing price in all models reaches the same value: 16.4533 $/MW. Table 12 reports total expected revenues, total expected costs, and net expected profits for each GENCO under the uniform pricing mechanism. Recall that with uniform pricing, each bidder, if it has a bid accepted, is paid the equilibrium price.

Table 9 GENCOs’ cost data.

GENCO 1 GENCO 2 GENCO 3 GENCO 4 GENCO 5

ei $/MW

fi $/MW2

2.00 1.75 3.00 3.00 1.00

0.0200 0.0175 0.0250 0.0250 0.0625

Table 10 Adjusting coefficients.

 ˇ wi1 wi2

GENCO 1

GENCO 2

GENCO 3

GENCO 4

GENCO 5

0 1 1 0

0 1 1 0

0 1 1 0

0 1 1 0

0 1 1 0

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

825

Table 11 Market price and quantity.

q1 (MW) q2 (MW) q3 (MW) q4 (MW) q5 (MW) Q (MW) Price ($/MW)

Naive

Forward expectations

Adaptive expectations

Moving average expectations

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

Table 12 GENCOs’ revenues, costs and profits. GENCO

Revenues

Costs

Profits

1 2 3 4 5

4100.1 4564.5 3255.2 3255.2 1778.0

1740.4 1832.3 1572.1 1572.1 837.9

2359.7 2732.2 1683.1 1683.1 940.1

Table 13 Adjusting coefficients.

 ˇ wi1 wi2

Fig. 8. GENCOs’ outputs.

Fig. 9. GENCOs’ outputs.

GENCO 1

GENCO 2

GENCO 3

0.2 0.2 0.2 0.8

0.6 0.0 0.5 0.5

−0.3 0.5 0 01

GENCO 4 −0.8 0.5 1 0

GENCO 5 0.1 1.0 0 1

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G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827

Table 14 Market equilibrium and gencos’ quantities.

q1 (MW) q2 (MW) q3 (MW) q4 (MW) q5 (MW) Q (MW) Price ($/MW)

Naive

Forward expectations

Adaptive expectations

Moving average expectations

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

189.6413 83.1370 243.2361 321.3592 89.3959 926.7695 18.3181

292.9966 242.8704 235.2030 235.2030 125.8308 1132.1038 14.6221

249.1957 277.4217 197.8434 197.8434 108.0654 1030.3696 16.4533

It can be observed in Table 12 that GENCO 2 earns higher profits. Since GENCO 3 and GENCO 4 have similar units, their profits are also similar. Next, we arbitrarily set the adjusting coefficients different from zero (see Table 13) to observe the effect on market equilibrium. The expected GENCOs’ outputs are reported numerically in Table 14 and graphically in Fig. 9. Table 14 shows that the GENCOs’ outputs are different in all models with respect to the naïve model due to adjusting factors. Consequently, the market presents multiple equilibria based on the GENCOs’ expectations. Some are socially better than others.

depends on the values of learning indexes. On the other hand, when production costs are not linear, system stability depends on market demand and production. The values of learning indexes, which represent the degree of competition, determine the market equilibrium falling between the two extreme market structures, Bertrand and monopoly. When the agent model and the interactions among the agents are simple, the market dynamic model can be represented by analytical equations. Thus, the properties of the market dynamics can be studied. However, when the agent is modeled in rather specific detail to more accurately represent the properties of practical agent decision-making, and the interactions among the agents are modeled in detail, the analytical description of the market dynamic process is far more complex. One appropriate way to solve the market dynamics problem is with Complex Agent System (CAS) simulation, which accurately captures the features of the agent and agents’ interactions. Such CAS simulations can be extended to include the market dynamic models described on this paper. However, this is a matter for future papers.

5. Adjusting factors, impact on market equilibria

Acknowledgements

The profit function of each player depends upon its rivals’ control variables in addition to its own control variable. The values of adjusting factors determine the two extreme market structures: Bertrand and monopoly, depicted in Fig. 10. The classic Cournot outcome is achieved if all of the adjusting factors in the forward expectation model are equal to zero. Perfect collusion, monopoly, occurs when the adjusting factors are set to −1, whereas the perfect competition is achieved when the adjusting factors are set to 1. Similarly, this occurs with the adjusting factors in the adaptive expectation model, but since we are considering that these factors move only within the range ˇ ∈ [0, 1], then the market equilibrium falls within Classical Cournot and Bertrand. Any other combination of adjusting coefficients will fall between monopoly and perfect competition models; eventually, one GENCO will act as leader in the market, as stated in the Stakelberg model. Cartel implies that the GENCOs collude, implicitly or explicitly, to optimize their total profits. Hence, the key goal of the learning process is to obtain accurate adjusting factors.

The research is supported by Cuerpo Academicos of the Government of Mexico (ANUIES-PROMEP), to whom sincere acknowledgement is expressed. This paper reports work partially funded by FCT (Portugal) under project reference PTDC/EEAENE/74550/2006 (PLAMS) and the National Science Foundation (NSF).

Fig. 10. Adjusting factors effect on market equilibria.

6. Conclusions This paper presented four basic types of learning in the decisionmaking process of generating companies in an oligopolistic market: naïve, forward expectation, moving average expectation, and adaptive expectation. The different expectations have different market equilibrium dynamic characteristics. In the case of naïve expectation, it always reaches the same equilibrium as the classical Cournot model. Although the equilibrium is the same as the equilibrium of the Cournot model, the naïve expectation model differs from the Cournot model because the naïve expectation is introduced. Market equilibrium is stable in all of the reported cases. When both GENCOs have linear cost function, the system’s stability

References [1] W. Yang, G.B. Sheblé, Modeling generation company decisions and electric market dynamics using control theory, Proc. of IEEE Power Engineering Society Summer Meeting (2002) 1385–1391. [2] K. Cuthbertson, Modelling expectations: a review of limited information estimation methods, Bull. Econ. Res. 1 (42) (1990) 1–34. [3] B.S. Montin, C.A. Nolder, An agent market model using evolutionary game theory. www.math.fsu.edu/∼bmontin. [4] H. Chen, K.P. Wong, D.H.M. Nguyen, C.Y. Chung, Analyzing oligopolistic electricity market using coevolutionary computation, IEEE Trans. Power Systems 21 (1) (2006) 143–152. [5] C. Athina, Tellidou, G. Anastasios, Bakirtzis, Agent-based analysis of capacity withholding and tacit collusion in electricity markets, IEEE Trans. Power Systems 22 (November (4)) (2007) 1735–1742. [6] A. Maiorano, Y.H. Song, M. Trovato, Dynamics of non-collusive oligopolistic electricity markets, in: IEEE Power Engineering Society Winter Meeting, 2000, pp. 838–844. [7] Z. Yu, F.T. Sparrow, T.L. Morin, G. Nderitu, A Stackelberg price leadership model with application to deregulated electricity markets, in: Proc. of IEEE Power Engineering Society Winter Meeting, 2000, pp. 1814–1819. [8] J.D. Liu, T.T. Lie, K.L. Lo, An empirical method of dynamic oligopoly behavior analysis in electricity markets, IEEE Trans. Power Systems 21 (2) (2006) 499–506. [9] A. Stothert, I. MacLeod, Competitive bidding as a control problem, IEEE Trans. Power Systems 15 (1) (2000) 88–94. [10] Y. Liu, F.F. Wu, Generator bidding in oligopolistic electricity markets using optimal control: fundamentals and application, IEEE Trans. Power Systems 21 (August (3)) (2006) 1050–1061. [11] Y.F. Liu, F.F. Wu, Risk management of generators’ strategic bidding in dynamic oligopolistic electricity market using optimal control, Generation, Transmission & Distribution, IET 1 (May (3)) (2007) 388–398. [12] C. Chen, Linear System Theory and Design, Oxford University Press, 1998.

G. Gutiérrez-Alcaraz, G.B. Sheblé / Electric Power Systems Research 80 (2010) 815–827 [13] E. Roth, I. Erev, Learning in extensive form games: experimental data and simple dynamic models in the intermediate term, Games Econ. Behav. 8 (1995) 164–212. [14] J.D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World, Irwin McGraw-Hill, 2000. [15] G. Gutierrez-Alcaraz, Sequential time-step generation companies decisions in oligopolistic electricity market, Electric Power Systems Res. 78 (5) (2008) 824–834. [16] G. Gutiérrez-Alcaraz, J.H. Tovar-Hernandez, E.L. Moreno-Goytia, Analysis of dynamic Cournot learning models for generation companies based on conjectural variations and forward expectation, Electric Power Systems Res. 79 (December (12)) (2009) 1698–1704. [17] Mostafa Sahraei-Ardakani, Ashkan Rahimi-Kian, A dynamic replicator model of the players’ bids in an oligopolistic electricity market, Electric Power Systems Res. 79 (May (5)) (2009) 781–788. [18] Mostafa Sahraei-Ardakani, Mahnaz Roshanaei, Ashkan Rahimi-Kian, Caro Lucas, A study of electricity market dynamics using invasive weed colonization

[19]

[20] [21]

[22]

[23] [24]

827

optimization, in: IEEE Symposium on Computational Intelligence and Games, 2008, pp. 276–282. Youfei Liu, Y.X. Ni, Felix F. Wu, Bin Cai, Existence and Uniqueness of Consistent Conjectural Variation Equilibrium in Electricity Markets, Int. J. Electrical Power & Energy Systems 29 (July (6)) (2007) 455–461. J. Friedman, Oligopoly Theory, Cambridge Surveys of Economic Literature, Cambridge University Press, 1983. Yiqun Song, Yixin Ni, Fushuan Wen, F.F. Wu, Conjectural variation based learning model for strategic bidding in spot market, Elect. Power Energy Systems 26 (2004) 797–804. L. Soto-Reyes, G. Gutierrez-Alcaraz, Energy Market Model Competition: A Discrete Event System Simulation Equivalent, in: 2007 IEEE Power Engineering Society General Meeting, Tampa, FL, USA, 2007 June. Howard E. Bell, Gershgorin’s Theorem and the zeros of polynomials, The American Mathematical Monthly 72 (March (3)) (1965) 292–295. Keith Cuthbertson, Modeling expectations: a review of limited information estimation methods, Bull. Econ. Res. 1 (42) (1990) 1–34.