Strategy Equilibrium in Stackelberg Model with Transmission Congestion in Electricity Market

  • Lee, Kwang-Ho (Department of Electrical Engineering, Dankook University)
  • Received : 2013.07.22
  • Accepted : 2013.09.23
  • Published : 2014.01.01


Nash Cournot Equilibrium (NCE) has been widely used in a competitive electricity market to analyze generation firms' strategic production quantities. Congestion on a transmission network may lead to a mixed strategy NCE. Mixed strategy is complicated to understand, difficult to compute, and hard to implement in practical market. However, Stackelberg model based equilibrium does not have any mixed strategy, even under congestion in a transmission line. A guide to understanding mixed strategy equilibrium is given by analyzing a cycling phenomenon in the players' best choices. This paper connects the concept of leader-follower in Stackelberg model with relations between generation firms on both sides of the congested line. From the viewpoint of social welfare, the surplus analysis is presented for comparison between the NCE and the Stackelberg equilibrium (SE).

1. Introduction

With the introduction of competition in the electricity market, competitive prices are of fundamental importance for designing and operating the electricity market. The prices are setbased on the market clearing rules and the market participants’ behaviors for pursuing their profit maximization. In order to predict and estimate the participants’ decisions under market rules, it is paramount to find equilibrium of generation firms’ strategies under a competition model. Also computing the equilibrium is essential in predicting market prices and analyzing performance of a trade between supply and demand in electricity market.

Using game theory to model generation firm’s strategic choices in production quantity has been attempted numerously in order to analyze competition in a market setting. The most extensively used among oligopoly market models has been the Cournot model in Nash Equilibrium (NE) analysis of generation wholesale markets [1-3]. In the Cournot model, generation firms choose a strategic quantity depending on the rivals’ expected production quantity. This format of assuming rivals’ behaviors is applicable equally to all the participants in the Cournot model. On the other hand in the Stackelberg model a certain player called “leader” assumes the others’ responses differently. This paper states that the Stackelberg model is more suitable to analyze the electricity market with transmission network that may have congestion.

The transmission line congestion status varies with the value of the generation quantities. Accordingly, the decision space can be divided into subsets depending on whether the transmission lines are congested or uncongested [4]. This leads to discontinuities in the reaction curves and causes the profit functions to be nondifferentiable and nonconcave [5, 6]. In this situation, they may not have a pure strategy of NE. Instead, the NCE has a form of mixed strategies where the players find it optimal to randomly choose between strategies [7].

In a basic representation of game theory, the mixed strategy is comprehensive and acceptable within the sample game. However, in the electricity market, the production quantity of a mixed strategy is complicated to understand conceptually, difficult to compute, and hard to implement practically in the market. This mixed strategy Nash equilibrium (MSNE) makes it harder to analyze the market and to work out the strategy of the generation firms. It may be desirable that the situation causing the MSNE be avoided [8]. Since transmission congestion is unavoidable in practical power systems, the market clearing mechanism for the generation firms needs to be adjusted to avoid the MSNE. This is the main reason that this paper suggests the Stackelberg model for finding equilibrium.

The mathematical programming method [9-11] can solve for the NE of pure strategy in a game that has differentiable and concave profit functions. When a transmission line is congested, however, this method has difficulties in determining equilibrium because the profit function can be nondifferentiable and nonconcave. The Nikaido-Isoda function and a relaxation algorithm (NIRA) are combined in [12, 13] to calculate the NE for a noncooperative game.

The Stackelberg model focuses on a differential or inequitable position in an oligopolistic competition, and utilizes the differences into the concept of “leader-follower”. This paper proposes a corresponding relation between the leader-follower and the generation firms in a transmission network. The generation quantity of each firm influences discriminately on the congested line. For example, the power injection in the receiving end of the congested line relieves congestion; on the other hand, the power in the supplying end intensifies congestion. This differential effect on congestion makes the firms behave differently in competition [14], where a certain firm plays the role of a leader, and the others play the role of followers. Based on this suggestion, the Stackelberg model is applied to analyze the competition in the electricity market with congestion.

The equation for solving the Stackelberg equilibrium (SE) is given, and the surplus analysis is derived comparatively in the NCE and the SE. In the numerical example, a “cycling” phenomenon is shown in a sample with congested network, and compared with the MSNE and the SE. The numerical result shows that the SE has pure strategy equilibrium and may result in better social welfare than in the NCE.


2. Cournot Nash Equilibrium

2.1 Optimization of cournot model

Nash equilibrium (NE) is the most widely used equilibrium concept in economics and in the electricity market. A set of strategies is called a NE if, holding the strategies of all other firms constant, no firm can obtain a higher profit by choosing a different strategy. Thus, in a NE, no firm wants to change its strategy.

Augustin Cournot presented a model of noncooperative oligopoly. The Cournot model assumes that each firm acts independently and attempts to maximize its profits by choosing its output [8]. A NE based on Cournot model is called Nash Cournot equilibrium (NCE) and widely used in studying an electricity market.

Assume there are N strategic firms having N generating units each. Firm i ∈ {1, …, Ng} has a unit i with a generating cost function Ci(qi) = biqi +0.5 miqi2 and a marginal cost ci(qi) = bi + miqi , where qi represents firm i’s generation power quantity, bi and mi are cost coefficients. The inverse demand function at bus j ∈ {1,…, Nd} is pj = aj − rjdj , where dj is the power demand, aj and rj are its coefficients.

The market operator (MO) is responsible for running the electricity market while settling the accepted bids and offers, and keeping a balance between load demand and generation supply in real time. The objective of the MO is quite different from the other participants. The MO attempts to maximize the demand side benefit as defined by an integration of the inverse demand function instead of maximizing profits. Formulating the MO’s objective into a quadratic program, we have

where D and G are the sets of all demand buses and generating units respectively. L is the set of all transmission lines and Tl acts as the power flow on the transmission line l. The constraint consists of a power balance equality (2), and transmission power inequalities (3).

Each rational strategic generation firm maximizes its profits (revenue minus generating costs) by selecting its own generation parameter accepting as given the strategic parameters of other firms [3]. Formulating firm i’s objective into a quadratic program with generation capacity limits, we have

where πi is the profit from unit i, and pi is a nodal price determined by the local demand, di at the bus unit i, and qMi is the capacity limit of qi .

When a transmission network has congestion, the nodal prices change differently at each node. In this paper, locational marginal pricing is adopted as a pricing method of the MO. As generation firms’ profit maximization needs to be solved with the MO’s problem simultaneously, these optimizations of decentralized decision makers require hierarchical coordination.

2.2 Congestion and cycling of choices

For understanding transmission congestion on generation firms’ strategies, an electricity power market is given a simple network as shown in Fig. 1. The system consists of 2 firms, and 2 buses with a local market at each bus. The marginal cost functions of firm F1, F2, and the inverse demand functions are respectively, MC1 = 0.4q1 , MC2 = 0.2q2 , p1 =100 − d1 , p2 =100 − d2 . The transmission lines are assumed to be lossless and have the power flow limit Tmax=5.

Fig.1.Duopoly system considering transmission congestion

The generation firm F2 produces more power than F1, since the two markets have identical demand function and F1 has higher cost function. The power flow is q2-d2, or d1-q1, less than 5.0 with a direction from F2 to F1. Under no congestion, the price at both markets is identical, so d1=d2=(q1+q2)/2. Since q2-d2≤5.0, if the gap between q1 and q2 is greater than 10, the line is congested. Let’s consider the firms’ profit maximization alternatively.

A. Case ①: Assume a congestion on the line, then demand and generation power has a relation as d1=q1+5. And the price p1=95-q1, the profit of F1 as in (4), π1=(95-q1)∙q1-0.2q12. Therefore F1 will choose q1=39.58 which is from an optimization of the π1 under congestion. Let the quantity symbolized as ‘q1c’ of F1’s critical point that means F1 always choose this value when the line is congested.

B. Case ②: Let’s look into F2’s strategy under F1’s pick of q1c. When F2 picks q2greater than 49.58, congestion arises on the line, since the gap between q1 and q2 is bigger than 10. Then d2=q2-5, and p2=105-q2, the profit of F2, π2=(105-q2)∙q2-0.1q22. The crest point of π2 is at q2=47.73 that is out of the boundary q2≥49.58.

When F2 picks q2 smaller than 49.58, congestion will not arise. Under no congestion, d2=(q1+q2)/2, the price of market 2, p2=100-d2=80.2-0.5q2. Then the crest point of π2 is at q2=66.84 that is out of boundary q2≤49.58. Fig. 2(a) shows the profit of F2 with respect to q2 when q1=q1c. Therefore the best choice of F2 is q2=49.58 which is the boundary point between congestion and uncongestion.

Fig. 2.Profit functions of (a) F2 at q1=39.58 and (b) F1 at q2=49.58

C. Case ③: Let’s observe F1’s response to F2’s pick q2=49.58. As checked before, when q1≤q1c, the line is congested, and the optimal choice of F1 is q1c. However under no congestion, the demand is d1=(q1+q2)/2, and the price is p1=75.1-0.5q1. Then the crest of theπ1 is at q1=53.72 that is in the boundary q1≥q1c. Fig. 2(b) shows the profit functions and the optimum point at q1=53.72.

D. Case ④: The best response of F2 with q1=53.72 is as follows. When p1>p2, the line is congested, and d1

E. Case ⑤: The F1’s best reaction under q2=60.95 is also based on the profit functions with respect to q1. While the line is not congested, d1-q1≤5, d1=(q1+q2)/2, and q1≥50.95. The profit of F1 under no congestion has a crest at q1=49.6, which is not feasible. The attainable optimum point is q1=50.95 and π1=1725.1 on the boundary case. As checked before, F1’s choice under congestion is q1c and π1=1880.2 larger than 1725.1. Therefore F1’s strategic response under q2=60.95 is to pick the q1c=39.58 discarding the pick before 53.72. This means to return to the case ①, and it will change the situation to cases ②, ③, ④, and ⑤ successively. These choices are not called equilibrium strategy, since they keep changing their choices when opposite player changes. This phenomenon is called “cycling”. Table 1 shows this cycling by summarizing their strategies.

Table1.Summarized best response and cycling of F1 and F2’s reactions

2.3 Mixed strategy equilibrium

The cycling phenomenon results from the discontinuities in their reaction curves. The transmission line congestion makes the decision space divide into subsets depending on whether they are congested or uncongested. This leads to discontinuities and nondifferentiable profit functions.

The hierarchical optimization of (1)~(6) can be solved easily by the mathematical programming method, once no inequality binds underlying decision space of differentiable functions in the Cournot model. Let the solved quantity parameters be q* . At a pure strategy NE, the strategies of all participants satisfy

where is the solved quantity parameter of firm i, qi is the possible quantity parameter firm i can choose, and is the solved quantity parameter set of all participants excluding firm i. By unilaterally altering their choices, none of the generating firms can improve their profits.

It may be that none such pure strategies satisfy the definition of Nash equilibrium (7). Instead, the firms may discover that they must play a combination of pure strategies, choosing amongst them randomly. This is a “mixed strategy,” which is specified by the probability distribution of the choice of pure strategies [7, 15].

Solving the two-level optimization for the sample case as summarized in Table 1 gives a mixed strategy Nash equilibrium (MSNE); q2=54.89, q1=[39.58, 51.83] with probability [0.657, 0.343]. At the NE, the firm, F1 chooses a mixed strategy consisting of two pure strategies with the probabilities, while F2 chooses a pure strategy. One of the two choices of F1 gives rise to congestion on a line, for example q1=51.83. The other, q1=39.58, on the contrary, makes no congestion. The former can be called a congestion strategy, the latter an uncongestion strategy. F1 uses more strategic choices, sometimes congestion strategy; other times uncongestion strategy, whereas F2 takes only one choice.

This paper asserts that the player choosing a mixed strategy plays the role of a leader, and the other plays the role of a follower from the viewpoint of the Stackelberg model. The leading player locates in a receiving area over a congested line, whereas the follower locates in a sending area. The leader, F1, is at bus 1on the receiving end in Fig. 1. This assumption is very useful in calculating a MSNE, and its usefulness has been verified over a variety of case studies.


3. Stackelberg Model

3.1 Stackelberg equilibrium at duopoly

It is assumed in Cournot model that all the competitors in a market have equal opportunity. However, in some industries, historical, institutional, or legal factors put the competitors into a differential or inequitable position in the market. For example, the firm that discovers and develops a new product has a natural first-mover advantage.

Heinrich von Stackelberg presented an important oligopoly model in 1934. In the Stackelberg model, one firm acts before the others. The leader firm picks its output level and then the other firms are free to choose their optimal quantities given their knowledge of the leader’s output. The follower’s best response is determined as in the Cournot model that competitors’ output is assumed to be fixed. However, the leader’s action is different from the followers’ actions in Cournot model [8].

The leader picks the output to maximize its profit subject to the constraint that the follower firm chooses its corresponding output using its Cournot best response function [16, 17]. In Stackelberg equilibrium (SE), the leader is better off and the follower is worse off than in a Cournot equilibrium (CE).

Stackelberg pointed out history, institution, law, discovery, and development as factors affect the determination of the leader and the followers in general industries. But in an electricity market, the transmission network is an important factor. Because the physical limits of transmission lines can restrict the economic dispatch of the generation power, the generation firms change their strategies by depending on the site with respect to the congested line. This paper postulates that the firm at the receiving area of a congested line has a leading opportunity, and the firm at the sending area has a follower position. When it is not clear if a firm locates in a sending area or a receiving area, PTDF(Power Transfer Distribution Factor) is a useful index to clear the vagueness [14,18,19].

Let’s look into the duopoly system and derive the Stackelberg equilibrium (SE). Firms have generating marginal cost; MCi=miqi, and the inverse demand functions of the market is pi=a-r(q1+q2). The firm F1 is assumed a leader, F2 follower. The follower’s best response condition is as follows;

The leader’s best response condition is derived by considering the follower reaction.

where, ∂q2 ⁄ ∂q1 is obtained from (8).

The SE is derived from (7), (8) and the demand function is as follows;

where, ΔA = m1m2 + 2r(m1 + m2) + 2r2 , ΔB = m1m2 + r(2m1 + m2) + r2 . qs1 is the leader’s output, qs2 is the follower’s output, and ps is the market price. The subscript‘s’ denotes for the SE and the subscript ‘c’ is used for the NCE.

where, ΔA and ΔB are the same as before.

3.2 Social welfare comparison between NCE and SE

The NCE and the SE are compared from the viewpoint of social welfare which is used universally as an index for the trading value in a market in microeconomics. This section shows that social welfare in the SE is greater than in the NCE in no congestion situation by comparing price and total quantity supplied.

The ratio of the prices between the NCE and the SE is as follows;

where, ΔA> ΔB =ΔA – r · m2 – r2 . Therefore, the price in SE is lower than that in the NCE.

The generation quantity of F1 in the SE and the NCE is compared by the following ratio;

The total quantity of generation, q1+q2, is derived from (8) in the SE and the NCE in common as follows;

The total quantity is proportional to the leader’s output, q1. Since the quantities, qs1>qc1 from the previous equation, the total quantity in the SE is greater than in the NCE. From the viewpoint of “Benefit” in demand function in microeconomics, the SE is better than the NCE, because the price in the SE is lower and the total demand in the SE surpasses that in the NCE. But the generation cost in the SE is larger than in the NCE because of greater demand in the SE.

Instead of the benefit in demand, the social welfare is suitable to evaluate a trade in a market. Since the social welfare is computed by benefit minus generation cost, the greater quantity may result in less social welfare.

Let’s compare quantity A and quantity B in Fig. 4. Even though the higher quantity B gives larger benefit, the social welfare in B is less than in A. The social welfare increases with increasing quantity until point C. On the other hand it decreases with increasing quantity after point C. So it is the point at issue whether the SE with greater quantity locates in region less or greater than C.

Fig. 3.Relation between market price and marginal cost

Fig. 4.Diagram of 3-bus system with a congested line

In the region A less than C, the market price(pa) is higher than the marginal cost(MCa). On the contrary, pb is lower than MCb in region B. Therefore the prices in the SE need to compare with the marginal cost of the two firms for observing the SW.

For an easy comparison, the market price(ps) derived in 3.1 is rearranged here;

Both the leader and the follower have lower marginal cost than the market price in the SE. So it is shown that the SE with a lower price and a larger total quantity has greater social welfare than in the NCE.


4. Results and Comparison

4.1 Equilibrium in duopoly competition

A. Uncongestion Case

In the duopoly system as Fig. 1, letting the transmission capacity be limitless leads to the NCE and the SE as in Table 2. In the SE, more demand quantity is supplied at a lower price than in the NCE. Therefore the SE gives more social welfare 7271.6 rather than 7188.6 of the CNE.

Table 2.Results of NCE and SE in duopoly system without congestion

B. Congestion Case

When the limit value, T, affect the strategies of the generation companies, the equilibria of Cournot and Stackelberg in Table 2 will not be retained. The physical limit of the power flow on a transmission line might be advantageous to a potential supplier. The change of the NCE and the SE with a consideration of T=5.0 is in the Table 3.

Table 3.Results of NCE and SE in duopoly system considering congestion

As mentioned in 2.2 in this paper, taking a consideration of the limit T=5.0 results in “cycling” of firms’ choices instead of a pure equilibrium. To avoid the cycling phenomenon, a mixed strategy of the NCE is to be solved, and the mixed strategy equilibrium of the duopolyis given with probability α=0.657, β=0.343 for uncongestion and congestion case respectively. In the uncongestion case, the power flow is 1.53 less than T, and the price is identical at both buses. In the congestion case, the flow is equal to T, and the price at bus 1 is 55.42, the price at bus 2 is 50.11. The social welfare is calculated as an expectation value with the probability.

In the SE, the state corresponds to the case 2 in Table 1 about the cycling reactions. From the viewpoint of the NCE, the firm F1 assumes F2 keeps the quantity q2=49.58, and changes q1 from 39.58 to 53.72 as in Table 1. However, in Stackelberg, F1 believes that F2 keeps following F1’s choice on the basis of F2’s profit maximization. No cycling happens in the SE as in Table 3, and the power flow is equal to the limit T with identical prices. So this state is a borderbetween uncongestion and congestion. In the case of Table 2, the profit of F1 is computed to 1730 in the SE, however, in the case of Table 3, it is 1880.2. The leader, F1, can increase its profit by utilizing the equipment scarcity of transmission capacity.

In the case of Table 3, the social welfare in the SE is 6369.5 lower than 6851.6 in the NCE. This paper shows that social welfare is higher in the SE than in the NCE as in Table 2, but when transmission congestion is considered it may not be true as in Table 3. The next section shows another result that the SE gives higher social welfare than the NCE even under consideration of congestion. The verification of the equilibria is provided in appendix A.

4.2 Equilibrium in 3-Players’ competition

An electricity power market is given with a simple network as shown in Fig. 4. The system consists of 3 generation firms, and 3 buses with a local market at each bus. The marginal cost functions of firm F1, F2, F3, and the inverse demand functions are respectively, MC1 = 10 + 0.3q1 , MC2 = 20 + 0.4q2 , MC3 = 15 + 0.45q3 , p1 = 70 − 0.7d1 , p2 = 80 − 0.5d2 , p3 = 90 − 0.4d3 . The transmission lines are assumed to be lossless and have the reactance satisfying x12 = x13 = 2 x23 .

The transmission limit is assumed Tmax=15.0, and the flow direction from 1 to 2 is determined by the inverse demand function at each bus and the marginal cost functions. The results of a mixed strategy of the NCE and a pure strategy of the SE are given in Table 4 and verified in appendix B.

Table 4.Results of NCE and SE in 3 firm competition systems under congestion

Due to the transmission limit, the NCE does not have a pure strategy, rather a mixed strategy. The uncongestion strategy in the MSNE has a probability 0.49. On the other hand, the congestion strategy has a probability 0.51, and leads to the nodal prices 52.3, 54.3 and 53.7 in bus 1, 2, and 3 respectively. The node of the highest price is bus 2, and it is a receiving end on the congestion line. Therefore the F2 that locates in bus 2 plays a role of a leader and the F3 and F1are the followers.

This market system also gives a pure strategy of the SE. The SE state corresponds to a border between congestion and uncongestion states. The identical price at each node denotes an uncongestion state, and the power flow equal to the limit means a congestion state. The total generation is 173.93 in the SE, which is higher than 172.3 in the NCE computed using an expected value with the probabilities. The price in SE is 52.47, which is lower than 52.8 in the NCE calculated using weighted average over the buses and utilizing expected value with the probabilities. The more quantity and lower price leads to a higher social welfare in the SE than in the NCE.

In this 3-player market, whether congestion arises or not,the social welfare in the SE is 7374.6 higher 7329.2 than in the NCE. This is one of the reasons that this paper recommends Stackelberg model for analyzing an electricity market with congestion. As mentioned before, the main reason is that the SE does not have a mixed strategy, while the NCE may have a mixed strategy. In order to implement this idea to the electricity market, amarked effort is needed to implement the concept of leader- follower in the electricity market and to adjust the market rule to acknowledge the first-mover advantage for the firms in the receiving area of the congested line.


5. Conclusion

Transmission congestion may cause a mixed strategy equilibrium based on the Cournot model in an electricity market. A mixed strategy equilibrium is complicated to understand, difficult to compute, and hard to implement in a practical market. In order to avoid the mixed strategy Nash equilibrium, the Stackelberg model with a concept of leader-follower is suggested and analyzed in this paper. The leader’s advantage corresponds to a major influence to congestion of a generation firm which is at the receiving end. The Stackelberg equilibrium is identified not to have mixed strategies, and have superiority to the Cournot Nash equilibrium from the viewpoint of social welfare in an uncongestion situation. When there is congestion in a transmission network, the Stackelberg equilibrium may have less social welfare. However, congestion is less frequent than uncongestion, so the SE will give better social welfare. This analysis will help to suppress bothersome arguments about mixed strategy equilibria in an electricity market.


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