• The Transactions of the Korean Institute of Electrical Engineers A
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• v.51 no.4
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• pp.196-201
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• 2002

An important aspect of the study of power system markets involves the assessment of strategic behavior of participants for maximizing their profits. In models for imperfect competition of a deregulated system, the key tack is to find the Nash equilibrium. When the constraints are not considered in the power market, the equilibrium has the form of a pure strategy. However, the constraints are considered, the equilibrium has the form of a mired strategy. In this paper the bimatrix game approach leer finding a mixed equilibrium is analyzed. The Nash equilibrium of a mixed strategy will be used adequately for the analysis of market power.

• Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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• v.23 no.2
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• pp.182-189
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• 2009

Nash equilibrium is usually used to investigate a generator's strategic bidding in electricity markets. Some literatures show that the transmission constraints may induce no pure strategy equilibrium and make it hard to find the equilibrium. Using an analytical approach to find a mixed strategy Nash equilibrium in electricity market with transmission constraints, we analyze the influence of transmission capacity on the mixed strategy Nash equilibrium. Finally, a simple numerical example is provided to support the claims of this paper.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.61 no.2
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• pp.187-193
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• 2012

Nash Equilibrium (NE) is as useful tool for investigating a participant's strategic generation quantity in a competitive electricity market. Cournot model may give a mixed strategy NE instead of a pure strategy when transmission constraints are considered. A mixed strategy is difficult to compute, complicated to understand conceptually, and hard to implement in an electricity market practically. This paper presents that a mixed strategy does not appear in Stackelberg leader-follower model even under a transmission congestion. A solution method is proposed for the leader-follower model under a nondifferentiable space of a strategy variable. Based on the pure strategy NE with a transmission line congested, the merit of leader-follower model is shown from a social welfare point of view.

• Journal of Electrical Engineering and Technology
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• v.9 no.1
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• pp.90-97
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• 2014

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).

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.55 no.11
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• pp.492-497
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• 2006

Nash Equilibrium (NE) is essential to investigate a participant's bidding strategy in a competitive electricity market. The transmission line constraints make it difficult to compute the NE due to causing a mixed strategy NE instead of a pure strategy NE. Computing a mixed strategy is more complicated in a multi-player game. The competition among multi-participants is modeled by a two-level hierarchical optimization problem. A mathematical programming approach is widely used in finding this equilibrium. However, there are difficulties to solving a mixed strategy NE. This paper presents two propositions to add heuristics to the mathematical programming method. The propositions are based on empirical studies on mixed strategies in numerous sample systems. Based on the propositions a new formulation is provided with a set of linear and nonlinear equations, and an algorithm is suggested for using the prepositions and the newly-formulated equations.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.57 no.1
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• pp.34-39
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• 2008

Nash Equilibrium(NE) is essential to investigate a participant's bidding strategy in a competitive electricity market. Congestion on a transmission line makes it difficult to compute the NE due to causing a mixed strategy. In order to compute the NE of a multi-player game, some heuristics are proposed with concepts of a key player and power transfer distribution factor in other studies. However, generation capacity constraints are not considered and make it more difficult to compute the NE in the heuristics approach. This paper addresses an effect of generation capacity limits on the NE, and suggest a solution technique for the mixed strategy NE including generation capacity constraints as two heuristic rules. It is reported in this paper that a role of the key player who controls congestion in a NE can be transferred to other player depending on the generation capacity of the key player. The suggested heuristic rules are verified to compute the mixed strategy NE with a consideration of generation capacity constraints, and the effect of the generation constraints on the mixed strategy NE is analyzed in simulations of IEEE 30 bus systems.

• Journal of Electrical Engineering and Technology
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• v.8 no.4
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• pp.675-685
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• 2013

Generation capacities and transmission line constraints in a competitive electricity market make it troublesome to compute Nash Equilibrium (NE) for analyzing participants' strategic generation quantities. The NE can cause a mixed strategy NE rather than a pure strategy NE resulting in a more complicated computation of NE, especially in a multiplayer game. A two-level hierarchical optimization problem is used to model competition among multiple participants. There are difficulties in using a mathematical programming approach to solve a mixed strategy NE. This paper presents heuristics applied to the mathematical programming method for dealing with the constraints on generation capacities and transmission line flows. A new formulation based on the heuristics is provided with a set of linear and nonlinear equations, and an algorithm is suggested for using the heuristics and the newly-formulated equations.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.59 no.7
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• pp.1226-1231
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• 2010

This paper looks at the influence of Financial Transmission Rights (FTRs) on the market value(Social Welfare; SW) in the competitive electricity market. The transmission line constraints make it difficult to compute the Nash Equilibrium (NE) due to causing a mixed strategy NE instead of a pure strategy NE. Computing a mixed strategy is more complicated in a multi-player game. The aim of this paper are to compute a mixed strategy NE and analyze SW in power transaction with FTRs. This paper introduces a formula and a technique for solving NE of multi-player game with FTRs. In addition, it analyzes the influence of holding of FTRs by generation company on SW and it proposes the SW at NE is influenced by Power Transfer Distribution Factor (PTDF) where holder of FTRs are located. The assertion is verified by calculating the mixed strategy utilizing the Cournot model widely used for studies on FTRs.

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.52 no.6
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• pp.347-353
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• 2003

In models of imperfect competition of deregulated electricity markets, the key task is to find the Nash equilibrium(NE). The approaches for finding the NE have had two major bottlenecks: computation of mixed strategy equilibrium and treatment of multi-player games. This paper proposes a payoff matrix approach that resolves these bottlenecks. The proposed method can efficiently find a mixed strategy equilibrium in a multi-player game. The formulation of the m condition for a three-player game is introduced and a basic computation scheme of solving nonlinear equalities and checking inequalities is proposed. In order to relieve the inevitable burden of searching the subspace of payoffs, several techniques are adopted in this paper. Two example application problems arising from electricity markets and involving a Cournot and a Bertrand model, respectively, are investigated for verifying the proposed method.

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.52 no.1
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• pp.62-67
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• 2003

An important aspect of the study of power system markets involves the assessment of strategic behavior of participants for maximizing their profits. In models of imperfect competition of a deregulated electricity system, the key task is to find the Nash equilibrium. In this paper, the bimatrix approach for finding Nash equilibria in electricity markets is investigated. This approach determines pure and mixed equilibria using the complementarity pivot algorithim. The mixed equilibrium in the matrix approach has the equal number of non-zero property. This property makes it difficult to reproduce a smooth continuous distribution for the mixed equilibrium. This paper proposes an algorithm for adjusting the quantization value of discretization to reconstruct a continuous distribution from a discrete one.