Discretely constrained mixed complementary problems: Application and analysis of a stylised electricity market

Abstract

Recent research provides various methods to formulate and solve discretely constrained mixed complementary problems (DC-MCP) by relaxing complementarity. This paper provides insight into different areas of DC-MCPs. First, we look at three different solution methods for DC-MCPs from the literature and compare them in terms of solutions and usability. The methods discussed in this paper use disjunctive-constraints, SOS1 variables and an implementation of a certain median function. The methods are applied to a stylized electricity market including a minimum-generation constraint, making the problem a DC-MCP. Furthermore, the paper discusses the mathematical and economic implications of solutions. It is shown that a relaxed version of the discrete restrictions on variables combined with MCP conditions may not lead to an equilibrium where no player has a unilateral incentive to deviate. To overcome this problem, this paper presents a method to implement a two-stage DC-MCP to find solutions which are in line with the economic definition of an equilibrium.

Keywords:

• DC-MCP
• unit commitment
• nash-equilibrium
• welfare-maximisation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 GAMS Development Corporation. General Algebraic Modeling System (GAMS) Release 24.2.1. Washington, DC, USA, 2013. http://www.gams.com

2 IBM. ILOG CPLEX. http://www-01.ibm.com/software/integration/optimization/cplex

3 In case of multiple solutions $Q_u^{*}$ the previous solution must be part of that set.

4 105 divides the range of possible outputs of players well and covers all relevant points. The variables are bounded within a small range around these grid values.