Particle Swarm Optimization for Computing Nash and Stackelberg Equilibria in Energy Markets

Interactions among stakeholders in deregulated markets lead to complex interdependent optimization problems. The present study is motivated by load control programs in energy markets and more precisely by using the power supply interruption as a tool for reducing consumers’ demand voluntarily, also known as voluntary load curtailment programs. The problem is formulated as a Stackelberg game, specifically, as a bilevel optimization problem that belongs to the mathematical programs with equilibrium constraints. In this game, a player that acts as leader determines the actions of the players that act as followers and play a Nash game among them through a subsidy program. The corresponding equilibria need to be found and the presence of nonconvex functions makes the use of metaheuristic algorithms attractive. An extension of particle swarm optimization is proposed for solving such problems based on the unified particle swarm optimization that is a variation of the plain particle swarm optimization algorithm. The proposed algorithm is tested by solving some examples of the formulated games in order to study its efficiency and the interactions between the stakeholders of the market.