Solving oligopolistic equilibrium problems with convex optimization Abstract   The approach of choice to analyze markets with oligopolistic competition has traditionally been complementarity modeling. In this paper we show that the majority of partial equilibrium models under imperfect competition in the (energy)economic literature can in fact be cast as optimization models, not requiring the derivation and implementation of KarushKuhnTucker conditions. This is achieved by adding appropriate terms accounting for market power exertion to the wellknown social welfare maximization objective. The method is applicable to both spatial Cournot oligopoly models and hybrid competition forms often implemented using conjectural variation approaches. We show how optimization and complementarity problems are equivalent, and provide a rationale for the terms accounting for market power exertion. Resulting models are solved orders of magnitude faster using offtheshelf optimization software, compared to solving complementarity problems. Large problem instances take minutes rather than hours, and one instance solves 640 times faster. The drastically reduced solution times greatly enhance modeling capabilities as they allow increased geographical scope and represent economic, technical and other characteristics in much more detail in equilibrium problems with imperfect competition. We present practical implications for the partial and multilevel equilibrium modeling community.

Energy Trading and Market Equilibrium in Integrated HeatPower Distribution Systems The increasing utilization of cogeneration plants and energy conversion facilities accelerates the integration of energy systems, which introduces both opportunities and challenges into system operation and management. Thorough mathematical model and analytical method which take into account the energy flow interdependence and strategic behaviors in multicarrier energy systems are highly desired. This paper studies an integrated heat power distribution system from the market perspective. The nonlinear AC optimal power flow (OPF) model is used to clear the distribution power market, which exactly accounts for network losses. To facilitate computation, convex relaxation is applied and polyhedral outer approximation of secondorder cone is performed, leading to a linear market clearing model. An optimal thermal flow (OTF) model is employed to clear the heat market. Energy consumptions of heating devices are charged according to locational marginal prices. The market equilibrium is interpreted by a fixed point between OPF and OTF and reformulated as a mixedinteger linear program. Dedicated optimization models are set forth to investigate the market impact of strategic providers and elastic demands, in which the market equilibrium condition serves as constraint. Case studies validate the effectiveness of the proposed model and method.