# A MIP framework for non-convex uniform price day-ahead electricity auctions

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## Abstract

It is well known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the non-convex, uniform-price Pan-European day-ahead electricity market “PCR” (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. Extending previous results, we propose a new primal-dual framework for these auctions, which has applications in both economic analysis and algorithm design. The contribution here is threefold. First, from the algorithmic point of view, we give a non-trivial exact (i.e., not approximate) linearization of a non-convex ‘minimum income condition’ that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables, and allowing us to solve market clearing instances involving most of the bidding products proposed in PCR using off-the-shelf MIP solvers. Second, from the economic analysis point of view, we give the first MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected block bids. We first show on a toy example that these two objectives are distinct from maximizing welfare. Third, we provide numerical experiments on realistic large-scale instances. They illustrate the efficiency of the approach, as well as the economics trade-offs that may occur in practice.

### Keywords

Day-ahead electricity market auctions Non-convexities Mixed integer programming Market coupling Equilibrium prices### Mathematics Subject Classification

90C11 90-08 90C06## Notes

### Acknowledgments

We greatly thank APX, BELPEX, EPEX SPOT, OMIE and N-Side for providing us with data used to generate realistic instances. We also thank organizers of the 11th International (IEEE) Conference on European Energy Market (Krakow, May 2014), as well as organizers of the COST Workshop on Mathematical Models and Methods for Energy Optimization (Budapest, Sept. 2014), for allowing us to present partial results developed here. This text presents research results of the P7/36 PAI project COMEX, part of the IPA Belgian Program. The work was also supported by EC-FP7 COST Action TD1207. The scientific responsibility is assumed by the authors.

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