Besides long-term bilateral contracts, a large part of the produ= ction of electricity is traded in day-ahead markets where prices and exchan= ges of energy are determined for each time slot of the following day, typic= ally an hour. Intraday and balancing markets are then meant to ensure secur= ity of supply and to balance positions taken in the day-ahead market which = could not be maintained.

In Europe, the past decade has seen the emergence of a Pan-European day-= ahead electricity market in the frame of the Price Coupling of Regions proj= ect (PCR), which is cleared using a common algorithm called Euphemia, handl= ing peculiarities of the different kinds of bidding products proposed by na= tional power exchanges. In a classical microeconomic setting, using supply = and demand bid curves submitted by participants, a (convex) optimization pr= oblem for which strong duality holds, and aimed at maximizing welfare, yiel= ds a market equilibrium. The optimal dual variables then correspond to equi= librium market prices: one for each time slot and each bidding zone.

These day-ahead markets are non-convex in the sense that participants ar= e allowed to describe operational constraints such as minimum power output = levels, and economic constraints such as start-up costs which must be recov= ered if a unit is started, rendering the primal welfare maximizing problem = non-convex, mainly due to the introduction of binary variables.

It can then easily be shown that most of the time, no market equilibrium= with uniform prices could exist, where the use of uniform prices means tha= t every bid of a given bidding zone and time slot is cleared at the same co= mmon market clearing price. The general approach throughout Europe is to us= e uniform prices, but to allow some non-convex bids to be paradoxically rej= ected in the sense that they would be profitable for the computed prices bu= t are none the less rejected, ensuring the existence of feasible solutions,= while enforcing all other market equilibrium conditions. This is classical= ly modelled as an MPEC, and handled by advanced branch-and-cut algorithms (= such as Euphemia), see [10, 8, 9].

**References**

[1] Bertocchi, M.I., Consigli, G., Dempster, M.A.H. (Eds.) : Stochastic = optimization methods in finance and energy, Springer, New York, 2011.

[2] Car=C3=B8e, C.C.; Schultz R.: A Two-Stage Stochastic Program for Uni= t Commitment Under Uncertainty in a Hydro-Thermal Power System, ZIB Preprin= t 98-11, Konrad-Zuse-Zentrum fur Informationstechnik (ZIB) Berlin, 1998.

[3] Frank, S., Steponavice, I., Rebennack, S.: A primer on optimal power= flow: A bibliographic survey - (i) formulations and deterministic methods,= Energy Systems 3(2012), 221-258.

[4] Frank, S., Steponavice, I., Rebennack, S.: A primer on optimal power= flow: A bibliographic survey - (ii) non-deterministic and hybrid methods, = Energy Systems 3(2012), 259-289.

[5] Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz, C.: Co= mplementarity Mdeli in Energy Markets, Springer, New York 2013.

[6] Lavaei, J., Low, S. H. : Zero duality gap in optimal power flow prob= lem, IEEE Transactions on Power Systems 27(2011), 92-107.

[7] Lee, J., Leung, J., Margot, F.: Min-up/min-down polytopes, Discrete = Optimization 1(2004), 7785.

[8] Madani, M., Van Vywe, M.: Computationally efficient mip formulation = and algorithms for european day-ahead electricity market auctions. European= Journal of Operational Research, 242(2):580-593, 2015.

[9] Madani, M., Van Vywe, M.: A mip framework for non-convex uniform pri= ce day-ahead electricity auctions. EURO Journal on Computational Optimizati= on, pages 1-22, 2015.

[10] Martin, A., Muller, J.C., Pokutta, S.: Strict linear prices in nonc= onvex european day-ahead electricity markets. Optimization Methods and Soft= ware, 29(1):189-221, 2014

[11] Morales-Espana, G., Gentile, C, Ramos, A.: Tight MIP formulations o= f the power-based unit commitment problem, OR Spectrum (to appear 2015).

[12] Sanchez-Martin, P., Ramos, A.: . Modeling transmission ohmic losses= in a stochastic bulk production cost model, Instituto de Investigacion Tec= nologica, Universidad Pontificia Comillas, Madrid, 1997.

[13] Schultz, R.: Stochastic programming with integer variables, Mathema= tical Programming 97(2003), 285-309.

[14] Sheble, G.B., Fahd, G.N.: Unit commitment literature synopsis, IEEE= Transactions on Power Systems 9(1994), 128-135.

[15] Takriti, S., Birge, J.R., Long, E.: A stochastic model for the unit= commitment problem, IEEE Transactions on Power Systems 11(1996), 1497-1508= .

**Contributors**

Dr Michael Diekerhof, RWTH Aachen University

Dr Mehdi Madani, Universit=C3=A9 catholique de Louvain