This paper aims to formulate the probabilistic ODE problem using CCP and suggests some ways of solving it approximately.

A natural way, although one leading to mathematical difficulties, consists in formulating the ODE problem using Chance-Constraint Programming (CCP). Hence the problem can be stated as a decision problem, wherein we must guarantee a positive power margin with a given probability level. In particular, the load charge curve, which is climate sensitive (temperature and nebulosity), the hydraulic-inflows of each reservoir, the power plant availability and the electricity prices. Many uncertainties strongly impact the electrical power industry and should be taken into account in our problem. In order to guarantee a positive power margin, defined as the difference between the offer and the demand, the ODE problem consists in taking optimal decisions such as hydro and thermal production levels, modifying maintenance schedules of thermal units, trading electricity on markets and exercising customer interruption options. In the context of electricity markets, the offer due to physical assets is extended by financial contracts and the Offer-Demand Equilibrium (ODE) problem can be stated as an extension of the traditional unit commitment problem. Introduction In the electrical power industry, whatever the time horizon, it is important to guarantee at each time step, the equilibrium between offer and demand and hence avoid shortage supply. A formal link between joint and individual chance constraints is exhibited and may lead to a simplified processing of the problem.

Second, we suggest some ways for approximately solving the obtained combinatorial chance-constrained program, which is in fact a stochastic multi-knapsack problem. The obtained problem matrix is sparse, large scale and with random coefficients describing underlying uncertainty factors affecting the available power of assets.

By introducing maximal production capacities, the computation becomes tractable for thermal plants but still leads to a combinatorial problem for hydraulic production. Computing the coefficients of the problem matrix is easy for financial assets, but a challenging task for physical assets. First, we state the prob-abilistic model. This paper deals with a Chance-Constrained Programming formulation and approximate resolution of an Offer-Demand Equilibrium problem in the context of electricity markets.

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