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We p r e s e n t a dynamic multistage stochastic programming model for the cost-optimal generation of electric power in a hydro-thermal system under uncertainty i n load, in ow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixed-integer variables and individual power unit constraints, but relatively few coupling constraints. Hence we e m p l o y s t o c hastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagrangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal rst stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints.

Economic needs and the ongoing liberalization of European electricity markets stimulate the interest of power utilities in developing models and optimization techniques for the generation and trading of electric power under uncertainty. Utilities participating in deregulated markets observe increasing uncertainty in load (i.e., demand for electric power) and prices for fuel and electricity on spot and contract markets. The mismatched power between actual and predicted demand may be supplied by the power system or by trading activities. The competitive environment forces the utilities to rate alternatives within a few minutes.In this chapter, we describe a mathematical model for optimal short term operation and trading of a hydro-thermal based electric utility, which is usually called unit commitment problem because of the important role of the commitment or on/off decisions. Furthermore, we present a methodology for modelling the stochastic data process in form of a scenario tree and report on a Lagrangian-based decomposition strategy for solving the optimization model. We also provide some numerical experience obtained from test runs on realistic data from the German utility Vereinigte Energiewerke AG (VEAG). The optimization model has emerged from a collaboration with engineers of VEAG. For our tests we use a configuration of the VEAG system consisting of 25 (coal-fired, gas-burning) thermal units and seven pumped storage hydro units. Its total capacity is about 13,000 megawatts (MW), including a hydro capacity of 1,700 MW; the peak loads of the system are about 8,600 MW. In contrast to other hydro-thermal based utilities the amount of installed pumped storage capacity enables the inclusion of pumped storage plants into the optimization. It is an additional feature of the VEAG system that, for a weekly planning period, inflows to reservoirs are negligible.There is a growing number of contributions to stochastic power system optimization with emphasis on modelling aspects and solution methods. For stochastic models including commitment decisions see

We present a mathematical model with stochastic input data for mean-risk optimization of electricity portfolios containing several physical components and energy derivative products. The model is designed for the optimization horizon of one year in hourly discretization. The aim consists in maximizing the mean book value of the portfolio at the end of the optimization horizon and, at the same time, in minimizing the risk of the portfolio decisions. The risk is measured by the conditional value-at-risk and by some multiperiod extension of CVaR, respectively. We present numerical results for a large-scale realistic problem adapted to a municipal power utility and study the effects of varying weighting of risk