Power Management in a Hydro-Thermal System under Uncertainty by Lagrangian Relaxation

Abstract

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.

A stabilized model and an efficient solution method for the yearly optimal power management

“…Model (15) can be obtained using a Value-at-Risk approach on the dual problem (11) in which τ and D are random (see [14] for more details). This observation allows us to choose κ(ε) = 1−ε ε as in [9].…”

“…In this case, when there is one hydro plant and an arbitrary number of thermal plants, the efficient solution method proposed in Section 6 can be used for computing the optimal value. When there is an arbitrary number of thermal, hydro and EJP units, a price decomposition technique (see [11], [14] for instance) can be useful. It amounts to solve the dual problem obtained when dualizing the demand satisfaction constraints.…”

“…Production management problems (and particularly power management problems) have been widely studied both concerning the modelling and the solution methods ( [5], [7], [8], [17], [12] [1], [11], [19], [16], [18]). Generally, the evolution of the uncertain parameters over the management period is modeled by a scenario tree and the goal is to minimize the expected production cost over this set of scenarios.…”

“…Generally, the evolution of the uncertain parameters over the management period is modeled by a scenario tree and the goal is to minimize the expected production cost over this set of scenarios. Recent solution methods (see [1], [11], [19] for instance), use Lagrangian relaxation and various nondifferentiable optimization methods and tools to solve the master problem.…”

Mean-risk optimization models for electricity portfolio management

“…Classical (stochastic and deterministic) optimization models for power production and trading are focused on (expected) profit maximization; see, e.g., [1], [2]. Risk management is usually considered as a separate task.…”