A Two-Stage Planning Model for Power Scheduling in a Hydro-Thermal System Under Uncertainty

A two-stage stochastic programming model for the short- or mid-term cost-optimal electric power production planning is developed. We consider the power generation in a hydro-thermal generation system under uncertainty in demand (or load) and prices for fuel and delivery contracts. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints that couple power units. It is assumed that the stochastic load and price processes are given (or approximated) by a finite number of realizations (scenarios). Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the primal problem is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 400.000 binary and 650.000 continuous variables, and more than 1.300.000 constraints.