Talk Abstract:

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

Werner Römisch
Humboldt-Universität Berlin
Institut für Mathematik
D-10099 Berlin, Germany
romisch@mathematik.hu-berlin.de
http://www-iam.mathematik.hu-berlin.de/~romisch

 

Coauthors: N. Gröwe-Kuska, M.P. Nowak, and I. Wegner also of Humboldt-Universität Berlin, Institut für Mathematik, D-10099 Berlin, Germany.

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain load is developed. 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 coupling power units. The stochastic load process is approximated by a finite number of realizations (scenarios) in scenario tree form within three steps. First an approximation of the load process is developed by adapting a SARIMA-model to historical load data. Then empirical means and variances of the approximate load process are aggregated to an initial (binary) scenario tree which is finally reduced by a scenario deletion procedure based on a suitable probability distance. Solving the Lagrangian dual by a proximal bundle (subgradient) method leads to a successive decomposition into stochastic single (thermal or hydro) 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 first stage (primal) decisions starting from the optimal (stochastic) multipliers 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 from 7 to 9 days. The sizes of the corresponding optimization problems go up to 200.000 binary and 350.000 continuous variables, and more than 500.000 constraints.

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