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Annals of Operations Research

, Volume 100, Issue 1–4, pp 251–272 | Cite as

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

  • Matthias P. Nowak
  • Werner Römisch
Article

Abstract

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or 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. It is assumed that the stochastic load process is given (or approximated) by a finite number of realizations (scenarios) in scenario tree form. Solving the dual by a 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 of 168 hours. The sizes of the corresponding optimization problems go up to 200 000 binary and 350 000 continuous variables, and more than 500 000 constraints.

multistage stochastic programming mixed-integer Lagrangian relaxation power management stochastic unit commitment 

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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Matthias P. Nowak
    • 1
  • Werner Römisch
    • 2
  1. 1.Institute of MathematicsHumboldt University BerlinBerlinGermany
  2. 2.Institute of MathematicsHumboldt University BerlinBerlinGermany

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