# Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming

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## Abstract

Nested Benders decomposition is a widely used and accepted solution methodology for multi-stage stochastic linear programming problems. Motivated by large-scale applications in the context of hydro-thermal scheduling, in 1991, Pereira and Pinto introduced a sampling-based variant of the Benders decomposition method, known as stochastic dual dynamic programming (SDDP). In this paper, we embed the SDDP algorithm into the scenario tree framework, essentially combining the nested Benders decomposition method on trees with the sampling procedure of SDDP. This allows for the incorporation of different types of uncertainties in multi-stage stochastic optimization while still maintaining an efficient solution algorithm. We provide an illustration of the applicability of our method towards a least-cost hydro-thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the Panama power system incorporating both electricity demand and inflow uncertainties.

## Keywords

Stochastic dual dynamic programming Hydro-thermal power system Nested Benders decomposition Sampling Scenario tree Electricity demand and inflow uncertainty## Mathematics Subject Classification

90C15 90C05 90C39 90C90## Notes

### Acknowledgments

The author thanks Mario Pereira (PSR) for his discussions on this research. He also thanks Panos M. Pardalos (University of Florida), David P. Morton (The University of Texas at Austin) and Bruno Flach (IBM) for their comments; Steven Frank, Timo Lohmann, Gregory Steeger (all Colorado School of Mines) and Josef Kallrath (BASF) for proofreading of the paper. The author also thanks the editor and the two reviewers for their thoughtful comments and suggestions.

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