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We consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized. A general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy. By computing an inner approximation to future cost functions, we can evaluate an upper bound on the cost of an optimal policy, and an outer approximation delivers a lower bound. The approach we describe is particularly useful in sampling-based algorithms, and a numerical example is provided to show the e¢ cacy of the methodology when used in conjunction with stochastic dual dynamic programming.

This paper is concerned with tuning the Stochastic Dual Dynamic Programming algorithm to make it more computationally e¢ cient. We report the results of some computational experiments on a large-scale hydrothermal scheduling model developed for Brazil. We …nd that the best improvements in computation time are obtained from an implementation that increases the number of scenarios in the forward pass with each iteration and selects cuts to be included in the stage problems in each iteration. This gives an order of magnitude decrease in computation time with little change in solution quality.

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The long-term hydrothermal scheduling is one of the most important problems to be solved in the power systems area. This problem aims to obtain an optimal policy, under water (energy) resources uncertainty, for hydro and thermal plants over a multi-annual planning horizon. It is natural to model the problem as a multistage stochastic program, a class of models for which algorithms have been developed. The original stochastic process is represented by a finite scenario tree and, because of the large number of stages, a sampling-based method such as the Stochastic Dual Dynamic Programming (SDDP) algorithm is required. The purpose of this paper is two-fold. Firstly, we study the application of two alternative sampling strategies to the standard Monte Carlo-namely, Latin hypercube sampling and randomized quasi-Monte Carlo-for the generation of scenario trees, as well as for the sampling of scenarios that is part of the SDDP algorithm. Secondly, we discuss the formulation of stopping criteria for the optimization algorithm in terms of statistical hypothesis tests, which allows us to propose an alternative criterion that is more robust than that originally proposed for the SDDP. We test these ideas on a problem associated with the whole Brazilian power system, with a three-year planning horizon.

Abstract.We consider the optimal scheduling of hydropower plants in a hydrothermal interconnected system. This problem, of outmost importance for large-scale power systems with a high proportion of hydraulic generation, requires a detailed description of the so-called hydro unit production function. In our model, we relate the amount of generated hydropower to nonlinear tailrace levels; we also take into account hydraulic losses, turbine-generator efficiencies, as well as multiple 0-1 states associated with forbidden operation zones. Forbidden zones are crucial to avoid nasty phenomena such as mechanical vibrations in the turbine, cavitation, and low efficiency levels. The minimization of operating costs subject to such detailed constraints results in a large-scale mixed-integer nonlinear programming problem. By means of Lagrangian Relaxation, the original problem is split into a sequence of smaller and easy-to-solve subproblems, coordinated by a dual master program. In order to deal better with the combinatorial aspect introduced by the forbidden zones, we derive three different decomposition strategies, applicable to various configurations of hydro plants (with few or many units, which can be identical or different). We use a Sequential Quadratic Programming algorithm to solve nonlinear subproblems. We assess our approach on a real-life hydroelectric configuration extracted from the south sub region of the Brazilian hydrothermal power system.

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