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Hydropower producers rely on stochastic optimization when scheduling their resources over long periods of time. Due to its computational complexity, the optimization problem is normally cast as a stochastic linear program. In a future power market with more volatile power prices, it becomes increasingly important to capture parts of the hydropower operational characteristics that are not easily linearized, e.g. unit commitment and nonconvex generation curves.Stochastic dual dynamic programming (SDDP) is a stateof-the-art algorithm for long-and medium-term hydropower scheduling with a linear problem formulation. A recently proposed extension of the SDDP method known as stochastic dual dynamic integer programming (SDDiP) has proven convergence also in the nonconvex case. We apply the SDDiP algorithm to the medium-term hydropower scheduling (MTHS) problem and elaborate on how to incorporate stagewise dependent stochastic variables on the right-hand sides and the objective of the optimization problem. Finally, we demonstrate the capability of the SDDiP algorithm on a case study for a Norwegian hydropower producer.The case study demonstrates that it is possible but timeconsuming to solve the MTHS problem to optimality. However, the case study shows that a new type of cut, known as strengthened Benders cut, significantly contributes to closing the optimality gap compared to classical Benders cuts.

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Electric Power Generation Expansion Planning (GEP) is the problem of determining an optimal construction and generation plan of both new and existing electric power plants to meet future electricity demand. We consider a stochastic optimization approach for this capacity expansion problem under demand and fuel price uncertainty. In a two-stage stochastic optimization model for GEP, the capacity expansion plan for the entire planning horizon is decided prior to the uncertainty realized and hence allows no adaptivity to uncertainty evolution over time. In comparison, a multi-stage stochastic optimization model allows full adaptivity to the uncertainty evolution, but is extremely difficult to solve. To reconcile the trade-off between adaptivity and tractability, we propose a partially adaptive stochastic mixed integer optimization model in which the capacity expansion plan is fully adaptive to the uncertainty evolution up to a certain period and follows the two-stage approach thereafter. Any solution to the partially adaptive model is feasible to the multi-stage model, and we provide analytical bounds on the quality of such a solution. We propose an efficient algorithm that solves a sequence of partially adaptive models, to recursively construct an approximate solution to the multi-stage problem. We identify sufficient conditions under which this algorithm recovers an optimal solution to the multi-stage problem. Finally, we conduct extensive test of our algorithm on a realistic GEP problem. Experiments show that, within a reasonable computation time limit, the proposed algorithm produces a significantly better solution than solving the multi-stage model directly.

5We consider the economic lot-sizing game with general concave ordering cost functions. It is 6 well-known that the core of this game is nonempty when the inventory holding costs are linear. 7The main contribution of this work is a combinatorial, primal-dual algorithm that computes a cost 8 allocation in the core of these games in polynomial time. We also show that this algorithm can be 9 used to compute a cost allocation in the core of economic lot-sizing games with remanufacturing 10 under certain assumptions.