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.

This paper describes a method for optimal scheduling of hydropower systems for a profit maximizing, price-taking and risk neutral producer selling energy and capacity to separate and sequentially cleared markets. The method is based on a combination of stochastic dynamic programming (SDP) and stochastic dual dynamic programming (SDDP), and treats inflow to reservoirs and prices for energy and capacity as stochastic variables.The proposed method is applied in a case study for a Norwegian watercourse, quantifying the expected changes in schedules and water values when going from an energy-only market to a joint treatment of energy and reserve capacity markets.Arild Helseth (M'10) was born on Stord, Norway in 1977. He received the M.Sc. and Ph.D. degrees in electrical power engineering from the Norwegian University of Science and Technology. Currently he works at SINTEF Energy Research with hydro-thermal and hydropower scheduling models and methods.

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Abstract:The authors describe a method for long-term hydro-thermal scheduling allowing treatment of detailed large-scale hydro systems. Decisions for each week are determined by solving a two-stage stochastic linear programming problem considering uncertainty in weather and exogenous market prices. The overall scheduling problem is solved by embedding such two-stage problems in a rolling horizon simulator. The method is verified on data for the Nordic power system, studying the incremental changes in expected socio-economic surplus for expansions in both the transmission and generation systems. Comparisons are made with a widely used existing long-term hydro-thermal scheduling model. The results indicate that the model is well suited to valuate the flexibility of hydropower in systems with a high share of intermittent renewable generation. Nomenclature Index setsA set of price zones C o, t set of Benders cuts for scenario o and week t D a set of price-elastic demand steps in zone a G a set of thermal generators in zone a ℋ a set of hydropower modules in zone a K set of time steps within the week ℒ a set of interconnections connected to zone a ℳ set of exogenous markets N h set of efficiency-curve segments for module h P a set of pumps in zone a S set of N S scenarios S R reduced set of N R scenarios ω h

The authors analyse the operational profitability of a hydropower system selling both energy and reserve capacity in a competitive market setting. A mathematical model based on stochastic dynamic programming is used to compute the water values for the system considering different power plant configurations. The uncertainties in inflow and both energy and reserve capacity prices are considered through a discrete Markov chain. Subsequently, the system operation is simulated based on the obtained water values to assess system performance and expected revenues from the two markets. The model is applied in a case study for a Norwegian hydropower producer, showing how the power plant operation changes and profitability increases when considering sale of reserve capacity. The authors emphasise on how the water values are influenced by the opportunity to sell reserve capacity, and assess how the representation of non-convex relationships in the water value computations as well as simulation influence the profitability. P g max , P g min Max./Min. capacity, MW Q min minimum river flow, m 3 /s Q gm discharge in point m, m 3 /s T number of weeks in planning horizon V max , V min Max./Min. reservoir volume, Mm 3 V n reservoir volume at point n, Mm 3