Mixed-integer nonseparable piecewise linear models for the hydropower production function in the Unit Commitment problem

The hydro unit commitment (HUC) problem seeks to determine, for a short-term horizon with (semi)hourly discretization, the status (on/off) and generation level of each generating unit (GU) to meet plant and GUs constraints. In the HUC problem, the nonlinearities and non-convexities of the hydro production function (HPF), and the presence of binary variables that identify which GUs must be dispatched at each time step make the search for a solution challenging. Due to the recent developments in commercial mixed-integer linear programming (MILP) solvers, it is possible to approximate the nonlinear and nonconvex HPF through piecewise-linear (PWL) models with reasonable accuracy. Throughout this paper we emphasize the potential advantages of seven MILP formulations that can be categorized as parametric and non-parametric methods. Given the complexities of state-of-the-art solvers, it is hard to predict which formulation performs better. Although some guidelines can be found in literature, the formulation that performs best can be strongly dependent on the specific problem structure or data. In this context, we develop and compare seven multidimensional nonseparable PWL models for representing the HPF in the HUC problem. To assess the performance of each PWL model, we present results using a 6-GU hydro plant of two different types.