Convex Hull Model for a Single-Unit Commitment Problem with Pumped Hydro Storage Unit

The single-unit commitment (1UC) aims at maximizing the payoff within a time series of given electricity prices. 1UC is subject to generation constraints, which is formulated as a mixed-integer programming (MIP) optimization problem. To reduce the computational complexity of PSU-1UC, this paper constructs the convex hull of PSU-1UC. Focusing on the possible combinations of generating and pumping time intervals, we first establish a dynamic program (DP) model which is solved in polynomial time. Second, a set of public variables were defined to describe the coupling relationship between different consecutive time intervals. Then, we reformulate the DP model into a linear programming (LP) model in a higher-dimensional space, which provides the convex hull formulation of PSU-1UC. Finally, we theoretically prove that the optimal solutions of the proposed convex hull model and the original MIP are the same. The exponential time complexity of PSU-1UC is reduced to polynomial time complexity, which will speed up the PSU self-scheduling/bidding problems and the decomposed algorithm for large-scale hydrothermal UC problems. Numerical experiments demonstrate the effectiveness and efficiency of the proposed model for PSU-1UC.