Inexactness of the Hydro-Thermal Coordination Semidefinite Relaxation

Hydro-thermal coordination is the problem of determining the optimal economic dispatch of hydro and thermal power plants over time. The physics of hydroelectricity generation is commonly simplified in the literature to account for its fundamentally nonlinear nature. Advances in convex relaxation theory have allowed the advent of Shor's semidefinite programming (SDP) relaxations of quadratic models of the problem. This paper shows how a recently published SDP relaxation is only exact if a very strict condition regarding turbine efficiency is observed, failing otherwise. It further proposes the use of a set of convex envelopes as a strategy to successfully obtain a stricter lower bound of the optimal solution. This strategy is combined with a standard iterative convex-concave procedure to recover a stationary point of the original non-convex problem.