Medium-term hydrothermal coordination by semidefinite programming

Hydrothermal coordination (HTC) is a problem that has been solved using direct and decomposition solution methods. The latter has shown shorter solution times than the former. A direct solution method for the HTC problem that is based in semidefinite programming (SDP) is presented in this paper. SDP is a convex programming method with polynomial solution time. The variables of the problem are arranged in a vector, which is used to construct a positive-definite matrix; the optimal solution is then found in the cone defined by the set of positive-definite matrices. An HTC problem can be formulated as a convex optimization problem without explicitly stating the integer value requirements for the thermal-plants discrete variables. Thus, it is possible to replace the nonconvex integer-value constraints by convex quadratic constraints, and then use SDP. Due to its polynomial complexity, it is not necessary to use decomposition or other tools for discrete optimization, such as enumeration schemes or other exponential-time procedures. No initial relaxation is necessary when applying a SDP algorithm; the solution shows only minor mismatches in the integer variables, which are easily corrected by a heuristic method. Different size test cases are presented. The solution quality is assessed by comparing with that produced by a Lagrangian relaxation method.