A Riemannian subgradient algorithm for economic dispatch with valve-point effect

The economic load dispatch problem (ELDP) is a classical problem in the power systems community. It consists in the optimal scheduling of the output of power generating units to meet the required load demand subject to unit and system inequality and equality constraints. This optimization problem is challenging on three different levels: the geometry of its feasible set, the non-differentiability of its cost function and the multimodal aspect of its landscape. For this reason, ELDP has received much attention in the past few years and numerous derivative-free techniques have been proposed to tackle its multimodal and nondifferentiable characteristics. In this work we propose a different approach, exploiting the rich geometrical structure of the problem. We show that the (nonlinear) equality constraint can be handled in the framework of Riemannian manifolds and we develop a feasible (all iterates satisfy the constraints) subgradient descent algorithm to provide fast convergence to local minima. To this end, we show that Clarke's calculus can be used to compute a deterministic admissible descent direction by solving a simple, low-dimensional quadratic program. We test our approach on four real data sets. The proposed method provides fast local convergence and scales well with respect to the problem dimension. Finally, we show that the proposed algorithm, being a local optimization method, can be incorporated in existing heuristic techniques to provide a better exploration of the search space.