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

Abstract

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 better exploration of the search space.

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

“…In our implementation, we update the sequence of matrices {B k } by using the BFGS method; see [43]. Moreover, we compare the nonsmooth trust region algorithm with Riemannian subgradient descent algorithms presented in [15,25,21]. The number of function evaluations is used as a measure efficiency for the algorithms.…”

“…median calculation of positive semidefnite tensors), and low-rank optimization (matrix completion, collaborative filtering, source separation); see [32,52,53,51]. Furthermore, these algorithms have many applications in image processing, computer vision, nonsmooth constrained optimization problems on linear spaces; [5,15,21].…”

A new approach to solve Economic Dispatch problem using a Hybrid ACO–ABC–HS optimization algorithm

“…If the method is to be applied to some different power system, the network needs to be properly tuned by training which takes a long time. In [19], a Riemannian subgradient algorithm with valve point effect is considered. Han et al [20] have proposed two new solution methods for Dynamic Economic Dispatch.…”

A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds

“…Therefore, techniques of nonsmooth analysis have to be employed. Recently, some research has been started on nonsmooth optimization algorithms in a manifold setting; see [4,5,10,14,15,24] and references therein. In [4], a cyclic proximal point algorithm on a Hadamard manifold is employed to minimize a nonsmooth function in this setting.…”

“…In [4], a cyclic proximal point algorithm on a Hadamard manifold is employed to minimize a nonsmooth function in this setting. Riemannian versions of the subgradient method have been studied in [5,10]. In [14], a nonsmooth trust region method is generalized on Riemannian manifolds.…”