Lagrangian Relaxation for Continuous-Time Optimal Control of Coupled Hydrothermal Power Systems Including Storage Capacity and a Cascade of Hydropower Systems with Time Delays

Abstract

Due to the global large-scale development of renewable energy resources and the high uncertainty and variability of these sources, control optimization of energy supply systems has become crucial to use production facilities efficiently to balance the power demand. This work considers a setting characterized by a coupled power supply system controlled by a single provider and comprising a cascade of hydropower systems (dams), fossil fuel power stations, and a large battery for storage. We first propose a novel mathematical modeling framework for problems arising from short-term continuous control of large power systems, which are modeled through continuous differential equations. The proposed framework is original because previous works formulated the problem using a large discrete optimization system, which doesn't take advantage of the system structure and must solve NP-hard mixed-integer programming problems. Moreover, cascaded dams introduce time-delay effects to the controls, making it impossible to use dynamic programming algorithms. We design a primal-dual strategy by introducing a Lagrangian relaxation technique over continuous-time constraints to overcome this feature and construct a nearly optimal policy efficiently. This approach yields a convex nonsmooth optimization problem to define the optimal Lagrangian multipliers, which is numerically solved using a limited memory bundle method. Given the values of the Lagrangian multipliers, we solve an associated sequence of Hamilton-Jacobi-Bellman partial differential equations instead of the more computationally challenging task of solving a large nonconvex problem. We use a penalization technique for the admissible primal solution to smooth the controls while achieving a sufficiently small duality gap. The analysis and numerical results based on the Uruguayan power system demonstrate the efficiency of the proposed framework.