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Due to the global large-scale development of renewable energy resources andthe high uncertainty and variability of these sources, control optimization ofenergy supply systems has become crucial to use production facilitiesefficiently to balance the power demand. This work considers a settingcharacterized by a coupled power supply system controlled by a single providerand comprising a cascade of hydropower systems (dams), fossil fuel powerstations, and a large battery for storage. We first propose a novelmathematical modeling framework for problems arising from short-term continuouscontrol of large power systems, which are modeled through continuousdifferential equations. The proposed framework is original because previousworks formulated the problem using a large discrete optimization system, whichdoesn't take advantage of the system structure and must solve NP-hardmixed-integer programming problems. Moreover, cascaded dams introducetime-delay effects to the controls, making it impossible to use dynamicprogramming algorithms. We design a primal-dual strategy by introducing aLagrangian relaxation technique over continuous-time constraints to overcomethis feature and construct a nearly optimal policy efficiently. This approachyields a convex nonsmooth optimization problem to define the optimal Lagrangianmultipliers, which is numerically solved using a limited memory bundle method.Given the values of the Lagrangian multipliers, we solve an associated sequenceof Hamilton-Jacobi-Bellman partial differential equations instead of the morecomputationally challenging task of solving a large nonconvex problem. We use apenalization technique for the admissible primal solution to smooth thecontrols while achieving a sufficiently small duality gap. The analysis andnumerical results based on the Uruguayan power system demonstrate theefficiency of the proposed framework.

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