Assessment of Lagrangian Relaxation with Variable Splitting for Hydrothermal Scheduling

Lagrangian relaxation (LR) is one of the most widely used techniques to solve the short-term hydrothermal scheduling problem (HTS) for large power systems. The classical form of LR relaxes coupling constraints such as demand and reserve requirements for the whole mix. However, since with this form of relaxation the number of dual variables increases with the number of coupling constraints, the approach becomes inapplicable if a large electrical network is represented in the problem. To overcome this drawback, we apply a LR technique based on variable splitting for the HTS. More precisely, by duplicating variables corresponding to hydro and thermal generation, the relaxation yields three separable local problems - one for the hydro plants, one for the thermal plants, and one with constraints coupling all the plants. The resulting nondifferentiable dual problem is solved by a bundle method. An augmented Lagrangian technique gives a final feasible solution. We assess the performance of our approach in terms of accuracy of the lower bound obtained for the optimal cost and of convergence speed. We also analyze the quality of the pseudo- optimal primal point (corresponding to the dual solution), in relation to feasibility. Different configurations of the actual Brazilian power system were taken for the test cases. Our results show that the proposed approach yields near feasible primal points in reasonable computational time, and is suitable for combining unit commitment constraints with an accurate representation of the electrical network.