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Augmented Lagrangean Relaxation and Decomposition Applied to the Short-Term Hydrothermal Coordination Problem

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Publication TypeConference Paper
Year of Publication1999
AuthorsBeltran, C.; Heredia, F. J.
Conference Name19th IFIP TC7 Conference on System Modelling and Optimization
Conference Date12-16/07/1999
Conference LocationCambridge, U.K.
Type of WorkContributed oral presentation
Key Wordsaugmented lagrangian relaxation; generalized unit commitment; block coordinated descent method; auxiliary principle problem; research
AbstractThe problem dealt with is called the Short-Term Hydrothermal Coordination (SHTC) problem. The objective of this problem is the optimization of electrical production and distribution, considering a short-term planning horizon (from one day to one week). Hydraulic and thermal plants must be coordinated in order to satisfy the customer demand of electricity at the minimum cost and with a reliable service. The model for the STHC problem presented here considers the thermal system, the hydraulic system and the distribution network. Nowadays the Lagrangean Relaxation (LR) method is the most widespread procedure to solve the STHC problem. The initial Classical Lagrangean Relaxation (CLR) method was improved by the Augmented Lagrangean Relaxation (ALR) method, although recent advances in the multiplier updating for the CLR method (cutting plane, bundle methods, etc.) have brought this classical method back into fashion. Two main advantages of the ALR method over the CLR method: (1) the ALR method allows us to obtain a saddle-point even in cases where the CLR method presents a duality gap. The solution of the STHC problem by the CLR method usually yields an infeasible primal solution $x_k$ due to the duality gap, whereas in the ALR method a solution of the dual problem provides a feasible primal solution. (2) The second advantage is that, using the CLR method, the differentiability of the dual function cannot be ensured and therefore nondifferentiable methods must be applied in the CLR method. This difficulty can be overcome if an augmented Lagrangean is used, since the dual function $q_c$ is differentiable for an appropriate c. Thus, the multipliers can be updated using ``large steps''. The main weakness of the ALR method is that the quadratic terms introduced by the augmented Lagrangean are not separable. If we want to solve the STHC problem by decomposition, some methods, such as the Auxiliary Problem Principle, or, as in our case, the Block Coordinate Descent method, must be used. However, the CLR method gives a separable Lagrangean. The starting point is the paper by Batut and Renaud [1] and therefore we use Variable Duplication plus the Augmented Lagrangean Relaxation (ALR) method. The method used by Batut and Renaud is improved theoretically and practically. From the theoretical point of view, the conservative Auxiliary Problem Principle is replaced by the Block Coordinate Descent Method that shows to be faster. From the practical point of view, an effective software package designed to solve the Optimum Short-Term Hydrothermal Coordination Problem, is incorporated in order to speed up the whole algorithm. Several medium to large scale instances of this problem have been solved showing the applicability of the proposed procedure.
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