augmented lagrangian relaxation

Generalized unit commitment by the radar multiplier method

Publication TypeThesis
Year of Publication2001
AuthorsCesar Beltran
Academic DepartmentDept. of Statistics and Operations Research. Prof. F.-Javier Heredia, advisor.
Number of Pages147
UniversityUniversitat Politècnica de Catalunya
DegreePhD Thesis
Key Wordsresearch; radar multiplier; generalised unit commitment; teaching
AbstractThis operations research thesis should be situated in the field of the power generation industry. The general objective of this work is to efficiently solve the Generalized Unit Commitment (GUC) problem by means of specialized software. The GUC problem generalizes the Unit Commitment (UC) problem by simultane-ously solving the associated Optimal Power Flow (OPF) problem. There are many approaches to solve the UC and OPF problems separately, but approaches to solve them jointly, i.e. to solve the GUC problem, are quite scarce. One of these GUC solving approaches is due to professors Batut and Renaud, whose methodology has been taken as a starting point for the methodology presented herein. This thesis report is structured as follows. Chapter 1 describes the state of the art of the UC and GUC problems. The formulation of the classical short-term power planning problems related to the GUC problem, namely the economic dispatching problem, the OPF problem, and the UC problem, are reviewed. Special attention is paid to the UC literature and to the traditional methods for solving the UC problem. In chapter 2 we extend the OPF model developed by professors Heredia and Nabona to obtain our GUC model. The variables used and the modelling of the thermal, hydraulic and transmission systems are introduced, as is the objective function. Chapter 3 deals with the Variable Duplication (VD) method, which is used to decompose the GUC problem as an alternative to the Classical Lagrangian Relaxation (CLR) method. Furthermore, in chapter 3 dual bounds provided by the VDmethod or by the CLR methods are theoretically compared. Throughout chapters 4, 5, and 6 our solution methodology, the Radar Multiplier (RM) method, is designed and tested. Three independent matters are studied: first, the auxiliary problem principle method, used by Batut and Renaud to treat the inseparable augmented Lagrangian, is compared with the block coordinate descent method from both theoretical and practical points of view. Second, the Radar Sub- gradient (RS) method, a new Lagrange multiplier updating method, is proposed and computationally compared with the classical subgradient method. And third, we study the local character of the optimizers computed by the Augmented Lagrangian Relaxation (ALR) method when solving the GUC problem. A heuristic to improve the local ALR optimizers is designed and tested. Chapter 7 is devoted to our computational implementation of the RM method, the MACH code. First, the design of MACH is reviewed brie y and then its performance is tested by solving real-life large-scale UC and GUC instances. Solutions computed using our VD formulation of the GUC problem are partially primal feasible since they do not necessarily fulfill the spinning reserve constraints. In chapter 8 we study how to modify this GUC formulation with the aim of obtaining full primal feasible solutions. A successful test based on a simple UC problem is reported. The conclusions, contributions of the thesis, and proposed further research can be found in chapter 9.
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Augmented Lagrangean Relaxation and Decomposition Applied to the Short-Term Hydrothermal Coordination Problem

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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Unit Commitment by Augmented Lagrangian Relaxation: Testing Two Decomposition Approaches

Publication TypeJournal Article
Year of Publication2002
AuthorsBeltran, C.; Heredia, F. J.
Journal TitleJournal of Optimization Theory and Applications
Pages295 - 314
Journal Date02/2002
PublisherSpringer Netherlands
Key Wordsaugmented lagrangian relaxation; generalized unit commitment; block coordinated descent method; auxiliary principle problem; research; paper
AbstractOne of the main drawbacks of the augmented Lagrangian relaxation method is that the quadratic term introduced by the augmented Lagrangian is not separable. We compare empirically and theoretically two methods designed to cope with the nonseparability of the Lagrangian function: the auxiliary problem principle method and the block coordinated descent method. Also, we use the so-called unit commitment problem to test both methods. The objective of the unit commitment problem is to optimize the electricity production and distribution, considering a short-term planning horizon.
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