Predispatch Linear System Solution With Preconditioned Iterative Methods
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Predispatch Linear System Solution With Preconditioned Iterative Methods Silvia M. S. Carvalho1
· Aurelio R. L. Oliveira2 · Mayk. V. Coelho3
Received: 22 May 2020 / Revised: 21 August 2020 / Accepted: 23 October 2020 © Brazilian Society for Automatics–SBA 2020
Abstract Primal-dual interior point method is used to minimize the predispatch generation costs and transmission losses on short-term operation planning of hydrothermal power systems with previously scheduled maneuvers and ramp rate constraints. Despite the efficiency shown by interior point methods for very large-scale problems, they generally perform only reasonably when applied to multiple-dimensional flow problems, the new specialized interior point algorithm performed here, overcomes this disadvantage. This specialization uses the preconditioned conjugated gradient method with an incomplete Cholesky factorization to solve a linear system in each iteration of the algorithm. The preconditioner developed exploring the structure of the problem is fundamental to ensure the efficiency of the method. Keywords Interior points methods · Preconditioners · Brazilian interconnected system · Predispatch · Maneuvers · Ramp rate constraints
1 Introduction Linear systems solver methods can be classified into direct methods that provide the exact solution for a certain system and iterative methods that generate a sequence of approximate solutions from an initial solution, based on preestablished tolerances (Campos 1995). The main problem in using direct methods for large-scale problems or problems that present poorly conditioned matrices lies in the fact that the latter become less efficient as the number of equations increases, due to the amount of operations that have to be performed in order to arrive at the solution of the linear system.
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Silvia M. S. Carvalho [email protected] Aurelio R. L. Oliveira [email protected] Mayk. V. Coelho [email protected]
1
Federal University of São Carlos - UFSCar, Rodovia João Leme dos Santos, Sorocaba, Brazil
2
University of Campinas - UNICAMP, Sérgio Buarque de Holanda-651, Campinas, Brazil
3
Federal University of Alfenas, Rua Gabriel Monteiro da Silva, 700, Alfenas, Brazil
Thus, iterative methods present a greater computational advantage; however, they are subject to failures directly related to the conditioning of the main system matrix. Therefore, in general, re-ordering and preconditioning processes are necessary to make iterative methods more efficient. An example of the most commonly used iterative methods in the solution of large linear equation systems is the Conjugate Gradient Method, considered very efficient for problems with a positive definite symmetric matrix (Golub and Van Loan 1996). One important line of research considers specific problem classes and explores the peculiarities of the matrix structure, in order to obtain even more efficient implementations (Carvalho et al. 1988; Oliveira et al. 1991; Carvalho and Oliveira 2015). This exploitation of the structure can lead to much diver
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