Numerical Investigation on the Solution of Ill-Conditioned Load Flow Linear Equations
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Numerical Investigation on the Solution of Ill‑Conditioned Load Flow Linear Equations Matheus Maia Roque1 · Lucas Bertoldo Rezende1 · Alam Pablo Silva Belfort1 · José Eduardo Onoda Pessanha1 Received: 24 September 2018 / Revised: 10 January 2019 / Accepted: 7 March 2019 © Brazilian Society for Automatics–SBA 2019
Abstract Solving load flow problems via Krylov subspace iterative methods is not a new subject in the power system industry. The early methods worked with symmetric positive definite matrices only, but new ones have been developed for general-purpose matrices, such as the generalized minimum residual method (known by the acronym GMRES). When compared with direct methods, their implementation demands too much effort and their performance is unaffordable without proper preconditioning and other strategies. However, important numerical issues that may justify the low (or high) efficiency and robustness of such methods have been usually uncovered in the power systems literature. This paper investigates some of these issues based on a multiuse incomplete LU preconditioner (known by the acronym ILU) focusing on ill- and very ill-conditioned real power systems aiming to provide important insights into ILU-GMRES performance and into the dilemma: iterative versus direct methods. Keywords Iterative methods · Linear systems · Load flow · Sparse matrices
1 Introduction There are several papers focusing on the solution of load flow sublinear problems via Krylov subspace preconditioned iterative methods (Flueck and Chiang 1998; Alves et al. 1999; Kulkarni et al. 2001; de Leon and Semlyen 2002; Chaniotis and Pai 2003; Dag and Semlyen 2003; Chen and Shen 2006; Mori and Iizuka 2007; Xu and Li 2013; Eidiani 2011; Pessanha et al. 2013; Pessanha et al. 2011; Idema et al. 2012). Some of these papers have proposed preconditioners for parallel or distributed computational architectures, * Matheus Maia Roque [email protected] http://www.lqe.iee.ufma.br Lucas Bertoldo Rezende [email protected] Alam Pablo Silva Belfort [email protected] José Eduardo Onoda Pessanha [email protected] 1
Power Quality Laboratory, Department of Electrical Engineering, Federal University of Maranhão, Av. dos Portugueses, 1966 ‑ Vila Bacanga, São Luís, MA 65080‑805, Brazil
neglecting serial computation. Furthermore, just a few devote the necessary attention to efficiency (low CPU time and the number of linear iterations as well) and robustness (solution of ill- and very ill-conditioned real power systems), and no comparisons are made with any direct method to justify its application. Besides, efficiency and robustness are unaffordable without a proper preconditioner, and the addition of one or more strategies such as reordering, is, in general, necessary if an incomplete LU factorization preconditioner is used. With the purpose of providing fundamental perceptions into the application of iterative methods in the solution of load flow problems, this paper provides a detailed numerical analysis involving a specific iterative sol
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