A Sparse Linear Systems Implementation for Electric Power Systems Solution Algorithms
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A Sparse Linear Systems Implementation for Electric Power Systems Solution Algorithms Leandro Ramos de Araujo · Débora Rosana Ribeiro Penido · José Luiz Rezende Pereira · Sandoval Carneiro Jr.
Received: 30 July 2012 / Revised: 12 November 2012 / Accepted: 21 December 2012 / Published online: 9 May 2013 © Brazilian Society for Automatics–SBA 2013
Abstract This paper presents a new object-oriented programming algorithm, which is applied to determine the minimum degree elimination ordering and to solve sparse large scale electrical network problems. The main contribution of the proposed algorithm lies in the procedure to cluster nodes having the same degree, at each step of the ordering scheme. Object-Oriented Programming (OOP) has been applied to insure a flexible and modular design. The proposed algorithm is compared to existing implementations in the literature and the results have shown its effectiveness. Keywords Ordering schemes · Sparse linear systems · Object-oriented programming · Electrical power systems · Large-scale power systems
1 Introduction Many engineering problems require the solution of sparse linear systems of equations (Pissanetzky 1984; Duff et al. 1986) that are characterized by coefficient matrices of very high dimensions, in which the majority of elements are null. Such is the case of electric power systems analysis algorithms (Penido et al. 2008; Penido et al. 2013; Schilling et al. 2009; Ciric et al. 2003), and many authors have addressed the problem of developing efficient sparse linear systems solvers (SparseLib++ 2012; SuperLU 2012; IMSL Numerical 2012; L. R. de Araujo (B) · D. R. R. Penido · J. L. R. Pereira Electrical Engineering Department, Federal University of Juiz de Fora, Juiz de Fora, Brazil e-mail: [email protected] S. Carneiro Jr. Electrical Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
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Ashcraft and Liu 2012; Pandit et al. 2001). Gupta and Muliadi (2001) have applied several software packages dedicated to linear systems solutions and have concluded that these solvers are very much application-dependent, which means that they can be very efficient for a particular problem, but may not be as efficient when solving for other types of problems. A major challenge in the design of efficient sparse systems routines is to avoid as much as possible the number of fillins in the factorization process (Duff et al. 1986). One of the first approaches to solve sparse asymmetric systems has been proposed in the 1950s by Markowitz (1957). Markowitz proposed to pre-determine the order of elimination in such way as to minimize the introduction of fill-in terms. Tinney and Walker (1967) introduced sparse-oriented programming techniques to solve large scale electric network problems. In particular, the well known Tinney-II ordering scheme that is widely used today in production grade programs. This method is also known as the minimum degree (MD) technique, and will be discussed in detail in this paper. The degree is defined for each diagonal term
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