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We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.

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The computational time required by interior-point methods is often dominated by the solution of linear systems of equations. An efficient specialized interior-point algorithm for primal block-angular problems has been used to solve these systems by combining Cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking constraints. In some problems this power series preconditioner resulted to be inefficient on the last interior-point iterations, when the systems became ill-conditioned. In this work this approach is combined with a splitting preconditioner based on LU factorization, which is mainly appropriate for the last interior-point iterations. Computational results are provided for three classes of problems: multicommodity flows (oriented and nonoriented), minimum-distance controlled tabular adjustment for statistical data protection, and the minimum congestion problem. The results show that, in most cases, the hybrid preconditioner improves the performance and robustness of the interior-point solver. In particular, for some block-angular problems the solution time is reduced by a factor of 10.

The primal-dual and predictor-corrector versions of interior point methods are developed for an optimal DC power flow model where Kirchhoff law's are represented by a network flow model with surrogate constraints. The resulting matrix structure is explored reducing the linear system to be solved either to the number of buses or to the number of independent loops, leading to very fast iterations. Either matrix is invariant and can be factored off-line. As a consequence of such matrix manipulations, a linear system which changes at each iteration must be solved; its size, however, reduces to the number of generating units. Numerical results with C implementation are presented for IEEE test systems and large scale Brazilian systems. The interior point method shows to be robust, achieving fast convergence in all instances tested.KEYWORDS: Electrical networks, optimal power flow, interior point methods, quadratic programming, network flow models. RESUMOOs métodos de pontos interiores primal-dual e preditorcorretor são desenvolvidos para um modelo de fluxo de potênciaótimo DC onde as leis de Kirchhoff são repre- sentadas por um problema de fluxo em redes com restrições adicionais. A estrutura matricial resultanteé explorada reduzindo o sistema linear a ser resolvido a um sistema da dimensão do número de barras ou, opcionalmente, do número de laços independentes, cuja matrizé invariante ao longo das iterações permitindo que o mé-todo tenha uma iteração bastante rápida. Como conseqüência, um sistema linear cuja matriz varia a cada iteração deve ser resolvido. A dimensão deste sistema se reduz ao número de geradores. Resultados numé-ricos com implementação em C são apresentados para sistemas testes do IEEE e sistemas brasileiros de grande porte. O método de pontos interiores se mostra bastante robusto convergindo rapidamente para todos os casos testados. PALAVRAS-CHAVE:Redes elétricas, fluxo de potênciá otimo, métodos de pontos interiores, programação quadrática, fluxo em redes. INTRODUÇÃOO problema de fluxo de potênciaótimo tem aplicação em diversos problemas de análise e operação de sistemas de potência, tais como despacho econômico, aná-lise de confiabilidade de geração e transmissão, análise de segurança, planejamento da expansão da geração e transmissão, e programação da geraçãoà curto prazo. Na grande maioria dessas aplicações, a representação linearizada (DC) do fluxo de potência tem sido adotada