Iterative methods and parallel computation for power systems

Many power system problems such as the load flow problem, the transient stability analysis problem and the state estimation problem require as part of their analysis the solution of a large sparse set of linear equations. For such linear systems sparsity-oriented Gaussian elimination based direct solvers have been widely used. However, as the dimension of linear equations becomes larger direct methods become impractical even if sparsity-preserving techniques are used.; This thesis introduces iterative methods as alternatives to direct methods and studies parallel computation aspects of power system problems. Specifically, conjugate gradient type iterative methods are described in detail for power system problems that are symmetric and positive definite. The generalized minimum residual and similar iterative methods such as bi-conjugate gradient are briefly described for power system problems that are nonsymmetric. But the primary attention is given to the conjugate gradient method.; The success of iterative methods lies in the preconditioning phase. The difficulty associated with the forward and back substitution phase of preconditioned conjugate gradient method when used in parallel environments is overcome by the introduction of a blocked bordered diagonal ordering based ILU preconditioner.; A class of partitioned incomplete inverse preconditioners is designed to make the sequential nature of triangular solvers amenable to parallel processing. The computation of partitioned incomplete inverses, however, can become expensive depending on both the structure and the sparsity of the matrix. To alleviate this problem, a class of computation-free partitioned inverses are also introduced.; The incomplete Cholesky factorization of a symmetric and positive definite matrix is not necessarily positive definite. A topological incomplete factorization, denoted as the XD method, is proposed. The XD method guarantees the positive definiteness of the incomplete factors.; It is proven that a class of matrices arising from power flow problem using Fast Decoupled Load Flow and DC Load flow formulation are M-matrices whose incomplete factorization remains positive definite.