This paper surveys in a tutorial fashion theoretical and applications aspects of the confirmation method for the solution of large scale system engineering problems. The continuation method is motivated and defined. The existence of the method is formulated in terms of degree theory a la Algebraic Topology. Examples are given throughout the paper and applications to engineering and economic problems are cited and described.

Continuation methods: Theory and applications

This paper surveys in a tutorial fashion theoretical and applications aspects of the continuation method for the solution of large scale system engineering problems. The continuation method is motivated and defined. The existence of the method is formulated in terms of degree theory a la Algebraic Topology. Examples are given throughout the paper and applications to engineering and economic problems are cited and described.

The continuation of the invariant subspaces algorithm combined with the projected Arnoldi method is presented to trace the critical (right-most or least damping ratio) eigenvalues for the power system stability analysis. A predictor-corrector method is applied to calculate the critical eigenvalues trajectories as power system parameters change. The method can handle eigenvalues with any multiplicity and very close eigenvalues during the tracing processes. The subspace dimension inflation and deflation is applied to deal with eigenvalue overlap. The subspace update is proposed by an efficient projected Arnoldi method that can utilize known information from the traced eigenvalues and subspaces

Critical Eigenvalues Tracing for Power System Analysis via Continuation of Invariant Subspaces and Projected Arnoldi Method

The continuation of invariant subspaces algorithm combined with projected Arnoldi method is presented to trace the critical (rightmost or least damping ratio) eigenvalues for the power system stability analysis. A predictor-corrector method is applied to calculate the critical eigenvalues trajectories as power system parameters change. The method can handle eigenvalues with any multiplicity and very close eigenvalues during the tracing processes. The subspace dimension inflation and deflation is applied to deal with eigenvalue overlap. The subspace update is proposed by an efficient projected Arnoldi method which can utilize known information from the traced eigenvalues and subspaces.

Three algorithms for the solution of the eigenvalue problem for a continuously parameterized family of sparse matrices are presented; a continuousLU (orLR) algorithm, a continuousQR algorithm, and a continuous Hessenberg algorithm Each of the three algorithms may be implemented recursively and the sparsity of the given matrices is preserved throughout the numerical process

K. S. Chao

Papers

Continuation algorithms for the eigenvalue problems