Arnoldi algorithms with structured orthogonalization

TL;DR

This paper introduces a stability-preserving Arnoldi algorithm tailored for large-scale power network simulations, utilizing structured orthogonalization to improve convergence and accuracy in computing matrix exponential functions.

Contribution

The paper develops a modified Arnoldi algorithm with structured orthogonalization for DAE systems, providing theoretical convergence analysis and validation through RLC network simulations.

Findings

The algorithm maintains stability in Krylov subspace computations.

Numerical ranges of the Krylov operator are confined to the right half plane.

Simulations demonstrate improved accuracy and efficiency in power network analysis.

Abstract

We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDN), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution can be decomposed to a sum of two projections, one in the range of the system operator and the other in its null space. The range projection can be computed with one shift-and -invert Krylov subspace method. The other projection can be computed with the algebraic equations. Differing from the ordinary Arnoldi method, the orthogonality in the Krylov subspace is replaced with the semi-inner product induced by the positive semi-definite system operator. With proper adjustment, numerical ranges of the Krylov operator lie in the right half plane, and we obtain theoretical convergence analysis for the modified Arnoldi algorithm in computing phi-functions. Lastly, simulations on RLC networks are demonstrated to validate the effectiveness of the Arnoldi algorithm with structured-orthogonalization.