Stability and Convergency Exploration of Matrix Exponential Integration on Power Delivery Network Transient Simulation

Abstract

We propose a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDNs), which are formulated as semi-explicit differential-algebraic equations (DAEs). The matrix exponential and vector products (MEVPs) compose the solution of DAEs in multistep integration methods and can be efficiently approximated with the rational Krylov subspace. To produce stable simulation results for the ill-conditioned system from semi-explicit DAEs, the revised Arnoldi algorithm introduces a new structured orthogonalization process to construct the Krylov subspace. We demonstrate the performance of the new algorithm with theoretical proof and experiments. In the computation of MEVPs, we utilize the exponential related $\varphi $ functions to improve the numerical accuracy. We further explore the optimal ratio to confine the spectrum in the rational Krylov subspace. Finally, the transient framework is tested on a group of system-level PDNs, showing that matrix exponential-based algorithms could achieve high efficiency and accuracy.