Jacobian-Free Poincaré-Krylov Method to Determine the Stability of Periodic Orbits of Electric Power Systems

A first application of a time domain Poincaré-Krylov approach for the study of the all-important stability of periodic solutions of nonlinear power networks is reported in this work. Whilst a Newton method solves the nonlinear algebraic equations resulting from the Poincaré map method, an iterative Krylov-subspace method based on a GMRES algorithm is applied for solving the Newton correction equations. More importantly, a QR factorization of the Hessenberg matrix involved in the GMRES least square problem is implemented using Givens rotations in order to avoid the linear growth of the computational complexity in large-scale power networks. Further, the stability of periodic solutions is determined by computing the Floquet multipliers using Ritz values and the Hessenberg matrix. Numerical tests carried out on a modified three-phase version of the IEEE 118-node system with a hydro-turbine synchronous generator and a grid-tied power converter demonstrate that speedup factors up to 8 are attainable with the Krylov-Subspace approach with respect to the standard Poincaré map method. An outstanding outcome of the comparative study is that the incorporation of QR factorization to the Hessenberg least square problem outperformed the classic periodic steady-state solvers, providing further computational savings up to 50%.