Convergence characteristics of the variable projection method for mode extraction

This paper reviews the variable projection method (VPM) for power system modal analysis and analyzes the method's convergence characteristics. The purpose of the VPM and other modal analysis tools is to decompose time series data into damped or undamped sinusoidal components, which provide insights into the dynamics of a measured or simulated disturbance. The paper gives five example cases of modal analysis with VPM, varying in size from a single synthetic signal to a 30-signal, 7-mode data set from simulations of a large actual power system. The analysis provides an initial indication that the VPM often finds a solution close to the matrix pencil initialization, and that the method's convergence speed can vary. While the inner loop of the method, the line search, is robust and quick, the outer VPM loop sometimes converges linearly or slower, requiring hundreds of iterations. Simpler cases with fewer modes tend to have a more consistent convergence, and are less sensitive to the initial modes selected.