Stabilizing Phase-balanced or Phase-synchronized Trajectories of Van der Pol Oscillators in Uniform Electrical Networks

We study the dynamics of Van der Pol oscillators in a class of electrical networks with the goal of synthesizing feedback strategies to stabilize either phase-synchronized or phase-balanced motions. The electrical networks are composed of transmission lines with series R-L circuit models that are uniform, by which we mean that the R-to-L ratios of all lines are the same. The oscillators are coupled through linear filters on their output currents to nodes in the network. Our main results illustrate how the signs of the local feedback gains determine the stability of either phase-synchronized or phase-balanced trajectories. The results have implications on (and indeed, the paper is motivated by) decentralized control of power converters: synchronized solutions are of interest in parallel-connected dc-ac inverters and phase-balanced solutions are of interest in interleaving switching waveforms for multiphase dc-dc converters.