A stability theorem for networks containing synchronous generators

Abstract

We revisit a by now well known stability result for power networks containing several synchronous generators, due to F. Dörfler and F. Bullo, SIAM Journal on Control and Optimization, 2012. The approximate (or reduced) model that we work with is called the network reduced power system (NRPS) model. The mentioned result from 2012 is that under certain conditions, the state trajectory of the NRPS model is close to the trajectory of the corresponding Kuramoto model (that is obtained by setting the inertias of the generators to zero), if both start from the same initial angles. The Kuramoto model is locally exponentially stable, so that this gives us useful indications on the behavior of the power network. We show that the mentioned result can be generalized and strengthened in several aspects. For instance, we show that the Kuramoto model has a unique asymptotically stable equilibrium point and we give an explicit characterization of its domain of attraction. We also characterize the region in the state space of the NRPS model where the asymptotic approximation by the state trajectories of the Kuramoto model is valid. We also significantly enlarge the class of NRPS models for which the result applies, and we prove the local exponential stability of the NRPS model for small enough inertia constants. We briefly indicate how this result can be useful for analyzing the stability of microgrids containing virtual synchronous machines.