Droop-Controlled Inverters are Kuramoto Oscillators

Motivated by the recent interest in smart grid technology and by the push towards distributed and renewable energy, we study the parallel operation of DC/AC inverters in a lossless microgrid. We show that the parallel interconnection of DC/AC inverters equipped with conventional droop controllers is precisely described by the Kuramoto model of coupled phase oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and exponentially stable. Remarkably, we find that the existence of such a synchronized solution does not depend on the selection of droop coefficients. We prove that the inverters share the network power demand in proportion to their power ratings if and only if the droop coefficients are selected proportionally, and we characterize the set of feasible loads which can be serviced. These results hold without assumptions on identical line characteristics or voltage magnitudes.