  Motivated by the recent and growing interest in smart grid technology, we
study the operation of DC/AC inverters in an inductive microgrid. We show that
a network of loads and DC/AC inverters equipped with power-frequency droop
controllers can be cast as a Kuramoto model of phase-coupled oscillators. This
novel description, together with results from the theory of coupled
oscillators, allows us to characterize the behavior of the network of inverters
and loads. Specifically, we provide a necessary and sufficient condition for
the existence of a synchronized solution that is unique and locally
exponentially stable. We present a selection of controller gains leading to a
desirable sharing of power among the inverters, and specify the set of loads
which can be serviced without violating given actuation constraints. Moreover,
we propose a distributed integral controller based on averaging algorithms
which dynamically regulates the system frequency in the presence of a
time-varying load. Remarkably, this distributed-averaging integral controller
has the additional property that it maintains the power sharing properties of
the primary droop controller. Our results hold without assumptions on identical
line characteristics or voltage magnitudes.
