  This paper examines the dynamics of power-electronic inverters in islanded
microgrids that are controlled to emulate the dynamics of Van der Pol
oscillators. The general strategy of controlling inverters to emulate the
behavior of nonlinear oscillators presents a compelling time-domain alternative
to ubiquitous droop control methods which presume the existence of a
quasi-stationary sinusoidal steady state and operate on phasor quantities. We
present two main results in this work. First, by leveraging the method of
periodic averaging, we demonstrate that droop laws are intrinsically embedded
within a slower time scale in the nonlinear dynamics of Van der Pol
oscillators. Second, we establish the global convergence of amplitude and phase
dynamics in a resistive network interconnecting inverters controlled as Van der
Pol oscillators. Furthermore, under a set of non-restrictive decoupling
approximations, we derive sufficient conditions for local exponential stability
of desirable equilibria of the linearized amplitude and phase dynamics.
