Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

TL;DR

This paper reveals that droop control laws are inherently embedded within the nonlinear dynamics of Van der Pol oscillators used in inverter control for microgrids, offering a new time-domain perspective.

Contribution

It demonstrates the intrinsic presence of droop laws in Van der Pol oscillator dynamics and proves global convergence and stability conditions for inverter networks.

Findings

Droop laws are embedded in Van der Pol oscillator dynamics.

Global convergence of amplitude and phase dynamics is established.

Conditions for local exponential stability are derived.

Abstract

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.