Almost global asymptotic stability of a grid-connected synchronous generator

TL;DR

This paper analyzes the global asymptotic stability of a simple grid-connected synchronous generator model, providing conditions for stability and convergence, with implications for designing synchronverters.

Contribution

It introduces easily checkable conditions ensuring almost global stability of a simplified SG model and develops an analytical framework using an integro-differential equation.

Findings

Existence of exactly two periodic trajectories, one stable and one unstable.

Almost all initial states lead to the stable periodic trajectory.

Application of the theory to a 500 kW synchronverter.

Abstract

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an "infinite bus", i.e. a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $2\pi$). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the {\em exact swing equation}, which resembles a forced pendulum equation and is equivalent to our fourth order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500 kW synchronverter.