Testing the Local Stability of a Multi-Machine Power System with Constant Power Loads

With the widespread adoption of switched power converters within the power grid as both sources and sinks, the necessity arises for establishing verifiable stability conditions. We assume here that power sources can be modelled as synchronous generators - this encompasses virtual synchronous generators - while power sinks are operating as constant power loads. We provide a small-signal stability criterion for a model that uses time-varying phasors, characterized by a lossless transmission network and constant power loads distributed throughout the network. A key concept in this study is the matrix $H$, representing the partial derivatives of the generator powers with respect to their relative angles, computed at an equilibrium point. Our main result is that $H > 0$ implies the local asymptotic stability of this equilibrium point. Moreover, the equilibrium point is unstable if $H$ is full-rank and not positive definite. To demonstrate the practical significance of our theorem, we conduct a case study, highlighting its applicability in real-world scenarios.