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Traditional synchronous generators with rotational inertia are being replacedby low-inertia renewable energy resources (RESs) in many power grids andoperational scenarios. Due to emerging market mechanisms, inherent variabilityof RESs, and existing control schemes, the resulting system inertia levels cannot only be low but also markedly time-varying. In this paper, we investigateperformance and stability of low-inertia power systems with stochastic systeminertia. In particular, we consider system dynamics modelled by a linearizedstochastic swing equation, where stochastic system inertia is regarded asmultiplicative noise. The $\mathcal{H}_2$ norm is used to quantify theperformance of the system in the presence of persistent disturbances ortransient faults. The performance metric can be computed by solving ageneralized Lyapunov equation, which has fundamentally differentcharacteristics from systems with only additive noise. For grids with uniforminertia and damping parameters, we derive closed-form expressions for the$\mathcal{H}_2$ norm of the proposed stochastic swing equation. The analysisgives insights into how the $\mathcal{H}_2$ norm of the stochastic swingequation depends on 1) network topology; 2) system parameters; and 3)distribution parameters of disturbances. A mean-square stability condition isalso derived. Numerical results provide additional insights for performance andstability of the stochastic swing equation.

A Performance and Stability Analysis of Low-inertia Power Grids with Stochastic System Inertia