Explicit formulas for the asymptotic variance of a linearized model of a Gaussian disturbance driven power system with a uniform damping-inertia ratio

Serious fluctuations caused by disturbances may lead to instability of power systems. With the disturbance modeled by a Brownian motion process, the fluctuations are often described by the asymptotic variance at the invariant probability distribution of an associated Gaussian stochastic process. Here, we derive the explicit formula of the variance matrix for the system with a uniform damping-inertia ratio at all the nodes, which enables us to analyze the influences of the system parameters on the fluctuations and investigate the fluctuation propagation in the network. With application to systems with complete graphs and star graphs, it is found that the variance of the frequency at the disturbed node is significantly bigger than those at all the other nodes. It is also shown that adding new nodes may prevent the propagation of fluctuations from the disturbed node to all the others. Finally, it is proven theoretically that larger line capacities accelerate the propagation of the frequency fluctuation and larger inertia of synchronous machines help suppress the fluctuations of the phase differences, however, these acceleration and suppression are quite limited.