Explicit formulas for the Variance of the State of a Linearized Power System driven by Gaussian stochastic disturbances

TL;DR

This paper derives explicit formulas for the covariance matrix in linearized power systems with Gaussian disturbances, revealing how network parameters influence fluctuations and identifying key nodes most affected by disturbances.

Contribution

It provides novel explicit formulas for the covariance matrix in power systems with uniform damping-inertia ratio, enhancing understanding of fluctuation behavior under stochastic disturbances.

Findings

Frequency variance at the disturbed node is significantly larger.

Adding nodes does not reduce variance at the disturbance source.

Line capacity and inertia influence fluctuations and phase differences.

Abstract

We look into the fluctuations caused by disturbances in power systems. In the linearized system of the power systems, the disturbance is modeled by a Brownian motion process, and the fluctuations are described by the covariance matrix of the associated stochastic process at the invariant probability distribution. We derive explicit formulas for the covariance matrix for the system with a uniform damping-inertia ratio. The variance of the frequency at the node with the disturbance is significantly bigger than the sum of those at all the other nodes, indicating the disturbance effects the node most, according to research on the variances in complete graphs and star graphs. Additionally, it is shown that adding new nodes typically does not aid in reducing the variations at the disturbance's source node. Finally, it is shown by the explicit formulas that the line capacity affect the variation of the frequency and the inertia affects the variance of the phase differences.