  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.
