  The energy transition towards an increased supply of renewable energy raises
concerns that existing electricity grids, built to connect few centralized
large power plants with consumers, may become more difficult to control and
stabilized with a rising number of decentralized small scale generators. Here,
we aim to study therefore, how local phase perturbations which may be caused by
local power fluctuations, affect the AC grid stability. To this end, we start
from nonlinear power balance equations and map them to complex linear wave
equations, yielding stationary solutions with phases $\varphi_i$ at generator
and consumer sites $i$. Next, we study deviations from these stationary
solutions. Starting with an initially localized perturbation, it is found to
spread in a periodic grid diffusively throughout the grid. We derive the
parametric dependence of diffusion constant $D$. We apply the same solution
strategy to general grid topologies and analyse their stability against local
perturbations. The perturbation remains either localized or becomes
delocalized, depending on grid topology, power capacity and distribution of
consumers and generators $P_i$. Delocalization is found to increase the
lifetime of perturbations and thereby their influence on grid stability, while
localization results in an exponentiallyfast decay of perturbations at all grid
sites. These results may therefore lead to new strategies to control the
stability of electricity grids.
