Delocalization of disturbances and the stability of ac electricity grids.

In order to study how local disturbances affect the ac grid stability, we start from nonlinear power balance equations and map them to complex linear wave equations. Having obtained stationary solutions with phases φ_{i} at generator and consumer nodes i, we next study the dynamics of deviations. Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid. We find the parametric dependence of diffusion constant D. We apply the same solution strategy to general grid topologies and analyze their stability against local perturbations. The perturbation remains either localized or becomes delocalized, depending on grid topology, power capacity, and distribution of consumers and generator power P_{i}. Delocalization is found to increase the lifetime of perturbations and thereby their influence on grid stability, whereas localization results in an exponentially fast decay of perturbations at all grid sites. These results may therefore lead to new strategies to control the stability of electricity grids.