Analysis of the dynamics and topology dependencies of small perturbations in electric transmission grids.

We study the phase dynamics in power grids in response to small disturbances and how this depends on the grid topology. To this end, we consider the swing equations in linear order in phase disturbances and solve the resulting linear wave equation, deriving the eigenmodes of the weighted graph Laplacian. A linear response expression for the deviation of frequency is given in terms of these eigenvalues and eigenvectors, which it is argued to be the basis for future power system stabilizers and other control measures in power systems. As an example, we present results for random networks based on the Watts-Strogatz model, where we observe a transition to localized eigenstates as the randomness in the degree distribution grows. Moreover, it is found that localization leads to faster decay rates. Thereby, disturbances are found to remain localized on a few nodes where they decay faster. Finally, we also consider the German transmission grid topology, where the eigenstate of the lowest eigenfrequency, the Fiedler vector, is found to be extended, with large intensities at the northwestern and southern boundaries.