Analysis of the Dynamics and Topology Dependencies of Small Perturbations in Electric Transmission Grids

Abstract

We study the dynamics of disturbances in unregulated electric transmission grids by adopting a Synchronous Motor model. We start our analysis with linearized system equations, which we represent as complex Fourier series to find their eigenmodes and eigenfrequencies. This reduces the problem to the diagonalization of a finite dimensional matrix, which depends on the stationary phase solutions of the grid, and is thereby inherently conditioned by the topology and the power distribution. This matrix is found to belong to the generalized Laplacian matrices, which relates the analysis of perturbation dynamics to a graph theory problem. We consider three networks: Small-world, Random and German transmission grid. We find that the density of eigenfrequencies highly depends on the topology. For the Random Network, it resembles the Marchenko-Pastur distribution. For the Small-world Network and the German transmission grid, we find strongly peaked densities with long tails. Moreover, we find that the algebraic connectivity of the generalized Laplacian matrix, determines the long-time transient behavior of perturbations, whereas the intensity of its eigenvector, the generalized Fiedler vector, discloses information of the modes localization. Finally, we find strong indications that the degree of localization tends to increase with a decrease of the network average clustering coefficient.