Stability and control of power grids with diluted network topology

We consider sparse random networks of Kuramoto phase oscillators with inertia in order to mimic and investigate the dynamics emerging in high-voltage power grids. The corresponding natural frequencies are assumed to be bimodally Gaussian distributed, thus modeling the distribution of both power generators and consumers, which must be in balance. Our main focus is on the theoretical analysis of the linear stability of the frequency-synchronized state, which is necessary for the stable operation of power grids and the control of unstable synchronous states. We demonstrate by numerical simulations that unstable frequency-synchronized states can be stabilized by feedback control. Further, we extend our study to include stochastic temporal power fluctuations and discuss the interplay of topological disorder and Gaussian white noise for various model configurations and finally demonstrate that our control scheme also works well under the influence of noise. Results for synthetic Erdös-Renyi random networks with low average connectivity and with symmetric or asymmetric bimodal frequency distributions are compared with those obtained by considering a real power grid topology, namely, the grid of Italy.