Power grid stability under perturbation of single nodes: Effects of heterogeneity and internal nodes

Non-linear equations describing the time evolution of frequencies and voltages in power grids exhibit fixed points of stable grid operation. The dynamical behaviour after perturbations around these fixed points can be used to characterise the stability of the grid. We investigate both probabilities of return to a fixed point and times needed for this return after perturbation of single nodes. Our analysis is based on an IEEE test grid and the second-order swing equations for voltage phase angles $\theta_j$ at nodes $j$ in the synchronous machine model. The perturbations cover all possible changes $\Delta\theta$ of voltage angles and a wide range of frequency deviations in a range $\Delta f=\pm1$~Hz around the common frequency $\omega=2\pi f=\dot\theta_j$ in a synchronous fixed point state. Extensive numerical calculations are carried out to determine, for all node pairs $(j,k)$, the return times $t_{jk}(\Delta\theta,\Delta \omega)$ of node $k$ after a perturbation of node $j$. We find that for strong perturbations of some nodes, the grid does not return to its synchronous state. If returning to the fixed point, the times needed for the return are strongly different for different disturbed nodes and can reach values up to 20 seconds and more. When homogenising transmission line and node properties, the grid always returns to a synchronous state for the considered perturbations, and the longest return times have a value of about 4 seconds for all nodes. The neglect of reactances between points of power generation (internal nodes) and injection (terminal nodes) leads to an underestimation of return probabilities.