Synchronization of cyclic power grids: equilibria and stability of the synchronous state

Abstract

Synchronization is essential for proper functioning of the power grid. We investigate the synchronous state and its stability for a network with a cyclic topology and with the evolution of the states satisfying the swing equations. We calculate the number of stable equilibria and investigate both the linear and nonlinear stability of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. The nonlinear stability, which we calculated by comparing the potential energy of the type-1 saddles with that of the stable synchronous state, depends on the network size ($N$) in a more complicated fashion. In particular we find that when the generators and consumers are evenly distributed in an alternating way, the energy barrier, preventing loss of synchronization approaches a constant value. For a heterogeneous distribution of generators and consumers, the energy barrier will decrease with $N$. The more heterogeneous the distribution is, the stronger the energy barrier depends on $N$. Finally, we found that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of $N\to\infty$. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.