Linear Approximations Provide Rigorous Conditions for Phase Locking in Oscillator Networks and Power Grids

Synchronization is a fundamental collective phenomenon in many natural and engineered networks. The stable operation of oscillator networks, such as electric power grids, requires phase locking with phase differences across edges below a given threshold. A central question is whether a given network supports such a state. In this contribution, we extend a convex optimization formulation of the nonlinear fixed point equations of the Kuramoto model, that also describe lossless power grids. Based on this formulation, we derive rigorous and tight bounds on the error introduced by an arbitrary approximation of the original model, in particular linear approximations. The bounds yield edge-wise regions of trust around the approximation, within which the true nonlinear solution must lie. They further provide sufficient and necessary conditions for phase locking under prescribed phase difference limits. Numerical experiments show that the bounds are tight over a broad range of parameters.