In this work, we explore a new approach to synchronization of coupledoscillators. In contrast to the celebrated Kuramoto model we do not work inpolar coordinates and do not consider oscillations of fixed magnitude. Wepropose a synchronizing feedback based on relative state information and localmeasurements that induces consensus-like dynamics. We show that, under a mildstability condition, the combination of the synchronizing feedback with adecentralized magnitude control law renders the oscillators' almost globallyasymptotically stable with respect to set-points for the phase shift,frequency, and magnitude. We apply these result to rigorously solve an openproblem in control of inverter-based AC power systems. In this context, theproposed control strategy can be implemented using purely local information,induces a grid-forming behavior, and ensures that a network of AC powerinverters is almost globally asymptotically stable with respect to apre-specified solution of the AC power-flow equations. Moreover, we show thatthe controller exhibits a droop-like behavior around the standard operatingpoint thus making it backward-compatible with the existing power systemoperation.
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