Shifted-Passivity Certification of Synchronous Generators with Rotor-Angle-Dependent Current Injection

This paper develops a local shifted-passivity certificate for a synchronous generator operating on a synchronous periodic orbit. The main difficulty is that the natural shifted energy balance contains an angle-frequency coupling term and a residual terminal-voltage coupling, which prevent a direct passivity inequality. We resolve these obstructions by augmenting the storage with an angle-frequency cross term and by introducing a rotor-angle-synchronized terminal current injection. The resulting algebraic conditions certify local strict dissipation of the augmented generator storage. As an application, we interconnect the certified generator models with dynamic transmission lines, shunt capacitors, and monotone static loads, and obtain a compositional sufficient condition for local asymptotic stability of the synchronous orbit.