Elizabeth L. Ratnam

Maryam Khodabakhshloo, Elizabeth L. Ratnam, Ian R. Petersen

This paper investigates equilibrium points and stability in two synchronous machine configurations: (i) a single generator with an impedance load and (ii) two interconnected machines with co-located loads. We consider both abc and dq reference frames to show that the equilibrium condition reduces to a cubic polynomial in the single-machine case and to an 18th- degree polynomial in the two-machine case. For the single-machine system, Lyapunov stability analysis and linearization based stability analysis are carried out. For the two-machine system, local stability is assessed through linearization and eigenvalue analysis. Illustrative examples confirm the existence of multiple equilibria and illustrate the impact of parameter variation on stability. Our results provide insight into the stability of synchronous machine systems.

Maryam Khodabakhshloo, Elizabeth L. Ratnam, Ian R. Petersen

The recent rapid proliferation of renewable energy is fundamentally changing the dynamic operations of power systems, necessitating new approaches to assess stability for these highly nonlinear systems. In this paper, we prove that synchronous machine systems, modeled in the nonlinear dq-frame, possess fundamental dissipativity properties. Specifically, we show passivity from current input to voltage output and a nonlinear negative imaginary property from torque input to rotor angle output. For the nonlinear system shifted around an equilibrium point, we derive explicit conditions for both passivity and the NI property to hold. Finally, we demonstrate that interconnection with passive droop controllers preserves these dissipativity properties with identical supply rates, thereby ensuring closed-loop stability.