Geometric optimal control for the second-order synchronous-machine model

Abstract

This paper studies the transient regulation problem for the classical second-order synchronous-machine infinite-bus model from the viewpoint of geometric optimal control. The swing dynamics are formulated as a nonlinear affine-in-control system on a cylindrical state space, where the rotor-angle dynamics and the speed deviation are coupled through a sinusoidal electrical power relation. After recalling the local linearization around a synchronous operating point, the discussion is connected with the classical equal-area criterion in order to interpret first-swing stability in geometric terms. The control problem is then reformulated on the cotangent bundle by means of Pontryagin's Maximum Principle, leading to a canonical Hamiltonian system whose extremals determine the optimal mechanical-power and damping-type control actions. A fault scenario is incorporated through a piecewise variation of the transferable power, and the resulting two-point boundary value problem is solved numerically by a collocation-based method. The simulations provide the power--angle curves, optimal control profiles, state responses, and phase portraits, showing how the Hamiltonian extremal steers the machine from a pre-fault operating regime toward a desired post-fault synchronous state while penalizing control effort and state deviations. In this way, the paper connects classical transient-stability concepts with a geometric optimal-control framework for the regulated swing equation.