An Optimal Control Interpretation of Path Dependent Energy Integrals in Power System Stability Analysis

A long standing challenge in applying Lyapunov or energy function methods in power system stability studies arises when key components of the nonlinear state space model do not form an exact function. This is typical when the model includes realistic effects such as transmission line losses or voltage dependent active loads. When the function in question is exact, its associated potential function yields a natural choice for a system Lyapunov function. When this condition does not hold, a pragmatic approach in the literature has been to simply pick a particular path (often the state trajectory of the "faulted" power system), and integrate along this path. The "transient energy function" thus obtained is not a Lyapunov function, but has been successfully used to predict the boundary of the region of attraction in practical problems. This paper will show that the function obtained is closely related to the cost of control in an optimal control problem derived from the power system dynamics. This leads to the papers key observation: there exists a path of integration, uniquely defined by the endpoint, such that the resulting function of state does yield a true Lyapunov function.