Approximating power system dynamics and energy functions by quasi-gradient models

Power systems models that possess a well-defined Lyapunov function frequently share the following structure: the normal form of their system state equations may be expressed as a constant matrix multiplying the gradient of a scalar function of state. This structure has previously been labeled as quasi-gradient and the scalar function shown to be a natural Lyapunov function for the system. Unfortunately, as detailed load/network representations or control loops are added to the model, the quasi-gradient structure if often lost. An approach is proposed for defining energy functions for models lacking the quasi-gradient structure. A modified Lyapunov function is derived for the quasi-gradient with the property that its derivative along trajectories of the system is a strictly negative definite quadratic form. This in turn guarantees that the property of the Lyapunov function being nonincreasing along trajectories is preserved for certain perturbations in the vector field. In particular, this property is shown to be preserved under the condition that transfer conductances are perturbed within a sufficiently small ball about zero