SINGULARLY PERTURBED HOPF BIFURCATION AND OSCILLATIONS IN POWER SYSTEMS

An electric power system normally functions at a stable operating point. However, the operating point can lose its stability due, say, to a disturbance and the subsequent change in system parameters. When this happens, it is not uncommon for the power system to exhibit oscillatory behavior. We use the Hopf bifurcation theorem to study this phenomenon, paying particular attention to the stability of the ensuing oscillations. We show that the classical swing equation for a power generator undergoes a Hopf bifurcation to periodic solutions if it is augmented to include any of the following effects: variable damping, frequency dependence of the electrical torque, lossy transmission line, or excitation control.; The case of a synchronous generator equipped with a fast excitation system motivates us to study Hopf bifurcation for singularly perturbed systems of ordinary differential equations. Using a result of the geometric theory of singular perturbations, we study the asymptotic behavior of the bifurcated periodic solutions of the perturbed system. A condition is given under which the stability of the bifurcated periodic solutions of the perturbed system can be determined from a computation on the unperturbed system. An extension is made to multiparameter singular perturbation problems, where the assumption of D-stability of the boundary layer system is used to study the stability of the periodic solutions. These results are then applied to the case of a synchronous generator with a fast excitation system.