Unstable limit cycles in an electric power system and basin boundary of voltage collapse

It has been reported that a saddle node bifurcation or a blue sky bifurcation causes voltage collapse in an electric power system. In these references, computer simulations are carried out with the voltage magnitude of the generator bus terminal held constant. The generator model described by differential equations of internal flux linkages allows the voltage magnitude of the generator bus terminal to change. By using this model, we have carried out computer simulations of the power system to determine the cause of voltage collapse. It is a cyclic fold bifurcation of the stable limit cycle caused by an unstable limit cycle that leads to the voltage collapse. The involvement of complicated sequences of unstable limit cycles with cyclic fold bifurcations is confirmed, and the voltage collapse which is caused by perturbation for steady states is related to these unstable limit cycles. This is very interesting from the point of view of a nonlinear problem. From the point of view of a power system, the power system will fluctuate in practice even in normal operation, and may sometimes operate beyond the limit of its stability in recent year. It is very important in this situation that we clarify bifurcations of limit cycles on the power system.