Limit-induced stable limit cycles in power systems

Heavily loaded power systems are susceptible to Hopf bifurcations, and consequent oscillatory instability. The onset of instability can be predicted by small disturbance (eigen-value) analysis, but the ensuring behaviour may depend strongly on nonlinearities within the system. In particular, physical limits place bounds on the divergent behaviour of states. This paper explores the situation where generator field-voltage limits capture behaviour, giving rise to a stable (though non-smooth) limit cycle. It is shown that shooting methods can be adapted to solve for such non-smooth limit-induced limit cycles. By continuing branches of limit-induced and smooth limit cycles, the paper established the co-existence of equilibria, smooth and non-smooth limit cycles. Furthermore, it is shown that when branches of limit-induced and smooth limit cycles merge, the limit cycles are annihilated at a grazing bifurcation.