Complementary limit induced bifurcation theorem and analysis of Q limits in power-flow studies

Complementary limit describes a change in system structure when two variables and constants in the system model exchange their status. Complementary limits have been used in the power system literature in the past for modeling the Q limit problem in power-flow equations. This paper presents a general theorem about static bifurications induced by complementary limits. A direct application of the theorem is a powerful characterization of the static bifurcations induced by Q-limits in any general large set of power-flow equations.Specifically, we prove that the Q-limit induced static bifurcation occurs whenever there is a change in the sign of the determinants of the respective power-flow Jacobian matrices when the critical generator bus changes from a PV bus to a PQ bus.The paper also discusses another application of complementary limits in power-flow equations for a system with phase shifting transformers.