Analytical Result on Local Optimal Solutions and Its

This paper presents analytical results on local optimal solutions; in particular, it establishes a close relationship between a local optimal power flow solution and a Type-one unstable equilibrium manifold (UEM) of the associated nonlinear dynamical system. This paper also shows three possible situations where a Type-one UEM can link with two nearby stable equilibrium manifolds. If the Type-one UEM links with two nearby regular stable equilibrium manifolds (i.e., two feasible components), then there is a one-to-one correspondence between the constraint violations at the Type-one UEM and the constraint boundary points of one local optimal power flow solution located in one of the two feasible components. In addition, constraints violated by a Type-one UEM can be used to comprehensively explain why the feasible region is divided into multiple feasible components.