Regions of Attraction Approximation Using Individual Invariance

Approximating regions of attraction in nonlinear systems require extensive computational and analytical efforts. In this paper, nonlinear vector fields are recasted as sum of vectors where each individual vector is used to construct an artificial system. The theoretical foundation is provided for a theorem in individual invariance to relate regions of attraction of artificial systems to the original vector field's region of attraction which leads to significant simplification in approximating regions of attraction. Several second order examples are used to demonstrate the effectiveness of this theorem. It is also proposed to use this theorem for the transient stability problem in power systems where an algorithm is presented to identify the critical clearing time through sequences of function evaluations. The algorithm is successfully applied on the 3-machine 9-bus system as well as the IEEE 39-bus New England system giving accurate and realistic estimations of the critical clearing time.