On the estimation of stability boundaries of nonlinear dynamic systems

The expansion methodology monotonically enlarges the estimated stability region of a general system so that the true stability boundary can be approached. It has advantages to reduce the conservativeness of the general system and to construct feasible algorithms for large-scale and complicated systems. Our paper provides fundamentals for the expansion theory of stability region estimation, locally and globally. It shows how the initial level set of energy function can be expanded via the diffeomorphism defined by the flow map so that the level set can approach to the exact boundary, even when the initial level set is not entirely within the true stability region. Such a generalization can broaden the valid scope to which the expansion approach can be applied. In addition, the rigorous topological characteristics of the determined local and global boundaries by such expansion are presented substantially. This provides a theoretical justification for the use of expansion schemes for those local estimation methods based on energy function, such as the Controlling Unstable Equilibrium Point (CUEP) method. The theoretical results also indicate that the estimation conservativeness of the conventional methods based on energy function can be significantly reduced. Finally, the proposed theory is tested on a simplified three-machine power system to verify its validation.