A branch and bound methodology for matrix polytope stability problems arising in power systems

The authors propose a formulation representing a family of linearizations in a power system dynamic model as a polytope of matrices. Using this polytope structure, a simple sufficient condition for robust stability of the matrix polytope is developed. An optimization scheme is proposed to provide a sufficient condition for instability of the polytope. These two tests are combined in a branch-and-bound algorithm that divides the polytope into smaller and smaller sub-polytopes, repeatedly testing for instability or robust stability. The algorithm terminates when any sub-polytope proves unstable or when all sub-polytopes prove robustly stable