On Multivariable Cell Structures and Leonov Functions for Global Synchronization Analysis in Power Systems
Power systems are a paramount example for dynamical systems, which are periodic with respect to several state variables. This periodicity typically leads to the coexistence of multiple invariant solutions (equilibria or limit cycles). As a consequence, while there are many classical techniques for analysis of boundedness and stability of such systems, most of these only permit to establish local properties. To overcome this limitation, a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented. The proposed method is inspired by the cell structure approach developed by Leonov and Noldus in the 70s and characterized by two main advances. First, the conventional cell structure framework is extended to the case of dynamics, which are periodic with respect to multiple states. Second, by introducing the notion of a Leonov function the usual definiteness requirements of standard Lyapunov functions are relaxed to sign-indefinite functions. Subsequently, the proposed approach is applied to the problem of global synchronization in power systems. The analysis is illustrated via numerical examples.