Analytical analysis of power network stability: necessary and sufficient conditions

The investigation of the synchronization of Kuramoto oscillators is a crucial applied model for studying harmonization in oscillating phenomena across physical, biological, and engineering networks. This chapter builds on previous studies by exploring the synchronization of Kuramoto oscillators while also conforming to more realistic models. Using the LaSalle Invariance Principle and contraction property, we introduce the necessary and sufficient conditions for frequency synchronization and phase cohesiveness. The novelty of this chapter’s contents lies in three key areas: First, we consider a heterogeneous second-order model with non-uniformity in coupling topology. Second, we apply a non-zero and non-uniform phase shift in coupling function. Third, we introduce a new Lyapunov-based stability analysis technique. Our findings demonstrate that heterogeneity in the network and the phase shift in the coupling function are key factors in network synchronization. We present the synchronization conditions based on network graph-theoretical characteristics and the oscillators’ parameters. Analysis of the results reveals that an increase in the phase shift and heterogeneity of oscillators will complicate the synchronization conditions. Numerical simulations confirm the validity of our theoretical results. One of the main applications of this study is the development of stability conditions for smart grids with Lossy-Power Network.

Mots clés

power network; smart grid; transmission line; transient stability; kuramoto oscillator; LaSalle's invariance principle; contraction property; synchronization; lyapunov function