Inertial power balance system with nonlinear time-derivatives and periodic natural frequencies

In this paper, we study the asymptotic behavior of the macroscopic power grid system derived from energy conservation. We assume that its coupling is all-to-all and that the energy supply and consumption are periodic. This system contains nonlinear time-derivative terms representing inertia and energy dissipation. We provide a sufficient condition for the existence of a special solution. This solution is a basin of attraction and its phase difference is periodic. To prove the existence of this solution, we employ a transformation to extract periodic variables. We consider nine functionals to show the asymptotic stability of the solution. A nine -dimensional vector whose components are the nine functionals satisfies a new nonlinear system. This property with the a priori estimate method yields the desired stability. We also obtain a sharp asymptotic formula for general solutions by combining this result and the existence of the solution with periodicity.

Automated summary

In this paper, the asymptotic behavior of a macroscopic power grid system derived from energy conservation is studied. A sufficient condition for the existence of a special solution as well as the stability of the solution are provided.