SIAM Journal on Applied Mathematics, Volume 81, Issue 3, Page 1202-1225, January 2021.
We consider synchronization of the all-to-all power grid model with inertia and nonlinear terms of the first- and second-order derivatives. This model was derived from the energy conservation law in [G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Eur. Phys. J. B, 61 (2008), pp. 485--491]. Since the model contains a nonlinear differential operator, we cannot use a Lyapunov functional and an a priori estimate method directly. In this paper, we provide a new framework to obtain complete phase and frequency synchronization of the power grid model. We assume that the coupling strength and the moment of inertia are positive and the initial data are near the synchronization state. After applying an appropriate transform, we derive a system of differential equations whose coefficient matrix is negative definite. The initial data assumption and the negative definite coefficient matrix lead to the global existence and synchronization of the power grid model with a nonlinear differential operator. For both identical and nonidentical cases, the exponential decay for the frequency difference is determined and the square of frequencies converges to the average of the normalized energy feeding rates.

中文翻译：

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具有惯性和非线性导数的功率平衡系统中的同步

SIAM 应用数学杂志，第 81 卷，第 3 期，第 1202-1225 页，2021 年 1 月。
我们考虑具有一阶和二阶导数的惯性和非线性项的全面电网模型的同步。该模型源自[G. Filatrella、AH Nielsen 和 NF Pedersen，Eur。物理。J. B, 61 (2008), pp. 485--491]。由于模型包含非线性微分算子，我们不能直接使用李雅普诺夫泛函和先验估计方法。在本文中，我们提供了一个新的框架来获得电网模型的完整相位和频率同步。我们假设耦合强度和转动惯量为正，初始数据接近同步状态。应用适当的变换后，我们推导出系数矩阵为负定的微分方程组。初始数据假设和负定系数矩阵导致具有非线性微分算子的电网模型的全局存在和同步。对于相同和不相同的情况，确定频率差的指数衰减并且频率的平方收敛到归一化能量馈送率的平均值。