Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators (Q2910918)

The authors aim to close a gap between the areas of power network synchronization, Kuramoto oscillators and consensus protocols. They consider a network-reduced power model, given by the swing equations NEWLINE\[NEWLINEM_i\ddot{\theta}_i + D_i\dot{\theta}_i = \omega_i - \sum_{j=1}^{n}a_{ij}\sin (\theta_i - \theta_j -\varphi_{ij}), \tag{1} NEWLINE\]NEWLINE where \(M_i, D_i >0\), \(\omega_i\in\mathbb{R}\), \(a_{ij}=a_{ji}\geq 0\) and \(\varphi_{ij}\in [0,\pi/2)\). Applying singular perturbation analysis (Tikhonov's method), they establish a relation between (1) and the corresponding nonuniform Kuramoto model NEWLINE\[NEWLINED_i\dot{\bar{\theta}}_i = \omega_i - \sum_{j=1}^{n}a_{ij}\sin (\bar{\theta}_i - \bar{\theta}_j -\varphi_{ij}). \tag{2}NEWLINE\]NEWLINE If (2) achieves synchronization and if \(\epsilon = M_i/D_i\) is sufficiently small, the differences in grounded coordinates stay small for all times, i.e., \((\theta_i(t)-\theta_n(t))=(\bar{\theta}_i - \bar{\theta}_n)+O(\epsilon)\) uniformly in \(t\geq 0\). By methods from consensus and synchronization theory, sufficient conditions for phase cohesiveness and frequency synchronization in the nonuniform Kuramoto model are derived, which then also apply to the power network model. They are paraphrased as: ``The network connectivity has to dominate the network's nonuniformity and the network's losses.''