On synchronization of third-order power systems governed by second-order networked Kuramoto oscillators incorporating first-order magnitude dynamics (Q6584173)

The authors consider the system of \(N\) amplitude/phase oscillators described by \N\begin{align*} \Nm\ddot{\theta}_i+\dot{\theta}_i & = p_i+\frac{K}{N}\sum_{l=1}^N b_ib_l\sin{(\theta_l-\theta_i)} \\\N\dot{b}_i+kb_i & = \omega_i+\frac{K}{N}\sum_{l=1}^N b_l\cos{(\theta_l-\theta_i)} \N\end{align*} \Nwith either \(m=0\) or \(m\neq 0\), modelling power system dynamics. They define frequency synchronization as all angular velocities \(\{\dot{\theta}_i\}\) becoming equal as \(t\to\infty\). They provide sufficient conditions for frequency synchronization in both models. Roughly speaking, these conditions are that \(k>K\), the ratio of the diameter of natural frequencies to \(K\) is sufficiently small, and the initial oscillator configuration is confined to a sector with the arc length less than \(\pi/2\) and the initial amplitudes \(\{b_i(0)\}\) are positive. Numerical results are shown to demonstrate the theory.