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\section{Fixed points and synchonous state  (Deb)}\label{fixed-points-and-synchonous-state-deb} (Deb)}  Stable operation of the power grid is described by $\dot{\theta_i}=0$ in the Swing equation \eqref{eq:swing}  such that each node oscillates with the reference frequency $\Omega = 50$ Hz.  Thepower grid operates stably at 50Hz in the model this relates to  \omega =0 in the swing equation. The  fixed point then is given by  \omega=0 and certain angles \theta which depend depends  on the power network connectivities $K_{ij}$  and the coupling powers at each node $P_j$.   For example,  inthe network Example: Two nodes  \theta\emph{1-\theta}2=arcsin(\frac{P}{K}) Define the critical coupling  as the minimal K so that there exists  a fixed point (2 node diagram) (2  node phase plot) 2-node system with one generator and one consumer: