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\section{Derivation of swing equation  (Xiaozhu)}  Energy conservation leads to swing equation \begin{align}  \forall\,\text{node}\,i, \hspace{30pt} P_i^{\,\text{source}} = P_i^{\,\text{diss.}} + P_i^{\,\text{acc.}} + \sum_jP_{ij}^{\,\text{trans.}}  \end{align}  \begin{itemize}  \item$P_i^{\,\text{source}}$: input mechanical power, generators \textquotedblleft $+$\textquotedblright, consumers \textquotedblleft $-$\textquotedblright\\  \item$P_i^{\,\text{diss.}}$: power dissipated through friction, $\kappa_i(\dot\phi_i)^2$\\  \item$P_i^{\,\text{acc.}}$: power accumulated in rotation, $\frac{d}{dt} E_k=\frac{1}{2}I_i\frac{d}{dt}(\dot\phi_i)^2$\\  \item$\displaystyle\sum_jP_{ij}^{\,\text{trans.}}$: the power transmitted between node $i$ and node $j$ \begin{align}  P_{ij}^{\,\text{trans.}}=P_{ij}=\dfrac{U^2}{X}\sin(\phi_i-\phi_j)=P_{ij}^{\,\text{max}}\sin(\phi_i-\phi_j)  \end{align}  \end{itemize}  \begin{align}  \text{with}\quad \phi_i=\Omega t+\theta_i\nonumber\hspace{4.8cm}  \end{align}    \begin{equation}  \ddot{\theta_i}=\underbrace{\dfrac{P_i^{\,\text{source}}-\kappa_i\Omega^2}{I_i\Omega^2}}_{P_i(+/-)}-\underbrace{\dfrac{2\kappa_i}{I_i}}_{\alpha_i}\dot\theta_i+\sum_j\underbrace{\dfrac{P_{ij}^{\,\text{max}}}{I_i\Omega}}_{\highlight{K_{ij}}}\sin(\theta_j-\theta_i)  \end{equation}