## 2 Autonomous Microgrid Model  
The considered MG is shown in Figure 1. Three CPLs are fed from three inverter based DGs through coupling inductances, LC filters, and transmission lines. LC filters are imposed to filter out the high frequency switching noises. Current, voltage, and power controllers are employed to control the DGs. Emulating the operation of the synchronous generator, the droop controller illustrated in Figure 2 is designed to accurately share the DGs output powers, which depend mainly on the power angle and amplitude of the output voltage, respectively. The DGs output voltage and current (\(v_{o}\) and \(i_{o}\)) are measured firstly to obtain the instantaneous real and reactive powers (\(P_{m}\) and \(Q_{m}\)) as illustrated in (1). Secondly, the average real and reactive powers (\(P_{c}\) and \(Q_{c}\)) are obtained from the instantaneous powers using a low pass filter as in (2). Finally, the average powers are used to obtain the frequency (\(\omega\)) and d-axis reference voltage (\(v^{*}_{od}\)) as given in (3). \[P_{m}=v_{od}i_{od}+v_{od}i_{od},\;Q_{m}=v_{od}i_{od}-v_{od}i_{od} \tag{1}\]  
\[P_{c}=\frac{\omega_{c}}{\omega_{c}+s}P_{m},\;Q_{c}=\frac{\omega_{c}}{\omega_{ c}+s}Q_{m} \tag{2}\]  
\[ \begin{array}{l}\omega=\omega_{n}-m_{p}P_{c}\;,\quad\quad\overset{\bullet}{ \theta}=\omega\\ v^{*}_{od}=V_{n}-n_{q}Q_{c}\;,\quad v^{*}_{od}=0\end{array}  \tag{3}\]  
where \(i_{odq}\) and \(v_{odq}\) are the dq components of the DGs output current and voltage respectively, \(\omega_{c}\) is the cut-off frequency, \(\omega_{n}\) is the nominal angular frequency of DG, \(V_{n}\) is the nominal magnitude of the DG voltage, and \(m_{p}\) and \(n_{q}\) are the droop controller gains. trollers, coupling inductance, line and load currents, LC filter, and load voltages are given respectively in Equations (4)-(11). \[ \begin{array}{l}i_{ld}^{*}=F_{lod}-\omega_{n}C_{f}v_{0q}+K_{pv}(v_{od}^{*}-v_{ od})+K_{iv}\int{(v_{od}^{*}-v_{od})dt}\\ i_{lq}^{*}=F_{i0q}+\omega_{n}C_{f}v_{od}+K_{pv}(v_{od}^{*}-v_{od})+K_{iv}\int{(v_{ od}^{*}-v_{od})dt}\\ \end{array}  \tag{4}\]  
\[ \begin{array}{l}v_{id}^{*}=-\omega_{n}L_{f}i_{lq}+K_{pc}(i_{ld}^{*}-i_{ld})+ K_{ic}\int{(i_{ld}^{*}-i_{ld})dt}\\ v_{lq}^{*}=\omega_{n}L_{f}i_{ld}+K_{pc}(i_{lq}^{*}-i_{lg})+K_{ic}\int{(i_{lq}^{ *}-i_{lg})dt}\\ \end{array}  \tag{5}\]  
\[ \begin{array}{l}\frac{d{i_{ld}}}{dt}=-\frac{r_{f}}{L_{f}}i_{ld}+\omega i_{ lq}+\frac{1}{L_{f}}(v_{id}-v_{od})\\ \frac{d{i_{ld}}}{dt}=-\frac{r_{f}}{L_{f}}i_{ld}-\omega i_{ld}+\frac{1}{L_{f}}( v_{iq}-v_{qq})\\ \end{array}  \tag{6}\]  
\[ \begin{array}{l}\frac{d{v_{od}}}{dt}=\omega v_{0q}+\frac{1}{C_{f}}(i_{ld}-i_ {od})\\ \frac{d{v_{od}}}{dt}=-\omega v_{od}+\frac{1}{C_{f}}(i_{ld}-i_{qq})\\ \end{array}  \tag{7}\]  
\[ \begin{array}{l}\frac{d{i_{ld}}}{dt}=-\frac{r_{c}}{L_{c}}i_{od}+\omega i_{ oq}+\frac{1}{L_{c}}(v_{od}-v_{bd})\\ \frac{d{i_{dq}}}{dt}=-\frac{r_{c}}{L_{c}}i_{od}-\omega i_{od}+\frac{1}{L_{c}}( v_{od}-v_{blq})\\ \end{array}  \tag{8}\]  
\[ \begin{array}{l}\frac{d{i_{lu}}{D_{i}}}{dt}=-\frac{r_{luid}}{l_{luid}}i_{ lineD_{i}}+\omega i_{lineQt}+\frac{1}{l_{luid}}(v_{bDj}-v_{bDk})\\ \frac{d{i_{lu}}{D_{i}}}{dt}=-\frac{r_{luid}}{l_{luid}}i_{lineQi}-\omega i_{ lineQi}+\frac{1}{l_{luid}}(v_{bQj}-v_{bQk})\\ \end{array}  \tag{9}\]  
\[ \begin{array}{l}\frac{d{i_{lu}}{D_{i}}}{dt}=-\frac{R_{luid}}{l_{load}}i_{ lloadDi}+\omega i_{loadQi}+\frac{1}{l_{luid}}v_{bDi}\\ \frac{d{i_{lu}}{D_{i}}}{dt}=-\frac{R_{luid}}{l_{load}}i_{lloadDi}-\omega i_{ loadDi}+\frac{1}{l_{luid}}v_{bQi}\\ \end{array}  \tag{10}\]  
\[ \begin{array}{l}\frac{d{v_{bDj}}}{dt}=\omega v_{bQi}+\frac{1}{C_{f}}(i_{oDi }-i_{loadDi}\pm i_{lineD_{i}j})\\ \frac{d{v_{bDi}}}{dt}=-\omega v_{bDi}+\frac{1}{C_{f}}(i_{oQi}-i_{loadQi}\pm i_ {lineQi,j})\\ \end{array}