# Dynamic Interactions between Parallel Grid-Forming Inverters in a Microgrid
## 3 Impedance Derivation  
In the impedance model derivation, the droop controls are ignored. The impedance is derived firstly by assuming that the virtual impedance unit and the feed-forward unit are not applied, and then by including the two units. Based on Figure 2, the \(LC\)-filter output can be modeled by a two-input, two-output model. This model has been derived in [22] by assuming the inverter output voltage and \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameters** & **Value** \\ \hline Rated power & \(S_{b}\) & 10 kW \\ \hline Rated voltage & \(V_{b}\) & 400 V \\ \hline \multirow{4}{*}{Converter filter} & \(R_{f,i}\) & 0.1 \(\Omega\) \\ \cline{2-3}  & \(L_{f,i}\) & 1.5 mH \\ \cline{2-3}  & \(C_{f,i}\) & 25 \(\mu\)F \\ \hline \multirow{2}{*}{Distribution feeder (\(Z_{l},i\))} & \(L_{l,i}\) & 0.45 mH \\ \cline{2-3}  & \(R/X\) ratio & 3 \\ \hline Local load (\(Z_{L}\)) & \(R_{load}\) & 80 \(\Omega\) \\ \hline Sampling period & \(T_{si}\) & \(10^{-4}\) \\ \hline Current control & \(K_{pc,i}\) & 5 \\ \hline \multirow{4}{*}{PR voltage control} & \(K_{pv,i}\) & 0.06 \\ \cline{2-3}  & \(K_{rv,i}\) & 10 \\ \cline{2-3}  & \(\omega_{c,i}\) & 8 rad/s \\ \cline{2-3}  & \(\omega_{o}\) & 120 \(\pi\) rad/s \\ \hline P-f droop & \(m_{i}\) & \(10^{-5}\) \\ \hline V-Q droop & \(n_{i}\) & \(10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 2: Control structure for the \(i\)-th \(VSC\) (\(i=1,2\)). the current flowing to the load as two inputs, while the converter current flowing to the point of the common coupling (PCC) bus and the PCC bus voltage are two outputs. \[ \begin{bmatrix}I_{L,i}(s)\\ V_{o,i}(s)\end{bmatrix} = \begin{bmatrix}G_{lo,i}(s)&Y_{L,i}(s)\\ -Z_{o,i}(s)&G_{Vo,i}(s)\end{bmatrix}\begin{bmatrix}I_{o,i}(s)\\ V_{PWM,i}(s)\end{bmatrix}  \tag{1}\]  
\[G_{lo,i}(s)=\frac{Z_{C_{f},i}(s)}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)},Y_{L,i}(s)= \frac{1}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)} \tag{2}\]  
\[Z_{o,i}(s)=\frac{Z_{L_{f},i}(s)Z_{C_{f},i}(s)}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)}, G_{Vo,i}(s)=\frac{Z_{C_{f},i}(s)}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)} \tag{3}\]  
where \(V_{PWM,i}(s)\) is the \(i\)-th inverter output voltage, and \(Z_{L_{f},i}(s)\) and \(Z_{C_{f},i}(s)\) are the impedance of the inductor and capacitor filter, respectively. Based on (1), the inner current control loop can be simplified in Figure 3.  
The inner loop dynamic can be obtained as follows:  
\[I_{L,i}(s)=G_{cl,i}(s)I_{L,i}^{s}(s)+G_{loc,i}(s)I_{o,i}(s) \tag{4}\]  
where \(G_{cl,i}(s)\) is the closed-loop transfer function of the current controller, and \(G_{loc,i}\) is the load current closed-loop transfer function. They are given as:  
\[G_{cl,i}(s)=\frac{T_{c,i}(s)}{1+T_{c,i}(s)},\ G_{loc,i}(s)=\frac{G_{lo,i}(s)}{ 1+T_{c,i}(s)} \tag{5}\]  
where the open-loop gain of the inner loop is denoted by \(T_{c,i}(s)\), and \(G_{del,i}(s)\) is the approximated switching delay:  
\[T_{c,i}(s)=K_{pc,i}G_{del,i}(s)Y_{L,i}(s), \tag{6}\]  
and  
\[G_{del,i}(s)=e^{-1.5T_{c}s}. \tag{7}\]  
The outer and the inner loop control can be simplified as in Figure 4. Therefore, the output voltage control can be described as follows  
\[V_{o,i}(s)=G_{clv,i}V_{o,i}^{s}(s)-Z_{ov,i}(s)I_{o,i}(s), \tag{8}\]  
\[G_{clv,i}(s)=\frac{T_{v,i}(s)}{1+T_{v,i}(s)},\ Z_{ov,i}(s)=\frac{Z_{C_{f},i}( s)}{1+T_{v,i}(s)}(1-G_{loc,i}(s)),\ T_{v,i}(s)=G_{v,i}(s)G_{cl,i}(s)Z_{C_{f},i}(s) \tag{9}\]  
\[Z_{o,i}(s)=\frac{Z_{L_{f},i}(s)Z_{C_{f},i}(s)}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)}, G_{Vo,i}(s)=\frac{Z_{C_{f},i}(s)}{Z_{L_{f},i}(s)+Z_{C_{f},i}(s)} \tag{3}\]0 Simplified structure for the inner current control. where \(G_{clv,i}(s)\) is the closed-loop transfer function of the outer-loop control, \(Z_{ov,i}(s)\) is the output impedance transfer function, and \(T_{v,i}(s)\) is open-loop gain of the outer-loop control.