# Low-Loss Active Grid Impedance Cancellation in Grid-Connected Inverters with LCL Filter
## 3 System Control and Modelling with the Proposed Compensator
### Modeling Analysis of the Whole System  
By combining Equations (5) and (11), the equivalent circuit is obtained as shown in Figure 9a, in which  
\[G_{d}(s)=e^{-1.5 sT_{S}} \tag{12}\]Figure 9a can be further simplified into Figure 9b by combining the two separated voltage sources based on KVL, as described in Section 2. Based on Figure 9b, the transfer function of the plant from the inverter output voltage \(u_{i}\) to grid current \(i_{o}\) is derived. \[G_{vi}(s)=\frac{\Delta I_{o}(s)}{\Delta U_{i}(s)}=\frac{s\ b_{1}+1}{s^{3}\ a_{3}+s^{2}\ a_{2}+s\ a_{1}} \tag{13}\]  
where \(b_{1}=R_{d}\ C_{f}\ G_{d}(s)\), \(a_{1}=L_{1}+L_{2}+L_{g}\ [1-G_{d}(s)]\), \(a_{2}=R_{d}\ C_{f}\ d(s)\ \left\{L_{1}+L_{2}+L_{g}\ [1-G_{d}(s)]\right\}\), and \(a_{3}=L_{1}C_{f}\big{\{}L_{2}+L_{g}[1-G_{d}(s)]\big{\}}\). Thus, based on the parameters tabulated in Table 1, the corresponding frequency characteristics of the original LCL filter, and the reshaped one with and without the digital delay are compared and plotted in Figure 10. It indicates that the original filter resonant frequency varies widely with the change of grid impedance while the effect of grid impedance would be cancelled only if the digital delay is not considered. The negative virtual grid impedance, \(-Z_{g}G_{d}(s)\), with the digital delay, only counteracts the effect of grid impedance within the filter resonant frequency. It is noted that the damping behavior of the compensator with \(-Z_{g}G_{d}(s)\) in Figure 10 is brought in by the digital delay, which has been studied in reference [34]. \begin{table}
\begin{tabular}{c c c} \hline  & **System Parameter** & **Value** \\ \hline  & \(L_{1}\) & 3.8 mH \\ LCL filter & \(L_{2}\) & 1.2 mH \\  & \(C_{f}\) & 3.3 μF \\ \hline Filter resonant frequency & \(f_{f}\) & 2.9 kHz \\ Grid inductance & \(L_{g}\) & 0–0.1 pu \\ DC capacitor & \(C_{dc}\) & \(3\times 1000\) μF \\ \hline Switching frequency & \(f_{s\text{2-1}}\) (Inverter) & 20 kHz \\  & \(f_{s\text{2-2}}\) (Compensator) & 40 kHz \\ \hline DC bus voltage & \(u_{in}\) & 220 V \\ DC capacitor voltage & \(u_{dc}\) & 30 V \\ Grid voltage & \(u_{\text{pcc}}\) & 125 V \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the components used in the inverter prototype. Figure 9: Equivalent circuit of the grid impedance compensator with the effect of digital delay. (**a**) The equivalent physical circuit of the compensator. (**b**) The simplified circuit of Figure 9a  
To evaluate the system performance in the case of digital delay, a proportional-resonant-plus-harmonic-compensation (PR+HC)-based grid current control is adopted by the reshaped inverter (CASE II), and compared with the original one by using typical capacitor current-based active damping scheme with unit PCC VFFC (CASE I). The equation of PR+HC regulator is given as  
\[G_{\text{C}}(s)=K_{P}+\sum_{h=1,3,5,7,9}\left(K_{ih}\frac{s\;\omega_{B}}{s^{2}+ s\;\omega_{B}+\omega_{h}{}^{2}}\right) \tag{14}\]  
where \(K_{P}\) is the proportional gain of current regulation, and \(K_{ih}\) and \(\omega_{B}\) are the gain and band of individual resonant frequency \(\omega_{h}\). Thus, by combining Figure 8 with (13) and (14), the open loop transfer function of the reshaped inverter is obtained,  
\[T_{i}(s)=G_{vi}(s)\;G_{\text{C}}(s)\;G_{d}(s) \tag{15}\]  
where the controller parameters are chosen based on the design criterion in [36] and the virtual resistance is designed as \(R_{v}=30\;\Omega\). The open loop transfer function of the inverter by using the capacitor current-based active damping function with unit PCC VFFC (CASE I) can be found in reference [11]. Then, comparative frequency response of \(T_{i}(s)\) with and without the grid impedance compensator is plotted in Figure 11. As seen, without the compensator, although the cutoff frequency of the main inverter is sustained above 0.9 kHz, the phase margin decreases sharply from 60 deg to 16 deg, when the grid impedance increases to 0.1 pu.