## 2 Command Feedforward Control for a GCI Current Control System  
Figure 1 shows a graphical representation of multiple DG systems interfaced with the main grid. As shown in Figure 1, the DG system consists of renewable energy generation system (solar power, wind power, etc. ), an energy storage system, LCL filters, and GCIs. An LCL filter is used to improve power quality. Time differential equations for the LCL filter and a grid voltage at the stationary reference frame (\(\alpha\) and \(\beta\) vector) are derived from Equations (1)-(3). \[\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{i}_{\mathrm{i}\alpha\beta}(\mathrm{s})= \frac{1}{L_{1}}(\mathrm{v}_{\mathrm{i}\alpha\beta}(\mathrm{s})-\mathrm{v}_{ \mathrm{cf}\alpha\beta}(\mathrm{s})) \tag{1}\]  
\[\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{v}_{\mathrm{cf}\alpha\beta}(\mathrm{s}) =(\frac{1}{C_{\mathrm{f}}}+\mathrm{R}_{\mathrm{d}}\frac{\mathrm{d}}{\mathrm{dt }})(\mathrm{i}_{\mathrm{i}\alpha\beta}(\mathrm{s})-\mathrm{i}_{\mathrm{g} \alpha\beta}(\mathrm{s})) \tag{2}\]  
\[\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{i}_{\mathrm{i}\alpha\beta}(\mathrm{s})= \frac{1}{L_{2}}(\mathrm{v}_{\mathrm{cf}\alpha\beta}(\mathrm{s})-\mathrm{v}_{ \mathrm{g}\alpha\beta}(\mathrm{s})) \tag{3}\]  
Variables used in (1)-(3) are defined in Table 1.  
A GCI is responsible for converting DC current coming from the battery (energy storage or PV panels, etc.) to AC current at the grid's side. A system's stable and consistent operation depends on the effectiveness of the grid current control in the CGI. Our proposed control algorithm controls the CGI's stationary reference frame current (AC), rather than the synchronous reference frame current (DC). The typical drawbacks of a controlled AC signal are steady state error and phase lag. To compensate for these drawbacks, command feedforward control (CFFC) was applied. In Figure 2, a system block diagram of a GCI current control system applying the CFFC is shown. For the purposes of this paper, the symbols \({}^{*}\) and \({}^{*}\) indicate command signal and estimated value, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{2}{c|}{**System Variables and Definition**} & \multicolumn{2}{c}{**System Variables and Definition**} \\ \hline L\({}_{1}\) & Inverter side LCL filter inductance & \(\mathrm{i}_{\mathrm{g}}\) & Grid current \\ \hline L\({}_{2}\) & Grid side LCL filter inductance & \(\mathrm{v}_{\mathrm{g}}\) & Grid voltage \\ \hline C\({}_{\mathrm{f}}\) & LCL filter capacitance & \(\mathrm{v}_{\mathrm{cf}}\) & Voltage across a filter capacitor \\ \hline R\({}_{\mathrm{d}}\) & Damping resistance & \(\mathrm{v}_{\mathrm{i}}\) & Output voltage from an inverter \\ \hline \end{tabular}
\end{table} 
Table 1: Definition of variables in a grid-connected inverter (GCI) current control system. Figure 1: Graphical representation multiple distributed generation (DG) systems interfaced with the main grid [23]. As seen in Figure 2, the physical system of the GCI current control system is composed of an LCL filter, grid voltage, and a PI controller used for stationary reference frame grid current control. A CFF controller is typically designed using an inverse model of the physical system. The inverse model of the LCL filter is simplified by neglecting \(\mathrm{R_{d}}\) and \(\mathrm{C_{f}}\) in developing the CFF controller. Using the CFFC developed with the simplified inverse model of the LCL filter, the transfer functions of a GCI current control system with and without applying the CFFC can be derived in Equations (4) and (5) respectively. In this paper, the symbol'indicates estimated value of the system parameter.