## 2 Materials and Methods
### Model of the Injection System
#### 2.1.3 Dual-Buck Inverter Model  
The inverter's output is connected to the electrical grid. The inverter consists of two DC-AC buck converters (Figure 4) operating in one of the half cycles of the line-voltage AC \(V_{LN}\). The MOSFETs \(Q_{6}\), \(Q_{3}\), the diode \(D_{1}\), and the inductor \(L_{4}\) make up the first buck converter when \(V_{LN}\) is positive. The MOSFETs \(Q_{5}\) and \(Q_{4}\), the diode \(D_{2}\), and the inductor \(L_{3}\) form the second buck converter when \(V_{LN}\) is negative. The main inductors, \(L_{3}\) and \(L_{4}\), the capacitor \(C_{f}\), and a second pair of small inductors, \(L_{1}\) and \(L_{2}\), make up an LCL filter at the inverter's output. In each half cycle, the corresponding upper MOSFET, \(Q_{6}\) or \(Q_{5}\), remains fully on. The lower MOSFETs, \(Q_{3}\) and \(Q_{4}\), operate at a high PWM frequency (\(f=50\) kHz) when the buck converter changes in the respective half cycle [19]. Figure 5 summarizes the behavior of the PWMs.  
As only one buck converter is active at each half cycle of \(V_{LN}\), the model of the circuit can be simplified to that of an LCL filter (Figure 6).  
\(R_{i}\) and \(R_{g}\) are the internal resistances of the inductors (\(L_{i}=L_{3}=L_{4}\) and \(L_{g}=2L_{1}\)), and \(V_{in}\) is the steady-state voltage of the buck converter. Then, \(V_{i}=dV_{bus}\), with \(d\) as the duty cycle.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(L_{s}\) & 4 \(\mu\)H \\ \(L_{p}\) & 0.1 \(\mu\)H \\ \(L_{m}\) & 0.632 \(\mu\)H \\ \(C_{ac}\) & 9.4 \(\mu\)F \\ \(C_{out}\) & 101.5 \(\mu\)F \\ \(C_{2}\) & 1.5 \(\mu\)F \\ \(n\) & 5.4 \\ \(u_{2 ss}\) & 37.4 V \\ \(x_{1 ss}\) & 8.6 A \\ \(x_{2 ss}\) & 275 V \\ \(u_{1 ss}\) & 0.78 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the flyback.  
Figure 4: DC-AC circuit of the dual-buck inverter. First buck converter: \(Q_{6}\), \(Q_{3}\), \(D_{1}\), and \(L_{4}\). Second, buck converter: \(Q_{5}\), \(Q_{4}\), \(D_{2}\), and \(L_{3}\)[19].  
Applying [PERSON]'s voltage and current laws to the LCL filter, the following equations are obtained:  
\[V_{i}=L_{i}\frac{di_{L}}{dt}+R_{i}i_{L}+V_{\text{C}}, \tag{22}\]  
\[V_{\text{C}}=L_{g}\frac{di_{g}}{dt}+R_{g}i_{g}+V_{\text{LN}}, \tag{23}\]  
\[i_{L}=i_{c}+i_{g}, \tag{24}\]  
\[i_{c}=C_{f}\frac{dV_{\text{C}}}{dt}. \tag{25}\]  
Therefore, the dynamic behavior of the system is given by:  
\[\frac{di_{L}}{dt}=-\frac{R_{i}}{L_{i}}i_{L}-\frac{1}{L_{i}}V_{C}+\frac{1}{L_{i }}V_{i}, \tag{26}\]  
\[\frac{di_{g}}{dt}=-\frac{R_{g}}{L_{g}}i_{g}+\frac{1}{L_{g}}V_{\text{C}}-\frac{1 }{L_{g}}V_{\text{LN}}, \tag{27}\]  
Figure 5: Behavior of the PWMs for each half cycle \(V_{LN}\).  
Figure 6: LCL inverter filter.  
\[\frac{dV_{C}}{dt}=\frac{1}{C_{f}}i_{L}-\frac{1}{C_{f}}i_{g}. \tag{28}\]  
Using the following expression as state variables,  
\[x=\begin{bmatrix}i_{L}\\ i_{g}\\ V_{C}\end{bmatrix}, \tag{29}\]  
and using \(u=\begin{bmatrix}V_{i}&V_{LN}\end{bmatrix}^{T}\) as input variables, a linear representation of the system in state space is:  
\[\begin{split} A&=\begin{bmatrix}-\frac{R_{i}}{L_{i}}&0&- \frac{1}{L_{i}}\\ 0&-\frac{R_{g}}{L_{g}}&\frac{1}{L_{g}}\\ \frac{1}{Cf}&-\frac{1}{Cf}&0\end{bmatrix},B=\begin{bmatrix}\frac{1}{L_{i}}&0\\ 0&-\frac{1}{L_{g}}\\ 0&0\end{bmatrix},\\ C&=\begin{bmatrix}1&0&0\end{bmatrix},\quad D=\begin{bmatrix}0&0\end{bmatrix}.\end{split} \tag{30}\]  
Alternatively, by applying the Laplace transform to Equations (22)-(25), we can obtain the transfer function for \(i_{L}\), which is given by:  
\[G_{p}(s)=\frac{I_{L}(s)}{V_{i}(s)-V_{LN}(s)}=\frac{b_{2}s^{2}+b_{1}s+b_{0}}{a_ {3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}}, \tag{31}\]  
where  
\[\begin{split} b_{0}&=1,\quad b_{1}=C_{f}R_{g},\quad b_{2}=C _{f}L_{g},\\ a_{0}&=R_{g}+R_{i},\quad a_{1}=L_{g}+C_{f}R_{g}R_{i}+L_{i},\\ a_{2}&=C_{f}L_{g}R_{i}+C_{f}L_{i}R_{g},\quad a_{3}=C_{f}L_{g}L_{i}. \end{split} \tag{32}\]  
Table 2 shows the parameters used in the model.