# Dual-Boost Inverter for PV Microinverter Application--An Assessment of Control Strategies
## 2 Operation and Modeling of the DBI
### Individual Boost Converter Switching Strategy  
In this case, each boost converter is operated individually through switching signals \(S_{1}\) and \(S_{3}\) (\(S_{2}\) and \(S_{4}\) being their respective complementary). As summarized in Table 2, the four possible switching states result in eight conduction states depending on the grid current polarity.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \begin{tabular}{c} **Switching** \\ **State** \\ \end{tabular} & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) & \(S_{4}\) & \
\begin{tabular}{c} **Conduction** \\ **State** \\ \end{tabular} &
\begin{tabular}{c} **Output** \\ **Current** \\ \end{tabular} & **Inductor Currents** \\ \hline
1 & 0 & 1 & 1 & 0 & 1 & \(i_{\text{s}}>0\) & \(i_{\text{L1}}<0\), \(i_{\text{L2}}>0\) \\  & & & & & 2 & \(i_{\text{s}}<0\) & \\ \hline
2 & 0 & 1 & 0 & 1 & 3 & \(i_{\text{s}}>0\) & \(i_{\text{L1}}<0\), \(i_{\text{L2}}<0\) \\  & & & & & 4 & \(i_{\text{s}}<0\) & \\ \hline
3 & 1 & 0 & 1 & 0 & 5 & \(i_{\text{s}}>0\) & \(i_{\text{L1}}>0\), \(i_{\text{L2}}>0\) \\  & & & & & 6 & \(i_{\text{s}}<0\) & \\ \hline
4 & 1 & 0 & 0 & 1 & 7 & \(i_{\text{s}}>0\) & \(i_{\text{L1}}>0\), \(i_{\text{L2}}<0\) \\  & & & & & 8 & \(i_{\text{s}}<0\) & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dual-boost inverter switching and conduction states.  
Figure 3: Dual-boost inverter topology.  
Considering the operation of the inverter, the averaged model is described by:  
**Boost converter \(x\):**  
\[L_{\mathrm{x}}\frac{di_{\mathrm{Lx}}(t)}{dt} =v_{\mathrm{pv}}-v_{\mathrm{cx}}(t)(1-d_{x}(t)) \tag{1}\] \[C_{\mathrm{x}}\frac{dv_{\mathrm{cx}}(t)}{dt} =(1-d_{x}(t))i_{\mathrm{Lx}}(t)\mp i_{\mathrm{s}}(t) \tag{2}\]  
where \(x\) is 1 or 2, depending on boost converter 1 or 2, \(i_{\mathrm{Lx}}\) is the current through the inductor \(L_{\mathrm{x}}\), \(i_{\mathrm{s}}\) is the grid current, \(v_{\mathrm{cx}}\) is the voltage of the output capacitors, \(v_{\mathrm{pv}}\) is the input voltage, and \(d_{x}(t)\) is the duty cycle of boost converter x.  
Moreover, the DBI is integrated with the grid through an inductive filter \(L_{\mathrm{s}}\) (\(R_{\mathrm{s}}\) being its equivalent resistance), as shown in Figure 4. The differential voltage equation can be obtained by  
\[L_{\mathrm{s}}\frac{di_{\mathrm{s}}(t)}{dt}=v_{\mathrm{o}}(t)-i_{\mathrm{s}}(t )R_{\mathrm{s}}-v_{\mathrm{s}}(t) \tag{3}\]  
where the output voltage of the inverter is defined as  
\[v_{\mathrm{o}}(t)=v_{\mathrm{c1}}(t)-v_{\mathrm{c2}}(t) \tag{4}\]  
In general, the output voltage of the inverter is a sinusoidal waveform, which is generated through the difference between the voltage output of each boost converter. These outputs do not necessarily have to be sinusoidal, as long as the difference between them is a sinusoidal output.  
In this case, the output of each DC-DC converter is chosen as a sinusoidal output voltage with a DC-bias (\(V_{\mathrm{DC}}\) required for proper operation as boost), as shown below:  
\[v_{\mathrm{c1}}(t) =V_{\mathrm{DC}}+\frac{V_{\mathrm{s}}}{2}\sin(\omega_{\mathrm{s} }t+\delta) \tag{5}\] \[v_{\mathrm{c2}}(t) =V_{\mathrm{DC}}-\frac{V_{\mathrm{s}}}{2}\sin(\omega_{\mathrm{s} }t+\delta) \tag{6}\]