# Dual-Boost Inverter for PV Microinverter Application--An Assessment of Control Strategies
## 3 Control Strategies of the DBI
### Linear Control  
The linear control scheme is based on the approach presented in [7; 23], where the individual DC-DC converter control strategy is used. For this reason, the explanation of the structure control is presented only for one boost converter. Each boost converter is regulated by two cascaded control loops, as shown in Figure 6, which include compensations to achieve the variable operation of the inverter. The inner control loop regulates the inductor current, while the external control loop regulates the output capacitor voltage. Both control loops are composed by a proportional resonant (PR) controller since the reference of the capacitor voltage control loop is a sinusoidal waveform, and this type of controller tracks sinusoidal references with zero steady-state error [26]. Each stage of the control scheme is described as follows. _Output voltage reference calculation:_ The active and reactive power transfer between the DBI and grid can be calculated through the model of the grid connection of the inverter and the equivalent complex phasor [27; 28]. In this analysis, the resistance of the inductive filter is not considered in the equivalent circuit of the grid-connected inverter, as seen in Figure 7a. Thus, the voltage equation can be rewritten as,  
\[v_{\mathrm{s}}(t)=v_{\mathrm{c1}}-v_{\mathrm{c2}}-L_{\mathrm{s}}\frac{d\dot{ i}_{\mathrm{s}}(t)}{dt} \tag{11}\]  
Considering the complex phasor diagram shown in Figure 7b and small variations of the phase angle \(\delta\), the active and reactive power are calculated as  
\[P\approx \frac{V_{\mathrm{s}}V_{\mathrm{o}}}{2\omega L_{\mathrm{s}}}\delta \tag{12}\] \[Q\approx \frac{V_{\mathrm{s}}(V_{\mathrm{o}}-V_{\mathrm{s}})}{2\omega L_{ \mathrm{s}}} \tag{13}\]  
where \(V_{\mathrm{s}}\) is the amplitude of the grid voltage, \(V_{\mathrm{o}}\) is the amplitude of the inverter's output voltage, and \(\delta\) is the phase angle of the output voltage of the inverter regarding the grid voltage. Based on Equations (12) and (13), the active and reactive power are regulated through the amplitude of the inverter's output voltage (\(V_{o}\)) and its phase angle (\(\delta\)) [28]. Since the main goal of the inverter is to inject AC current into the grid in phase with  
Figure 6: Linear control scheme of the grid-connected DBI. the grid voltage, the active power reference is regulated. The reactive power is not considered (\(Q^{*}=0\)). Therefore, the capacitor voltage references can be calculated as  
\[v_{\text{cl}}^{*}= V_{\text{DC}}+\frac{V_{\text{s}}}{2}\sin\left(\omega_{\text{s}}t+ \frac{2P^{*}\omega_{\text{s}}L_{\text{s}}}{V_{\text{s}}V_{\text{o}}}\right) \tag{14}\] \[v_{\text{cl}}^{*}= V_{\text{DC}}-\frac{V_{\text{s}}}{2}\sin\left(\omega_{\text{s}}t+ \frac{2P^{*}\omega_{\text{s}}L_{\text{s}}}{V_{\text{s}}V_{\text{o}}}\right) \tag{15}\]  
where \(P^{*}\) is the active power reference obtained from the PI controller of the input voltage capacitor \(C_{\text{pv}}\). Furthermore, the DC voltage reference is calculated as  
\[V_{\text{DC}}=kv_{\text{pv}}+\frac{V_{\text{s}}}{2} \tag{16}\]  
where \(k\) is a boost factor (\(k>1\)), which ensures the minimum DC-bias (boost behavior), because the bottom waveform of the AC component must be greater than the input voltage in order to achieve the minimum elevation requirement. _Inner current control loop:_ The transfer function for the inner current control loop is based on Equation (1), with two considerations. The first consideration is that the capacitor's voltage \(v_{\text{cl}}\) in the variable operation is considered as a variable gain, because the inductor current loop bandwidth is higher than the capacitor's voltage loop bandwidth. Therefore, this voltage can be included as an inverse gain in the current loop, as shown in Figure 8a. The second consideration is that the PV voltage can be interpreted as a perturbation, which is compensated through a gain with the opposite value of the PV voltage.