## 2 System Description and Its Mathematical Modeling
### Controller Design for the DC PV Voltage  
Once the grid current control is designed with the specified settling time, the DC PV voltage controller can be designed by assuming a perfect average grid current tracking and synchronization to the grid voltage. Under this condition, the grid current \(i_{g}\) can be substituted by its reference \(i_{g,\text{ref}}\) and the \(v_{\text{dc}}\)-to-\(i_{g,\text{ref}}\) transfer function can be obtained from the following reduced-order model  
\[\frac{d\overline{v}_{\text{dc}}}{dt}=\frac{1}{C_{\text{dc}}}\left(2I_{\text{ MPP}}-\frac{I_{\text{mpp}}}{V_{\text{mpp}}}\overline{v}_{\text{dc}}-i_{g, \text{ref}}\frac{2D-1}{(1-D)D}\right) \tag{28}\]  
Assuming a slowly varying irradiance \(S\) and temperature \(\Theta\), one has \(\hat{I}_{\text{MPP}}=0\) and \(\hat{V}_{\text{mpp}}=0\) and therefore by performing Laplace transformation, the following small-signal model in the \(s\)-domain is obtained  
\[\frac{\hat{V}_{\text{dc}}(s)}{\hat{I}_{g,\text{ref}}(s)}=-\frac{ 2D-1}{s+\frac{1}{C_{\text{dc}}(1-D)D}} \tag{29}\]  
Therefore, being a first order, without an excessive phase lag, a PI controller with a transfer function \(H_{\text{pi}}(s)=K_{p,\text{v}}(\tau s+1)/(s\tau)\) is enough for controlling the PV voltage. For a simplified design, the cut-off frequency \(\omega_{z,\text{pi}}=1/\tau\) of the PI compensator zero is selected to approximately cancel the pole of the transfer function (29). This cancellation cannot be exact for many reasons. First, the MPP voltage \(V_{\text{mpp}}\) and current \(I_{\text{mpp}}\) vary with weather conditions. Second, even with invariant weather  
Figure 9: Phase margin and crossover frequency of the loop gain frequency response for the differential boost inverter for \(S=1000\) W/m\({}^{2}\) (\(P_{\text{max}}\approx 1.4\) kW) and \(\Theta=25\)\({}^{\circ}\)C when the duty cycle is varied in the range \((0.28,0.72)\).  
conditions, the tolerance in the components will make the cancellation not perfect. By ignoring tolerances in the components and assuming that the MPP voltage \(V_{\rm mpp}\) does not significantly change with the weather conditions, an adaptive approach depending on the MPP extracted power and the resulting MPP current can be used for selecting the zero \(\omega_{z,\rm pi}\) of the PI controller to accurately cancel the pole of (29). Therefore, taking into account this approximate pole-zero cancellation, the loop gain corresponding to the DC PV voltage control can be approximated by the transfer function of an integrator as follows  
\[\frac{\hat{V}_{\rm dc}(s)}{\hat{I}_{g,\rm ref}(s)}H_{\rm pi}(s)=-\frac{K_{p,v} (2D-1)}{(1-D)D}\frac{1}{C_{\rm dc}s} \tag{30}\]  
In reality, the DC averaged PV voltage is not constant and has an AC component at the double frequency (\(2\omega_{g}\)) of the grid as will be explained in the next section. Therefore, a low pass filter with a cut-off frequency smaller than \(2\omega_{g}\) must be added either after the voltage sensor or at the output of the PI controller. Taking into account this filter, the resulting expression of the small-signal loop gain corresponding to the DC PV voltage control can be expressed as follows  
\[\frac{\hat{V}_{\rm dc}(s)}{\hat{I}_{g,\rm ref}(s)}H_{\rm pi}(s)\frac{2\omega_{ g}}{s(s+2\omega_{g})}=-\frac{K_{p,v}(2D-1)}{(1-D)D}\frac{2\omega_{g}}{C_{\rm dc }s(s+2\omega_{g})} \tag{31}\]  
The gain \(K_{p,v}\) can be adjusted for a suitable system response at a constant value of \(D\) corresponding to the maximum and minimum conversion gain. The system settling time corresponding to the DC PV voltage can be selected in the order of magnitude of some grid periods. For effective tracking of the maximum power, the MPPT perturbation period must be selected enough larger than the selected settling time.