## 5 Numerical Simulation  
The values of the control parameters are depicted in Table 2. First, the different controller design is performed according to the theoretical analysis provided in the previous section. In particular, using the bode plot shown in Figure 9, a crossover frequency \(f_{c}=1.8\) kHz at \(D=0.28\) was selected with a corresponding phase margin \(\varphi_{m}\) above 40\({}^{\circ}\) within the entire range of the operating duty cycle. For achieving these values, the gain, the poles and zeros of the type-III controller are selected at the values depicted in Table 2. The value of the amplitude of the ramp compensator is selected in such a way to avoid subharmonic oscillation within the entire range of the considered duty cycle values. The used PV generator consists of a string of four series-connected modules each one with a maximum power point voltage \(V_{\rm mpp}=38.4\) V, a maximum power current \(I_{\rm mpp}=9.13\) A and a maximum power 350 (4 \(\times\) 350 W= 1.4 kW) at standard test conditions. The manufacturer specifications for one single module are detailed in [32] and summarized in Table 3.  
A PV generator from the Renewable Energies library of PSIM\({}^{\copyright\copyright\) software was used as an input to the inverter. The PV generator block has two inputs one corresponds to irradiance \(S\) in W/m \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Current sensors gains \(R_{sL}\) and \(R_{sg}\) & 0.1 \(\Omega\) and 1 \(\Omega\) \\ Time constant \(\tau\) of the PV voltage PI controller & 0.0247 s \\ Proportional gain \(K_{p,p}\) of the PV voltage PI controller & 0.2 \\ Cut-off frequency of the PV voltage filter \(\omega_{c}/(2\pi)\) & 50 Hz \\ Zeros \(\omega_{z}/(2\pi)\) and poles \(\omega_{p}/(2\pi)\) of the type-III controller \(\omega_{z}/(2\pi)\) & 500 Hz and 50 kHz \\ Proportional gain \(K_{p,c}\) of the grid current type-III controller & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The used parameters for the different loop controllers of the DC–AC differential boost inverter. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Number of series-connected cells in a module & 72 \\ Open-circuit voltage of a module \(V_{\alpha c}\) & 46.5 V \\ Short-circuit current \(I_{sc}\) & 9.60 A \\ PV voltage at maximum power \(V_{\rm mpp}\) & 38.4 V \\ PV current at maximum power \(I_{\rm mpp}\) & 9.13 A \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of one single module forming the used PV panel. and another to temperature \(\Theta\) in \({}^{\circ}\)C. The temperature was maintained constant and the irradiance profile was defined by a piecewise linear signal generator which is connected to its corresponding PV generator input. The PV source is connected to a 50 Hz grid via the DC-AC differential boost inverter which converts the 154 V DC link PV voltage to 230 V RMS AC with power factor control. The MPPT was implemented in the inverter by means of a C block model using the P&O algorithm which provides the reference for the PV DC voltage. A PI regulator is used to control the PV voltage to its desired reference value. The design of the PV DC-link voltage controller was performed analytically based on the reduced-order model (31). Using this expression, the parameters of this controller have been selected to provide a settling time of one grid voltage cycle. The sampling period of the MPPT controller has been chosen to be equal to five times this settling time. The sampling period is \(T_{\rm mppt}=0.1\) s and the perturbation amplitude applied to \(V_{\rm mpp}\) is \(\Delta V_{\rm mpp}=4\) V.  
According to (36), with value of the extracted power and the grid frequency and the value of the DC-link capacitance \(C_{\rm dc}\), the AC-ripple of the PV voltage amplitude is about 7.5% of its nominal value 154 V at irradiance \(S=1000\) W/m\({}^{2}\) and temperature \(\Theta=25\)\({}^{\circ}\)C with a maximum power 1.4 kW.