## 2 Structure of the Grid-Connected PV System
### Boost Converter Design and Control  
MPPT schemes drive an operating point at maximum power peak to maximize the tracking and power efficiency of a PV system. The operating point is maintained close proximity to the GMPP by adjusting the duty cycle and switching behavior of the boost converter [28]. A continuous conduction mode (CCM) boost converter was designed \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models/Parameters** & **MPPs** & \(\mathbf{V_{oc}}\) (\(\mathbf{V}\)) & \(\mathbf{V_{mp}}\) (\(\mathbf{V}\)) & \(\mathbf{I_{cc}}\) (\(\mathbf{A}\)) & \(\mathbf{I_{mp}}\) (\(\mathbf{A}\)) & \(\mathbf{P_{mp}}\) (\(\mathbf{W}\)) \\ \hline \multirow{3}{*}{Simulation (Isoltech 1 STH-250-WH)} & MPP & 223.7 & 183.9 & 8.71 & 8.14 & 1497 \\  & GMPP\({}_{\text{r}}\) & 222.8 & 189.6 & 8.71 & 6.72 & 1275 \\  & GMPP\({}_{\text{m}}\) & 219.7 & 123.7 & 8.71 & 7.47 & 924.4 \\  & GMPP\({}_{\text{l}}\) & 215.9 & 59.98 & 8.71 & 8.13 & 487.64 \\ \hline \multirow{3}{*}{Experiment (Atlantis Energy System SS125 LM)} & MPP & 55.5 & 43.5 & 5.22 & 4.91 & 213.6 \\  & GMPP\({}_{\text{r}}\) & 54.96 & 47.03 & 5.22 & 3.05 & 143.6 \\ \cline{1-1}  & GMPP\({}_{\text{m}}\) & 53.85 & 30.71 & 5.22 & 2.53 & 77.72 \\ \cline{1-1}  & GMPP\({}_{\text{l}}\) & 52.99 & 13.18 & 5.22 & 4.88 & 64.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the PV array. Figure 4: Patterns for uniform and partial shading conditions are listed as: (**a–d**) Shading patterns for PV array of Isoltech 1 STH-250-WH; (**e–h**) Shading patterns for Atlantis Energy System SSI25 LM. because it has no power degradation in its circuit elements. The CCM converter boosts up the output voltage from the PV module voltage based on the duty cycle perturbation value. The duty cycle is fed by the pulse width modulation (PWM) comparator to generate pulses for switching a MOSFET or IGBT OFF and ON [29]. Equation (2) shows that the output voltage is larger than the input voltage whenever the duty cycle step size is regulated from 0.2 to 0.9 [30]. Figure 5a shows a simplified network of the boost converter. Figure 5b,c show the ON and OFF states of the boost converter configurations respectively. A state-space model is derived from the equivalent circuits of the converter to find a transfer function. The nonlinear averaged state-space equations are obtained by ignoring the terms formed by the product of two small signal quantities. The derived equations are Equations (2)-(5). \[\hat{\mathrm{x}}_{1}=-\ \frac{\mathrm{R}*\mathrm{R}_{\mathrm{c}}*\mathrm{D}}{ \mathrm{L}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\hat{\mathrm{x}}_{1}-\ \frac{\mathrm{R}*\mathrm{D}}{\mathrm{L}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\hat{\mathrm{x}}_{2}+\left[\frac{\mathrm{R}* \mathrm{R}_{\mathrm{c}}}{\mathrm{L}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\ \mathrm{x}_{10}\ +\ \frac{\mathrm{R}}{\mathrm{L}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\ \mathrm{x}_{20}\right] \tag{2}\]  
\[\hat{\mathrm{x}}_{2}=\frac{\mathrm{R}*\mathrm{D}}{\mathrm{C}\left(\mathrm{R}+ \ \mathrm{R}_{\mathrm{c}}\right)}\hat{\mathrm{x}}_{1}\ -\ \frac{\mathrm{x}_{2}}{\mathrm{C}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}-\ \frac{\mathrm{R}}{\mathrm{C}\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\ \mathrm{x}_{10}\ \hat{\mathrm{d}} \tag{3}\]  
\[\mathrm{Y}=\frac{\mathrm{R}\ *\hat{\mathrm{x}}_{2}}{\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\ +\ \frac{\mathrm{R}*\mathrm{R}_{\mathrm{c}}*\mathrm{D}}{\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\hat{\mathrm{x}}_{1}-\ \frac{\mathrm{R}*\mathrm{R}_{\mathrm{c}}}{\left(\mathrm{R}+\ \mathrm{R}_{\mathrm{c}}\right)}\ \mathrm{x}_{10}\ \hat{\mathrm{d}} \tag{4}\]  
These sets of equations represent the averaged steady-state model for the boost converter linearized around an operating point (D, \(\mathrm{X}_{10}\) and \(\mathrm{X}_{20}\)). A lower equivalent series resistance \(\mathrm{R}_{\mathrm{c}}\) compared with R is considered.