## 2 Description the PV System Based on the NEC Boost Converter
### Mathematical Model  
The operation of the NEC boost converter, when the MOSFET is activated (\(u=1\)), produces Topology 1, depicted at the top of Figure 3. The figure shows that the connection of the MOSFET imposes the voltage of \(C_{cb}\) capacitor (\(v_{cb}\)) into the diode, thus reverse-biasing the diode to deactivate it. Moreover, in this topology, the \(C_{cb}\) capacitor provides a path for the output current, which corresponds to the \(L_{2}\) current \(i_{2}\), ensuring a continuous current condition. Finally, it is evident that the output current of Node A is equal to the sum of the inductor currents (\(i_{1}+i_{2}\)), thus imposing a continuous input current to the NEC circuit. Then, the differential equations describing the dynamic behavior of both inductor currents \(i_{1}\) and \(i_{2}\), the internal capacitor voltage \(v_{cb}\), and the PV voltage \(v_{pv}\) are as follows:  
\[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\] \[\frac{di_{2}}{dt}=\frac{v_{pv}+v_{cb}-v_{b}}{L_{2}}\] (2) \[\frac{dv_{cb}}{dt}=\frac{-i_{2}}{C_{cb}}\] (3) \[\frac{dv_{pv}}{dt}=\frac{i_{pv}-(i_{1}+i_{2})}{C_{pv}} \tag{4}\]  
When the MOSFET is deactivated (\(u=0\)), Topology 2 occurs, which is depicted at the bottom of Figure 3. In this topology, the \(C_{cb}\) capacitor provides a path for the current of \(L_{1}\); however, in this case, the diode is activated by the combined current flow of \(i_{1}+i_{2}\). Finally, as in topology 1, the output current of Node A is equal to the sum of the inductor currents  
Figure 3: Topologies of the PV system based on the NEC boost converter. (\(i_{1}+i_{2}\)), imposing a continuous input current to the NEC circuit. The differential equations describing the dynamic behavior in this second topology are as follows:  
\[\frac{di_{1}}{dt}=\frac{v_{pv}-v_{cb}}{L_{1}} \tag{5}\] \[\frac{di_{2}}{dt}=\frac{v_{pv}-v_{b}}{L_{2}}\] (6) \[\frac{dv_{cb}}{dt}=\frac{i_{1}}{C_{cb}}\] (7) \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]0  
Then, the switched model of the system is obtained by combining the differential equations for both topologies using the control signal of the MOSFET (\(u\)) as follows:  
\[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]1 \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]2 (10) \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]3 (11) \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]4  
Another useful representation corresponds to the averaged model, which averages the control signal \(u\) within the switching period \(T_{sw}\), as given in (13). Therefore, the average dynamic behavior of both currents and voltages is modeled using (14)-(17). \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]5 \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]6 (14) \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]7 (15) \[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]8  
\[\frac{di_{1}}{dt}=\frac{v_{pv}}{L_{1}} \tag{1}\]9 \[\frac{di_{2}}{dt}=\frac{v_{pv}+v_{cb}-v_{b}}{L_{2}}\]0 (24) \[\frac{di_{2}}{dt}=\frac{v_{pv}+v_{cb}-v_{b}}{L_{2}}\]1  
In the above expressions, \(\delta i_{1}\) is the current ripple in \(i_{1}\), \(\delta i_{2}\) is the current ripple in \(i_{2}\), and \(\delta v_{cb}\) is the voltage ripple in \(C_{cb}\). The voltage ripple in \(C_{pv}\) is calculated by using the second-order filter expression in Node A, as reported in [22], which results in the following equation for the ripple \(\delta v_{pv}\):  
\[\frac{di_{2}}{dt}=\frac{v_{pv}+v_{cb}-v_{b}}{L_{2}}\]2  
Finally, the voltage and current stresses on the semiconductor are also calculated. From the first topology in Figure 3, the voltage supported by the MOSFET is calculated as given in (27) and the current supported by the diode as given in (28). Similarly, from the second topology, the current supported by the MOSFET is calculated as given in (29) and the voltage supported by the diode as given in (30).