## 2 Description the PV System Based on the NEC Boost Converter
### Comparison with the Classical Boost Converter
The DC current is simple to calculate, since for a fair comparison both systems are considered operating with the same input current \(i_{pv}\) and voltage \(v_{pv}\), and the same output voltage \(v_{b}\). For this comparison, \(d=0.5\); thus, \(i_{dc}=i_{pv}/2\) and \(v_{pv}=v_{b}/2\). The RMS currents are calculated as \(\sqrt{\frac{1}{\tau_{sw}^{2}}\cdot\int_{0}^{T_{sw}}i\;dt}\), where \(i\) must be replaced by the output currents of the PV systems, i.e., \(i=i_{d}\) for the classical boost system and \(i=i_{2}\) for the NEC boost system. Figure 5 shows the waveforms of the output currents for both PVsystems, which enables defining the expressions of those waveforms as (35) for \(i_{d}\) (classical boost) and (36) for \(i_{2}\) (NEC boost). \[i_{d}=\left\{ \begin{array}{ll}0&\text{for}\quad 0\leq t<\frac{T_{ \text{sw}}}{2}\\ -\frac{4\cdot\delta i_{i_{2}}}{T_{\text{sw}}}\cdot t+\big{(}i_{pv}+\delta i_{i_ {2}}\big{)}&\text{for}\quad\frac{T_{\text{sw}}}{2}\leq t<T_{\text{sw}}\\ \end{array} \right. \tag{35}\] \[i_{2}=\left\{ \begin{array}{ll}\frac{4\cdot\delta i_{2}}{T_{ \text{sw}}}\cdot t+\big{(}\frac{i_{pv}}{2}-\delta i_{2}\big{)}&\text{for}\quad 0 \leq t<\frac{T_{\text{sw}}}{2}\\ -\frac{4\cdot\delta i_{2}}{T_{\text{sw}}}\cdot t+\big{(}\frac{i_{pv}}{2}+ \delta i_{2}\big{)}&\text{for}\quad\frac{T_{\text{sw}}}{2}\leq t<T_{\text{sw}} \end{array} \right. \tag{36}\]  
Taking into account that \(\delta i_{L_{b}}=2\cdot\delta i_{1}=2\cdot\delta i_{2}\) for this comparison (\(d=0.5\)), the RMS values for \(i_{d}\) (classical boost) and \(i_{2}\) (NEC boost) are given in (37) and (38), respectively. \[i_{RMS,d}=\sqrt{\frac{2\cdot\delta i_{2}^{2}}{3}+\frac{i_{pv}^{2} }{2}} \tag{37}\] \[i_{RMS,2}=\frac{\sqrt{3}}{6}\cdot\sqrt{4\cdot\delta i_{2}^{2}+3 \cdot i_{pv}^{2}} \tag{38}\]  
Diving the RMS expression for the classical boost (37) into the RMS expression for the NEC boost (38) results in a constant relation equal to \(\sqrt{2}\). This means that the RMS value of the output current provided by the classical boost converter is 41.4% higher than the RMS current provided by the NEC boost converter, which produces higher power losses on the elements of the second stage. Moreover, the AC components of the output currents can be calculated from \(i_{RMS}^{2}=i_{dc}^{2}+i_{ac}^{2}\). In this way, expression (39) provides the AC component of the output current introduced into the second stage by the classical boost solution, while expression (40) reports the AC components generated by the NEC boost solution. Those AC components introduce undesired current harmonics to the second stage, which perturb the control system and degrade the power quality of the DC-link. In fact, since the PV system produces DC power, any AC component is useless, and the presence of those AC components increment the power losses (due to the ohmic effect) and perturb both the control systems and measurement devices. \[i_{ac,d}=\sqrt{\frac{2\cdot\delta i_{2}^{2}}{3}+\frac{i_{pv}^{2} }{4}} \tag{39}\] \[i_{ac,2}=\frac{\delta i_{2}}{\sqrt{3}} \tag{40}\]  
The magnitude of those undesired AC components are contrasted by dividing the AC component produced by the classical boost solution (39) with the AC component produced  
Figure 5: Waveforms of the output currents of classical boost and NEC boost PV systems (\(d=0.5\)). by the NEC boost solution (40). Such an analysis is performed for different values of the current ripple \(\delta i_{2}\) (as percentage of the DC value \(i_{2}=i_{pv}/2\)), and the results are reported in Figure 6. Those results put into evidence the significant increment in the current harmonics introduced by the classical boost converter in comparison with the NEC converter: for an inductor current ripple of 5%, the classical boost PV system introduces an AC component 34.67 times higher than the NEC boost PV system; for an inductor current ripple of 20%, such a relation drops to 8.76 times, but the NEC boost PV system still introduces an AC component much smaller.