## 5 Discussion  
Only the PV module under examination has one type of progressive fault in the previous circuit simulations. The rest of the elements, such as PV modules, wires, bypass diodes, were assumed to work in a non-faulty condition. The aforementioned does not mean that the results are specific to the selected module location; if the faulty module is located at any other place in the array, the obtained charts would be equivalent to the graphs presented in this work. Based on the simulation results it is confirmed that when a progressive fault affects a photovoltaic module, its differential voltage changes according to the severity of the failure mode. Figure 3b confirms that the progressive fault unbalances the voltage in the string; this means that as the progressive fault progress, the faulty module's differential  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(R4\) & \(Vmpp\) (V) & \(Pmax  
voltage decrease, and the differential voltage of the affected modules increase. Hence, the relative percentage difference (\(\Delta_{V}\)) increases, and the output power of the array decreases; these facts are easily checked in Figure 3c,d. For all the fault types analyzed, the same performance was observed in sub-charts in Figures A1-A6. The degree of fault affectation in the module is calculated with the relative percentage difference (6), and this indicator is equivalent to the percentage of power loss if the modules work at the same string current. Table 5 presents the numerical results for the SRD fault. These results correspond to the evolution of the maximum power-point. In these maximum power points, it seems that the power loss indicator (\(\Delta_{V}\)) is highly correlated with the power loss of the module calculated at standard test conditions, MPL. Actually, for this fault condition, the estimated power \(\Delta_{V}\) and the MPL are highly correlated, the Pearson Coefficient [43] is, in this case, \(r=0.9999\); therefore, it is possible to calculate a linear regression as,  
\[MPL=1.0311\Delta_{V}-0.1082,\ \forall\ \Delta_{V}\ at\ Pmax. \tag{9}\]  
Moreover, the obtained linear regression has a very low variability, given a determination coefficient \(R^{2}\) of 99.99%. This result is not unique for this failure mode, the correlation between \(\Delta_{V}\) and MPL appears for all the fault types analyzed. Graphically, it can be appreciated in Figure 4 and the parameters for the best fit line equation presented in Table 6.  
In this PV array model of 1.6 kWp, as the progressive fault increase, the \(Vmpp\) moves affecting all the modules' power production; this means generating small losses in all the modules in the array. These small changes in power production caused by the progressive fault could be hidden because they are too small to be detected by the monitoring system. For the 1.6 kWp example, if \(R_{\mathrm{s}}\) double its value in whatever PV module, the total power loss of the array drops about 0.86%, which could be negligible. However, this percentage represents one faulty module losing around 6.26% of its energy (See the third row in Table 5). This hidden effect is even more drastic in large PV installations; let's suppose that a faulty module has a constant power loss, and if the number of PV modules in the array is high enough, the total power loss tends to zero. However, the faulty module is still losing the same percentage of energy and may evolve into a more severe fault condition. The correlation \(MPL\propto\Delta_{V}\) can be used to detect abnormal behavior. Moreover, it is possible to develop a simple criterion for fault detection and location in SP arrays. The Equation (10) is based on (6), and it checks if the relative percentage difference between two voltages is more than a threshold value called \(\delta\). Here \(V_{max}\) represents the maximum differential voltage of all the modules in the string \(j\), and \(V(i)\) is the differential voltage of the analyzed module.