## 2 Theoretical Framework: Detection Criterion Fundamentals  
The explicit expression of (1) for the module voltage is obtained using Lambert-\(\mathcal{W}\) transformation, if we have an equation with the form \(y=xe^{x}\), the term \(x\) could be obtained as \(x=\mathcal{W}(y)\) for any \(x>-1/e\). Therefore, the voltage is expressed as follows,  
\[V=(I_{ph}+I_{s})R_{p}-I(R_{p}+R_{s})-N_{s}\eta V_{t}\mathcal{W} \left(\frac{R_{p}I_{s}}{N_{s}\eta V_{t}}e\left(\frac{(I_{ph}+I_{s}-I)R_{p}}{N_ {s}\eta V_{t}}\right)\right). \tag{2}\]  
It is clear that if a failure affects the PV module performance, the parameters in (1) and (2) change their values and hence the voltage \(V\) in (2) also varies [28]. If the initial values of all the parameters are known it is possible to identify the degradation condition of one PV module using (2). Nevertheless, the aforementioned is seldom the case. Not only the initial values are unknown but also a PV installation is comprised of several PV modules connected in series and parallel and the relationships between the parameter variation in one PV module will affect the others.  
In a typical photovoltaic plant, PV modules are connected using a configuration known as series-parallel (SP). In an SP configuration, the PV modules are first connected in series forming strings which are then connected in parallel as depicted in Figure 1, forming a PV array of \(m\times n\) modules. The voltages in the \(j\)-string, formed by \(m\) modules, are governed by [PERSON]'s voltage law [20], as follows;  
\[\sum_{i=1}^{m}V(i,j)-V_{blk}(j)-V_{op}=0\qquad, \tag{3}\]  
where \(V(i,j)\) is the differential voltage of the module \(i\) in the string \(j\) as in Equation (2), \(V_{blk}(j)\) is the voltage of the blocking diode, and \(V_{op}\) is the operational voltage point of the array.  
When all the modules are identical (3) can be simplified as follows,  
\[mV(j)-V_{blk}(j)-V_{op}=0. \tag{4}\]  
Now consider the case in which at least one module is degraded or faulty. Due to the SP configuration the sum of all the voltages in the string with the faulty PV module does not change, see Equation (3). Therefore, the PV modules that are part of the string in which a faulty module is located will change their electrical operating point. We call the aforementioned elements _affected_ PV modules, as it can be seen in Figure 2.  
If we denote \(m_{f}\) as the number of modules with the same fault, (3) can be rewritten as,  
\[(m-m_{f})\cdot V_{a}(j)+m_{f}\cdot V_{f}(j)=V_{op}+V_{blk}(j) \tag{5}\]  
where \(V_{f}(j)\) is the voltage of the faulty module (_degraded_) and \(V_{a}(j)\) is the voltage of the affected modules.  
Notice that the sum of the voltages at the faulty and at the affected modules in a given string should be equal to the sum of the voltages of non-faulty modules in a string parallel to it. Hence, the variation in the voltages of the affected modules and the faulty modules must behave oppositely, i.e., when the voltage at the faulty decreases the voltage at the affected modules increases. Moreover, the difference between the voltages at the affected and the faulty modules can be expressed as follows,  
\[\Delta_{V}=\frac{V_{a}-V_{f}}{V_{a}}100,\qquad\forall I>0. \tag{6}\]  
Given that the current trough the string, \(I\), is shared by the modules, (6) can be rewritten as  
\[\Delta_{V}=\frac{IV_{a}-IV_{f}}{IV_{a}}100=\frac{P_{a}-P_{f}}{P_{a}}100,\qquad \forall I>0. \tag{7}\]  
where \(P_{a}\) and \(P_{f}\) are the power produced by the affected and faulty modules, and \(I\) is the current of the \(j\)-string (\(j\)). Therefore (6) represents a measure proportional to the power reduction in a PV module.  
Figure 1: Nomenclature for the SP PV array.