## 2 Methodology  
The proposed methodology to study the electrical behavior of PDS is based on the analysis of the electrical current signals. Since the electrical signals can be linear or nonlinear, stationary or nonstationary, depending on the connected loads and/or the presence of RES, there is no a unique optimal analysis method. So, the proposed methodology (as shown in the flowchart of Figure 1) is based on several checks (colored blocks) on the dataset in order to select, step by step, the most suitable method. The first check (blue block) is the linearity degree of the analyzed signal, which can be evaluated by means of the PC index (Equation (1)) that returns the coherence degree between two signals and is defined by [3]:  
\[\mathrm{PC}=\frac{\left|\sum_{i}(\mathrm{FFT}_{i}(\Phi_{\mathrm{a}})\cdot \mathrm{FFT}_{i}(\Phi_{\mathrm{b}}))\right|^{2}}{\sum_{i}\left|\mathrm{FFT}_{i }(\[PERSON]_{\mathrm{a}})\right|^{2}\sum_{i}\left|\mathrm{FFT}_{i}(\Phi_{\mathrm{b }})\right|^{2}} \tag{1}\]  
where FFT is the Fast Fourier Transform, while \(\Phi_{\mathrm{a}}\) and \(\Phi_{\mathrm{b}}\) represent the phases of the two electrical signals. The PC index varies in the range \(0\div 1\), where \(0\) indicates that the two signals are not coherent and \(1\) means that they are totally coherent. In particular, PC index is used for two different aims: (i) if one of the two signals used in (1) is a reference signal (for example a sinusoidal signal), then the index provides a measure about the discrepancy of the signal under test with respect to the reference one (in this case it is named \(\mathrm{PC_{ref}}\)); (ii) if the electrical signals deriving from two PDS are compared, the PC gives the information about the mutual similarity (here it is named \(\mathrm{PC_{sigs}}\)). In the first case, \(PC_{ref}=1\) says that the system under investigation is linear, otherwise it is nonlinear. If the system is nonlinear, then the most suitable analysis method to characterize an electrical signal is the HHT, which extracts the time-varying oscillations, named Intrinsic Mode Function (IMF), by means of the EMD. The Hilbert transform \(Y(t)=\mathrm{H}\{X(t)\}\) (Equation (2)) of a time-dependent signal \(X(t)\) is given by:  
\[Y(t)=\frac{1}{\pi}P\int_{-\infty}^{+\infty}\frac{X(t^{\prime})}{t-t^{\prime}} dt^{\prime} \tag{2}\]  
where \(P\) denotes the principal value. In this paper, the electrical signal \(X(t)\) is represented by the current \(i(t)\). The Hilbert complex extension of \(X(t)\) is \(Z(t)=X(t)+jY(t)=X(t)+j\,H\{X(t)\}=\,a(t)e^{i\theta(t)}\), where \(j\) is the imaginary unit and the amplitude \(a(t)\) and the phase \(\theta(t)\) are expressed by \(a(t)=\ \left[X^{2}(t)+Y^{2}(t)\right]^{\frac{1}{2}}\) and \(\theta(t)=arctan(Y(t))/X(t)\), respectively. The signal \(X(t)\) can be decomposed into a finite collection of IMFs [36], i.e., \(X(t)=\sum_{l=1}^{N}c_{l}(t)+r_{N}(t)\), with \(c_{l}(t)\) the time-dependent IMFs with variable amplitude and \(r_{N}(t)\) the non-oscillatory residue term, resulting from the sifting process. The number of oscillations of the IMFs decreases as the order of IMF increases; this depends on the iterative extraction procedure of each IMF. In fact, each step of EMD produces a new signal that is equal to the subtraction between the original one and the mean value of the envelope curves of its local maximum and minimum values. The complex representation (Equation (3)) of each IMF \(c_{l}(t)\) is the following:  
\[z_{l}(t)=c_{l}(t)+j\frac{1}{\pi}P\int_{-\infty}^{\infty}\frac{c_{l}(t^{\prime} )}{t-t^{\prime}}dt^{\prime}=a_{l}(t)e^{i\theta_{l}(t)} \tag{3}\]  
Figure 1: Flow-chart to choose the most suitable method. with \(a_{l}(t)\) the amplitude of \(z_{l}(t)\), and \(\theta_{l}(t)=\int\omega_{l}(t)dt\), \(l=1,2,\ldots,N\).