## 2 Methodology
The original amplitude \(X(t)\) (Equation (4)) of the electrical signal expanded in a Fourier representation via the IMF components can be written in the form:  
\[X(t)=Re\left\{\sum_{l=i}^{N}z_{l}(t)\right\}=Re\left\{\sum_{l=i}^{N}a_{l}(t)e^{ i\theta_{l}(t)}\right\}=Re\left\{\sum_{l=i}^{N}a_{l}(t)e^{2\pi j\int f_{l}(t)}\right\} \tag{4}\]  
with \(Re\) the real part and \(f_{l}\) the instantaneous frequency of the \(l\)-th IMF. The residue \(r_{N}(t)\) is neglected, when it is a monotonic function without the possibility to extract another IMF. Instead, if the system is linear, it needs a second check (orange block in Figure 1) to determine whether the signal is stationary or not. The signal is stationary when the summary statistics (e.g., mean and variance) are stable during a long time, i.e., they have no trend and no variation. The verifications are usually done in two alternative ways. The first one provides that the time series is divided into more partitions and the means and variances of the groups are compared each other; this technique is known as _runs test_. If the differences are not statistically significant, then the time series is likely stationary. The second one is based on statistical tests (e.g., the Augmented Dickey-Fuller's (ADF) test [37; 38] based on the unit root test) and on the null hypothesis that the time series is stationary. ADF returns the \(p\)-value; so, fixing the significance level \(\alpha=0.05\), if \(p-value<\alpha\), the null hypothesis is rejected, and the time series is non-stationary. In this case, the most suitable method to characterize an electrical signal in the time-frequency space is the WA, which is based on a function--called wavelet mother. Here, the Morlet wavelet mother [39], which is often chosen in the applications of power systems [28], is used. There are several orthogonal wavelet families, which are the Daubechies, Symlets, Coiflets, and Meyer, whereas the criteria to select the mother wavelet are the following [40]:  
* The wavelet should have an enough number of vanishing moments to represent the main features. * The wavelet should provide sharp cutoff frequencies to reduce the amount of leakage energy. * The total wavelet energy should be higher than that of the same feature of the same signal. * The wavelet basis should be orthonormal. When considering a time domain signal \(x(t)\), the continuous wavelet transform (Equation (5)) is a linear function given by:  
\[W(u,s)=\frac{1}{\sqrt{s}}\int_{-\infty}^{+\infty}x(t)\psi^{*}\left(\frac{t-u} {s}\right)dt \tag{5}\]  
with \(s\) and \(u\) the scale and translation parameters of the mother wavelet \(\psi(t)\) (Equation (6)) that in this case is:  
\[\psi_{u,s}=\frac{1}{\sqrt{s\pi f_{B}}}e^{j2\pi f_{C}(\frac{t-u}{s})}e^{-\left( \frac{t-u}{s}\right)\ /f_{B}} \tag{6}\]  
where \(f_{B}\) and \(f_{C}\) are two characteristic parameters [41]. In addition, the HHT can be used to study linear and nonstationary signals, but it does not return further information with respect to the one obtained by WA, and the extraction of the information is more complicated. Instead, if the signal is linear and stationary, then the most suitable method is surely the FFT, because it returns reliable and fast results. Also, in this case, it is possible but less convenient to apply WA or HHT. Finally, the penetration of PV plants can be evaluated (green block in Figure 1), calculating the PC\({}_{\text{sigs}}\) index by means of (1), where the two input signals are the current in the distribution circuit to analyze and the current flowing in a passive benchmark line. By calculating the PC\({}_{\text{sigs}}\) between other distribution circuit and the same passive benchmark, it is possible to estimate the different penetration level of PV plants in different PDS. Figure 2 resumes the applicability of the different methods.