Lightning Location and Peak Current Estimation From Lightning-Induced Voltages on Transmission Lines With a Machine Learning Approach

In this article, a machine-learning-based model for the regression of cloud-to-ground lightning location and peak current from time-domain waveforms of lightning-induced voltage measurements on overhead transmission lines is presented. A principal component analysis (PCA) procedure is applied for extracting significant features and decreasing the dimension of the input vector. Then, a shallow neural network is trained with the results of the PCA. The obtained results show that the proposed approach can be the base for a tool able to regress lighting location with an accuracy comparable to or even better than traditional methods [i.e., lightning location system (LLS)] and provide a peak current estimate more accurate than LLS and more actual and widespread than direct tower measurements (which are limited to a reduced number of recorded events in some specific regions). Such a tool would also have significant advantages in terms of costs, since it would not require a dedicated instrumentation.


Lightning Location and Peak Current Estimation From Lightning-Induced Voltages on Transmission
Lines With a Machine Learning Approach Martino Nicora , Member, IEEE, Mauro Tucci , Senior Member, IEEE, Sami Barmada , Senior Member, IEEE, Massimo Brignone , Member, IEEE, and Renato Procopio , Senior Member, IEEE Abstract-In this article, a machine-learning-based model for the regression of cloud-to-ground lightning location and peak current from time-domain waveforms of lightning-induced voltage measurements on overhead transmission lines is presented.A principal component analysis (PCA) procedure is applied for extracting significant features and decreasing the dimension of the input vector.Then, a shallow neural network is trained with the results of the PCA.The obtained results show that the proposed approach can be the base for a tool able to regress lighting location with an accuracy comparable to or even better than traditional methods [i.e., lightning location system (LLS)] and provide a peak current estimate more accurate than LLS and more actual and widespread than direct tower measurements (which are limited to a reduced number of recorded events in some specific regions).Such a tool would also have significant advantages in terms of costs, since it would not require a dedicated instrumentation.

I. INTRODUCTION
L IGHTNING strokes are natural phenomena representing a significant source of risk for people, structures (e.g., civil buildings and wind turbines), infrastructures (e.g., power transmission and distribution, and telecommunications systems), railway, aviation, and natural environments.Knowing the location of a lightning strike and its channel-base peak current (i.e., the peak value of the return stroke current waveform measured at the base of the lightning channel [1]) is relevant for several geophysical and electrical engineering applications.
Geophysical researchers are interested in lightning as a precursor of severe weather events.Lightning forecasting algorithms [2], [3], [4] can be applied to track the spatial evolution of convective structures in real time [5], [6], with the aim of preventing catastrophic weather effects.
On the other hand, electrical engineers need stroke location and peak current data in procedures aimed at quantifying the lightning risk for a structure or infrastructure and designing the appropriate lightning protection systems (e.g., the lightning performance assessment of an overhead power line [7], [8], [9], [10], [11]).
At present, cloud-to-ground lightning strokes are mainly localized with lightning location systems (LLSs).An LLS is a network of sensors separated by 50-400 km and detecting cloudto-ground discharge signals in the very low frequency/lowfrequency range [12].These sensors typically measure the radiation electric and/or magnetic field and localize the lightning stroke employing different techniques (magnetic direction finding, time-of-arrival, and hybrid methods) [12].LLS location accuracy can vary depending on the sensor coverage and technology.For the Italian network Sistema Italiano Rilevamento Fulmini, which is part of the European Cooperation for Lightning Detection (EUCLID), the median location accuracy is 250 m or even less in some regions [13], [14].
Furthermore, LLSs infer peak currents from measured peak fields.As discussed in [12], some issues exist and are summarized as follows.
1) The field-to-current linear model partially integrated into the lightning location algorithm to account for field attenuation is applied to all types of strokes although it has been validated with rocket-triggered lightning and direct tower measurements only for negative subsequent strokes with peak currents lower than 60 kA.2) LLSs have increased their capability of detecting weaker sources; this increments the possibility of misidentifying cloud discharges.Such a classification error may distort peak current statistics.Different studies have revealed that an increase of the flash (i.e., first stroke) detection efficiency leads to a reduction in mean and median peak current (e.g., according to [15] when the flash detection efficiency raises from 75% to 100%, the mean peak current value decreases by 13.6%).
However, for both lightning protection and lightning research, the current parameters provided in [16] are still used to a large extent as the primary reference source [11].
In this context, the aim of this article is to present an alternative method for cloud-to-ground lightning location and peak current estimation with a machine learning (ML) model trained with lightning-induced voltage measurements on overhead transmission lines.This is intended to be the first step of a broader project whose final outcome is hopefully a tool able to: 1) locate lightning strokes with an accuracy comparable to or even better than LLS and 2) perform a peak current estimate more accurate than LLS and more actual and widespread than direct tower measurements.Such a tool would also have significant advantages in terms of costs, since it would not require a dedicated instrumentation.
The first attempt to implement this method can be found in [22], whose authors presented an ML-based lightning location algorithm relying on data from preinstalled voltage measurement systems on power transmission lines.In spite of obtaining promising results in terms of location error, the main drawback of [22] is that the model of the physical system (lightning+power network) is really simplified.Indeed, the lightning-induced voltage database required for the ML model training and testing is generated with the Rusck's formula [23] (which is valid for a lossless single-wire transmission line above a perfectly conducting ground), and the excitation source is a step current propagating upward along the lightning channel.
Therefore, the aim of this work is to test the validity of the idea in a more realistic situation.To this extent, the Lightning Power Electromagnetic Simulator for Transient Overvoltages (LIGHT-PESTO) [24] code is used to simulate the voltage waveforms measured on the line.LIGHT-PESTO allows us to consider: 1) more complex and realistic lines and a detailed model of the power systems connected at the lines ends; 2) any value of soil conductivity and permittivity; and 3) different channel-base current waveforms and different propagation models, considering the most accurate literature references.
Furthermore, in [22], the voltage signals are normalized, thus losing information about the peak current.On the contrary, in this work, the peak current is inserted in the target of the ML model, allowing a peak current estimate from voltage measurements.
From a methodological point of view, the ML model has the task to predict three numerical values that represent the target, i.e., the (x, y) coordinates of the stroke impact point and the peak current I.A simulation with LIGHT-PESTO provides the input features to the ML model, which are represented by the induced voltage waveforms in some points of the line in the selected time samples (i.e., simulating the presence of voltage sensors).
Even in the case of a practical implementation of the procedure, where the induced voltage waveforms are measured, several hundreds of time samples would be required to accurately represent the waveforms.As a consequence, we are approaching a difficult regression problem with a high-dimensional input, and many ML methods, such as fully connected neural networks (NNs), are known to suffer the curse of dimensionality in such cases.
For this reason, in order to reduce the number of inputs, a principal component analysis (PCA) procedure is set up, with the aim of extracting significant features.Reducing the dimension of the input vector allows the use of a shallow NN that is trained with the results of the PCA applied to the simulated induced voltages.The results of the NN approach are encouraging, outperforming other ML methods such as support vector regression (SVR) and k-nearest neighbors (K-NN).
The rest of this article is organized as follows.Section II provides details on the properties of the lightning-induced voltage simulator LIGHT-PESTO [24], defines the physical system model, and describes the generated voltage database.In Section III, this voltage database is used to train an ML-based model able to regress the lightning location and peak current.Section IV shows the obtained results.Finally, Section V concludes this article.

II. DATA ACQUISITION
The ML model training and testing are performed with a database of simulated lightning-induced voltage measurements.Section II-A discusses the features of the employed simulator.Section II-B describes the simulation setup, i.e., the power system model, the lightning model, numerical assumptions, simulation parameters, and the generated voltage database.

A. Lightning-Induced Voltage Simulator
Lightning-induced voltage measurements are simulated with the dedicated tool LIGHT-PESTO.This simulator originated from a numerical code proposed in [25] that solves the fieldto-line coupling problem in a PSCAD-EMTDC environment by means of the second-order finite-difference time-domain scheme of the Agrawal's model [26].The approach of [25] allows us to model a realistic network, automatically distinguish between direct and indirect lightning strokes, and account for power system precontingency conditions.Validation has been performed against experimental results obtained at the highvoltage facility of the University of São Paulo, Brazil [27], [28].Several experimental measurements proved the accuracy of the model in different realistic scenarios, i.e., single-phase straight distribution line (see [28,), in the presence of discontinuities (see [28,Figs. 18 and 19]), and in the presence of surge arresters and multilateral configuration (see [28,Figs. 22 and 23]).
Finally, the LIGHT-PESTO interface was developed in the MATLAB-Simulink environment and presented in [24].The interface allows the user to select: 1) the lightning parameters, i.e., the channel-base current waveform and peak, the channel height, the return stroke speed, and the model of the propagation of the current along the channel; 2) any ground permittivity and conductivity; 3) the simulation parameters (duration, time step, and space step); and 4) the power system parameters, allowing multiline systems and discontinuities.

B. Lightning-Induced Voltage Simulation 1) Power System Model:
The configuration of the power system consists in a 10-km straight single line, with a conductor diameter of 1 cm and height from the ground of 10 m.It is assumed that termination resistances at the line ends are equal to the characteristic impedance of the line (i.e., 498 Ω).
2) Detection Domain: A 10 × 10 km 2 area is considered (see Fig. 1).The transmission line is located 100 m beside the detection domain.Two voltage sensors (named VS1 and VS2) are located on the line at 2 km from each other (x = 4 km and x = 6 km, respectively).An ambiguity in lightning location may be caused by considering only two sensors.Indeed, signals from (x, y) and (x, −y) are the same.However, in applications, the ambiguity can be solved by signals from a sensor installed on another line, or by taking into account the terrain topography, or the presence of objects and structures making the system asymmetric [22].A number (N = 2000) of lightning events are simulated into the detection domain.The coordinates of the lightning strokes are extracted according to uniform distributions [7], [8], [9], [10].
3) Lightning Model: A typical straight vertical lightning channel with height H = 8 km is considered.The return stroke propagation speed is assumed to be 0.4 c 0 , where c 0 is the speed of light in vacuum (3 × 10 8 m/s).For the propagation of the current along the channel, the well-known modified transmission line with exponential decay (MTLE) model is selected [29] .The channel-base current waveform is modeled with the Heidler function [30] with typical parameter values of the first stroke [31], except for the peak current I, which is extracted according to a log-normal distribution with median 31.1 kA and logarithmic standard deviation 0.485 kA [11].
4) Ground: The ground is supposed to be lossy, but uniform, with conductivity equal to 10 mS/m, whereas the ground relative permittivity is assumed to be 10.

5) Simulation Parameters:
A time step dt = 0.1μs is chosen.The considered time frame is [0, T ], where T = H 0.4 c 0 = 66.7μs is the limit of validity of the current propagation model [29].Thus, the simulated voltage measurements are arrays with n = 1 + T dt = 668 elements.The line discretization step is selected with a formula that combines the Courant's stability condition with the method of characteristics [32], i.e., dx = 3 c 0 dt = 90 m.Computational efficiency is assured by the use of LIGHT-PESTO (a Microsoft Windows 10 PC equipped with 16 GB of RAM and Intel Core i7-2600 CPU at 3.4 GHz needs about 3 s for the simulation of a single random stroke and the computation of the lightning-induced voltage on the power system [24]).
6) Voltage Database: As specified before, 4000 simulated voltage measurements with n = 668 elements are provided for the training and testing phases of the ML model.Moreover, to make a fair comparison with the results of [22], another voltage measurement database is generated considering a simplified case where a constant peak current of 30 kA and a perfect electric conductor (PEC) ground are assumed (the other characteristics remain unchanged with respect to what is described in the previous paragraphs).Therefore, in the following section, two different ML models will be presented, one for each case, i.e., the simplified one (constant I = 30 kA, PEC ground) and the advanced one (random I, finite ground conductivity of 10 mS/m), which are labeled Case 1 and Case 2, respectively.Fig. 2 reports an extract of the database generated for Case 2.

III. ML MODEL
As explained in Section II, two different databases have been created: for Case 1, only the stroke location is unknown, while for Case 2, the peak current should be determined too.The Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.criteria and choices made by the authors in the definition of the preprocessing and dimensionality reduction procedures are unique; however, different NNs have been created for different cases.In the following subsections, the main rationale behind the authors' choices is described in detail.

A. Waveform Preprocessing and Feature Extraction
Looking at the waveforms shown in Fig. 2, there are several issues that need to be addressed in order to build an efficient and accurate ML based tool.One aspect is related to the fact that the time-domain signals carry redundant information (for instance, the waveforms tails), and a more efficient and compact representation of the signals could be achieved.A second aspect is related to the so-called curse of dimensionality, which should be avoided when an NN is used.
A more physically related consideration is relative to the delay time between the induced voltage waveforms.Lightning strokes are simulated one by one, and all starting at time t = 0 (see Fig. 2).Then, the induced voltage starting time depends on the distance between the lightning impact point and the closest voltage sensor; this is a meaningful information that, however, could not practically be used by a real LLS since the time t = 0 could not easily measured with the required degree of accuracy.This is overcome by removing from both waveforms v 1 and v 2 the number of leading zero values present at the beginning of the closest waveform before performing the analysis; in this way, all the impulses start at the same time.This action creates vectors of different dimension; consequently, all the waveforms are afterward truncated, removing point from the tail, to the length of the shortest one, i.e., to n 1 < n time samples for waveforms acquired by VS1, and to n 2 < n time samples for waveforms acquired by VS2.After this initial processing, the discrete Fourier transform (DFT) is applied to each vector v 1 and v 2 resulting in a set of complex values; considering only values up to the Nyquist frequency, a vector U ∈ R n 1 +n 2 is created by stacking the real part {DFT(v i )} and the imaginary part {DFT(v i )} of both vectors i = 1, 2. As mentioned before, in order to avoid the curse of dimensionality, the input vector dimension (n 1 + n 2 after the truncation described above) must be reduced, and to reach this goal, a PCA has been set up.Among different approaches, the authors have verified that a PCA applied to the frequency content of the voltage signals gives good results and low computational costs (low number of principal components).In particular, the application of the PCA to the frequency-domain dataset, consisting in the N observations of vectors U, results in P principal components capturing the Γ% of the total variance.This yields a feature vector X ∈ R P , which is the representation of U in the principal component space.The complete preprocessing procedure is depicted in Fig. 3.

B. Case 1: NN for the Lightning Strike Location Regression
A shallow feedforward NN is chosen as the ML paradigm, with a sigmoidal activation function in the hidden layer and a linear activation function in the output layer.As shown in Fig. 4, the extracted PCA features are given as an input to the NN, whereas the output of the NN is the position vector p = (x, y) ∈ R 2 .In this simplified case, all the events share the same peak current; therefore, the focus in on the prediction of the lighting strike coordinates.
For Case 1, we compared the NN approach with other ML algorithms, such as SVR and K-NN, obtaining the best results using NNs.For this reason, we only adopt NNs in Case 2.

C. Case 2: NN for the Lightning Strike Location Regression
In the second and more advanced case, the events are simulated assuming a random peak current with log-normal distribution [11], and the extracted peak current values can be conveniently used as ground truth for training the regression model.Since the NN algorithm has shown good performance for Case 1, the same approach is selected also for Case 2. However, to better exploit the knowledge of peak currents at prediction time, the scheme depicted is Fig. 5

has been implemented.
A first NN, NN 1 , is trained to predict the lightning peak current associated with each voltage waveform.Then, at prediction time, the feature vector X is normalized by the predicted peak current and given as input to a second NN, NN 2 that shall predict the coordinates of the point of impact.The training of NN 2 is performed with a dataset where each waveform is normalized by the corresponding ground truth peak current.It is worth to note that the normalization can be applied to the PCA values X in place of the original waveforms v 1 and v 2 , because of the linearity of the preprocessing steps, consisting in fast Fourier transform and PCA operations.

D. Training and Model Selection
The selection of the significant hyperparameters of the various models, such as the best number of neurons in the hidden layer H, is performed by a grid search based on a tenfold cross validation, i.e., cyclically training on 90% of the data and testing on the resulting 10%.This method can provide an insight into how the model generalizes to an unknown dataset and contributes to improve model prediction performance.Applying the PCA to the training set results in P = 10 principal components capturing Γ = 99.52% of the total variance.The average location error e of a given model is calculated as the tenfold cross-validation mean value of the Euclidean distance (denoted by • ) between the predicted location p and the real location p: where N = 2000 is the number of observations.In particular, using cross validation means that each point is used in test once; then, the cross-validation error (1) is computed averaging the errors in all test folds.The NN was the best performing method in our experiments, among SVR and K-NN, and a special attention to the training parameters was needed to obtain the best results.In particular, the NNs were trained using Bayesian regularization within the Levenberg-Marquardt algorithm [33], where the weight vector w of the network is updated as where J is the Jacobian matrix that contains first derivatives of the network errors with respect to the weights and biases, and e is a vector of network errors.The Jacobian matrix can be computed through a standard backpropagation technique [33].
The positive scalar parameter μ controls the applied method: when μ is zero; (2) is just Newton's method.On the other hand, when μ is large, (2) becomes gradient descent with a small step size.Newton's method is faster and more accurate near a minimum of the error; thus, the adaptive value μ is decreased by μ dec after each successful step (reduction in performance function) and is increased by μ inc only when a tentative step would increase the performance function.In our application, the choice of the two hyperparameters μ dec and μ inc has proved to be fundamental, especially for Case 2. In fact, being the learning a difficult task, it was important to avoid early stopping, therefore choosing not to use a validation set.As a result, the most relevant condition that stops the training was that μ exceeds a maximum allowed value μ max , giving to this parameter a crucial role.For these reasons, the relevant hyperparameters μ dec , μ inc , and μ max were selected and fixed using a preliminary search based on cross validation.

A. Case 1: Constant Peak Current and PEC Ground
The results of the calculation of the error defined in (1) using a tenfold cross validation as a function of the number of hidden neurons for Case 1 are shown in Fig. 6.The proposed approach results in a very low location error of about 62 m, based on the averaging of N = 2000 test samples using cross validation, where each sample is used in the test once.This prediction is better than the median location accuracy of the European LLS EUCLID, which is 250 m or slightly less in some regions [13], [14].It is worth to note than these results have been obtained for Case 1 where different simplifying assumptions are considered.However, the average value of this prediction is comparable with the results of [22].Regarding the distribution of the location error, starting from the same system considered in Case 1, Karami et al. [22] obtained that in more than 65% of the cases, their ML model was able to predict the source with less than 100-m location error.In our case, the percentage of prediction errors below 100 m is 89%, indicating that the variance of the error of our model is much lower than the one presented in [22], despite that a similar average error is reported.Table I resumes the comparison between the two models.The promising result is a strong indication of the goodness of the preprocessing procedure described in Fig. 3    an average location error of 350 m and K-NN of 560 m.We used fine-tuning approaches to optimize the hyperparameters of both SVR and K-NN, but their performance lagged behind.The main results of the experiment performed for Case 1 are shown in Tables II and III and in Figs.7-10.The good R 2 values found in the regression analysis of the single x and y coordinates shown in Figs.7 and 8 attest both the goodness of the model and the fact that the simplified assumptions made in Case 1 make the problem easier to solve with respect to Case 2. The histogram shown in Fig. 9 suggests that the points for which the location error is high are statistically few and not able to affect the result.Indeed, the 95th percentile of the model location error is 220 m.On the other hand, the 95th percentile of LLS location accuracy is typically higher (e.g., 1600 m in [13]).The scatter plot of the stroke events in the spatial domain (see Fig. 10) shows that greatest errors are  made in the proximity of the line.This behavior is obtained also in Case 2, and it is discussed in detail in the following sections.

B. Case 2: Variable Peak Current and Non-PEC Ground
Regarding Case 2, we performed a cross validation of the model shown in Fig. 5, with a number of neurons fixed to 10, which was optimal for Case 1, both for NN 1 and NN 2 .In fact, a grid search conducted on the advanced dataset confirmed that 10 was the optimal number of hidden neurons for all the networks in Case 2 too.Special attention was paid to avoid data leakage, i.e., using training data during the test or test data during the training.In particular, NN 1 and NN 2 are trained simultaneously using training data, while the test is done in series, following the flow of information, as shown in Fig. 5.As a result, the tenfold cross-validation location error obtained for Case 2 is 290 m.The degradation of the performance with respect to Case 1 was expected, because of the more accurate simulation and the variable peak current.However, the result is encouraging because it is comparable with LLS location accuracy [13], [14].LLSs typically have a better accuracy near the sensors.However, the work presented in this article is intended as the first step of a broader project, which aims at developing a lighting detection system integrated with overhead power systems and exploiting measurements from voltage sensors.Since for Italy the density of power lines is very significant (14 and 136 km/km 2 for transmission lines and distribution lines, respectively, as declared by the Italian transmission system operator [34]), in principle, one can expect the presence of voltage sensors sufficiently close to the lightning strike.This would allow a prediction with performances of the order of magnitude of that obtained in the present work.
The average error e I % of the peak current Î predicted by NN 1 , with respect to the real current I, was calculated as a mean absolute percentage error (MAPE) resulting in an average value of e I % = 12.5%.The model of   is 930 m, which is higher with respect to Case 1 (220 m), but lower than that of typical LLS location accuracy (e.g., 1600 m for [13]).It can be observed from Fig. 14 that a small number of large errors are distributed for lighting strokes located near the line where the voltage sensors are placed, hence with low y values.This small number of large errors, that represent only the 0.45% of the total number of events, have a large impact on the global error e, which would reduce by 7% from 290 to 270 m if one excluded those samples.This behavior is due to physics of the system.Indeed, the exact analytical dependence of the lightning-induced voltage on the distance between the point of impact and the line is difficult to evaluate since it requires to compute lightning electromagnetic fields and then the solution of the Maxwell's equations describing the field-to-line coupling problem.However, simplified approaches have verified that, at least, the induced voltage peak can be approximated with an expression that is linear [23] or quadratic [35] with respect to the inverse of such a distance.Hence, near the line, small y variations Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.can produce significant variations in the induced voltage, and this may affect the capability of the ML model to regress the y coordinate.
Different cutoff values of the y-coordinate prediction for Case 2 are considered.As reported in Table IV, the average location error slightly decreases with increasing cutoff value, but does not approach the average location error obtained for Case 1.
Particular attention shall be paid to the fact that for Case 2, the output of NN 1 affects the input of NN 2 ; thus, error propagation may occur.In Fig. 15, a plot of the relative error on the peak current prediction for each simulated event is shown.The strong correlation with Fig. 14 can be clearly observed.In particular, the correlation coefficient between the pointwise location error and the corresponding current prediction error is 81%, indicating that a better peak current estimation could   greatly improve location prediction.Unfortunately, predicting the current is a difficult task, and this can be observed by the large dispersion of the regression error shown in Fig. 16.To assess the maximum theoretical improvement that could be accomplished with a perfect current estimation, the test phase of Case 2 was repeated by normalizing each test event with the corresponding real current, instead of the predicted one, and in this manner, the results were very close to those of Case 1.In particular, the average location error reduced from 290 to 68 m, indicating that the performance of Case 1 can be interpreted as theoretical minimum performance for more practical applications where the current has to be predicted.On the other hand, the increased error in the region near to the line shall not be imputed to error propagation between NN 1 and NN 2 as it is also manifested in Case 1, where the peak current is not predicted, indicating that this behavior is somehow intrinsic in the data-driven approach.Finally, some considerations on the effects of ground conductivity σ, noise, and sampling time are proposed.The ground conductivity mostly affects the radial component of the lightning electric field, which is the source term of the field-to-line coupling differential equations.To assess how this may affect the ML-based prediction, we retrained and retested the algorithm of Case 2 with a database of simulated voltage waveforms computed assuming σ = 5 mS/m.The average location error when σ = 5 mS/m is 270 m and, thus, slightly lower than that obtained with σ = 10 mS/m (290 m).
In order to accommodate the unknown and variable value of ground conductivity, we performed another training and cross-validation round using the composed dataset obtained by combining and shuffling the two aforementioned datasets with σ = 10 mS/m and σ = 5 mS/m.The average location error resulted in 280 m, suggesting that even in the case of variable ground conductivity, the prediction is accurate.
As far as the sampling frequency is concerned, from 10 to 5 MHz led to an unchanged location error of 290 m for the σ = 10 mS/m case.This result is supported by a frequencydomain analysis that indicates that most of the energy of the signals (99%) lies below 2 MHz.This consideration can affect the choice and the core of the measurement system, requiring at least a 5-MHz sampling frequency.
Regarding the effect of the noise, we conducted two experiments by injecting an addictive white Gaussian noise (AWGN) in the time-domain signals, considering first a signal-to-noise ratio (SNR) of 76 dB, corresponding to a noise of 1V RMS , and second an SNR of 56 dB, corresponding to a noise of 10V RMS .The location error with added noise and SNR equal to 76 dB was 324 m, while with the SNR of 56 dB, the location error was 520 m.Both of these noises levels, especially in the second case, are expected to be much larger than a realistic measurement noise; then, the performance in the presence of real noise is likely to be between 290 m (absence of noise) and 324 m (large noise).To perform properly in the presence of noise, the NNs needed to be trained with the noisy data.All of the aforementioned tests and analyses were performed using a tenfold cross-validation approach, and the reported errors are the cross-validation errors.
Future works will extend the proposed approach to more realistic configurations, and this will allow us to test the effect of some devices here neglected (e.g., voltage transformers and their high-frequency behavior, and nonlinear devices like arresters among others) and to assess multiple-sensor systems.In particular, a more realistic configuration with multiple lines equipped with voltage measurement systems and protections devices against overvoltages will be the object of a future work.For what concerns a larger domain of localization, this does not appear an issue, as in Italy the actual density of power lines (14 km/km 2 and 136 km/km 2 for transmission lines and distribution lines, respectively, as declared by the Italian transmission system operator) would guarantee the presence of voltage sensors sufficiently close to the lightning strike.This would allow a prediction with expected performances of the order of magnitude of the one obtained in the present work.

V. CONCLUSION
This article presented an ML-based model for the regression of cloud-to-ground lightning location and peak current from lightning-induced voltage measurements on overhead transmission lines.A shallow feedforward NN was chosen as the ML paradigm.After a first methodological case that allowed to choose the NN hyperparameters and to compare the proposed approach with the existing literature, a more realistic situation is considered, in which a first NN is trained to regress the lightning peak current; then, the feature vector is normalized by the predicted peak current and given as input to a second NN that shall predict lightning location.As far as localization is concerned, the tenfold cross-validation location error obtained was 290 m, which is comparable with the state of the art of LLS: the main advantage is that, contrarily to the traditional LLS, the proposed approach does not require any dedicated sensors' network.As regards the peak current prediction, a 12.5% MAPE was obtained outperforming traditional systems that have been validated against direct measurements only for negative subsequent strokes.Furthermore, an 81% correlation with the location error was found, suggesting that location regression can be improved with better peak current estimation.Future works will extend the proposed approach to more realistic configurations, and this will allow us to test the effect of some devices here neglected (e.g., voltage transformers and their high-frequency behavior, and nonlinear devices like arresters among others) and to assess multiple-sensor systems.Moreover, the possibility of developing a deep learning algorithm will be investigated.

Fig. 1 .
Fig. 1.Top view of the detection domain.The power system is represented by the bold solid line along the x-axis.The two voltage sensors VS1 and VS2 are located on the line 2 km from each other.

Fig. 2 .
Fig. 2. Extract of the simulated lightning-induced voltage measurement database for Case 2 (random I, finite ground conductivity of 10 mS/m).Solid lines represent signals at the voltage sensor VS1 (x = 4 km) and dashed lines represent signals at VS2 (x = 6 km).The location of the stroke and the peak current are reported for each lightning event.

Fig. 4 .
Fig. 4. NN for predicting the strike position in Case 1: all the events have the same peak current.

Fig. 5 .
Fig. 5. NN model for predicting the strike position in Case 2. For the training of NN 2 , the PCA features X are normalized by means of the target current, while at prediction time, the PCA features are normalized by means of the current predicted by NN 1 .

Fig. 6 .
Fig. 6.Location error as a function of the number of hidden neurons for Case 1.

Fig. 7 .
Fig. 7. Regression scatter plot of the predicted x-coordinate for Case 1, showing 2000 test points.

Fig. 8 .
Fig. 8. Regression scatter plot of the predicted y-coordinate for Case 1, showing 2000 test points.

Fig. 10 .
Fig. 10.Scatter plot of the 2000 simulated events in the detection domain.The size of the marker is proportional to the location error (Case 1).

Fig. 5
was also evaluated substituting NN 2 with an SVR model and a K-NN model, while keeping NN 1 to predict the peak currents.The results of the experiments performed are shown in TableII.The mean absolute errors (MAEs) of the location prediction along the x and y coordinates are shown in TableIII.With the two sensors aligned along the x-direction, a better location performance is expected along the x axis.The prediction error of the proposed model in Case 2 is further analyzed in Figs.11-16.As has been observed for Case 1, the model location error distribution represented by the histogram of Fig.13significantly decreases at high values: the 95th percentile

Fig. 11 .
Fig. 11.Regression scatter plot of the predicted x-coordinate for Case 2, showing 2000 test points.

Fig. 12 .
Fig. 12. Regression scatter plot of the predicted y-coordinate for Case 2, showing 2000 test points.

Fig. 14 .
Fig. 14.Scatter plot of the 2000 simulated events in the detection domain.The size of the marker is proportional to the location error (Case 2).

Fig. 15 .
Fig. 15.Scatter plot of the 2000 simulated events in the detection domain.The size of the marker is proportional to the relative error of the predicted peak current (Case 2).

Fig. 16 .
Fig. 16.Regression scatter plot of the predicted peak current for Case 2, showing 2000 test points.
and of the prediction capabilities of the NN.Regarding the other methods applied, SVR gave Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I CASE 1
[22]L ASSUMPTIONS AND RESULTS COMPARED WITH THE METHOD OF KARAMI ET AL.[22]

TABLE IV AVERAGE
LOCATION ERROR FOR CASE 2 WITH DIFFERENT PREDICTED y-COORDINATE CUTOFF VALUES