Channel-base Current Research Articles

Applying High‐Speed Video Images to Inverse Channel Base Current Based on NARX Neural Network

AbstractThe high‐speed video (HSV) images were taken as the input data of the nonlinear autoregressive network (NARX) to reconstruct the return‐stroke channel base current (CBC). The integrated luminosity (IL) of discharge channel is firstly obtained by summing the pixel values in HSV images, and the correlation between the IL and CBC is analyzed as well. Then, a NARX model is trained by taking the IL and CBC as the input and output data, respectively. With the input of the IL of another discharge channel, the trained NARX model is applied to inverse the corresponding CBC. The inversed result suggests that the proposed method can be used to infer the CBC that was not yet measured.

The Analytical Expression of Lightning Channel Base Current by Lorentzian Pulse Fitting

Aiming for the robustness improvement in a noisy environment, we propose a new representation method based on Lorentzian pulse fitting for the analytical expression process of the measured lightning channel base current (LCBC) in this article. The Lorentzian pulse train is determined only by a finite number of free parameters, which can be estimated from the high-energy part of the LCBC signal spectrum. To obtain Fourier coefficients from the LCBC measurements, we design a sum-of-sinc digital filter. In addition, we propose a denoising algorithm based on nuclear norm minimization to improve the accuracy of the parameter estimation process. By using only the high-energy spectrum region, our method is more resistant to noise during the fitting process. Furthermore, although the Lorentzian pulse fitting method cannot strictly achieve the value of LCBC as 0 at the time of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t=0_+$</tex-math></inline-formula> , it provides a simple and integrable expression form in the time domain, which has great application value in the analysis of lightning magnetic field characteristics. Simulation results demonstrate that our proposed method achieves higher analytical expression accuracy for LCBC signal in noisy environments than the classical versions, both in terms of the signal itself and its first derivative.

The Application of the Cholesky-Based Monte Carlo Method to Evaluate Lightning Channel Base Current

An application of the Cholesky-based Monte Carlo method is introduced to generate the lightning characteristics for the statistical analysis of lightning. The method explains how to use this approach to generate the lightning base current characteristics close to the measured parameters of channel base current. Then for about 20 000 different channel base current characteristics generated by this method, the Heidler function parameters are calculated by the genetic algorithm. The obtained Heidler parameters are statistically analyzed. After that, the Cholesky decomposition of Heidler parameters is determined. In the statistical investigations of lightning, this matrix generally allows for the removal of the part of calculations related to the determination of Heidler parameters. The calculation time and the calculation burden are significantly reduced in statistical studies. The obtained results show the effectiveness of this method.

Estimation of lightning channel-base current from far electromagnetic field in the case of inclined channel

• Expressions to estimate the current peak from the far-field peak considering channel tilt are derived. • Expressions to reconstruct the entire current waveform from the field waveform are derived. • The expressions are derived for three most widely used engineering return-stroke models. • This is the first attempt to solve the inverse problem (infer sources from fields) for inclined channels. The inclination of a lightning return-stroke channel significantly affects the waveform of the associated electromagnetic field. In this paper, we have derived mathematical expressions to estimate the peak and waveshape of lightning return-stroke current at the channel base from the waveform of electromagnetic field observed more than some tens of kilometers away from an inclined lightning channel. Three models of the lightning return stroke are considered, which are the transmission-line (TL) model, the modified TL model with linear current decay with height (MTLL), the modified TL model with exponential current decay with height (MTLE). The peak of the lightning return-stroke current at the channel base is estimated from the peak of observed lightning electric or magnetic field using an analytical relation between the peak of channel-base current and that of the radiation field component for an inclined channel. The waveform of the channel-base current is reconstructed from the electric or magnetic field waveform using a relation between the channel-base current and the sum of induction and radiation field components for an inclined channel. It has been shown that the peak and waveshape of current at the base of an inclined channel are sufficiently accurately estimated with the proposed expressions. This work is the first attempt to solve the inverse problem (infer source parameters from electromagnetic fields) for the case of inclined lightning channel. Solving this problem is important for developing methodology of remote measurements of lightning currents.

On the Fourier Transform of Measured Electric Fields Radiated by a Lightning Return Stroke

The Fourier transform (FT) of the electric field (EF) radiated by a lightning return stroke is shown to include a distributional term at zero frequency, which prevents numerical algorithms, such as fast FT, from working properly. Inspired by the finite duration of any field measurement in real applications, we introduce a time-windowing strategy, whereby the integration time-interval of the FT becomes bounded. As a result, the distributional term is removed from the FT, whereas the functional part of the FT is modified throughout the whole range of frequencies, in such a way that a good performance of the numerical algorithms can be restored. The dependence of the modified spectra of the EFs on the length of the time window is also investigated and three numerical experiments with different channel-base currents are performed to validate the whole resulting framework.

Impact of Lightning Channel Base Current (CBC) Function Modeling on Computed Lightning Induced Overvoltage Waveshapes

Adverse effects of lightning strike are studied through the modeling of the lightning-induced over-voltage (LIOV) using the channel-base current (CBC) functions. A lightning current is characterized by the parameters like current peak value (Im), time-taken by the current to reach Im (tm), maximum rise-rate of the current (di/dt)max (steepness), and charge transferred at the contact point of the strike (Q). Using the CBC functions, the lightning electromagnetic fields (LEMFs) are calculated to compute the LIOV using a coupling model. Since the choice of CBC function plays an important role in LIOV computation, the present work studies its effect on the computed LIOV for matching and zero load impedance. As a case study, two functions: Heidlers and NCBC functions were examined. It was shown that a-sort-of delay characteristic from the NCBC function is reflected in the computed LIOV and also the tripolar nature of the LIOV for the case of imperfect ground. The difference between the LIOV computed using NCBC and Heidler?s function shows space-time reflections of the order of <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$10^4$</tex></formula> magnitude.

On the reconstruction of the attenuation function of a return-stroke current from the Fourier Transform of finite-duration measurements

A correct representation of the lightning current is crucial when the electromagnetic field radiated to a point of interest has to be computed. Based on the engineering models of Transmission Line type, such representation involves the knowledge of the return-stroke speed, the channel-base current, the channel height and the attenuation function. Whereas the first three quantities can be measured in different ways, no measurement technique can directly provide reliable information on the attenuation function. In the past decades, researchers have applied various strategies to address this problem. These strategies are all based on a common feature, i.e., the unknown function is postulated a priori and then validated through the comparison of the computed electromagnetic fields at one or more observation points with the corresponding measured waveforms. In this paper, we propose an alternative approach for the identification of the attenuation function: starting from appropriate measurements of the return-stroke speed, the channel-base current, the channel height and the radiated electromagnetic field, we first formulate an algebraic inverse and ill-posed problem, obtained from the discretization of integral equations relating the source to the radiated field in the frequency domain, and then we solve it by means of a Tikhonov regularization technique. The proposed framework is preceded by a detailed theoretical analysis, with special emphasis on the description and filtering of measurement noise and on the minimum duration of the measurement time-windows ensuring reliable results.

Computation of Lightning Current from Electric Field Based on Laplace Transform and Deconvolution Method

A method for computation of the lightning channel base current from the corresponding vertical component of lightning electric field was presented. The algorithm was developed by applying Laplace transform. The lightning current was estimated from its deconvolution with a special transfer function. The transfer function includes information about geometry and physical properties of entire lightning impulse generation system. The method was verified for a Heidler-type base current and a MTLL model of its propagation within the lightning channel. Research was done for close, middle, and far distance to the lightning strike point. Optimum performance was obtained for the middle distance of several kilometers where the electrostatic, induction, and radiation components of the transfer function were of the same range. An analysis was done for input electric field with and without noise superimposed on its time domain waveform. Relative uncertainties for the electric field and calculated lightning channel base current were similar each other. The presented approach can substantially increase a number of lightning current parameters which can be identified on the basis of its electric field signature. This method can be applied by the lightning location systems using preprocessing which increases the timing efficiency of the transfer function estimation.

A survey on analytical solutions and tools for lightning-induced voltages calculations

In this paper, that is an extended version of a work [1] presented at the 2019 International Symposium on Lightning Protection (XV SIPDA), a survey of analytical solutions available in the literature for the evaluation of the lightning-induced voltages in case of both step and linearly rising channel-base currents is presented. Two tools, CiLIV and LIV, based on some of the above mentioned analytical solutions are also presented. Finally, a validation of both analytical solutions and tools is presented.

Modified Transmission Line Model with a Current Attenuation Function Derived from the Lightning Radiation Field—MTLD Model

In return strokes, the parameters that can be measured are the channel base current and the return stroke speed. For this reason, many return stroke models have been developed with these two parameters, among others, as inputs. Here, we concentrate on the current propagation type engineering return stroke models where the return stroke is represented by a current pulse propagating upwards along the leader channel. In the current propagation type return stroke models, in addition to the channel base current and the return stroke speed, the way in which the return stroke current attenuates along the return stroke channel is specified as an input parameter. The goal of this paper is to show that, within the confines of current propagation type models, once the channel base current and the return stroke speed are known, the measured radiation field can be used to evaluate how the return stroke current attenuates along the channel. After giving the mathematics necessary for this inverse transformation, the procedure is illustrated by extracting the current attenuation curve from the typical wave shape of the return stroke current and from the distant radiation field of subsequent return strokes. The derived attenuation curve is used to evaluate both the subsequent and first return stroke electromagnetic fields at different distances. It is shown that all the experimentally observed features can be reproduced by the derived attenuation curve, except for the subsidiary peak and long zero-crossing times. In order to obtain electromagnetic fields of subsequent return strokes that are in agreement with measurements, one has to incorporate the current dispersion into the model. In the case of first return strokes, both current dispersion and reduction in return stroke speed with height are needed to obtain the desired features.

Waveform Reconstruction of Lightning Current Based on Output‐Error Model and Finite‐Difference Time‐Domain Method

AbstractA simple method combining Output‐Error (OE) model and Finite‐Difference Time‐Domain (FDTD) method is proposed to reconstruct lightning current in this paper. First, the FDTD method is adopted to calculate electromagnetic fields of any desired point, and the transfer function between the obtained fields and channel base current is built by using OE model. Then, the lightning current is reconstructed by putting the actual measured electromagnetic field into the corresponding transfer function. The validity of the proposed method is demonstrated through the measured data given in the literature. Moreover, three influence factors on the reconstruction performance of this method are analyzed as well, including the effect of distance from field observation point to lightning channel, ground conductivity and the velocity of return stroke.

Evaluation of an Approximate Channel-Base Current and Its Analytical Function Parameters Based on the Measured Lightning Magnetic Field

In this paper, a new method is proposed for directly evaluating approximate lightning channel-base-current (CBC) parameters ( $I_{m}$ , ( di / dt )max, t FD, and t SD) and analytical function parameters (AFPs) of the CBC based on the measured lightning magnetic fields using a deconvolution and differential evolution (DE) method. This proposed method comprises two sequential steps. The first step is for determining the lightning CBC based on the measured magnetic field using modified transmission line model with exponential current decaying with height (MTLE) through deconvolution, while the second step is for evaluating the AFPs (Heidler's function) corresponding to the lightning CBC obtained from the first step. Data from the measured magnetic field above ground level ( z = 10 m) at different horizontal distances ( r = 9 km, 4.6 km, 2 km, and 50 m) from the lightning channel are considered for testing the proposed method. The sum of two Heidler's functions is used to represent the lightning CBC; and its parameters, namely I 01, I 02, τ 11, τ 12, τ 21, τ 22, n 1 , and n 2 are identified using DE as a tool. Further, this methodology is helpful in the evaluation of the CBC parameters and mathematical modeling of the lightning CBC and the estimation of the lightning current through the channel using data from the measured magnetic field for different engineering return stroke models.

Parametric and statistical investigations of lightning-induced voltages on overhead lines by exact analytical solutions

In this paper, which is an extended version of the paper presented by the authors at the 34th International Conference on Lightning Protection (ICLP 2018), a parametric and a statistical analysis of lightning-induced voltages on overhead lines has been carried out by means of exact analytical solutions, for both step-function and linearly rising channel-base currents. The role played by the various parameters affecting the induced voltages has been analyzed. Further, the statistical investigations has allowed us to identify the parametric statistical distributions which best fit the data obtained by simulations. Based on this statistical investigation, an analysis of the lightning performance of an overhead line has been carried out too.

A New Inversion Algorithm for Estimating the Return-Stroke Velocity

To study the lightning return-stroke process, this paper proposed a new inversion algorithm for evaluating the average return-stroke velocity. As the Gaussian function covers a frequency range wider than the typical channel-base current, the main step of this method was to treat the channel-base current representing using the Gaussian function and the corresponding azimuthal magnetic field at close distance as the inputs and outputs, respectively, to establish the transfer functions. We have verified the proposed method using the azimuthal magnetic field with/without noises at different distances from the lightning channel. In addition, we compared the evaluated results using different return-stroke models including the transmission line (TL) model, the modified TL model with linear (MTLL) current decay with height, and the modified TL model with exponential (MTLE) current decay with height. The evaluated results are in agreement with the simulated return-stroke velocity, indicating the effectiveness and robustness of this method for predicting the return-stroke velocity. The proposed inversion method is effective in estimating the return-stroke velocity, which can be applied in lightning physics research.

Lightning Channel-Base-Current Estimation Using Engineering Return-Stroke Models From Measured Magnetic Field Based on Deconvolution Method

Estimation and analysis of the lightning channel-base-current (CBC) are essential requirements in the process of designing protection systems against lightning strokes. Thus, a new method is proposed to evaluate the lightning CBC based on the measured magnetic field using the deconvolution method. Engineering return-stroke models, such as transmission line (TL), modified TL with linear current decaying with height (MTLL), and modified TL with exponential current decaying with height (MTLE) models are effectively utilized in the evaluation of lightning CBC from the measured magnetic field. The data of measured magnetic field above the ground surface at four radial distances from the lightning channel, reported in the literature, are considered as input to the proposed method. The peak of lightning CBC is calculated for return-stroke velocities in range one-third to two-third the velocity of light in free space, specifically at 100, 125, 150, 175, and 200 m/μs for aforementioned measured magnetic field data using MTLE, MTLL, and TL models. The variations in the peak currents, corresponding to the variation in return-stroke velocities, are also observed. Closeness in the peak of simulated CBC between each of the models used in the study is analyzed and reported. Furthermore, this approach will be helpful for better understanding while designing protection systems against lightning.

Broadband Characteristics of Chaotic Pulse Trains Associated With Sequential Dart Leaders in a Rocket‐Triggered Lightning Flash

AbstractBased on simultaneous measurements of fast E‐field change, very high frequency radiation and channel mapping, channel base current, and high‐speed video, the characteristics of chaotic pulse trains (CPTs) during propagation of dart leaders in a rocket‐triggered lightning were analyzed. The triggered lightning involved 16 subsequent leader‐return strokes traversing in the same grounding channel. At least five dart leaders showed significant CPT signals during its propagation, with much larger intensity than that in normal leader stage at broadband of low‐frequency–medium‐frequency, very high frequency, and visible light band. All CPTs occurred in the final stage of the leader development, within 150 μs prior to the return strokes. The leader velocity and the charge transfer are both related to the instantaneous intensity of CPT, but the relationship is not always solid when the two parameters are considered individually. Nevertheless, the large product of return stroke charge to 1 ms and the velocity of the leader (QV) could be a reliable proxy/indicator for CPT. All CPTs correspond to a larger local QV with a threshold higher than 1 × 107 C · m/s when CPT is determined as chaotic electric field pulses with magnitude larger than 3 std of the magnitude of E‐fluctuation waveform of the whole flash.

Evaluating Lightning Channel-Base-Current Function Parameters for Identifying Interdependence of Wavefront and Tail by PSO Method

Lightning channel-base-current (CBC) function parameters for negative first return stroke (NFS), negative subsequent return stroke (NSS), and positive first return stroke (PFS) are evaluated using particle swarm optimization (PSO) technique. In this paper, parameters ( I01 , I02 , τ 11 , τ12 , τ21 , τ22 , n1, and n2 ) in the sum of two Heidler's CBC function are identified using PSO as a tool for the typical (50%) and severe (5%) cases of NFS, NSS, and PFS reported by IEC Standard 62305-1, 2010. The result of PSO tuned CBC, the error between computed and measured as given in the standard was calculated and a good agreement was observed. Heidler's function parameters (HFPs) are also tuned to keep four constants with systematic variation of one out of all prominent parameters ( I m , (di/dt) max, tFD , tSD , and Q ) of full (front and tail) CBC wave of NFS and NSS. Interdependence between these parameters is scrutinized during resemblance of the computed and the measured CBC. This is an attempt to show that a PSO can be used for identifying HFPs to straightforwardly obtain the CBC full wave. Thus, the data generated and reported including extreme severe case (1%) can produce a tailor-made full wave of CBC. Furthermore, this approach will be needful in research related to the lightning CBC characteristics.

Use of PSO to determine lightning channel‐base‐current function parameters for standard severe negative first and subsequent return stroke approximation

In this study, particle swarm optimisation (PSO) is applied to determine the Heidler's function parameters (HFPs) for approximating the measured full lightning channel-base-current (CBC) of the severe (5% data in IEC standard 62305-1, 2010) negative first return stroke (NFS) and subsequent return stroke (NSS). The parameters in the sum of two Heidler's functions, namely I 01 , I 02 , τ 11 , τ 12 , τ 21 , τ 22 , n 1 , and n 2 , are identified using PSO as a tool. The HFPs for severe NFS and NSS are tuned to keep four parameters constant with systematic variation of one out of all prominent parameters (i.e. peak current ( I m ), maximum time rate of change of current ((d i /d t ) max ), front duration ( t FD ), stroke duration ( t SD ), and total charge ( Q )) of full CBC wave and vice versa. The corresponding data is reported, and an independent controlling of HFPs is analysed to achieve the desired full CBC wave. Also, an attempt has been made in this study to examine the interdependence between the wavefront and tail of CBC using PSO. The data thus generated and reported for severe NFS and NSS can be used to generate the suitable full CBC waveforms. This methodology will be helpful in lightning-related simulation study and research.

Representation of Severe Negative Subsequent Return Stroke by Optimization based Channel-Base-Current Function Parameters

In this paper, the lightning channel-base-current (CBC) function parameters are identified using four different optimization techniques (such as genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), and sine cosine algorithm (SCA)) to representing the measured data given in IEC standard 62305-1, 2010 for the severe (5%) case negative subsequent return stroke (NSS). The sum of two Heidler’s functions (HFs) is used for representing the desired severe NSS, and its parameters (namely, I01, I02, τ11, τ12, τ21, τ22, n1, and n2) are identified using GA, PSO, DE, and SCA. The data thus generated and reported for approximating the measured severe NSS. During this process, the data obtained by SCA showing a good agreement between the computed and measured CBC as compared to the data obtained by GA, PSO, and DE. However, the data provided by all above mentioned optimization algorithms can be used to observe the behavior of the return stroke current at different heights through a lightning channel, and this will useful in calculation of lightning electromagnetic field (LEMF).

Fast evaluation of lightning‐induced voltages of overhead line and buried cable considering the lossy ground

In this study, an efficient method for the evaluation of lightning-induced voltages on a single-conductor overhead line and a single-core buried cable is proposed. In this method, the numerical evaluation of the temporal convolution due to the lossy ground is simplified by a full consideration of the inherent relation between the frequency-dependent ground impedance per-unit-length (p.u.l.) and admittance p.u.l., based on which, the evaluation of the lightning-induced voltages on an overhead line or buried cable system can be implemented in an efficient way. To further improve the efficiency, the frequency-domain method of evaluation of the lightning horizontal electric fields above or below the ground surface is modified by dividing the lightning channel base current into high-frequency and low-frequency components. Some numerical examples are conducted under MATLAB platform to validate the proposed method and the efficiency is discussed.