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An identification procedure for lightning return strokes
Journal of Electrostatics 51}52 (2001) 326}332
An identi"cation procedure for lightning return strokes Amedeo Andreotti , Federico Del"no *, Paola Girdinio, Luigi Verolino Dipartimento di Ingegneria Elettrica, Universita% degli studi di Napoli Federico II, via Claudio 21, I-80125 Napoli, Italy Dipartimento di Ingegneria Elettrica, Universita% degli studi di Genova, via Opera Pia 11a, I-16145 Genova, Italy
Abstract In this paper an inverse procedure for the identi"cation and the reconstruction of the waveform of the lightning return stroke current is presented. It is based on the acquisition of the vertical component of the electric "eld radiated by the discharge channel at di!erent locations on the ground and it does not require any information about the channel base current. The approach has been validated by means of the numerical simulation of the classical Transmission Line, Modi"ed Transmission Line Linear and Modi"ed Transmission Line Exponential engineering models, showing good accuracy in all cases. 2001 Elsevier Science B.V. All rights reserved. Keywords: Lightning return stroke model; Recording of lightning; Electric "eld
1. Introduction Return stroke models are mathematical expressions, which allow to determine the observed properties of the lightning return stroke [1,2]. There are several categories of return stroke models, namely gas or physical models, electromagnetic models, distributed circuit models and engineering models. In this paper, we shall deal with the last ones. Engineering models allow to calculate the lightning channel current i(z, t) at any time t and at any height z, relating it to the channel base current i(0, t) by means of a suitable height dependent attenuation function P(z). * Corresponding author. E-mail address: federico.del"[email protected] (F. Del"no). 0304-3886/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 0 1 ) 0 0 0 9 7 - 3
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
327
Our purpose is to describe the possibility of identifying P(z) by means of an inverse procedure making use of only the measured vertical component of the electric "eld. We will show that the developed procedure can be performed without any information about the channel base current, whose measurements are generally very noisy. In this paper, a static application of the identi"cation algorithm is "rstly presented in order to highlight its main peculiarities.
2. Electrostatic case Let us consider the situation sketched in Fig. 1, where a vertical linear distribution of charges lies above a perfectly conducting plane. Indicating with P(0, 0, z) the source point, with P(x, y, z) the observation point and with (z) the linear charge density, the vertical component of the electrostatic "eld can be obtained by the following expression:
& 1 z!z z#z E (r, z)" (z) ! dz, X 4 [r#(z!z)] [r#(z#z)]
(1)
being the dielectric constant of the free space and r"x#y. In particular, at the ground level one has
(uH) ! u
E (r, 0)" du, X 2 H (u#r/H)
(2)
where u"z/H and is the maximum value of the charge density.
Our aim is to determine (z) having measured E (r, 0) using electrostatic "eld X sensors placed on the ground. It is therefore necessary to solve (2) with respect to F(u)"(uH)/ , but this equation, namely a Fredholm equation of the "rst kind
[3], exhibits numerical problems, which suggest us to face it by means of an adequate
Fig. 1. Geometry of the static problem.
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A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
expansion. To this purpose, let F(u) be solved as F(u)" p (u), I I I
(3)
where (u) is a complete basis in the considered functional space and p are the I I expansion coe$cients. Calling "r/H, using N "eld sensors and arresting to N the series (3), (2) can be rewritten as follows:
!2 H , u E (r , 0) (u) du p , X Q I I (u#)
Q I
(4)
which represents a N;N linear system whose unknown vector is p"[p , p , 2 , p ]. , Solving (4) one can obtain the p coe$cients and inserting them into (3), the expresI sion for F(u). Several bases can be chosen for the representation (3), as the classical piecewise constant functions or the Fourier expansions, but, due to the fact that inverse problems are ill-posed [4], it frequently happens that a theoretical available basis can provide extremely large number of expansion functions or directly unfruitful results in the reconstruction process. For instance, let us try and reconstruct the function F(u)"e\S, with H"19 km. We decided to adopt a modi"ed basis of Chebyshev polynomials [5] of the kind (u)"¹ (u)"cos[(k!1)arccos(u)]. This basis I I\ allows to get a good accuracy in the identi"cation of F(u)"e\S using only few terms, provided that the "eld sensors are equally distanced on the ground and placed in the range (50m}15 km). With such provisos, only N"10 "eld sensors are required to obtain the exponential waveform (Fig. 2).
Fig. 2. Reconstruction of the linear charge density.
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
329
3. Dynamic case In strict analogy with the static case, let us examine now the "eld problem associated to a lightning discharge. In order to settle a mathematical model of the problem, we start making the following two reasonable assumptions: (i) the lightning channel is considered as a vertical antenna above a perfectly conducting plane (Fig. 3); (ii) the return stroke wavefront starts traveling up at time zero. According to (i) and (ii), the vertical component of the radiated electric "eld is given, in the frequency domain, by the following equation:
& 1 R E (r, z, )" G (r, z, z,)exp !j I(z,) dz, X 4 c \&
(5)
where is the angular frequency, c is the speed of light in free space, G is a suitable X Green function depending on the physical and geometrical parameters of the problem and I(z, ) is the return stroke current, expressed by the following relation [2,6]:
z I(z, )"I ()exp !j P(z), v
(6)
where v is the current wave propagation speed and I () is the channel base current. If the observation point P is at ground level, E becomes X
I () & R z E (r, 0, )" G (r, 0, z, )exp !j # X 4 v c \&
Fig. 3. Geometry of the dynamic problem.
P(z) dz.
(7)
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A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
Eq. (7) is, as in the static case, a Fredholm integral equation of the "rst kind in the unknown P(z). If we can measure E we can derive the expression for P(z), provided X that it is suitably expanded. According to the same arguments described in the previous section, we propose to expand the function P(z) directly as
z z P(z)" p ¹ " p cos (k!1)arccos . (8) I I\ H I H I I Now, we imagine to dispose N sensors on the ground at distances r (s"1, 2,2, N) Q from the lightning channel in order to measure E . Let us indicate with the ratio X between the current wave propagation speed, v, and the speed of light in the free space, c. If the series (8) is truncated to N and is de"ned, for shortness, as "I ()p , (7) I I I becomes
& , z z 4 E () G (r , z, )exp !j R# ¹ dz. (9) XQ I X Q Q I\ H c \& I For assigned , (9) represents a linear system of N complex equations in the unknowns (k"1, 2, 2, N). I It is worth noting that (9) allows to determine both the attenuation function P(z) and I (). As a matter of fact, one can be easily convinced that the value of the channel base current at the given working frequency can be reconstructed noting that P(z) is a real function and that P(0)"1. As a consequence, it must follow that , I ()" (0). (10) I I I Once the value of I () has been determined, the expansion coe$cients p can be I extracted from the solution vector in the following way: I I p " . (11) I , ¹ (0) I I\ I The approach above described has been validated for a wide class of return strokes, with heights varying from 15 km up to 30 km and with a propagation speed v"1.9;10 m/s. The "eld sensors have been equally spaced on the ground in the distance range 1}2.5 km from the lightning channel (for longer distances the assumption of perfectly conducting ground fails) and the working frequency has been set to 100 MHz, in order to show that the algorithm works well over the typical lightning frequencies. The dependence of the procedure on the chosen distance interval for sampling the "eld has been studied. In this case, the algorithm has shown a good stability, provided that the extension of the interval is not lower than 800 m. No dependence on the position of the initial and the "nal sampling point has been highlighted.
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
331
Fig. 4. MTLE reconstruction (H"19 km) with N"12 expansion functions.
The proposed method has been applied to the reconstruction of standard models of the attenuation function P(z)"1 (Transmission Line model, TL); P(z)"1!z/H (Modixed Transmission Line Linear model, MTLL); P(z)"e\ XY (Modixed Transmission Line Exponential model, MTLE). TL and MTLL models have been identi"ed respectively with one and two expansion functions, since they are contained in the adopted functional basis. As far as the most complicated MTLE model is concerned, a good level of accuracy is reached with N"12 expansion functions (Fig. 4) and is maintained till N"20. After this threshold value, the reconstruction error starts growing again, showing the typical behavior of an inverse problem's error. The proposed inverse procedure correctly reconstructs the attenuation function for any height of the lightning channel in the range 15}30 km. The identi"cations of the MTLE model for return strokes at the boundary of this interval, i.e., for H"15 and for H"30 km, have shown errors (calculated as the di!erence between the imaginary part of the reconstructed function and the imaginary part of the real function, which is identically zero) varying, respectively, from 10\ (H"15 km) to 10\ (H"30 km).
4. Conclusions We have presented a new method to identify the return stroke characteristics by means of "eld measurements at di!erent locations on the ground. The method has permitted to reconstruct the height-dependent attenuation function for a wide class of return strokes and for a given working frequency, showing good accuracy with few expansion terms.
332
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
References [1] C. Gomes, V. Cooray, Concepts of lightning return stroke models, IEEE Trans. Electromag. Comp. 42 (2000) 82}96. [2] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromag. Compat. 40 (1998) 403}426. [3] M.L. Krasnov, A.I. Kisselev, G.I. Makarenko, Equations Integrales, MIR, Moscow, 1977, French translation. [4] A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, DC, 1977. [5] E. Isaacson, H.B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. [6] A. Andreotti, U. De Martinis and L. Verolino, A procedure for the return stroke identi"cation, IEEE Trans. Electromag. Compat. 43 (2) (2001).
An identi"cation procedure for lightning return strokes Amedeo Andreotti , Federico Del"no *, Paola Girdinio, Luigi Verolino Dipartimento di Ingegneria Elettrica, Universita% degli studi di Napoli Federico II, via Claudio 21, I-80125 Napoli, Italy Dipartimento di Ingegneria Elettrica, Universita% degli studi di Genova, via Opera Pia 11a, I-16145 Genova, Italy
Abstract In this paper an inverse procedure for the identi"cation and the reconstruction of the waveform of the lightning return stroke current is presented. It is based on the acquisition of the vertical component of the electric "eld radiated by the discharge channel at di!erent locations on the ground and it does not require any information about the channel base current. The approach has been validated by means of the numerical simulation of the classical Transmission Line, Modi"ed Transmission Line Linear and Modi"ed Transmission Line Exponential engineering models, showing good accuracy in all cases. 2001 Elsevier Science B.V. All rights reserved. Keywords: Lightning return stroke model; Recording of lightning; Electric "eld
1. Introduction Return stroke models are mathematical expressions, which allow to determine the observed properties of the lightning return stroke [1,2]. There are several categories of return stroke models, namely gas or physical models, electromagnetic models, distributed circuit models and engineering models. In this paper, we shall deal with the last ones. Engineering models allow to calculate the lightning channel current i(z, t) at any time t and at any height z, relating it to the channel base current i(0, t) by means of a suitable height dependent attenuation function P(z). * Corresponding author. E-mail address: federico.del"[email protected] (F. Del"no). 0304-3886/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 0 1 ) 0 0 0 9 7 - 3
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
327
Our purpose is to describe the possibility of identifying P(z) by means of an inverse procedure making use of only the measured vertical component of the electric "eld. We will show that the developed procedure can be performed without any information about the channel base current, whose measurements are generally very noisy. In this paper, a static application of the identi"cation algorithm is "rstly presented in order to highlight its main peculiarities.
2. Electrostatic case Let us consider the situation sketched in Fig. 1, where a vertical linear distribution of charges lies above a perfectly conducting plane. Indicating with P(0, 0, z) the source point, with P(x, y, z) the observation point and with (z) the linear charge density, the vertical component of the electrostatic "eld can be obtained by the following expression:
& 1 z!z z#z E (r, z)" (z) ! dz, X 4 [r#(z!z)] [r#(z#z)]
(1)
being the dielectric constant of the free space and r"x#y. In particular, at the ground level one has
(uH) ! u
E (r, 0)" du, X 2 H (u#r/H)
(2)
where u"z/H and is the maximum value of the charge density.
Our aim is to determine (z) having measured E (r, 0) using electrostatic "eld X sensors placed on the ground. It is therefore necessary to solve (2) with respect to F(u)"(uH)/ , but this equation, namely a Fredholm equation of the "rst kind
[3], exhibits numerical problems, which suggest us to face it by means of an adequate
Fig. 1. Geometry of the static problem.
328
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
expansion. To this purpose, let F(u) be solved as F(u)" p (u), I I I
(3)
where (u) is a complete basis in the considered functional space and p are the I I expansion coe$cients. Calling "r/H, using N "eld sensors and arresting to N the series (3), (2) can be rewritten as follows:
!2 H , u E (r , 0) (u) du p , X Q I I (u#)
Q I
(4)
which represents a N;N linear system whose unknown vector is p"[p , p , 2 , p ]. , Solving (4) one can obtain the p coe$cients and inserting them into (3), the expresI sion for F(u). Several bases can be chosen for the representation (3), as the classical piecewise constant functions or the Fourier expansions, but, due to the fact that inverse problems are ill-posed [4], it frequently happens that a theoretical available basis can provide extremely large number of expansion functions or directly unfruitful results in the reconstruction process. For instance, let us try and reconstruct the function F(u)"e\S, with H"19 km. We decided to adopt a modi"ed basis of Chebyshev polynomials [5] of the kind (u)"¹ (u)"cos[(k!1)arccos(u)]. This basis I I\ allows to get a good accuracy in the identi"cation of F(u)"e\S using only few terms, provided that the "eld sensors are equally distanced on the ground and placed in the range (50m}15 km). With such provisos, only N"10 "eld sensors are required to obtain the exponential waveform (Fig. 2).
Fig. 2. Reconstruction of the linear charge density.
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
329
3. Dynamic case In strict analogy with the static case, let us examine now the "eld problem associated to a lightning discharge. In order to settle a mathematical model of the problem, we start making the following two reasonable assumptions: (i) the lightning channel is considered as a vertical antenna above a perfectly conducting plane (Fig. 3); (ii) the return stroke wavefront starts traveling up at time zero. According to (i) and (ii), the vertical component of the radiated electric "eld is given, in the frequency domain, by the following equation:
& 1 R E (r, z, )" G (r, z, z,)exp !j I(z,) dz, X 4 c \&
(5)
where is the angular frequency, c is the speed of light in free space, G is a suitable X Green function depending on the physical and geometrical parameters of the problem and I(z, ) is the return stroke current, expressed by the following relation [2,6]:
z I(z, )"I ()exp !j P(z), v
(6)
where v is the current wave propagation speed and I () is the channel base current. If the observation point P is at ground level, E becomes X
I () & R z E (r, 0, )" G (r, 0, z, )exp !j # X 4 v c \&
Fig. 3. Geometry of the dynamic problem.
P(z) dz.
(7)
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A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
Eq. (7) is, as in the static case, a Fredholm integral equation of the "rst kind in the unknown P(z). If we can measure E we can derive the expression for P(z), provided X that it is suitably expanded. According to the same arguments described in the previous section, we propose to expand the function P(z) directly as
z z P(z)" p ¹ " p cos (k!1)arccos . (8) I I\ H I H I I Now, we imagine to dispose N sensors on the ground at distances r (s"1, 2,2, N) Q from the lightning channel in order to measure E . Let us indicate with the ratio X between the current wave propagation speed, v, and the speed of light in the free space, c. If the series (8) is truncated to N and is de"ned, for shortness, as "I ()p , (7) I I I becomes
& , z z 4 E () G (r , z, )exp !j R# ¹ dz. (9) XQ I X Q Q I\ H c \& I For assigned , (9) represents a linear system of N complex equations in the unknowns (k"1, 2, 2, N). I It is worth noting that (9) allows to determine both the attenuation function P(z) and I (). As a matter of fact, one can be easily convinced that the value of the channel base current at the given working frequency can be reconstructed noting that P(z) is a real function and that P(0)"1. As a consequence, it must follow that , I ()" (0). (10) I I I Once the value of I () has been determined, the expansion coe$cients p can be I extracted from the solution vector in the following way: I I p " . (11) I , ¹ (0) I I\ I The approach above described has been validated for a wide class of return strokes, with heights varying from 15 km up to 30 km and with a propagation speed v"1.9;10 m/s. The "eld sensors have been equally spaced on the ground in the distance range 1}2.5 km from the lightning channel (for longer distances the assumption of perfectly conducting ground fails) and the working frequency has been set to 100 MHz, in order to show that the algorithm works well over the typical lightning frequencies. The dependence of the procedure on the chosen distance interval for sampling the "eld has been studied. In this case, the algorithm has shown a good stability, provided that the extension of the interval is not lower than 800 m. No dependence on the position of the initial and the "nal sampling point has been highlighted.
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
331
Fig. 4. MTLE reconstruction (H"19 km) with N"12 expansion functions.
The proposed method has been applied to the reconstruction of standard models of the attenuation function P(z)"1 (Transmission Line model, TL); P(z)"1!z/H (Modixed Transmission Line Linear model, MTLL); P(z)"e\ XY (Modixed Transmission Line Exponential model, MTLE). TL and MTLL models have been identi"ed respectively with one and two expansion functions, since they are contained in the adopted functional basis. As far as the most complicated MTLE model is concerned, a good level of accuracy is reached with N"12 expansion functions (Fig. 4) and is maintained till N"20. After this threshold value, the reconstruction error starts growing again, showing the typical behavior of an inverse problem's error. The proposed inverse procedure correctly reconstructs the attenuation function for any height of the lightning channel in the range 15}30 km. The identi"cations of the MTLE model for return strokes at the boundary of this interval, i.e., for H"15 and for H"30 km, have shown errors (calculated as the di!erence between the imaginary part of the reconstructed function and the imaginary part of the real function, which is identically zero) varying, respectively, from 10\ (H"15 km) to 10\ (H"30 km).
4. Conclusions We have presented a new method to identify the return stroke characteristics by means of "eld measurements at di!erent locations on the ground. The method has permitted to reconstruct the height-dependent attenuation function for a wide class of return strokes and for a given working frequency, showing good accuracy with few expansion terms.
332
A. Andreotti et al. / Journal of Electrostatics 51}52 (2001) 326}332
References [1] C. Gomes, V. Cooray, Concepts of lightning return stroke models, IEEE Trans. Electromag. Comp. 42 (2000) 82}96. [2] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromag. Compat. 40 (1998) 403}426. [3] M.L. Krasnov, A.I. Kisselev, G.I. Makarenko, Equations Integrales, MIR, Moscow, 1977, French translation. [4] A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, DC, 1977. [5] E. Isaacson, H.B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. [6] A. Andreotti, U. De Martinis and L. Verolino, A procedure for the return stroke identi"cation, IEEE Trans. Electromag. Compat. 43 (2) (2001).