An Algorithm for the Exact Evaluation of the ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 2, MAY 2007

401

An Algorithm for the Exact Evaluation of the Underground Lightning Electromagnetic Fields Federico Delfino, Member, IEEE, Renato Procopio, Member, IEEE, Mansueto Rossi, Farhad Rachidi, Senior Member, IEEE, and Carlo Alberto Nucci, Fellow, IEEE

Abstract—The number of power installations lying underground has been increasing in the last few years, and such devices are very sensitive to the effect of the lightning electromagnetic fields, due to the massive presence of power electronics. As a consequence, the scientific community has devoted much effort in the direction of a more accurate modeling of underground lightning fields and their coupling to cables. The exact expressions of the underground lightning fields have been derived by Sommerfeld decades ago. However, their numerical evaluation has always been a hard task because of the presence of slowly converging improper integrals. In the past, some approximate formulas have been derived, which have been included in field-to-transmission line coupling models to estimate the effect of lightning on buried cables. In this paper, an efficient algorithm for the evaluation of the Sommerfeld expression for underground fields is presented, and its mathematical features are discussed. The numerical treatment of the Sommerfeld integrals is based on a proper subdivision of the integration domain, the application of the Romberg technique, and the definition of a suitable upper bound for the error due to the integral truncation. The remarkable efficiency in terms of CPU time of the developed algorithm makes it possible to use it directly in field-to-buried cable coupling simulation codes. Finally, the developed algorithm is used to test the validity of the Cooray’s simplified formula for the computation of underground horizontal electric field. It is shown that predictions of the Cooray’s formula are in good agreement with exact solutions for large values of ground conductivity (0.01 S/m). However, for poor conductivities (0.001 S/m or so), Cooray’s expression yields less satisfactory results, especially for the late time response. Index Terms—Electromagnetic propagation in absorbing media, lightning, modeling.

I. INTRODUCTION N THE past, much attention had been paid to the problem of the interaction between lightning electromagnetic fields and overhead transmission lines. This had led to the formulation of different reliable field-to-transmission line coupling models [1], [2]. All these models require an accurate evaluation of the lightning electromagnetic fields along the line, taking into account the effect of the ground conductivity [3]. Unfortunately, the presence of a lossy ground in the model implies the computation of the slowly converging Sommerfeld integrals [4], thus

I

Manuscript received April 13, 2006; revised November 27, 2006. F. Delfino, R. Procopio, and M. Rossi are with the Electrical Engineering Department, University of Genoa, Genoa I-16145, Italy (e-mail: federico. [email protected]; [email protected]; [email protected]). F. Rachidi is with the Electromagnetic Compatibility Group, Swiss Federal Institute of Technology, Lausanne CH-1015, Switzerland (e-mail: [email protected]). C. A. Nucci is with the Department of Electrical Engineering, University of Bologna, Bologna I-40136, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TEMC.2007.897127

making the performance of the algorithms prohibitive in terms of CPU time, especially for long lines. As a result, many simplified approaches have been derived to overcome this problem [5]–[8], and some efficient algorithms have been proposed to evaluate the exact expressions in order to test the validity of the approximate formulas [9], [10]. On the other hand, a large number of power and telecommunication lines are laid underground, and since the electronic devices present in such installations are very sensitive toward the effect of lightning-induced electromagnetic fields, a huge effort has recently been made to obtain models which can describe adequately the phenomenon of the coupling of lightning fields to underground cables [11]–[14]. Obviously, the inputs for these models are the underground lightning fields. Their expressions have been derived originally by Sommerfeld [4], and later studied by Banos [15] and Bannister [16], but from a numerical point of view, they exhibit the same general features as the above-ground fields. Again, as for the case of overhead lines, an approximate formula has been proposed [17] and adopted for the simulation of lightning induced disturbances in buried cables [18], [19]. In this paper, an efficient algorithm for the evaluation of the exact expressions for the lightning underground electromagnetic fields is presented, and its mathematical features are thoroughly discussed. Moreover, simulations are performed with the aim of comparing this approach with the approximate formulas and testing their accuracy. The paper is organized as follows. In Section II, the derivation of the exact expression for the underground fields is presented. Then, the main features of the algorithm developed for the Sommerfeld integrals’ numerical treatment are outlined. In Section III, the approximate formula is presented and some numerical experiments are performed in order to test its validity. Finally, some conclusive remarks are drawn in Section IV. II. DERIVATION OF THE EXACT FORMULAS FOR THE UNDERGROUND FIELDS In this section the derivation of the exact expression for the underground lightning electromagntic fields is presented. In Section II-A, the solution of the hertzian dipole Sommerfeld problem inside the ground is briefly recalled (see [4] for its complete derivation). Such solution, known as Green’s function, is useful to obtain the lightning fields, as explained in Section II-B. A. Green’s function The derivation of the Green’s functions for the electromagnetic fields (i.e., the expression of the fields due to a vertical

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Fig. 2.

Fig. 1.

Model geometry of the electric dipole radiation in the subsoil.

dipole lying at height z
over a lossy ground [4]) is presented. In Fig. 1, a vertical dipole is placed at source point P
(0, 0, z
), while the observation point is P(r, φ, z), with z < 0. The upper half-space is air, which is assumed to be lossless and characterized by magnetic permeability µ0 and permittivity ε0 . The lossy ground (lower half space) has a conductivity σ and a permittivity ε. As is well known [4], the set of equations solving this problem is   = −µ0 δ(P − P
)e3 z > 0  + k2 A  ∆A     E + k2 A   ∆A E E =0 z 0.

Fig. 3.

Path of integration for the Sommerfeld integrals.

decay constant (MTLE model [21], [22]), the expressions for the lightning field components become 

jI(0, ω) +∞ λ3   = J0 (λr)Q(λ)eµE z dλ E  zL  2πωε0 0 n2 µ + µE    jI(0, ω) +∞ λ2 µE J1 (λr)Q(λ)eµE z 2 dλ ErL = −  2πωε0 0 n µ + µE     n2 I(0, ω) +∞ λ2   HϕL = − J1 (λr)Q(λ)eµE z dλ. 2µ + µ 2π n E 0 (7) The function Q appearing in (7) is the result of the integration along the channel, namely   ω 1   1 ω j − µ − H

H j −µ− v α e −1 α dz
=   e v . Q(λ) = 1 ω 0 j −µ− v α (8) The integral

+∞ µm I= Jυ (λr)eµE z 2 E λn Q(λ)dλ (9) n µ + µ E 0 appearing in (7) with different values of n, m, and υ is known as the Sommerfeld integral. In this section, the main mathematical features of this integral are analyzed in order to find out a fast and reliable procedure for their numerical evaluation. C. Branch Points The integrand in (9) is a function of the complex variable λ and is not uniquely determined because of the square roots µg and µE that appear in it. Corresponding to the four combinations of signs of µg and µE , the integral (9) is four valued, and its Riemann surface has four sheets. In order to ensure the convergence of the integral and its vanishing for z to ± ∞, one must take µ = λ2 − k 2 with positive real part. As shown in [4], µE also has to be taken with positive real part in order to make convergent the integrals expressing the vector potential and the fields inside the earth. Stating this rule of signs, only one of the four sheets is singled out as a “permissible sheet.” This means that in performing the integration, it is necessary that the chosen path lies only on the permissible sheet. This is achieved by joining the two branch points µ and µE by two branch cuts, which may not be intersected by the path of integration. The chosen path is the one depicted in Fig. 3, in which the integration variable is u = λ/k, so as the branch points occur at u = 1 and u = kE /k [23].

(10)

Recalling the definition of µ2E , it readily follows that  2 Im µE < 0. So, indicating with φ as the phase angle of µ2E , one has that π < φ < 2π. As a consequence, if θ is the phase angle of µE , either θ = (φ/2) or θ = (φ/2) + π can be chosen. In the second case, we have (3π/2) < θ < 2π while in the first we have (π/2) < θ < π. So, to meet the requirement in (10), θ = (φ/2) + π must be chosen, and as a consequence, it results in

  φ 1 + cos φ φ + π = − cos = + (11) cos θ = cos 2 2 2

  φ φ 1 − cos φ sin θ = sin + π = − sin = − . (12) 2 2 2 Now, expressing µE in polar form, it follows that  

1 + cos φ 1 − cos φ −j µE = |µE | 2 2

(13)

in which λ2 −

cos φ = 

λ − 2

 |µE | = 4

ε 2 ε0 k

ε 2 ε0 k

2

ε λ − k2 ε0 2

+

(14) σ 2 k 2 µε00

2 + σ2 k2

µ0 ε0

(15)

where the square roots are all real functions and j 2 = −1. As far as µ is concerned, one can observe that µ = lim µE , σ→0+ thus resulting in ε→ε0  2 for λ > k λ − k 2 , (16) µ= 2 2 for λ < k. −j k − λ , D. σ-Dependent Terms µE 1 The terms gsr (λ) = n2 µ+µ and gsz (λ) = n2 µ+µ appearing E E in (5) and (7) are the terms in which the dependence of the fields on the ground conductivity is contained, since they would be zero if the ground were a perfect conductor. As a matter of fact, for a fixed k, they are functions of λ. It can be verified that their behavior shows some sort of resonance for λ = k, i.e., their first derivative in the neighborhood of k is very high. By way of example, in Figs. 4 and 5, the real and the imaginary parts of the term gsr (λ) are represented, respectively, in order to verify what has been previously stated. As a consequence, the integrand of (9) in the neighborhood of k, in spite of being continuous, is not smooth. This fact has an important consequence: the integral (9) is a Hankel transform, but the above mentioned property makes it impossible for the traditional Hankel transform algorithms (which are Gaussian quadrature methods) to correctly evaluate the integral in (9).

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having posed u = λ/k and indicated with f as the integer and function of the first integral of (7). The integral between 1.01 and ∞ requires to approximate the infinity, i.e., to find out a number M such that the difference between the original integral and the one between 1.01 and M is sufficiently small. Typically the number M is found out with iterative procedures; here, it is possible to derive an upper bound for the error E in the so-called integral tail for each frequency and to estimate M in an analytical way, thus reducing the computational costs. Let   ∞   4 3 µE z  J0 (kru)k u gsz (ku)e Q(ku)du . (17) E(M ) =  M

Real part of the term gsr (λ) for k = 2.09 × 10−5 .

Fig. 4.

It readily follows that

∞   J0 (kru)k 4 u3 gsz (ku)eµE z Q(ku) du. E(M ) ≤

(18)

M

We observe the following. 1) The term eµE z can be treated as       |eµE z | = eRe(µE )z  ejIm(µE )z 

  4ε  Re(µE )z kz u2 − εε0 k u2 z = e ≤e for u≥ . ≤e 3ε0 2) The absolute values of both Q and gs are decreasing functions for u > 1. 3) The asymptotic expansion of the Bessel function states that by

2 |J0 (kru)| < πkru

Imaginary part of the term gsr (λ) for k = 2.09 × 10−5 .

Fig. 5.

as in [26], one has

E. Vertical Component of the Electric Field The expression for the vertical component of the electric field is the first equation in (7). As stated before, such formula holds in the frequency domain, and unfortunately, it is not possible to convert it back into the time domain analytically. So, the inverse Fourier transform [24] must be carried out numerically, which requires the numerical evaluation of the first equation of (7) for many different values of frequency in an assigned range (in our case, 10 000 samples in the range [0 Hz, 107 Hz]). Moreover, one observes the following for each frequency. 1) The integrand contains both the Bessel function (which is highly oscillating) and the σ-dependent term gsz (λ) (whose behavior is shown in Figs. 4 and 5). 2) The integral must be carried out over a semiinfinite domain, it readily following that such calculation requires a huge computational effort. To overcome the first problem, a Romberg method [25] was used, dividing the interval of integration into subintervals with particular attention to the neighborhood of k, namely

k 0



f (ku)du = k

∞

0



0.99

1

f (ku)du+k

f (ku)du+k

0.99



1.01 1

∞

f (ku)du + k

f (ku)du 1.01



√ 4 ∞ u 5 2k E(M ) ≤ |gsz (kM )Q(kM )| √ ek 2 z u 2 du. πkr M (19)

Finally, since the integral in (19) is known analytically [26], it follows that √ 4 − 7   7 −kM z kz 2 2k , Γ − E(M ) ≤ |gsz (kM )Q(kM )| √ 2 2 2 πkr (20) where Γ is the incomplete Gamma function [26]. The last problem to be faced is the one relevant to the dc field. As a matter of fact, the position u = λ\k is not possible for f = 0 Hz. This implies that another method must be used to perform the so-called static term. Let us reconsider the Sommerfeld integral appearing in the first of (5); if k approaches 0  jσ   n2 →   ωε0   2 µE → λ2 (21)  µ2 → λ2    ωε0 1   → n2 µ + µE jσλ

DELFINO et al.: ALGORITHM FOR THE EXACT EVALUATION OF UNDERGROUND LIGHTNING EM FIELDS

405

one has that [26]

+∞

+∞ 1 j λ3 −µz
+µE z J (λr)e dλ → 0 2πωε0 0 n2 µ + µE 2πσ 0   2 r 1 3
, −1, 1, 2 (22) J0 (λr)eλ(z−z ) λ2 dλ = F 3 πσR 2 R

G. Azimuthal Component of the Magnetic Field

where F is the hypergeometric function [26]. As a consequence, the static term for the Sommerfeld integral appearing in the first of (7) is given by

With some algebraic manipulations, one has √ 3 ∞ u 3 2k ek 2 z u 2 du E(M ) ≤ |gsz (kM )Q(kM )| √ πkr M

jI(0, ω) +∞ λ3 J0 (λr)eµE z Q(λ)dλ limk→0 2πωε0 0 n2 µ + µE  

I(0, 0) H 1 r2 3 = , −1, 1, 2 P (z
)dz
. F πσ R3 2 R 0

(23)

∞ M

  J1 (kru)k 3 u2 gsz (ku)eµE z Q(ku)du .

(29)

(30)

√ − 3    2 2k 3 −kz 2 5 −kM z . Γ , E(M ) ≤ |gE(kM )Q(kM )| √ 2 2 2 πkr (31)

As observed in the first and second equation of (7), the problems to be faced are the same as before, in order to get the time-domain expressions of the radial component of the electric field. The only differences are relevant to the integral tail and the static term, since the formulas are slightly different. Indicating again with E the upper-bound for the integral tail and with M the last point of the interval on which the integral is taken, one has   ∞   3 2 µE z  J1 (kru)k u gsr (ku)e Q(ku)du . (24) E(M ) =  M

With considerations similar to the ones done in the previous section, it follows that √ 3 ∞ u 3 2k ek 2 z u 2 du πkr M



 E(M ) = 

and finally [26]

F. Radial Component of the Electric Field

E(M ) ≤ |gsr (kM )Q(kM )| √

In this case, the upper bound for the integral tail can be found as

(25)

and again [26] √ 3  − 5   5 −kM z kz 2 2k , . Γ − E(M ) ≤ |g(kM )Q(kM )| √ 2 2 2 πkr (26) As far as the static term is concerned, now one has

+∞

+∞ −1 µE −j −µz
+µE z 2 J (λr)e λ dλ → 1 2πωε0 0 n2 µ + µE 2πσ 0   2 r −3r 1 −λ(z−z
) 2 J1 (λr) e λ dλ = F 2, − , 2, 2 . (27) 2πσR4 2 R As a consequence, the static term for the Sommerfeld integral appearing in the second of (7) is given by

jI(0, ω) +∞ µE limk→0 − J1 (λr)eµE z λ2 Q(λ)dλ 2πωε0 0 n2 µ + µE  

−3rI(0, 0) H 1 r2 1 = , 2, P (z
)dz
. (28) F 2, − 2πσ R4 2 R2 0

As far as the static term is concerned, recalling the third equation of (5), one can observe that n2 2π



+∞ 1 λ2 −µz
+µE z J (λr)e dλ → 1 n2 µ + µE 2π 0 0   r2 r 3 λ (z−z
) , 0, 2, 2 . J1 (λr)e λdλ = F 2πR3 2 R +∞

(32)

As a consequence, the static term for the Sommerfeld integral appearing in the third equation of (7) is given by

n2 I(0, ω) +∞ λ2 limk→0 J1 (λr)eµE z Q(λ)dλ 2π n2 µ + µE 0  

rI(0, 0) H 1 r2 3 = , 0, 2, 2 P (z
)dz
. F 2π R3 2 R 0

(33)

III. NUMERICAL CALCULATIONS We focused on the radial component of the electric field, since this is the predominant component of the lightning electric field that penetrates the ground and appears as the source term in the field-to-buried cables coupling equations [18], [19]. In this section, a detailed comparative analysis will be carried out with the most popular existing approach proposed by Cooray [17]. Such formula provides the electric field produced by a vertical lightning channel below the ground surface, as a function of the electric field at the air–soil interface. In the time domain, one has

erL (t, r, z) =

t

erL (t − s, r, 0)y(s)ds

(34)

0

having indicated with lowercase letters time-domain functions. The function y is defined as e−at/2 atz  2 2  I0 a t − tz u(t − tz ) + e−atz /2 δ(t − tz ) y(t) = 2 t2 − t2z (35)

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TABLE I PARAMETERS OF THE TWO HEIDLER’S FUNCTIONS USED TO REPRODUCE THE CHANNEL-BASE CURRENT WAVESHAPE

√ where a = σε , tz = −z µ0 ε, δ is the Dirac function, u is the step function, and I0 is the modified Bessel function of first type and zeroth order. To compute the convolution integral (34) one must know the horizontal field at ground level, which can be calculated using the Cooray–Rubinstein formula [6]–[8]. The approximate formula (34) is a time-domain expression, which is of great advantage especially if only few temporal samples are needed. In our simulations, we are interested in examining the horizontal electric field behavior for a time window ranging typically from 0 to 10−5 s, with a time step of 10−8 s. As far as the channel base current is concerned, we have used the Heidler’s function [27] with appropriate numerical values representing waveshapes associated with typical first and subsequent strokes, as reported in Table I. As mentioned in Section II-B, the adopted return stroke model is the MTLE one [21], [22]. The decay constant α in such model is assumed to be equal to 2 km, and the current propagation speed v within the lightning channel is fixed at 1.5 × 108 m/s while the channel height is assumed to be equal to 8 km. The last number is useless whenever all the calculations are carried out in the time domain; but, since the Sommerfeld integrals evaluation is in the frequency domain, the knowledge of the channel height becomes important. Moreover, the calculations of the exact expressions for the radial electric field require to find out a number M such that the upper bound for the error E(M ) defined by (26) is small enough. The choice of M depends basically on two parameters, the depth z and the wavenumber k. This can be shown by the second equation of (7) containing the product k ∗ z in the exponential term, which determines the vanishing speed of the integrand function for large λ. This means that the more k ∗ z grows up, the smaller is M . After some experiments, we have chosen M = π/k for z = −10 m and M = 3π/k for z = −5 m. In Fig. 6, the quantity 100 ∗ E

π

k  ε% =  2  +∞  µE µ dλ  0 J1 (λr)Q(λ)e E z n2λµ+µ  E

which represents an upper bound for the percent error in the integral tail, has been plotted as a function of the frequency, assuming a depth of 10 m. It should be observed that for the frequency range of interest, ε% is smaller than 6 · 10−3 %. In this case, the calculation takes about 1 min on a Pentium IV 2.66-GHz 512-MB RAM. Decreasing the value of M leads to excessive errors while doubling the value of M results in a mean value for ε% of about

Fig. 6.

Upper bound of the error in the integral tail.

Fig. 7. Horizontal electric field generated by the first stroke current (r = 50 m, z = −5 m and σ = 0.01 S/m).

Fig. 8. Horizontal electric field generated by the first stroke current (r = 50 m, z = −5 m, and σ = 0.001 S/m).

10−10 , but requires about 15 s more in the calculations. Thus, the proposed value seems to be a good compromise between solution accuracy and procedure efficiency. Let us consider a field observation point P at a distance r = 50 m from the lightning discharge, and a current waveshape typical of first return-strokes (30-kA peak, 12-kA/µs max steepness). In Figs. 7 and 8, the results of the comparison between the exact solution [second equation of (7)] and the Cooray’s simplified formula (34) are shown for z = −5 m and for a ground conductivity equal to 0.01 and 0.001 S/m, respectively. As can be seen, the approximate formula proposed by Cooray appears to be very accurate for the larger value of conductivity.

DELFINO et al.: ALGORITHM FOR THE EXACT EVALUATION OF UNDERGROUND LIGHTNING EM FIELDS

407

Fig. 9. Horizontal electric field generated by the first stroke current (r = 50 m, z = −10 m, and σ = 0.01 S/m).

Fig. 11. Horizontal electric field generated by a subsequent stroke current (r = 50 m, z = −5 m, and σ = 0.01 S/m).

Fig. 10. Horizontal electric field generated by the first stroke current (r = 50 m, z = −10 m, and σ = 0.001 S/m).

Fig. 12. Horizontal electric field generated by a subsequent stroke current (r = 50 m, z = −5 m, and σ = 0.001 S/m).

However, for poor ground conductivity (0.001 S/m), Cooray’s expression yields less satisfactory results, especially for the late time response. This is probably due to the fact that Cooray’s expression is derived considering only the radiation component of the underground field. For r = 50 m, the radiative term is dominant only for the first microsecond or so, which explains why for the late time response wide differences between the two approaches appear. It should be noted that Cooray’s formula acts like a high pass filter on the channel base current and expected under ground electromagnetic fields. Similar simulations have been preformed considering an observation point located at a depth z = −10 m. The results obtained for ground conductivities equal to 0.01 and 0.001 S/m are plotted in Figs. 9 and 10, and again the overall agreement of the Cooray’s formula with the exact approach can be noted only for larger values of conductivity. The last four cases are relevant to the case of a typical subsequent return stroke current (12-kA peak current, 40-kA/µs max steepness). The fields are computed at z = −5 m for ground conductivities of 0.01 S/m (Fig. 11) and 0.001 S/m (Fig. 12); and at z = −10 m for ground conductivities of 0.01 S/m (Fig. 13) and 0.001 S/m (Fig. 14). For the case of subsequent stroke current (characterized by a faster rise time and larger steepness), the Cooray’s approximate formula seems to provide better results, though for poor conductivities and late time, some differences still persist. In light of these results and in order to verify the effectiveness of the Cooray approximate formula at higher distances from the

Fig. 13. Horizontal electric field generated by a subsequent stroke current (r = 50 m, z = −10 m, and σ = 0.01 S/m).

Fig. 14. Horizontal electric field generated by a subsequent stroke current (r = 50 m, z = −10 m, and σ = 0.001 S/m).

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Fig. 15. Horizontal electric field generated by the first stroke current (r = 100 m, z = −10 m, and σ = 0.01 S/m).

Fig. 16. Horizontal electric field generated by the first stroke current (r = 100 m, z = −10 m, and σ = 0.001 S/m).

channel, the comparison between the two approaches were also performed at observation points r = 100 m and r = 200 m, with a depth of 10 m. Fig. 15–22 show the obtained results. It can be seen, as expected, that differences become less significant for larger distances where the radiation term plays a more dominant role. In [18], a comparison between the results provided by the application of (34) and (35) and the ones obtained by Zeddam in [28] is presented and a better agreement is found between the two approaches. The reason for this better agreement is the choice of the channel-base current waveform adopted in [28], i.e., a double-exponential function with higher frequency content. In Fig. 23, the magnitude spectrum of the double-exponential waveform adopted in [28] is presented. For comparison, the spectra associated with the Heidler’s waveforms adopted in this study that correspond to typical first and subsequent return-strokes are also shown on the same plot. As can be seen, the double-exponential waveform exhibits the largest high-frequency content, resulting to a more predominant

Fig. 17. Horizontal electric field generated by the first stroke current (r = 200 m, z = −10 m, and σ = 0.01 S/m).

Fig. 18. Horizontal electric field generated by the first stroke current (r = 200 m, z = −10 m, and σ = 0.001 S/m).

Fig. 19. Horizontal electric field generated by a subsequent stroke current (r = 100 m, z = −10 m, and σ = 0.01 S/m).

DELFINO et al.: ALGORITHM FOR THE EXACT EVALUATION OF UNDERGROUND LIGHTNING EM FIELDS

Fig. 20. Horizontal electric field generated by a subsequent stroke current (r = 100 m, z = −10 m, and σ = 0.001 S/m).

409

Fig. 23. Spectrum of the three-channel base current models. The double exponential adopted in [28] and the Heidler’s model corresponding to a first and a second stroke.

Fig. 24. Vertical electric field generated by a subsequent stroke current (r = 50, z = −10 m, and σ = 0.01. Fig. 21. Horizontal electric field generated by a subsequent stroke current (r = 200 m, z = −10 m, and σ = 0.01 S/m).

radiation component, and a better accuracy of the Cooray’s formula. In order to show the ability of the proposed approach also for evaluating other components of the lightning fields, the waveform of the vertical electric field is depicted in Fig. 24, for the case of a subsequent return stroke. As expected, the amplitude of such component is much smaller than that of the radial one (see Fig. 13 for comparison). IV. CONCLUSION AND PERSPECTIVES

Fig. 22. Horizontal electric field generated by a subsequent stroke current (r = 200 m, z = −10 m, and σ = 0.001 S/m).

In this paper, an efficient algorithm for the evaluation of the exact expression for the underground fields generated by a lightning discharge was presented. Simulations were performed in order to test its effectiveness and reliability. The developed algorithm to compute underground electromagnetic fields was fast enough to be directly included in field-to-buried cable coupling simulation codes. It was used to test the validity of the Cooray’s simplified formula for the computation of underground horizontal electric field, and it was shown that predictions of the Cooray’s formula are in good agreement with exact solutions for large values

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of ground conductivity (∼0.01 S/m). However, for poor conductivities (∼0.001 S/m), Cooray’s expression may yield less satisfactory results, especially for the late time response. Future work will concern the insertion of such algorithm into a suitable field-to-buried cable coupling code in order to accurately compute the overvoltages induced in buried cables, and to assess the validity of the simplified methods. This will be done by projecting the horizontal field onto the line, and evaluating it in the points required for the solution of the coupling equations, as explained in [18]. REFERENCES [1] F. M. Tesche, M. V. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models. New York: Wiley, 1997. [2] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 1994. [3] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Influence of a lossy ground on lightning-induced voltages on overhead lines,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 250–264, Aug. 1996. [4] A. Sommerfeld, Partial Differential Equations in Physics. New York: Academic, 1949. [5] F. Rachidi, M. Ianoz, C. A. Nucci, and C. Mazzetti, “Calculation methods of the horizontal component of lightning return stroke electric fields,” presented at the Int. Wroclaw Symp. Electromagn. Compat., Wroclaw, Poland, Sep. 1992. [6] V. Cooray, “Horizontal fields generated by return strokes,” Radio Sci., vol. 27, no. 4, pp. 529–537, Jul./Aug. 1992. [7] M. Rubinstein, “An approximate formula for the calculation of the horizontal electric field from lightning at close, intermediate and long ranges,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, Aug. 1996. [8] V. Cooray, “Some considerations on the “Cooray–Rubinstein” formulation used in deriving the horizontal electric field of lightning return strokes over finitely conducting ground,” IEEE Trans. Electromagn. Compat., vol. 44, no. 4, pp. 560–566, Nov. 2002. [9] V. Cooray, “On the validity of several approximate theories used in quantifying the propagation effects on lightning generated electromagnetic fields,” presented at the 8th Int. Symp. Lightn. Prot. SIPDA, S˜ao Paulo, Brazil, Nov. 2005. [10] F. Delfino, R. Procopio, and M. Rossi, “Evaluation of lightning electromagnetic fields over a lossy ground,” in Proc. 7th Int. Conf. Comput. Exp. Methods Electr. Eng. Electromagn., Orlando, FL, Mar.2005, pp. 619– 629. [11] F. M. Tesche, A. W. Kalin, B. Brandly et al., “Estimates of lighntinginduced voltage stresses within buried shielded conduits,” IEEE Trans. Electromagn. Compat., vol. 40, no. 4, pp. 492–504, Nov. 1998. [12] A. Galvan and V. Cooray, “Lightning induced voltages on bare and insulated buried cables,” presented at the 13th Zurich Int. Symp. EMC, Zurich, Switzerland, 1999. [13] A. Zeddam and P. Degauque, “Current and voltage induced on telecommunication cables by a lightning stroke,” Electromagnetics, vol. 7, pp. 541– 564, 1987. [14] E. F. Vance, Coupling With Shielded Cables. New York: Wiley, 1978. [15] A. Ba˜nos, Dipole Radiation in the Presence of a Conducting Half-Space. New York: Pergamon, 1966. [16] P. R. Bannister, Simplified Expressions for Electromagnetic Fields of Elevated, Surface, or Buried Dipole Antennas. New London, CT: Naval Underwater System Centre, New London Lab., 1987, vol. I. [17] V. Cooray, “Underground electromagnetic fields generated by the return stroke of lightning flashes,” IEEE Trans. Electromagn. Compat., vol. 43, no. 1, pp. 75–84, Feb. 2001. [18] E. Petrache, F. Rachidi, M. Paolone, C. A. Nucci, V. Rakov, and M. Uman, “Lightning-induced disturbances in buried cables part I: Theory,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 498–508, Aug. 2005. [19] M. Paolone, E. Petrache, F. Rachidi et al., “Lightning-induced disturbances in buried cables part II: Experiment and model validation,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 509–520, Aug. 2005. [20] V. A. Rakov and M. A. Uman, “Review and evaluation of lightning return stroke models including some aspects of their application,” IEEE Trans. Electromagn. Compat., vol. 40, no. 4, pp. 403–426, Nov. 1998.

[21] C. A. Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, “On lightning return stroke models for LEMP calculations,” presented at the 19th Int. Symp. Lightn. Prot., Graz, Austria, Apr. 1988. [22] F. Rachidi and C.A. Nucci, “On the Master, Lin, Uman, Standler and the modified transmission line lightning return stroke current models,” J. Geophys. Res., vol. 95, pp. 20389–20394, Nov. 1990. [23] F. Delfino, R. Procopio, M. Rossi, and L. Verolino, “Lightning current identification over a conducting ground plane,” Radio Sci., vol. 38, no. 3, pp. 15-1–15-11, 2003. [24] E. O. Brigham, The Fast Fourier Transform and Its Applications. London, U.K.: Prentice-Hall, 1988. [25] P. Linz, Theoretical Numerical Analysis. New York: Wiley, 1974. [26] I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products. San Diego, CA: Academic, 1980. [27] F. Heidler, “Analytische blitzstromfunktion zur LEMP-berechnung,” presented at the 18th Int. Conf. Lightn. Prot., Munich, Germany, 1985. [28] A. Zeddam, “Couplage d’une onde electromagn´etique rayonn´ee par une d´echarge orageuse a` un cable de t´el´ecommunications” Ph.D. dissertation, Univ. Sci. Technol. Lille, Lille, France, 1988.

Federico Delfino (M’03) was born in Savona, Italy, on February 28, 1972. He received the graduate (cum laude) and Ph.D. degrees in electrical engineering from the University of Genoa, Genoa, Italy, in 1997 and 2001, respectively. He is currently a Researcher in the Department of Electrical Engineering, University of Genoa. His research interests include electromagnetic field theory, numerical techniques applied to electromagnetic compatibility, and lightning electromagnetics. He is the author or coauthor of more than 60 reviewed journal and international conference proceeding papers. Dr. Delfino is a member of the Italian Electrical Engineering Association (AEIT).

Renato Procopio (M’03) was born in Savona, Italy, on March 6, 1974. He received the graduate (cum laude) and Ph.D. degrees in electrical engineering from the University of Genoa, Genoa, Italy, in 1999 and 2004, respectively. He is currently a Researcher in the Department of Electrical Engineering, University of Genoa. His research interests include lightning return stroke current modeling, TL theory, and power quality improvement in distribution networks. He is the author or coauthor of more than 50 reviewed journal and international conference proceeding papers. Dr. Procopio is a member of the Italian Electrical Engineering Association (AEIT).

Mansueto Rossi was born in Savona, Italy, on April 10, 1974. He received the graduate (cum laude) and Ph.D. degrees in electrical engineering from the University of Genoa, Genoa, Italy, in 1999 and 2004, respectively. He is currently a Researcher in the Department of Electrical Engineering, University of Genoa, Genoa. His research interests include integral equations in electromagnetic field theory and numerical techniques applied to electromagnetic compatibility. He is the author or coauthor of more than 50 reviewed journal and international conference proceeding papers.

DELFINO et al.: ALGORITHM FOR THE EXACT EVALUATION OF UNDERGROUND LIGHTNING EM FIELDS

Farhad Rachidi (M’93–SM’02) was born in Geneva, Switzerland, in 1962. He received the M.S. and Ph.D. degrees in electrical engineering from the Swiss Federal Institute of Technology, Lausanne, in 1986 and 1991, respectively. He worked at the Power Systems Laboratory, Swiss Federal Institute of Technology, until 1996. In 1997, he joined the Lightning Research Laboratory, University of Toronto, Canada. From April 1998 to September 1999, he was with Montena Electromagnetic Compatibility (EMC), Switzerland. He is currently the Head of the EMC Group, the Swiss Federal Institute of Technology, Lausanne. His research interests include electromagnetic compatibility, lightning electromagnetics, and electromagnetic field interactions with transmission lines. He is the author or coauthor of more than 200 reviewed journal and international conference proceeding papers. Dr. Rachidi is a member of the scientific committees of the International Conference on Lightning Protection, International Zurich Symposium on EMC, etc. He is the Convener of the joint Conference Internationale des grands Reseaux Electriques (CIGRE)-Congres Internationale des Reseaux Electriques de Distribution (CIRED) working group on protection of MV and LV networks against lightning. He was the recipient of the 2005 IEEE EMC Society Technical Achievement Award and of the 2005 CIGRE Technical Committee Award.

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Carlo Alberto Nucci (M’91–SM’02–F’07) was born in Bologna, Italy, in 1956. He received the graduate degree (with honors) in electrical engineering from the University of Bologna, Bologna, Italy, in 1982. He was a Researcher with the Power Electrical Engineering Institute, University of Bologna, in 1983, where he became an Associate Professor in 1992, and a Full Professor and Chairperson of power systems, in 2000. He is the Chairman of the Study Committee C4 ‘System Technical performance’ of CIGRE. His research interests include power systems transients and dynamics, with particular reference to lightning impact on power lines, system restoration after black-out, and distributed generation. He is the author or coauthor of more than 200 reviewed journal and international conference proceeding papers. Dr. Nucci is a Fellow of the Institution of Engineering and Technology (IET). He serves on the chair of the International Steering Committee of the IEEE PowerTech and IEEE PES Italian chapter PE31 in region 8. Since January 2005, he has been Regional Editor of the Electric Power System Research Elsevier, for Africa and Europe.