On the Transmission-Line Approach for the Evaluation ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 30, NO. 2, APRIL 2015

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On the Transmission-Line Approach for the Evaluation of LEMP Coupling to Multiconductor Lines Fabio Napolitano, Fabio Tossani, Carlo Alberto Nucci, Fellow, IEEE, and Farhad Rachidi, Fellow, IEEE

Abstract—This paper provides some clarification on the subject of the response of multiconductor lines to lightning-originated electromagnetic fields. Specifically, the effect of the presence of more conductors on the line response is analyzed with and without taking into account the ground losses in the surge propagation. The effect of the boundary conditions is also discussed. The presented analysis shows, in particular, that the response of a matched multiconductor lossless line differs from that of a single-conductor line at the same height only in the line currents. The effect of the nearby line conductors on the induced voltages is, indeed, noticeable when the ground losses are taken into account in the surge propagation. Index Terms—Ground losses, lightning electromagnetic pulse, lightning-induced voltages, multiconductor lines.

I. INTRODUCTION

T

HE RESPONSE of a multiconductor overhead line to an external incident electromagnetic (EM) field, such as the one originated by a nearby lightning return stroke, has been treated by several authors [1]–[13]. In case of lines with no grounded conductors (shield wires, neutrals), this effect has been considered as insignificant [3], [4], [11] or null [1], [7], [9], [13] while, in other studies, it has been found that the magnitude of the induced voltages on a multiconductor line is higher [2] or could be higher [3] than that on a single-conductor line of the same height above ground. Rachidi et al. [12] helped to clarify some of the mentioned disagreements which, in most cases, were due to the use of a particular coupling model [2], [3], which has been demonstrated not to be adequate [14]. The aim of this paper is to further clarify some issues which were not fully covered in [12] and to add some more general considerations on the subject by making reference to the coupling models proposed in the literature. In particular, the shielding capabilities of the line assumed by Rachidi et al. [12], where each wire is terminated on a resistance equal to its surge impedance that is calculated in the absence of the other

Manuscript received October 30, 2013; revised March 06, 2014; accepted April 07, 2014. Date of publication May 06, 2014; date of current version March 20, 2015. Paper no. TPWRD-01233-2013. F. Napolitano, F. Tossani, and C. A. Nucci are with the Department of Electrical, Electronic, and Information Engineering, University of Bologna, Bologna 40136, Italy (e-mail: [email protected]; [email protected]; [email protected]). F. Rachidi is with the Electromagnetic Compatibility (EMC) Group, Swiss Federal Institute of Technology, Lausanne, Vaud CH-1015, Switzerland (e-mail: farhad.rachidi@epfl.ch). Digital Object Identifier 10.1109/TPWRD.2014.2318515

conductors is worthy of further discussion, that the present paper is aimed at providing. The second point that this paper is aimed at addressing is the effect of the ground losses on the surge propagation along a multiconductor line illuminated by an external field. Such an issue has been treated by several authors (e.g., [15]–[17]) by representing the ground losses as an additional longitudinal term in the coupling equations (the so-called ground impedance in the frequency domain, or its time-domain counterpart the transient ground resistance). One conclusion of these studies is that the finite ground conductivity decreases the magnitude of the induced voltage at the line terminations for a stroke location equidistant from them. Further, in [16], Kannu and Thomas have correctly observed that the transient ground resistance effect is to slightly increase the induced voltage at the midpoint of the line for the same type of stroke location. In this paper, we provide some further elements on the transient ground resistance effect, by extending the analysis to a matched multiconductor line. This paper is organized as follows: in Section II, we provide some clarification on the so-called shielding capabilities ascribed to the line conductors when each one is terminated on its characteristic impedance calculated in the absence of the other ones. In Section III, the results described in Section II are discussed by making reference to the field-to-line coupling models proposed in the literature. Furthermore, the maximum amplitude of the overvoltages along a multiconductor line is analyzed to better illustrate the effect of the presence of multiple conductors. In Section IV, the inclusion of losses in the surge calculation is analyzed in order to evaluate their effect on the induced voltages along a multiconductor line. Finally, summary and conclusions are presented in Section V. II. VOLTAGES INDUCED ON A THREE-PHASE LINE BY AN EXTERNAL INCIDENT ELECTROMAGNETIC FIELD We assume the same two configurations of [3] and [12], which are shown in Fig. 1. In the first one (vertical configuration), the three conductors are disposed along the same vertical axis at different heights above ground [Fig. 1(a)]; in the second (horizontal configuration), the conductors are located at the same height above the ground surface [Fig. 1(b)]. The presence of one and two ground (or shielding) wires for configurations (a) and (b), respectively, is also taken into account. The radius of each conductor is 9.14 mm for phase conductors and 3.96 mm for the ground wires. The length of the line is assumed to be 1 km, if not otherwise specified. The lightning return-stroke

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Fig. 1. Considered power-line configurations. (a) Vertical. (b) Horizontal. Conductors numbered 1, 2, and 3 refer to phase conductors 4 and 5 to ground wires.

location is assumed to be at 50 m from the center of the line and equidistant to the line terminations. The channel-base current is typical of subsequent return strokes [18] with a peak value of 12 kA and a maximum time derivative of 40 kA s, represented by using the sum of two Heidler's functions [19]. The return stroke velocity is assumed to be 1.3 m/s, and the adopted return stroke model is the modified transmission-line model with exponential decay (MTLE) [20], [21] with a decay constant 2 km. The incident electromagnetic field is evaluated by assuming an ideal ground (perfectly conducting), if not otherwise specified. The induced voltages can be evaluated using the Agrawal et al. model [22], which, for the case of a lossless line is given by

(1) (2) where is the vector of the horizontal component of • the incident electric field (calculated in absence of the line conductors) along the axis at the conductor's height where the subindex denotes the particular wire; • and are the matrices of the line per-unit-length inductances and capacitances, respectively; • is the line current vector; • is the scattered voltage vector, from which the total voltage vector can be evaluated as (3) where •

is the vertical component of the incident electric field (calculated in absence of the line conductors).

Fig. 2. Voltages induced at the three-conductor line ends of configuration (a) for surge diagonal matrix terminations (disregarding the presence of ground wire).

Let us assume that the line is terminated on pure resistive terminations represented by the matrices and . The boundary conditions are given by (4)

(5) The results shown hereafter are obtained by using a secondorder finite-difference time-domain solution of the field-to-line coupling equations [23], [24]. A common practice in lightning induced overvoltage calculations is to match the line ends with the line surge impedance in order to avoid reflections. Diversely from the case of nonilluminated lines, a matched impedance at the line ends does not prevent the presence of “bumps,” that are not to be ascribed to reflections, but to the effect of the so-called “risers” [19]. Fig. 2 shows the voltages induced at the terminations of the three line conductors for the vertical configuration disregarding the presence of the ground wire. Each wire is terminated on a resistance equal to its characteristic impedance determined in the absence of the other conductors (henceforth called surge diagonal matrix terminations). Hence, the line is not matched and some reflections arise in agreement with what was previously found in [12] and [16]. The same case is shown in Fig. 3 but, this time, for a matched line, namely, using a full matrix termination impedance as detailed in what follows. For the considered case of a lossless, homogeneous multiconductor line, the characteristic impedance matrix can be simply expressed as the multiplication of the per-unit-length inductance matrix and the surge propagation speed. Therefore,

NAPOLITANO et al.: ON THE TRANSMISSION-LINE APPROACH FOR THE EVALUATION OF LEMP COUPLING

Fig. 3. Voltages induced at the three-conductor line ends of configuration (a) for matched terminations (disregarding the presence of ground wire).

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Fig. 4. Voltages induced at the three-conductor line ends of configuration (b) for surge diagonal matrix terminations (disregarding the presence of ground wire).

for the case of Fig. 3, the termination matrices of resistances are given by (6) where is the speed of light. For the case of Fig. 2, all offdiagonal terms of the termination resistance matrices are set to zero, while the diagonal term ii corresponds to the characteristic impedance of the th conductor, evaluated in absence of other conductors (corresponding to the so-called surge diagonal matrix terminations). The difference between the results of Figs. 2 and 3 is due to the effect of the offdiagonal terms of the surge impedance matrix, disregarded in the induced voltage calculation shown in Fig. 2. As a matter of fact, the same partial differential equation problem defined by (1) and (2) leads to different solutions depending on the boundary conditions. Fig. 4 shows the voltages induced on the three conductors of configuration (b) (in the absence of the shield wires) when the offdiagonal terms of the matrix terminations are set to zero. In Fig. 5, the results are shown, instead, for the case of matched terminations. Now, for such a case, as well as for the case of an infinitely-long line, the boundary conditions relevant to the terminations do not introduce any dependence among the conductors' voltages. It is interesting to observe that the same applies for the case of an open line for which the currents in (4) and (5) are equal to zero and, therefore, no dependencies among the conductors' voltages are introduced. Note that the results presented in Figs. 2 and 4 correspond to the same configuration assumed in [12] with surge diagonal matrix terminations. In Figs. 6 and 7, the currents relative to the cases of Figs. 4 and 5 are shown, respectively. Considering that the distance between the line conductors is much smaller than the distance to the lightning strike location, the resulting differences between the induced voltages will be negligible, hence, before they could

Fig. 5. Voltages induced at the three-conductor line ends of configuration (b) for matched terminations (disregarding the presence of ground wire).

be affected by the boundary conditions, the induced voltages on the three conductors are equal, while the currents in each conductor differ from one another due to mutual coupling. In the next section, we provide some further discussion on this matter. III. ON THE RESPONSE OF A MULTICONDUCTOR LINE TO AN EXTERNAL INCIDENT EM FIELD A. Case of an Infinitely Long Line The voltages induced by an external EM field along an infinitely long lossless line without discontinuities are not affected

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Fig. 8. Equivalent differential circuit of Agrawal et al. coupling model for the case of a lossless line.

For the case of the horizontal configuration, equations in (7) share the same forcing functions, while (8) can be written for each conductor as

(9)

Fig. 6. Currents flowing at the termination of the three-conductor line of configuration (b) for surge diagonal matrix terminations (disregarding the presence of ground wire).

For the assumed geometry, the capacitance matrix is

(10) Equations (9) and (10) clearly show that the forcing function in the D'Alembert equation for the middle conductor is smaller than those corresponding to the external conductors. It is interesting now to consider the coupling model by Rachidi [25] where only the incident magnetic-field components appear explicitly as forcing functions in the equations

(11)

(12) Fig. 7. Currents flowing at the termination of the three-conductor line of configuration (b) for matched terminations (disregarding the presence of ground wire).

by the presence of other conductors [1], [6], [11], while the currents are. This is shown by the two sets of three D'Alembert inhomogeneous (7) and (8), which can be derived from the Agrawal et al. coupling equations (1) and (2) (the equivalent circuit for a single conductor line is shown in Fig. 8)

where • is the scattered current vector, related to the total current vector, by the following expression:

(13) •

(7)

(8)

,

and are the horizontal components of the incident magnetic field. The equivalent circuit of Rachidi model (Fig. 9) contains only currents sources. Nevertheless, we will show that the predicted voltage on each conductor is again not affected by the presence of the others, while the current flowing in the line conductors is.


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Fig. 9. Equivalent differential circuit of the Rachidi coupling model for the case of a lossless line.

As for the case of the Agrawal model, the two coupled equations of Rachidi model (11) and (12) can be combined to derive uncoupled inhomogeneous D'Alembert equations

Fig. 10. Voltage peak amplitudes along the line of each conductor, for the case of perfectly matched terminations (solid lines) and surge diagonal matrix terminations (dotted lines). Vertical configuration.

(14)

(15) Assuming the same incident magnetic field for each conductor, the induced overvoltages given by (14) are equal while, according to (15), the currents will generally be different from one conductor to another which is, by the way, in perfect agreement with the fact that the two coupling models—Agrawal and Rachidi—are equivalent [26]. Needless to say, the same development can be repeated by adopting the Taylor et al.. model [27], leading to the same conclusion.

Fig. 11. Induced voltages at 0 m (500 m far from the point of the line facing the stroke location); comparison between an infinite line and a matched one. Horizontal configuration.

B. Case of a Line of Finite Length Let us consider, without losing generality, the vertical configuration [Fig. 1(a)] in the absence of the shield wire. The distributions of the voltage peak amplitudes along the line for the case of matched terminations (solid lines) and surge diagonal matrix terminations (dashed line) are shown in Fig. 10. For the considered case, the profile of the overvoltage peak amplitudes in the internal part of the line are the same, independent from the type of the terminations. The peak distribution plots of Fig. 10 show some minor sags near the line ends for the case of matched terminations too, which, however, have to be ascribed to the effect of the so-called “risers.” To better illustrate this effect, Fig. 11 presents a comparison, for the case of a horizontally configured line, between the overvoltage measured at 0 m (500 m away from the position of the line “facing” the strike location) of a matched line and an infinitely long one (the considered length is 5 km so that, in the observed time window, there is no influence of the portions beyond 5 km on the presented overvoltages). In Fig. 12, the same comparison is presented, this time concerning the voltage peak

Fig. 12. Induced voltage peak amplitude along the line; comparison between an infinite line and a matched one. Horizontal configuration.

amplitudes along the line, again for a horizontally configured line. As expected, the “risers” effect results in some further sag of the amplitude distribution plots in the immediate vicinity of the terminations.

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Fig. 14. Equivalent differential circuit of the Agrawal et al. coupling model for the case of a lossy line (the symbol denotes the convolution product).

Fig. 13. Distribution of the voltage peak amplitudes along the line of each conductor for the case of matched terminations (solid lines) and surge diagonal matrix terminations (dotted lines) with the presence of a shield wire grounded at the line terminations. Vertical configuration.

For the assumed stroke location, the effect of the presence of the other conductors for the case of surge diagonal matrix terminations or, more in general, nonmatched terminations appears only at the neighborhood of the line ends. This result is of interest for lightning performance assessment of distribution lines [28], in particular, when the lightning performance has to be evaluated at a specific pole [29], since, for some stroke locations, there is no effect due to the presence of other conductors in the central portion of the line. Of course, the situation is different for a lightning return stroke located in the vicinity of one of the line ends. In Fig. 13, the same case of Fig. 10 is shown, but this time, it is assuming the presence of a grounded wire. Since the upper wire is assumed to be grounded only at the line ends, the profile of the overvoltage peak amplitudes in the inner part of the line is the same as Fig. 10. This is an expected result since the reflected voltages/effect of risers at the line ends will reach these central positions at a time longer than the zero-to-peak time of the induced voltages. To obtain a significant shielding effect, multiple groundings along the line are required [28], [30]–[33]. IV. SURGE PROPAGATION LOSSES Accounting for the presence of a lossy ground characterized by its conductivity and its relative permittivity , the Agrawal et al.. coupling equations in the time domain read [17]

(16) (17) where is the transient ground resistance matrix [34]. In our analysis, the wire impedance, the ground admittance, and the line conductance have been disregarded (an approximation

valid for typical overhead lines a few meters above the ground [15]). Following the same procedure adopted for the case of a lossless line, (16) and (17) can be combined to obtain

(18)

(19) where (20) The corresponding equivalent circuit is shown in Fig. 14. The above equations show that the presence of the transient ground resistance matrix is introduced on the right side of (18) with a term , which is a function of the currents flowing into the line conductors and in the ground return. Therefore, even if the line is infinitely long or matched, the voltage induced by an external EM field in a lossy line is affected by the presence of other conductors of the line. A comparison between the overvoltage induced on an ideal (lossless) single-conductor line, a lossy single-conductor line, an ideal (lossless) three-phase line, and a lossy three-phase line are shown in Fig. 15, for a horizontal configuration and a symmetrical stroke location. The observation points are: 1) at the line termination; 2) 500 m away from the midpoint of the line; and 3) in the middle of the line. The ground conductivity is assumed to be 0.001 S/m and its relative permittivity is set to 10; the incident horizontal EM field is evaluated with the Cooray–Rubinstein (CR) formula [35], [36]. The return stroke model and the channel-base current are the same as described in Section II. The line is 2 km long, and it is matched by taking into account the frequency dependence of the characteristic impedance. For the case of a homogeneous lossy ground, the characteristic impedance of the line is frequency dependent, so the matrix of the termination becomes (21)


where

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is the ground impedance matrix [37] (22)

and (23)

Fig. 15. Total voltages. Comparison between an ideal line (single phase and three phase), a lossy single-conductor line and a lossy three-phase line; 10. Horizontal configuration, 2 km long line matched at both 0.001 S/m, ends. The incident horizontal EM field is evaluated using the CR formula for all of the considered cases (lossy and ideal).

For the case of frequency-dependent terminations, the product of the termination matrix and the current vector in (4) and (5) turn into a convolution product, which has to be solved numerically. As can be seen from Fig. 15, the induced voltages for the case of a three-phase ideal line are the same as those along a single-phase ideal line, in agreement with the analysis presented in the previous section. On the other hand, the induced voltage at the termination of a lossy single conductor line has lower peak amplitude and longer risetime compared to the case of an ideal line. Such an effect is even more significant for the case of a three-phase lossy line. Contrary to what occurs at the line terminations at the middle point [Fig. 15(c)] the effect of the transient ground resistance matrix in the coupling model is to slightly increase the peak of the induced voltage amplitude compared to the ideal case, in agreement with the conclusions reported by Kannu and Thomas for a nonmatched line [16]. Note that all of the simulation results presented in Fig. 15 have been obtained by using the same incident field. As a result, the observed increase in Fig. 15(c) for the case of lossy lines can only be ascribed to the effect of the transient ground resistance, which in the Agrawal et al. model (Fig. 14) is represented by (20), whose effect is to alter the scattered voltage with respect to the ideal case, as shown in (18). The observed aforementioned effect is stronger as the number of conductors grows. Let us now analyze the different effect that the transient ground resistance has on the total voltage in different positions along the line. Fig. 16 refers to the same cases of Fig. 15 but, this time, the incident and scattered voltages are shown individually. As (3) shows, the total voltage is given by the sum of the scattered voltage and the incident one; while the incident voltage remains practically unchanged when the ground is lossy [38], the scattered voltage is clearly affected by the line losses. The effect of the transient ground resistance on the scattered voltage depends on the position along the line. At the point in front of the stroke location, the effect is to weaken the reaction of the line to the external LEMP; therefore, a decrease of the scattered voltage magnitude results in an increase of the total voltage as scattered and incident voltages have different polarity. At the line terminations (for the considered configuration), a reduction of the scattered voltage peak results in the observed decrease of the total voltage peak as scattered and incident voltages have the same polarity. Figs. 15 and 16 further show that the transient ground resistance effect on the induced voltage varies as a function of time. Initially, the line losses effect is to decrease the voltage peak and to increase its rise time; as the time goes by, the effect of losses is to increase the voltage with respect to the ideal case.

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In the case of a lossy line, the induced overvoltages along a horizontally configured line are the same in all of the wires as in the case of an ideal line; however, the multiconductor line response is different from the single-conductor one. This result is in agreement with those presented in [16]. V. SUMMARY AND CONCLUSIONS We discussed some issues relevant to the LEMP response of multiconductor lines above a lossy ground which, according to the authors' opinion, were deserving some further clarifications. In particular, the differences between the behavior of matched and nonmatched lines concerning the effect of the various line conductors as well as the effect of the ground losses have been addressed making use of coupling models based on transmission-line theory. The cases for which the induced voltage on one conductor is not affected by the presence of other conductors are those of an infinitely long lossless line, of a lossless matched line, and of a lossless open line. For the second case, a reduction of the amplitude of the induced voltages with respect to the case of an infinitely long line is observed, but only near the line terminations, and this is not due to the presence of the other conductors but to the so-called “risers,” which describe the effect of the missing portion of the illuminated line beyond the line terminations. The response of a matched lossless multiconductor line differs from that of a single-conductor line at the same height only in the line currents. The effect of the nearby line conductors on the induced voltages is, indeed, noticeable when the ground losses are taken into account in the surge propagation. Note, finally, that the conclusions of this paper remain valid for multiconductor lines illuminated by any arbitrary incident electromagnetic fields. REFERENCES

Fig. 16. Scattered and incident voltages. Comparison between an ideal line (single phase and three phase), a lossy single-conductor line and a lossy three0.001 S/m, 10. Horizontal configuration, 2 km long phase line; line matched at both ends. The incident horizontal EM field is evaluated using the CR formula for all the considered cases (lossy and ideal).

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Fabio Napolitano, photograph and biography not available at the time of publication.

Fabio Tossani, photograph and biography not available at the time of publication.

Carlo Alberto Nucci (F’10), photograph and biography not available at the time of publication.

Farhad Rachidi (F’10), photograph and biography not available at the time of publication.