A. Borghetti, C.A. Nucci, M. Paolone

Transcription

1 Effect of ta instrumented towers on the statistica distributions of ightning current parameters and its infuence on the power system ightning performance assessment A. Borghetti, C.A. Nucci, M. Paoone Abstract Statistica distributions of ightning current ampitude, time-to-peak vaue and other ightning current parameters, used in power system insuation coordination, are based on experimenta data obtained by means of ta instrumented towers. It is, however, generay accepted that these distributions are affected by the presence of the tower due to its attractive radius. Current ampitudes, in particuar, are biased towards higher vaues with respect to those that woud refer to fashes at ground. In this paper we propose a procedure, based on the Monte Caro method, that aows to infer the statistica distributions of ightning current parameters at ground eve starting from the cassica ones, i.e. those obtained from data measured using ta instrumented towers. The procedure is more genera than others proposed in the iterature for the same purpose, in that it can be appied whatever attractive radius expression is used. The procedure is appied to quantify the tower bias on the cassica statistica distribution of ightning current ampitude for a number of avaiabe attractive radius expressions. Additionay, the comparison between the indirect-ightning performances of an overhead ine, inferred by adopting both the cassica, toweraffected, and the unaffected statistica distributions at ground of the ightning current ampitude, is given. Index Terms Power system ightning protection, Lightning statistics, Monte Caro method, Induced overvotages. I. NOMENCLATURE Let X be a random variabe with ognorma distribution; µ denotes the median vaue of X, which corresponds to the antiogarithm of the mean vaue of variabe og(x); δ denotes the standard deviation of variabe og(x). δ vaues are given with reference to common ogarithm, i.e., base 10. T II. INTRODUCTION he probabiistic approach to power system insuation coordination requires the knowedge of the statistica distributions of ightning current parameters [1]. Nowadays, the distributions adopted by power engineers are basicay those derived from the experimenta data gathered by means of ee- vated instrumented towers in the ast decades [2-7]. We sha refer to these distributions as to the conventiona ones. There is, however, genera concern on the fact that these distributions are affected by the presence of the tower; ightning current ampitudes, in particuar, are biased towards higher vaues [8-13], as the so-caed attractive radius of the tower tends to increase for fashes with arger currents 1. There are indeed severa expressions for such an attractive radius [15-20], a predicting its increase with the return-stroke current. In view of the above, the conventiona distributions shoud not be used as such for power system insuation coordination studies. One shoud first eiminate the eary-mentioned tower effect to obtain distributions at ground, and then appy the obtained distributions to the specific structure of interest (ine poes, ine conductors) by taking into account the reevant direct-stroke exposure mode (given, in turns, by the attractive radius - or atera distance - expression). Note, additionay, that for the case of overhead distribution ines, for which it is very important to take into account the overvotages induced by strokes hitting the ground in their vicinity (indirect strokes), to accompish appropriate insuation coordination the statistica distributions of interest are indeed those of the ightning current parameters at ground. Pettersson [11] aready studied the probem and proposed an anaytica formua that aows obtaining the statistica distribution of the ightning current ampitude at ground starting from that obtained from eevated instrumented towers. Such a formua, which appies ony when the reationship between the attractive radius and the current ampitude is exponentia, and ony to the current ampitude, has been afterwards appied by Sabot [12] to the Cigré ightning current ampitude distribution. In [44], has presented the reationship among the probabiity density functions of peak currents reevant to strokes hitting a mast, of strokes hitting a conductor, and of strokes to open ground. These reationships have been appied in [44] by to the IEEE ightning current distribution [6], having a median vaue of 31 ka (and assumed to be inferred ony from transmission ine A. Borghetti, C. A. Nucci and M. Paoone are with the Department of Eectrica Engineering, University of Boogna, Boogna, Itay. (e-mai: {aberto.borghetti, caroaberto.nucci, 1 The ightning current parameters are aso affected by the infuence of the refections at the top and at the basis of the tower (e.g. [14]). These effects are here disregarded. Aso, in this paper, we focus ony on downward negative fashes. 691

2 measurements), to obtain a median vaue for open ground equa to 23 ka. To perform this cacuation the atera distance expression proposed in [18] was appied, which is of the same exponentia type assumed by Pettersson in order to derive its formua. In order to overcome the above-mentioned imits of the Pettersson formua, and to aow for the treatment of additiona ightning current parameters different from the peak ampitude, we propose here an aternative approach based on the Monte Caro method. With it, it is possibe to infer the statistica distributions of any ightning current parameter at ground starting from the conventiona ones, for any of the exposure modes proposed in the iterature. Using the proposed approach, we infer a number of statistica distributions of various ightning current parameters, and this for different attractive radius expressions. We eventuay evauate the impact of the above-mentioned tower effect on the assessment of the ightning performance of distribution overhead ines. The different modes describing the exposure of a tower and/or an overhead ine to direct ightning strokes are briefy summarized in Section III where various expressions for the attractive radius (atera distance) are summarized and reviewed. The method that we propose is described in Section IV and, in Section V, is appied to the ightning current parameter distributions of current ampitude and front duration presented in [4], obtained from the experimenta records at Monte San Savatore [21]. Section VI contains a comparison between the indirect-ightning performance of an overhead ine inferred by adopting both the affected and the unaffected current statistica distributions. III. MODELS DESCRIBING THE EXPOSURE OF AN ELEVATED STRUCTURES TO DIRECT LIGHTNING STROKES As the ightning eader descends toward an eevated object, it reaches a point known as the striking point. At this point, it wi initiate a juncture either with the object or with the ground depending on its charge, its distance from the structure, on the type (vertica mast or horizonta conductor), and height of the structure. By assuming the eader channe perpendicuar to the ground pane, it is generay accepted that the fash wi stroke the structure if its prospective ground termination point, i.e. its stroke ocation in absence of the structure, ies within the so-caed attractive radius r (aso caed atera distance for the case of horizonta conductors, as those of overhead ines). Severa expressions are avaiabe to evauate such a distance. Some of them are based on the Eectrogeometric mode [22]; as shown in Fig. 1, the vaue r (in m) is determined from ( ) 2 r = r r h for h < r (1a) 2 s g g r = r for h r (1b) s g where h is the height of the structure (in m) and r s and r g are the so-caed critica distances (in m) to the structure and to the ground respectivey. These striking distances are reated to the ightning current by means of the foowing expressions rs α = I β p rg k rs = (2) where I p is the current ampitude in ka, and the vaues of α, β and k are independent of I p. Tabe I reports some of the vaues proposed in the iterature on transmission ine shieding. Expression 2 is an approximation of the formua proposed by Love [16] using the exponentia format [6]. direct stroke h r s r nearby stroke r g direct stroke h r s = r nearby stroke Fig. 1. Eectrogeometric mode: r s and r g are the striking distances to the structure (mast or horizonta conductor) and to ground respectivey; r is the attractive radius (or atera distance) of the structure. TABLE I VALUES OF CONSTANTS OF STRIKING DISTANCE EQUATIONS (2) PROPOSED BY DIFFERENT AUTHORS Exposure mode α β k 1. Armstrong and Whitehead [15] IEEE [6,16] r g 0.55* 0.9** * adopted by IEEE Std [22] for an average conductor height greater than 40 m. ** adopted by IEEE Std [23] for distribution ines Other expressions, namey those by [17], [18], Deera and Garbagnati [19,20] are avaiabe; they have been inferred more recenty by regression anaysis, from the resuts of more compex and physicay oriented modes than the Eectrogeometric one. For these expressions, a formua of the foowing type can be used for the attractive radius b r = c + a I (3) where the vaues of a, b and c depend on the specific expression, and are shown in Tabe II. TABLE II VALUES OF CONSTANTS OF ATTRACTIVE RADIUS AND LATERAL DISTANCE EQUATION (3) PROPOSED BY DIFFERENT AUTHORS p Exposure mode c A b 3. [17] 0 4. From [18] 0 5. From Deera and Garbagnati [19,20] * for towers ** for horizonta conductors h 0.6 * 0.67 h 0.6 ** 2.83 h 0.4 * 1.57 h 0.45 ** 0.7 h * 0.69 ** 3 h h 1 It is worth noting that concerning the expression, henceforth caed expression 3, in [17] two atera distance formuas are proposed, one for masts with heights up to 100 m, and another one for horizonta conductors, with an 80% reduction of parameter a (see Tabe II). Concerning the expression (expression 4), in [18] an anaytica formua is proposed for horizonta conductors with 692

3 height range of 10 m and 50 m and for ightning currents with I p in the range 5-31 ka. The parameters are those of Tabe II. For free standing structures, in [18] the two foowing formuas are given: 0.40 r = 24.6 h for I p =31 ka and h in the range m, r 0.63 = 12.4 I for h=40 m and I p in the range of 5-60 ka. From these two formuas, a first approximation for coefficient a of Tabe II is derived by dividing 24.6 by , then obtaining, for different tower heights, curves simiar to those shown in Fig. 5 of [18]. Concerning expression 5, the constant vaues have been inferred in [25] by interpoation of pots of the atera distance of a sim structure vs. its height (in the range 5 to 100 m), cacuated using the eader progression mode of Deera- Garbagnati [19,20]. IV. PROCEDURE FOR THE EVALUATION OF LIGHTNING CURRENT DISTRIBUTIONS TO GROUND To obtain the statistica distributions of ightning parameters at ground one shoud be abe to record the ightning currents of a arge number of ightning fashes hitting the ground within a certain area. However, to accompish that, one needs the presence of a ta instrumented tower, which, as earier mentioned, does affect the distributions. As a matter of fact, of a the strokes that, in absence of the tower, woud hit the ground in its vicinity, the tower attracts ony some of them, due to the aready described attractive radius concept. However, if we consider an area around the tower ocation, supposed circuar for convenience, such that its radius is equa to the attractive radius r * corresponding to the minimum peak current vaue I p * observed at the top of the tower, a the strokes with perspective stroke ocation within such an area wi be coected by the tower. The proposed approach consists of appying the Monte Caro method to generate a popuation of ightning events with perspective stroke ocation within such an area of radius r *, starting from the conventiona statistica distributions of the ightning currents coected by the tower, as described in what foows. We generate a significant number of ightning events (e.g ), each characterized by a number of random variabes (ampitude I p, time to peak vaue t f, etc.), and perspective radia distance x g from the tower ocation. For each event, the vaues of the various ightning current parameters are randomy seected from the corresponding statistica distributions reevant to the tower measurements. Correation coefficients between the ightning parameters can be aso taken into account by appying the inverse transform method [26], as shown for instance in [27]. The vaue of x g associated to each direct ightning event is generated assuming that the stroke ocations are uniformy distributed around the tower; for each ightning event, x g is then generated from a distribution with 2 probabiity density function equa to 2 x / r. From the popuation of direct ightning events generated as above described, we seect the set of stroke events having distance x g from the tower ocation ower than r *. The statistica distributions of the ightning parameters associated to these g events are then evauated, which, under the considered assumptions, are indeed the desired distributions of the ightning parameters to open ground, without the bias introduced by the tower. V. APPLICATION OF THE PROPOSED PROCEDURE TO THE LIGHTNING CURRENT STATISTICAL DISTRIBUTIONS BY BERGER ET AL. Let us now consider the statistica distributions of the ightning current parameters by Berger et a. [3], obtained from measurements on the 70 m high tower instaed at the top of Monte San Savatore in Switzerand (near Lugano, 912 m above sea eve) 2. In Tabe III the median µ t and standard deviation δ t vaues of the first peak and of the front duration (assumed to be ognormay distributed) as given in [4] are reported. TABLE III MEDIAN AND STANDARD DEVIATIONS OF FIRST PEAK AND FRONT DURATION OF NEGATIVE DOWNWARD FIRST STROKES RECORDED AT MONTE SAN SALVATORE [4] Parameter µ t δ t First Peak I p (ka) Front duration t f (µs) The median vaue and standard deviation of parameter t f (front duration) are obtained by those of parameter T 30, i.e. the time interva between the 30 percent and 90 percent ampitude intercepts ( t = T / ) [4]. Aso, a correation coefficient f ρ t = 0.47 is taken into account between peak vaue and front duration [4]. In Tabe IV we report the resuts obtained by appying our procedure to the experimenta distributions of Tabe III for a the modes of Tabes I and II describing the ightning exposure of the tower. For these cacuations, the experimenta data of Berger et a. have been assumed to be coected by a tower on a ground pane, assuming that the effect of the presence of the mountain can be disregarded in the expression of the attractive radius of the tower, a point that certainy requires additiona investigation [4,44]. The minimum vaue of current peak has been assumed equa to 2 ka, for a the cacuations. The distributions at ground of current ampitude have median vaues ranging from 27.4% (attractive radius expression 3) to 20.2% (attractive radius expression 5) ower than the median of the origina distribution. The median vaues of front times range from 15.8% to 10.5% ower than the median of the origina distribution, due to the correation between front time vaues and current ampitudes. 2 For a certain imited period of time, at the top of the mountain there were two towers of different height (70 and 90 m). In this paper we disregard the effect of the presence of the second tower on the statistica distributions of the ightning current parameters. 693

4 TABLE IV MEDIAN AND STANDARD DEVIATIONS OF CURRENT PARAMETER DISTRIBUTIONS TO GROUND FOR THE ATTRACTIVE RADIUS EXPRESSIONS OF TABLES I AND II. Parameter Peak I p (ka) Exposure mode µ g σ g Front µ g duration τ f σ g (µs) ρ g We now compare the resuts obtained with the proposed procedure with those that can be obtained by using the earier-mentioned anaytica formua derived by Pettersson [11]. Such a formua aows cacuating the µ g and σ g vaues of the ognorma distribution of the current ampitudes at ground, from the corresponding µ t and σ t vaues of the conventiona distribution coected by means of an instrumented tower: σ g = σ t (4) 2 µ = µ exp 2 b σ ( ) g t g where b is the exponent of the attractive radius expression assumed by Pettersson to have an exponentia form namey of type (3) with c=0 which means that (4) can be appied to exposure modes 3 and 4. For the case of the Eectrogeometric mode (modes 1 and 2), the attractive radius assumes an exponentia form ony if h>r g (equation (1b)) or [28] when both r g =r s and h<<r g 3. In this second case, the attractive radius can be written as r = h I β (5) α p To the best of our knowedge, equation (4) cannot be appied to exposure mode 5. Foowing [12] we have appied (4) to the current peak distributions by Berger et a. [3], by assuming the vaues of parameter b of (4) equa to the b vaues reported in Tabe II for exposure modes 3 and 4. For exposure modes 1 and 2 (eectrogeometrica) we have appied (4) by using for coefficient b both β and β/2. We have aso appied equation (4) to exposure mode 5, in order to quantify the effect of parameter c, not taken into account in (4), on the resuts. The median vaues µ g of Tabe V are then obtained. (Note, as earier mentioned, that by using (4), ony the parameters of the statistica distribution of ightning current ampitudes can be evauated.) The comparison of the resuts of Tabe IV and V shows that the proposed procedure gives practicay the same resuts as those obtained by appying (4), when exposure modes 3 and 4 are appied, which are indeed of the type assumed by Pettersson in order to derive (4). For exposure mode 2, the median vaue predicted by (4) matches with that of the proposed approach if b is assumed equa to β; this is supported by the fact that, for mode 2, the probabiity that r g be arger than 70 m is greater than 90%. For the case of exposure mode 3 At east for most of ightning current ampitudes [12]. 1, the resut of (4) differs from that of the proposed approach when b is set equa to β/2; in fact the probabiity that r g be much ower than 70 m is very ow (our cacuations show that the probabiity that r g be ower than 70/3 is ony 0.02%). For this mode, however, the resut predicted by (4) sighty differs form our resut even for b=β, as, for this case, the probabiity that r g be arger than 70 m is ony 28.8%. TABLE V MEDIAN AND STANDARD DEVIATIONS OF CURRENT AMPLITUDE DISTRIBUTION TO GROUND BY APPLYING EQ. (4) FOR THE DIFFERENT ATTRACTIVE RADIUS EXPRESSIONS (EXPOSURE MODELS) OF TABLES I AND II. Parameter Peak I p (ka) µ g 23.4 * Exposure mode * * 24.1* 21.0* * σ g * from (4) using b=β/2 (r g=r s and h<<r g) ** from (4) using b= β (h>r g) VI. APPLICATION OF THE RESULTS TO THE EV ALUATION OF INDIRECT LIGHTNING PERFORMANCE OF OVERHEAD LINES To evauate the impact of the proposed modification of the statistica distributions of ightning current parameters at ground, in this paragraph the indirect ightning performance of an overhead ine is cacuated by using both the ighting current statistica parameters of Tabe III, affected by the presence of the tower, and those, corrected according to the proposed procedure, of Tabe IV. To this purpose, we consider a 2 km ong, 10 m high overhead ine, matched at both end, and a striking area around the ine, wide enough to incude a the ightning events that can induce a votage aong the ine with maximum ampitude greater than the considered insuation eve (e.g. about 20 km 2 ). The procedure presented by the authors in [27], aso based on the Monte Caro method, is appied to generate a significant number of events (a east 10 4 ). Each event is characterised by four random variabes: the peak vaue of the ightning current I p, its front time t f (correated) and the two co-ordinates of the stroke ocation. Such events are generated, as above mentioned, assuming the statistica ognorma distributions of current peak and front time of both Tabe III and Tabe IV; the stroke ocations are uniformy distributed within the earier mentioned surface around the ine (see [25,27] for further detais). As we are cacuating the indirect ightning performance of the ine, a the events corresponding to direct strokes are disregarded. For each event the ightning-induced votages on the ine are cacuated by means the LIOV code [29-31]. 4 4 The LIOV code has been deveoped in the framework of an internationa coaboration invoving the University of Boogna (Department of Eectrica Engineering), the Swiss Federa Institute of Technoogy (Power Systems Laboratory), and the University of Rome La Sapienza (Department of Eectrica Engineering). It is based on the fied-to-transmission ine couping formuation of Agrawa et a. [32], suitaby adapted for the case of an overhead ine above a ossy ground iuminated by an indirect ightning eectromagnetic fied; the LEMP is cacuated by assuming the MTLE returnstroke engineering mode [33,34] and using the Cooray-Rubinstein formua [35,36] to take into account the finite vaue of the ground resistivity in the 694

5 The ightning performance of the ine is cacuated by using the different atera distance expressions of Tabes I and II, reevant to distribution ines, in order to distinguish between direct and indirect strokes. The resuts obtained by using the parameters of Tabe III are shown in Figs. 2 and 4 and those obtained by using the parameters of Tabe IV are shown in Figs. 3 and 5. Resuts of Figs. 2 and 3 refer to the case of perfecty conducting ground pane, whie those of Figs. 4 and 5 refer to the case of a ground with conductivity equa to S/m. It can be observed that the appication of the modified current statistica distributions resuts, as expected, in a better performance of the distribution ine to indirect ightning strokes, being these distributions characterized by a ower median vaue. Additionay, it is shown that the resuts differ very much depending on the expression adopted to evauate the atera distance. VII. CONCLUSIONS In this paper we have proposed a procedure, based on the Monte Caro method, that aows to infer the statistica distributions of ightning current parameters (peak ampitude, front time, etc.) at ground eve, starting from those obtained from measurements using ta instrumented towers. The procedure is more genera than others proposed in the iterature for the same purpose, in that it can be appied whatever attractive radius (atera distance) expression is assumed, and is not imited to the current peak ampitude ony. The distribution of the peak ampitude at ground exhibits median vaues ranging from 27.4% (attractive radius expression by ) to 20.2% (attractive radius by Deera- Garbagnati) ower than the median of the origina distribution. Other exposure modes (IEEE, Amstrong-Withehead and ) predict median vaues that are within the abovementioned range. The median vaues of current front times range from 15.8% to 10.5% ower than the median of the origina distribution, as a consequence of the generay assumed correation between front time vaues and current ampitudes. We have aso compared the indirect-ightning performance of an overhead distribution ine inferred by adopting both the affected and the unaffected current ampitude statistica distributions, and have found a difference in the two cases. The performance of the ine appears, however, more affected by the exposure mode that is used for the determination of indirect strokes. Such a difference tends to decrease for increasing vaues of the ground resistivity. The authors fee that the above concusions shoud be taken into account in power systems insuation coordination practice. fied cacuation with correction by Cooray [37] according to the remarks by Wait [38]. Concerning the effect of the ground resistivity in the cacuation of the ine parameters, with particuar reference to the ground transient resistance, the Carson expression [39] is used. Indeed, as in the LIOV code a above-mentioned modes are impemented in the time domain, the ground transient resistance formua derived by Timotin [40] which corresponds to the Carson formua is used. Recenty, the expression proposed in [41] has been introduced in the LIOV code, which corresponds to the genera Sunde s expression for the ground impedance [42]. VIII. ACKNOWLEDGEMENTS The authors gratefuy acknowedge Dr. A. Sabot whose hepfu suggestions have motivated this paper, and Dr. F. Rachidi for his usefu comments. No. of induced overvotages with magnitude exceeding the vaue in abscissa/(100 km year 0.10 IEEE Fig. 2. Indirect-ightning performances of an overheard ine above a perfecty conducting ground, by adopting the different atera distance expression of Tabes I and II and the ightning current distributions of Tabe III as distributions at ground. No. of induced overvotages with magnitude exceeding the vaue in abscissa/(100 km year) 0.10 IEEE Fig. 3. Indirect-ightning performances of an overheard ine above a perfecty conducting ground, by adopting the different atera distance expression of Tabes I and II and the reevant current statistica distributions at ground of Tabe IV. No. of induced overvotages with magnitude exceeding the vaue in abscissa/(100 km year) 0.10 IEEE Fig. 4. Indirect-ightning performances of an overheard ine above a ossy ground with conductivity equa to S/m, by adopting the different atera distance expression of Tabes I and II and the ightning current distributions of Tabe III as distributions at ground. 695

6 No. of induced overvotages with magnitude exceeding the vaue in abscissa/(100 km year 0.10 IEEE Fig. 5. Indirect-ightning performances of an overheard ine above a ossy ground with conductivity equa to S/m, by adopting the different atera distance expression of Tabes I and II and the reevant current statistica distributions at ground of Tabe IV. IX. REFERENCES [1] L. O. Barthod and L. Paris, The probabiistic approach to insuation coordination, Eectra, no. 13, pp , 1970 [2] F. Popoansky, Frequency distribution of ampitudes of ightning currents, Eectra, no. 22, [3] K. Berger, R. B. Anderson and H. Kroninger, "Parameters of Lightning Fashes", Eectra, no. 41, pp , Juy [4] R. B. Anderson and A.J., "Lightning Parameters for Engineering Appications", Eectra, no. 69, pp , March [5] R.B. Anderson and A.J., "A Summary of Lightning Parameters for Engineering Appications", in Proc. of CIGRE, paper no , [6] J. G. Anderson, Lightning performance of EHV-UHV ines, in Transmission ine reference book, 345 kv and above, Pao Ato: EPRI, [7] E. Garbagnati and G. B. Lopiparo, "Lightning Parameters - Resuts of 10 Years of Systematic Investigation in Itay", in Proc. Internationa Aerospace Conference on Lightning and Static Eectricity, Oxford, AL, pp. A1-1 to A1-12, [8] M.A. Sargent, "The Frequency Distribution of Current Magnitudes of Lightning Strokes to Ta Structures", IEEE Trans. Power Apparatus and Systems, PAS-91(5), pp , September/October [9] R. Gode, "The ightning conductor", in Lightning: Voume 2, Lightning protection, R. Gode, Ed., New York: Academic Press, pp , [10] A.M. Mousa, K.D. Srivastava, The impications of the eectrogeometric mode regarding effect of height of structure on the median ampitude of coected ightning strokes, IEEE Trans. Power Deivery, vo. 4, no. 2, pp , [11] P. Pettersson, A unified probabiistic theory of the incidence of direct and indirect ightning strikes, IEEE Trans. Power Deivery, vo. 6, no. 3, pp , Juy 1991 [12] A. Sabot, An engineering review on ightning, transient overvotages and the associated eements of eectrogeometric compatibiity, in Proc. Ninth Internationa Symposium on High Votage Engineering, Graz, Austria, [13] G. Diendorfer, W. Schuz, Lightning incidence to eevated objects on mountains, in Proc. Internationa Conference on Lightning Protection, Birmingham, UK, [14] S. Guerrieri, C.A. Nucci, F. Rachidi, M. Rubinstein, On the infuence of eevated strike objects on directy measured and indirecty estimated ightning currents, IEEE Trans. Power Deivery, vo. 13, no. 4, pp , [15] H.R. Amstrong and E.R. Whitehead, Fied and Anaytica Studies of Transmission Lines Shieding, IEEE Trans. Power Apparatus and Systems, PAS-87, pp , January [16] E.R. Love, Improvements on ightning stroke modeing and appications to the design of EHV and UHV transmission ines, M.Sc. Thesis, University of Coorado, [17] A.J., An improved eectrogeometric mode for transmission ine shieding anaysis, IEEE Trans. Power Deivery, vo. 2, pp , Juy [18] F.A.M., Modeing of Transmission Line Exposure to Direct Lightning Strokes, IEEE Trans. Power Deivery, vo. 5, no.4, pp , October [19] L. Deera, E. Garbagnati, Lightning stroke simuation by means of the eader progression mode, Parts I and II, IEEE Trans. Power Deivery, vo. 5, no. 4, pp , October [20] M. Bernardi, L. Deera, E. Garbagnati, G. Sartorio, "Leader progression mode of ightning: updating of the mode on the basis of recent test resuts", in Proc. Internationa Conference on Lightning Protection, Forence, pp , 1996 [21] K. Berger, Nove observations of ightning discharges: Resuts of research on Mount San Savatore, Jour. Frankin Institute, 283, [22] R.H. Gode, The Frequency of occurrence and the distribution of ightning fashes to transmission ines, AIEE Trans., 64, pp , [23] IEEE Guide for Improving the ightning performance of eectric power overhead distribution ines, IEEE Standard , June [24] IEEE Guide for Improving the ightning performance of transmission ines, IEEE Standard , Dec [25] A. Borghetti, C.A. Nucci, M. Paoone, M. Bernardi, Effect of the atera distance expression and of the presence of shieding wires on the evauation of the number of ightning induced votages, in Proc. 25th Internationa Conference on Lightning Protection, Rhodes, Greece, September [26] R.Y. Rubinstein, Simuation and the Monte Caro method, New York, Wiey, [27] A. Borghetti, C.A. Nucci, Estimation of the frequency distribution of ightning induced votages on an overhead ine above a ossy ground: a sensitivity anaysis, in Proc. Internationa Conference on Lightning Protection, Birmingham, UK, September [28] P. Pettersson, Probabiistic theory of ightning incidence, in Proc. 22 nd Internationa Conference on Lightning Protection, Budapest, UK, September [29] C.A. Nucci, F. Rachidi, M. Ianoz and C. Mazzetti, Lightning-induced votages on overhead power ines, IEEE Trans. on EMC, vo. 35, [30] F. Rachidi, C.A. Nucci, M. Ianoz and C. Mazzetti, Infuence of a ossy ground on ightning-induced votages on overhead ines, IEEE Trans. on EMC, vo. 38, No 3, pp , [31] F. Rachidi, C.A. Nucci, M. Ianoz, Transient anaysis of muticonductor ines above a ossy ground, IEEE Trans. Power Deivery, vo. 14, no. 1, pp , [32] A.K. Agrawa, Price H.J., Gurbaxani S.H., Transient response of a muticonductor transmission ine excited by a nonuniform eectromagnetic fied, IEEE Trans. on EMC, vo. 22, no. 2, pp , May [33] C.A.Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, On ightning return stroke modes for LEMP cacuations, in Proc. 19th Internationa Conference on Lightning Protection, Graz, Apri [34] F. Rachidi and C.A. Nucci, On the Master, Lin, Uman, Stander and the Modified Transmission Line ightning return stroke current modes, Journa of Geophysica Research, vo. 95, pp , Nov [35] V. Cooray, "Horizonta fieds generated by return strokes", Radio Science, Vo. 27, No. 4, pp , Juy-August [36] M. Rubinstein, "An approximate formua for the cacuation of the horizonta eectric fied from ightning at cose, intermediate, and ong range", IEEE Trans. on EMC, vo. 38, no. 3, pp , August [37] V. Cooray, Some consideration on the Cooray-Rubinstein approximation used in deriving the horizonta eectric fied over finitey conducting ground, in Proc. 24th Internationa Conference on Lightning Protection, Birmingham, UK, September [38] J.R. Wait, Concerning the horizonta eectric fied of ightning, IEEE Trans. on EMC, vo. 39, no. 2, pp. 186, May [39] J.R. Carson, Wave propagation in overhead wires with ground return, Be System Technica Journa, 5, , [40] A.L. Timotin, Longitudina transient parameters of a unifiar ine with ground return, Rev. Roum. Sci. Techn. Eectrotechn. et Energ., 12, 4, pp , Bucarest, [41] F. Rachidi, S.L. Loyka, C.A. Nucci, M. Ianoz, On the Singuarity of the Ground Transient Resistance of Overhead Transmission Lines, in Proc. 25th Internationa Conference on Lightning Protection, Rhodes, Greece, September [42] E.D. Sunde, Earth Conduction Effects in Transmission Systems, New York, Dover, [43] A. Borghetti, C.A. Nucci, M. Paoone, Lightning performances of distribution ines: sensitivity to computationa methods and to data, in Proc. IEEE Power Engineering Society Winter Meeting, Coumbus, OH, [44] F.A.M., Modeing of ightning incidence to ta structures, IEEE Trans. Power Deivery, vo. 9, no. 1, pp , January