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Evaluation of a first return stroke model considering the lightning channel tortuosity
Electric Power Systems Research 113 (2014) 25–29
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Evaluation of a first return stroke model considering the lightning channel tortuosity Juan Diego Pulgarín-Rivera a,∗ , Camilo Younes a , Mauricio Vargas b a b
Universidad Nacional de Colombia, Colombia Integral Consultancy on Energy and Sustainable Development, CINERGY S.A.S., Colombia
a r t i c l e
i n f o
Article history: Available online 24 March 2014 Keywords: Lightning Return stroke modeling Lightning channel tortuosity Linear charge density Return stroke current profile Return stroke current peak attenuation
a b s t r a c t In this paper, the influence of the lightning channel tortuosity on the spatial and temporal return stroke current distribution, is evaluated. A return stroke model, developed with the current generation concept, has been used. The random tortuosity of the lightning channel is represented by a numerical model which also enables the linear charge density modeling. It is shown, that when the tortuosity of the lightning channel is taken into account, the return stroke current profile exhibits an amplitude attenuation and a time delay with respect to the current profile of the associated straight and vertical channel. It is worth pointing that the results presented in this paper are theoretical and obtained by means of numerical simulation, with exception of the linear charge density model, which was obtained using both, numerical simulation as well as experimental data obtained previously by other authors, as described in the paper. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nowadays there are a variety of lightning return stroke models provided in the literature. According to [1], most of the published models can be assigned to one or sometimes two of the following four classes: gas dynamic or physical models, electromagnetic models, distributed-circuit models and engineering models. Most of the return stroke models do not consider the tortuous geometry of the lightning channel, which is inherent to lightning, and most of them represent the lightning channel as straight and vertical to a flat ground, however, real lightning channels exhibit tortuous and branched geometries. Some of the principal works conducted in order to analyze the effect of the random geometries of the lightning channel on the models for the different stages of the phenomena, have considered mainly the effect of the lightning channel geometry on the radiated electromagnetic fields [2,3,20,4–7], the effect of the tortuousity on the induced voltages [8], the characterization of tortuosity by means of laboratory experiments with short sparks [9] and the lightning propagation modeling by means of simulation [10]. One of the authors of the present paper has been involved with a research concerning the modeling of the lightning leader stage for channels with tortuosity and branching and as a result of this study,
∗ Corresponding author. Tel.: +57 68904493. E-mail addresses: [email protected] (J.D. Pulgarín-Rivera), [email protected] (C. Younes), [email protected] (M. Vargas). http://dx.doi.org/10.1016/j.epsr.2014.02.029 0378-7796/© 2014 Elsevier B.V. All rights reserved.
a new model has been developed. The model in question was proposed by Vargas and Torres and it is described in [11,12]. This model enables the calculation of the linear charge density and many other parameters, all of them considering the tortuosity of the lightning channel. The model proposed by Vargas and Torres uses the total electric charge stored in the stepped leader channel and the relationship between the total charge in the leader and the return stroke peak current, obtained by Cooray et al. in [21], by means of experimental measurements of current waveforms measured by other authors. Cooray et al constructed an idealized model for the stepped leader, thundercloud, and ground, and solved this model numerically by means of the charge simulation method. The unknowns were the electrical charges along the stepped leader channel. In this way, Cooray et al. estimated the distribution of charge along the stepped leader, predicting a major accumulation of charge near the tip of the descending leader. For the complete description of the Cooray et at. model, please refer to [21]. The model of Vargas and Torres includes a methodology for the computer generation of stepped leader random geometries with a macroscale tortuousity similar to the one recorded in nature and is a model based on the bipolar and bidirectional concept of the leader channel. In this way, as opposed to the Cooray et al. model [21], which represented the leader channel as straight and vertical to a flat ground, the model of Vargas and Torres is able to represent the leader channel including random geometries with tortuosity and with or without branches. Using this more realistic representation of the leader channel, Vargas and Torres made use of the
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J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
charge simulation method as well, in order to obtain the distribution of charge along the leader channel. Vargas and Torres also found that there is a major accumulation of charge near the tip of the descending leader. For a detailed description of the Vargas and Torres model, please refer to [11,12]. In this paper, the authors make use of the Vargas and Torres model [11,12], in order to simulate random lightning channels with a macroscale tortuosity similar to the one observed in nature [13–15], and then, these simulations are used to evaluate a return stroke model. In this manner, the authors compare the results obtained when the tortuosity of the channel is taken into account, with those that are obtained for the associated straight and vertical channel. The return stroke model used to evaluate the effect of the lightning channel geometry is a model that uses the concept of current generation and it is described in [16]. This work is an effort to comprehend the effect of lightning channel tortuosity on the linear charge distribution along the leader channel and on the distribution of currents along it.
y=
(z − zsM )(yfM − ysM ) (zfM − zsM )
+ ysM
(3)
For this case the variable is the total distance along the tortuous channel from ground to the point with height z. 2.3. Regression analysis With the previous definitions, the data of 1000 of the simulated channels were used to produce an equation by means of nonlinear regression analysis [17], and the remainder data of the other 1000 channels were used to test the validity of the model. A strong correlation was found between the linear charge density, , and the variables Q (total charge stored on each channel) and (the coordinate defined above). The results obtained are presented in Section 4.1. 3. Return stroke model
2. Linear charge density modeling 2.1. Data description In order to model the linear charge density along a tortuous lightning channel, the Vargas and Torres model was used. First, 2000 random lightning channels with tortuous geometries were generated and then, the model was used to calculate the linear charge density on each segment of each channel. The data obtained consist of the following variables: 1. ij : The linear charge density, for the segment j, in channel number i. 2. Qi : The total electric charge stored in the complete channel number i. 3. xsij , ysij , zsij : The Cartesian coordinates for the starting or initial point of the segment j in the channel i. 4. xfij , yfij , zfij : the Cartesian coordinates for the final point of the segment j in the channel i. 5. ij : The total distance along the tortuous channel from ground to the center of the segment j in the channel i. According to this notation, the limits for the variables i and j are the following: 1 ≤ i ≤ 1000, 1 ≤ j ≤ p. Where p is the number of segments composing the tortuous channel. 2.2. General definition and calculation for From the above definitions, and for a specific tortuous channel (constant value for i), the variable can be related to a height z over the associated straight channel. Thus, the variable is a function of height z ( = f(z)). This means that for each point in a straight channel, there is an equivalent point on the tortuous channel which has the same height and can be calculated as follows: M−1
(z) = [
2
2
The assumptions presented in [16] for a straight channel, remain for the tortuous channel but with the following modifications: (a) The linear charge density of the descending leader is variable and it depends on the coordinate, according to Eq. (10). (b) All the physical events which depend on z in a straight channel, are dependent on in a tortuous channel. 3.2. Mathematical modeling Since for the tortuous channel, all the physical events described in [16] depend on the coordinate, the mathematical modeling presented there, has to be expressed in terms of this variable. Thus, the velocity of the connecting leader can be expressed as:
vc () = v0 e(/c )
(xfj − xsj ) + (yfj − ysj ) + (zfj − zsj ) ]
(x − xsM )2 + (y − ysM )2 + (z − zsM )2
The velocity of the return stroke can be represented by:
(zfM − zsM )
+ xsM
(5)
The current per unit length injected into the channel at a given point, with coordinate can be represented for the tortuous channel by: (1)
where M is the channel segment number that satisfies the condition zfM ≥ z. The x and y can be calculated from the symmetric equations for a line in space: (z − zsM )(xfM − xsM )
(4)
2
x=
3.1. Assumptions for a tortuous channel
vr () = vi e(−−lc /r )
j=1
+
The first return stroke model presented in [16] has been selected in order to compute the current profile due to a lightning channel with tortuosity. This model enables the inclusion of different charge density profiles and it also enables the calculation of currents for a lightning channel with a known geometry, inasmuch as this model describes some physical aspects of the return stroke process. Following is a description of the assumptions made in order to include the lightning channel geometry in the model and the mathematical modeling needed to compute the currents according to these assumptions.
(2)
Ic (, t) =
(, Q ) (−t/()) e ()
(6)
Where the charge per unit length (, Q) is modeled by Eq. (10). The variables v0 , vi , c , r and lc can be calculated for the tortuous channel according to the procedure presented in [16] and using the equations described above.
J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
27
Now, according to the principles of the current generation type return stroke models [18,19], the total current in a point and a time t, for a tortuous channel can be written as:
M
I(, t) =
Ic
, t − ton −
( − ) c
d
(7)
where d is a differential element of the channel, with the coordinate equal to , and the activation time ton , is defined by the following expression:
ton =
0
d
v()
(8)
where v() represents the variation of the return stroke front velocity. This velocity has been represented for segments below and above the contact point between the upward connecting leader, and the descending leader, by means of Eqs. (4) and (5). The time t, velocity v() and length M can be related by the following expression for a tortuous channel:
t=
M
0
M − d + c v()
(9)
with these definitions, the function () can be evaluated for a tortuous channel, using the same technique described in [18,19], and replacing z by . Finally, for a known tortuous geometry, the return stroke current profile can be determined using the Eq. (7). In this manner, the current waveform for a point of height z can be determined for the associated straight channel and for the tortuous channel in a point with the same height (z), the last, obtained through Eq. (1). The results obtained are presented in Sections 4.2 and 4.3. 4. Results and analyses The results presented in this section were obtained by means of numerical simulation and therefore they are theoretical work. Even so, the linear charge density model has shown to be in agreement with the findings of other authors, as it is described to continue. The results for the distribution of currents along the leader channel are obtained by evaluating the model previously published in [16], changing two input parameters, namely the channel geometry and the linear charge density model. 4.1. Linear charge density The model obtained for the linear charge density along a tortuous channel is presented in Eq. (10). (, Q ) =
(0.003768 + Q )(0.005088e−0.0004386 ) 9.98 − 7.826e−10
ρ [C/m]
(10)
This equation describes the variation of the linear charge density along a tortuous channel, and it is a function of the distance from the ground along the tortuous channel, , and the total electric charge stored in the complete channel, Q. As it was mentioned in the Section 2.3, a strong correlation was found between the variables , Q and . This correlation allowed to obtain the Eq. (10). The results for the goodness of fit obtained are the following: the sum of squares due to error, SSE = 0.1521, the coefficient of multiple determination, R2 = 0.9329, the adjusted coefficient of multiple determination, AdjR2 = 0.9329 and the root mean squared error, RMSE = 0.001886. As an example, a value of Q = −0.894 C has been selected, and the variation of the linear charge density has been plotted as a function of . The result is shown in Fig. 1. In this same figure, the linear charge density profile obtained by Cooray et al in [21] for an expected return stroke peak current of 30 kA, is plotted in
γ [m ]
Fig. 1. Variation of the linear charge density along the leader channel. Solid line: by means of Eq. (10), dashed line: the charge density profile obtained by Cooray et al. in [21] for a return stroke peak current of 30 kA.
order to compare both methodologies. It is important to remember that the Cooray et al. model considers a straight leader channel, while the Vargas and Torres model considers a leader channel with tortuosity. Eq. (10) is a prediction of the Vargas and Torres model and Fig. 1 shows that both, the Vargas and Torres and Cooray et al. models, predict a major accumulation of electric charge near the tip of the descending leader, and smaller amounts of charge as one moves towards the cloud. It is worth noting that both models predict a similar distribution of charge, with some difference, because the Vargas and Torres model considers the tortuosity of the leader channel when calculating the distribution of charge, but the total electric charge stored in the leader channel is the same for both models. According to the relationship obtained by Cooray et al. in [21], a return stroke peak current of −30 kA corresponds to a total leader charge of −0.894 C (the value selected to be used with Eq. 10 to plot the solid line in Fig. 1). 4.2. Tortuous channel with homogeneous linear charge density vs. straight channel with homogeneous linear charge density In this section, the current profiles of tortuous channels are compared with those of the associated straight channels. Two specific geometries of tortuous channel are used, and the charge per unit length remains constant in both cases. A value for the total charge of −3 C has been uniformly distributed along the straight and the two tortuous channels. The values used for the parameters are the following: lc = 53 m, c = 13.548, r = 1500, vi = 2 × 108 m/s y v0 = 4 × 106 m/s. In Fig. 2 a peak current attenuation can be observed in the waveforms for the tortuous channel with respect to those for the straight channel. The results are summarized in Table 1. The results obtained for the second tortuous channel are shown in Fig. 3. Again, a peak current attenuation can be observed in Fig. 3 and the results are summarized in Table 2. The results obtained in this section, and presented in Figs. 2 and 3 have shown two important facts when the current waveforms for a straight channel are compared with those for a tortuous channel. Table 1 Peak current attenuation, tortuous channel 1 and homogeneous. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 1000 2000
21.06 16.2 9.27
20.7 15.43 8.28
1.71 4.75 10.67
28
J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
Fig. 2. First return stroke current profile for: a straight channel (solid line) with homogeneous charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 1000 m and (c) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4360 m.
Fig. 4. First return stroke current profile for: a straight channel (solid line) with variable charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4360 m. Table 3 Peak current attenuation, tortuous channel 1 and variable. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 2000
21.06 9.27
16.17 5.16
23.22 44.34
velocity decreases more rapidly than in the straight channel, then causing an attenuation in the peak currents value. 4.3. Tortuous channel with variable linear charge density vs. straight channel with homogeneous linear charge density
Fig. 3. First return stroke current profile for: a straight channel (solid line) with homogeneous charge density and a tortuous channel (dashed line) with homogeneous charge density, at diffzerent heights: (a) 500 m, (b) 1000 m and (c) 2000 m. Simulation case 2 of 2, with a total length of the tortuous channel equal to 4477 m. Table 2 Peak current attenuation, tortuous channel 2 and homogeneous. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 1000 2000
21.06 16.2 9.27
20.34 15.36 8.04
3.42 5.18 13.27
First, a time delay is present in the currents for the tortuous channel and second, a peak current attenuation is also present in the currents for the tortuous channel. These two factors can be associated to physical causes, which are described to continue: the time delay is due to the channel geometry, because the return stroke front has to travel longer in a tortuous channel than in a straight one to reach the same height. Also in this travel, the velocity of the return stroke front decreases more rapidly in the tortuous channel than in the straight channel, because the return stroke velocity is dependent on z in a straight channel and it is dependent on in a tortuous channel. The other important fact is the peak current attenuation. This result cannot be associated with the linear charge density, for in that part the simulations were made with a homogeneous distribution of electric charge equal to −3 C along each channel. Again, this attenuation is due to the tortuosity of the lightning channel, because the total current in a point depends on the velocity of the return stroke front, and because in the tortuous channel, this
In this section, the return stroke current profiles for tortuous channels are compared with those for the associated straight channels. In this case, it is included not only the tortuosity of the lightning channel, but also the variation of its linear charge density. The values of the parameters for the straight channel are: lc = 53 m, = 0.00075 C/m, c = 13.548, r = 1500, vi = 2 × 108 m/s y m/s, and for the tortuous channel the values are the same except for: lc = 105 m, c = 26.8403. The linear charge density is calculated by means of Eq. (10) with Q = −3 C. Again, two specific geometries of tortuous channel are used (the same of Section 4.2). In this case, Fig. 4 shows a much more notable attenuation in the peak currents value for the tortuous channel compared to the straight channel, the results are summarized in Table 3. The simulations for the second tortuous channel are shown in Fig. 5. Again, the attenuation is present and the results are summarized in Table 4. The results presented in Figs. 4 and 5 have shown that when the linear charge density along the channel is taken into account together with the tortuosity, the peak current attenuation and the time delay are much more notable than in the case of considering only the channel geometry. These results can be analyzed in Tables 3 and 4. As well as the physical causes already analyzed in the previous section, there are some other aspects to be considered about the results obtained in this section. Here, the major cause of Table 4 Peak current attenuation, tortuous channel 1 and variable. Height (m)
Straight channel (kA)
500 2000
21.06 9.27
Tortuous channel (kA) 15.73 5
Attenuation (%) 25.31 46.06
J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
Fig. 5. First return stroke current profile for: a straight channel (solid line) with variable charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4477 m.
the time delays and peak current attenuation is due to the expression for the linear charge density predicted by the Vargas and Torres model [11,12]. Considering that the length of the upward connecting leader (lc ) depends on the linear charge density stored near the tip of the descending leader, and that the Vargas and Torres model has predicted the major accumulation of charge near the tip of the descending leader, it is evident that the length of the upward connecting leader has to increase. This fact can be seen in the results of the simulations, where the value for the length of the connecting leader in the case of homogeneous charge density was calculated equal to 53 m and equal to 105 m for the case of variable charge density. According to Eq. (5), the return stroke front velocity depends on lc and that is the principal reason why in the case of variable charge density the time delays and peak current attenuations are more notable than in the case of homogeneous charge density. 5. Conclusions In this paper, a simple model for the linear charge density distribution along tortuous lightning channels has been developed. Also, a technique for the evaluation of the return stroke current profile has been developed using a current generation type return stroke model. It is important to point that the results obtained here are only theoretical work, with exception of the model for the linear charge density, which was obtained using both, numerical simulation and experimental measurements reported by other authors. The model obtained for the linear charge density predicts major accumulations of electric charge near the tip of the descending leader and less amounts of electric charge as one moves towards the thundercloud, which is in accordance to the findings of other authors, as the model presented in [21]. This behavior is a prediction of the Vargas and Torres model [11,12]. An important advantage of this model is that it allows the calculation of the linear charge density in a simple way, without having to simulate every time the charge density is required. The principal results have shown that when the currents are calculated using a homogeneous charge density for both, the straight and the tortuous channel, they are present time delays and peak
29
current attenuations in the currents for the tortuous channel with respect to the straight channel. When the linear charge density model is included and considered together with the channel geometry, the time delays and peak current attenuations are much more notable than in the case of considering only the channel tortuosity. It was found that the effect of tortuosity is to cause a time delay in the currents and to reduce the return stroke propagation velocity. The effect of the linear charge density was found to produce a larger upward connecting leader, thus reducing also the return stroke front velocity and causing a major peak current attenuation. Finally, and according to the previous conclusions, this work has shown that when modeling the return stroke currents, the model for the linear charge density is more important than the channel tortuosity, because the latter produces major time delays and peak current attenuations. References [1] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromagn. Compat. 40 (4) (1998) 403–426. [2] G. Lupò, C. Petrarca, V. Tucci, M. Vitelli, EM fields generated by lightning channels with arbitrary location and slope, IEEE Trans. Electromagn. Compat. 42 (1) (2000) 39–53. [3] Z. kuo Zhao, Q. lin Zhang, Influence of channel tortuosity on the lightning return stroke electromagnetic field in the time domain, Atmos. Res. 91 (2-4) (2009) 404–409. [4] D.K.L. Chia, A.C. Liew, Effect of tortuosity of lightning stroke path on lightning electromagnetic fields, in: 2008 Asia-Pacific Symposium on Electromagnetic Compatibility & 19th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore, 2008. [5] F. Guo, B. Zhou, Influence of branch geometry on radiation field characteristics from lightning return strokes in the time domain, in: CEEM 2009, 5th AsiaPacific Conference on Environmental Electromagnetics, 2009. [6] M. Saito, M. Ishii, N. Itamoto, Influence of geometrical shape of return stroke channel on associated electromagnetic fields, in: 2011 7th Asia-Pacific International Conference on Lightning, November 1–4, 2011, Chengdu, China, 2011. [7] M. Saito, M. Ishii, N. Itamoto, Influence of geometry of – GC strokes on associated electromagnetic waveforms, in: 2011 International Symposium on Lightning Protection (XI SIPDA), Fortaleza, Brazil, October 3–7, 2011, 2011. [8] A. Andreotti, U.D. Martinis, C. Petrarca, V.A. Rakov, L. Verolino, Lightning electromagnetic fields and induced voltages: influence of channel tortuosity, in: 2011 XXXth URSI General Assembly and Scientific Symposium, 2011. [9] D. Amarasinghe, U. Sonnadara, M. Berg, V. Cooray, Channel tortuosity of long laboratory sparks, J. Electrostat. 65 (8) (2007) 521–526. [10] A. Gulyás, N. Szedenik, 3D simulation of the lightning path using a mixed physical-probabilistic model – the open source lightning model, J. Electrostat. 67 (2-3) (2009) 518–523. [11] M. Vargas, H. Torres, On the development of a lightning leader model for tortuous or branched channels – Part I: Model description, J. Electrostat. 66 (2008) 482–488. [12] M. Vargas, H. Torres, On the development of a lightning leader model for tortuous or branched channels – Part II: Model results, J. Electrostat. 66 (2008) 489–495. [13] R. Hill, Tortuosity of lightning, Atmos. Res. 23 (3) (1988) 217–233. [14] R.D. Hill, Analysis of irregular paths of lightning channels, J. Geophys. Res. 73 (6) (1968) 1897–1906. [15] V.P. Idone, R.E. Orville, Channel tortuosity variation in Florida triggered lightning, Geophys. Res. Lett. 15 (7) (1988) 645–648. [16] V. Cooray, R. Montano, V. Rakov, A model to represent negative and positive lightning first strokes with connecting leaders, J. Electrostat. 60 (2004) 97–109. [17] D.M. Bates, D.G. Watts, in: Nonlinear Regression Analysis and Its Applications, John Wiley and Sons, Inc., Hoboken, New Jersey, 1988. [18] V. Cooray (Ed.), The Lightning Flash, The Institution of Electrical Engineers, IEE, London, United Kingdom, 2003. [19] V. Cooray, Predicting the spatial and temporal variaton of the electromagnetc fields, currents and speeds of subsequent return strokes, IEEE Trans. Electromagn. Compat. 40 (4) (1998) 427–435. [20] D.M. LeVine, R. Meneghini, Simulation of radiation from lightning return strokes: the effects of tortuosity, Radio Sci. 13 (5) (1978) 801–809. [21] V. Cooray, V. Rakov, N. Theethayi, The lightning striking distance-revisited, J. Electrostat. 65 (2007) 296–306.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Evaluation of a first return stroke model considering the lightning channel tortuosity Juan Diego Pulgarín-Rivera a,∗ , Camilo Younes a , Mauricio Vargas b a b
Universidad Nacional de Colombia, Colombia Integral Consultancy on Energy and Sustainable Development, CINERGY S.A.S., Colombia
a r t i c l e
i n f o
Article history: Available online 24 March 2014 Keywords: Lightning Return stroke modeling Lightning channel tortuosity Linear charge density Return stroke current profile Return stroke current peak attenuation
a b s t r a c t In this paper, the influence of the lightning channel tortuosity on the spatial and temporal return stroke current distribution, is evaluated. A return stroke model, developed with the current generation concept, has been used. The random tortuosity of the lightning channel is represented by a numerical model which also enables the linear charge density modeling. It is shown, that when the tortuosity of the lightning channel is taken into account, the return stroke current profile exhibits an amplitude attenuation and a time delay with respect to the current profile of the associated straight and vertical channel. It is worth pointing that the results presented in this paper are theoretical and obtained by means of numerical simulation, with exception of the linear charge density model, which was obtained using both, numerical simulation as well as experimental data obtained previously by other authors, as described in the paper. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nowadays there are a variety of lightning return stroke models provided in the literature. According to [1], most of the published models can be assigned to one or sometimes two of the following four classes: gas dynamic or physical models, electromagnetic models, distributed-circuit models and engineering models. Most of the return stroke models do not consider the tortuous geometry of the lightning channel, which is inherent to lightning, and most of them represent the lightning channel as straight and vertical to a flat ground, however, real lightning channels exhibit tortuous and branched geometries. Some of the principal works conducted in order to analyze the effect of the random geometries of the lightning channel on the models for the different stages of the phenomena, have considered mainly the effect of the lightning channel geometry on the radiated electromagnetic fields [2,3,20,4–7], the effect of the tortuousity on the induced voltages [8], the characterization of tortuosity by means of laboratory experiments with short sparks [9] and the lightning propagation modeling by means of simulation [10]. One of the authors of the present paper has been involved with a research concerning the modeling of the lightning leader stage for channels with tortuosity and branching and as a result of this study,
∗ Corresponding author. Tel.: +57 68904493. E-mail addresses: [email protected] (J.D. Pulgarín-Rivera), [email protected] (C. Younes), [email protected] (M. Vargas). http://dx.doi.org/10.1016/j.epsr.2014.02.029 0378-7796/© 2014 Elsevier B.V. All rights reserved.
a new model has been developed. The model in question was proposed by Vargas and Torres and it is described in [11,12]. This model enables the calculation of the linear charge density and many other parameters, all of them considering the tortuosity of the lightning channel. The model proposed by Vargas and Torres uses the total electric charge stored in the stepped leader channel and the relationship between the total charge in the leader and the return stroke peak current, obtained by Cooray et al. in [21], by means of experimental measurements of current waveforms measured by other authors. Cooray et al constructed an idealized model for the stepped leader, thundercloud, and ground, and solved this model numerically by means of the charge simulation method. The unknowns were the electrical charges along the stepped leader channel. In this way, Cooray et al. estimated the distribution of charge along the stepped leader, predicting a major accumulation of charge near the tip of the descending leader. For the complete description of the Cooray et at. model, please refer to [21]. The model of Vargas and Torres includes a methodology for the computer generation of stepped leader random geometries with a macroscale tortuousity similar to the one recorded in nature and is a model based on the bipolar and bidirectional concept of the leader channel. In this way, as opposed to the Cooray et al. model [21], which represented the leader channel as straight and vertical to a flat ground, the model of Vargas and Torres is able to represent the leader channel including random geometries with tortuosity and with or without branches. Using this more realistic representation of the leader channel, Vargas and Torres made use of the
26
J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
charge simulation method as well, in order to obtain the distribution of charge along the leader channel. Vargas and Torres also found that there is a major accumulation of charge near the tip of the descending leader. For a detailed description of the Vargas and Torres model, please refer to [11,12]. In this paper, the authors make use of the Vargas and Torres model [11,12], in order to simulate random lightning channels with a macroscale tortuosity similar to the one observed in nature [13–15], and then, these simulations are used to evaluate a return stroke model. In this manner, the authors compare the results obtained when the tortuosity of the channel is taken into account, with those that are obtained for the associated straight and vertical channel. The return stroke model used to evaluate the effect of the lightning channel geometry is a model that uses the concept of current generation and it is described in [16]. This work is an effort to comprehend the effect of lightning channel tortuosity on the linear charge distribution along the leader channel and on the distribution of currents along it.
y=
(z − zsM )(yfM − ysM ) (zfM − zsM )
+ ysM
(3)
For this case the variable is the total distance along the tortuous channel from ground to the point with height z. 2.3. Regression analysis With the previous definitions, the data of 1000 of the simulated channels were used to produce an equation by means of nonlinear regression analysis [17], and the remainder data of the other 1000 channels were used to test the validity of the model. A strong correlation was found between the linear charge density, , and the variables Q (total charge stored on each channel) and (the coordinate defined above). The results obtained are presented in Section 4.1. 3. Return stroke model
2. Linear charge density modeling 2.1. Data description In order to model the linear charge density along a tortuous lightning channel, the Vargas and Torres model was used. First, 2000 random lightning channels with tortuous geometries were generated and then, the model was used to calculate the linear charge density on each segment of each channel. The data obtained consist of the following variables: 1. ij : The linear charge density, for the segment j, in channel number i. 2. Qi : The total electric charge stored in the complete channel number i. 3. xsij , ysij , zsij : The Cartesian coordinates for the starting or initial point of the segment j in the channel i. 4. xfij , yfij , zfij : the Cartesian coordinates for the final point of the segment j in the channel i. 5. ij : The total distance along the tortuous channel from ground to the center of the segment j in the channel i. According to this notation, the limits for the variables i and j are the following: 1 ≤ i ≤ 1000, 1 ≤ j ≤ p. Where p is the number of segments composing the tortuous channel. 2.2. General definition and calculation for From the above definitions, and for a specific tortuous channel (constant value for i), the variable can be related to a height z over the associated straight channel. Thus, the variable is a function of height z ( = f(z)). This means that for each point in a straight channel, there is an equivalent point on the tortuous channel which has the same height and can be calculated as follows: M−1
(z) = [
2
2
The assumptions presented in [16] for a straight channel, remain for the tortuous channel but with the following modifications: (a) The linear charge density of the descending leader is variable and it depends on the coordinate, according to Eq. (10). (b) All the physical events which depend on z in a straight channel, are dependent on in a tortuous channel. 3.2. Mathematical modeling Since for the tortuous channel, all the physical events described in [16] depend on the coordinate, the mathematical modeling presented there, has to be expressed in terms of this variable. Thus, the velocity of the connecting leader can be expressed as:
vc () = v0 e(/c )
(xfj − xsj ) + (yfj − ysj ) + (zfj − zsj ) ]
(x − xsM )2 + (y − ysM )2 + (z − zsM )2
The velocity of the return stroke can be represented by:
(zfM − zsM )
+ xsM
(5)
The current per unit length injected into the channel at a given point, with coordinate can be represented for the tortuous channel by: (1)
where M is the channel segment number that satisfies the condition zfM ≥ z. The x and y can be calculated from the symmetric equations for a line in space: (z − zsM )(xfM − xsM )
(4)
2
x=
3.1. Assumptions for a tortuous channel
vr () = vi e(−−lc /r )
j=1
+
The first return stroke model presented in [16] has been selected in order to compute the current profile due to a lightning channel with tortuosity. This model enables the inclusion of different charge density profiles and it also enables the calculation of currents for a lightning channel with a known geometry, inasmuch as this model describes some physical aspects of the return stroke process. Following is a description of the assumptions made in order to include the lightning channel geometry in the model and the mathematical modeling needed to compute the currents according to these assumptions.
(2)
Ic (, t) =
(, Q ) (−t/()) e ()
(6)
Where the charge per unit length (, Q) is modeled by Eq. (10). The variables v0 , vi , c , r and lc can be calculated for the tortuous channel according to the procedure presented in [16] and using the equations described above.
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27
Now, according to the principles of the current generation type return stroke models [18,19], the total current in a point and a time t, for a tortuous channel can be written as:
M
I(, t) =
Ic
, t − ton −
( − ) c
d
(7)
where d is a differential element of the channel, with the coordinate equal to , and the activation time ton , is defined by the following expression:
ton =
0
d
v()
(8)
where v() represents the variation of the return stroke front velocity. This velocity has been represented for segments below and above the contact point between the upward connecting leader, and the descending leader, by means of Eqs. (4) and (5). The time t, velocity v() and length M can be related by the following expression for a tortuous channel:
t=
M
0
M − d + c v()
(9)
with these definitions, the function () can be evaluated for a tortuous channel, using the same technique described in [18,19], and replacing z by . Finally, for a known tortuous geometry, the return stroke current profile can be determined using the Eq. (7). In this manner, the current waveform for a point of height z can be determined for the associated straight channel and for the tortuous channel in a point with the same height (z), the last, obtained through Eq. (1). The results obtained are presented in Sections 4.2 and 4.3. 4. Results and analyses The results presented in this section were obtained by means of numerical simulation and therefore they are theoretical work. Even so, the linear charge density model has shown to be in agreement with the findings of other authors, as it is described to continue. The results for the distribution of currents along the leader channel are obtained by evaluating the model previously published in [16], changing two input parameters, namely the channel geometry and the linear charge density model. 4.1. Linear charge density The model obtained for the linear charge density along a tortuous channel is presented in Eq. (10). (, Q ) =
(0.003768 + Q )(0.005088e−0.0004386 ) 9.98 − 7.826e−10
ρ [C/m]
(10)
This equation describes the variation of the linear charge density along a tortuous channel, and it is a function of the distance from the ground along the tortuous channel, , and the total electric charge stored in the complete channel, Q. As it was mentioned in the Section 2.3, a strong correlation was found between the variables , Q and . This correlation allowed to obtain the Eq. (10). The results for the goodness of fit obtained are the following: the sum of squares due to error, SSE = 0.1521, the coefficient of multiple determination, R2 = 0.9329, the adjusted coefficient of multiple determination, AdjR2 = 0.9329 and the root mean squared error, RMSE = 0.001886. As an example, a value of Q = −0.894 C has been selected, and the variation of the linear charge density has been plotted as a function of . The result is shown in Fig. 1. In this same figure, the linear charge density profile obtained by Cooray et al in [21] for an expected return stroke peak current of 30 kA, is plotted in
γ [m ]
Fig. 1. Variation of the linear charge density along the leader channel. Solid line: by means of Eq. (10), dashed line: the charge density profile obtained by Cooray et al. in [21] for a return stroke peak current of 30 kA.
order to compare both methodologies. It is important to remember that the Cooray et al. model considers a straight leader channel, while the Vargas and Torres model considers a leader channel with tortuosity. Eq. (10) is a prediction of the Vargas and Torres model and Fig. 1 shows that both, the Vargas and Torres and Cooray et al. models, predict a major accumulation of electric charge near the tip of the descending leader, and smaller amounts of charge as one moves towards the cloud. It is worth noting that both models predict a similar distribution of charge, with some difference, because the Vargas and Torres model considers the tortuosity of the leader channel when calculating the distribution of charge, but the total electric charge stored in the leader channel is the same for both models. According to the relationship obtained by Cooray et al. in [21], a return stroke peak current of −30 kA corresponds to a total leader charge of −0.894 C (the value selected to be used with Eq. 10 to plot the solid line in Fig. 1). 4.2. Tortuous channel with homogeneous linear charge density vs. straight channel with homogeneous linear charge density In this section, the current profiles of tortuous channels are compared with those of the associated straight channels. Two specific geometries of tortuous channel are used, and the charge per unit length remains constant in both cases. A value for the total charge of −3 C has been uniformly distributed along the straight and the two tortuous channels. The values used for the parameters are the following: lc = 53 m, c = 13.548, r = 1500, vi = 2 × 108 m/s y v0 = 4 × 106 m/s. In Fig. 2 a peak current attenuation can be observed in the waveforms for the tortuous channel with respect to those for the straight channel. The results are summarized in Table 1. The results obtained for the second tortuous channel are shown in Fig. 3. Again, a peak current attenuation can be observed in Fig. 3 and the results are summarized in Table 2. The results obtained in this section, and presented in Figs. 2 and 3 have shown two important facts when the current waveforms for a straight channel are compared with those for a tortuous channel. Table 1 Peak current attenuation, tortuous channel 1 and homogeneous. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 1000 2000
21.06 16.2 9.27
20.7 15.43 8.28
1.71 4.75 10.67
28
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Fig. 2. First return stroke current profile for: a straight channel (solid line) with homogeneous charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 1000 m and (c) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4360 m.
Fig. 4. First return stroke current profile for: a straight channel (solid line) with variable charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4360 m. Table 3 Peak current attenuation, tortuous channel 1 and variable. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 2000
21.06 9.27
16.17 5.16
23.22 44.34
velocity decreases more rapidly than in the straight channel, then causing an attenuation in the peak currents value. 4.3. Tortuous channel with variable linear charge density vs. straight channel with homogeneous linear charge density
Fig. 3. First return stroke current profile for: a straight channel (solid line) with homogeneous charge density and a tortuous channel (dashed line) with homogeneous charge density, at diffzerent heights: (a) 500 m, (b) 1000 m and (c) 2000 m. Simulation case 2 of 2, with a total length of the tortuous channel equal to 4477 m. Table 2 Peak current attenuation, tortuous channel 2 and homogeneous. Height (m)
Straight channel (kA)
Tortuous channel (kA)
Attenuation (%)
500 1000 2000
21.06 16.2 9.27
20.34 15.36 8.04
3.42 5.18 13.27
First, a time delay is present in the currents for the tortuous channel and second, a peak current attenuation is also present in the currents for the tortuous channel. These two factors can be associated to physical causes, which are described to continue: the time delay is due to the channel geometry, because the return stroke front has to travel longer in a tortuous channel than in a straight one to reach the same height. Also in this travel, the velocity of the return stroke front decreases more rapidly in the tortuous channel than in the straight channel, because the return stroke velocity is dependent on z in a straight channel and it is dependent on in a tortuous channel. The other important fact is the peak current attenuation. This result cannot be associated with the linear charge density, for in that part the simulations were made with a homogeneous distribution of electric charge equal to −3 C along each channel. Again, this attenuation is due to the tortuosity of the lightning channel, because the total current in a point depends on the velocity of the return stroke front, and because in the tortuous channel, this
In this section, the return stroke current profiles for tortuous channels are compared with those for the associated straight channels. In this case, it is included not only the tortuosity of the lightning channel, but also the variation of its linear charge density. The values of the parameters for the straight channel are: lc = 53 m, = 0.00075 C/m, c = 13.548, r = 1500, vi = 2 × 108 m/s y m/s, and for the tortuous channel the values are the same except for: lc = 105 m, c = 26.8403. The linear charge density is calculated by means of Eq. (10) with Q = −3 C. Again, two specific geometries of tortuous channel are used (the same of Section 4.2). In this case, Fig. 4 shows a much more notable attenuation in the peak currents value for the tortuous channel compared to the straight channel, the results are summarized in Table 3. The simulations for the second tortuous channel are shown in Fig. 5. Again, the attenuation is present and the results are summarized in Table 4. The results presented in Figs. 4 and 5 have shown that when the linear charge density along the channel is taken into account together with the tortuosity, the peak current attenuation and the time delay are much more notable than in the case of considering only the channel geometry. These results can be analyzed in Tables 3 and 4. As well as the physical causes already analyzed in the previous section, there are some other aspects to be considered about the results obtained in this section. Here, the major cause of Table 4 Peak current attenuation, tortuous channel 1 and variable. Height (m)
Straight channel (kA)
500 2000
21.06 9.27
Tortuous channel (kA) 15.73 5
Attenuation (%) 25.31 46.06
J.D. Pulgarín-Rivera et al. / Electric Power Systems Research 113 (2014) 25–29
Fig. 5. First return stroke current profile for: a straight channel (solid line) with variable charge density and a tortuous channel (dashed line) with homogeneous charge density, at different heights: (a) 500 m, (b) 2000 m. Simulation case 1 of 2, with a total length of the tortuous channel equal to 4477 m.
the time delays and peak current attenuation is due to the expression for the linear charge density predicted by the Vargas and Torres model [11,12]. Considering that the length of the upward connecting leader (lc ) depends on the linear charge density stored near the tip of the descending leader, and that the Vargas and Torres model has predicted the major accumulation of charge near the tip of the descending leader, it is evident that the length of the upward connecting leader has to increase. This fact can be seen in the results of the simulations, where the value for the length of the connecting leader in the case of homogeneous charge density was calculated equal to 53 m and equal to 105 m for the case of variable charge density. According to Eq. (5), the return stroke front velocity depends on lc and that is the principal reason why in the case of variable charge density the time delays and peak current attenuations are more notable than in the case of homogeneous charge density. 5. Conclusions In this paper, a simple model for the linear charge density distribution along tortuous lightning channels has been developed. Also, a technique for the evaluation of the return stroke current profile has been developed using a current generation type return stroke model. It is important to point that the results obtained here are only theoretical work, with exception of the model for the linear charge density, which was obtained using both, numerical simulation and experimental measurements reported by other authors. The model obtained for the linear charge density predicts major accumulations of electric charge near the tip of the descending leader and less amounts of electric charge as one moves towards the thundercloud, which is in accordance to the findings of other authors, as the model presented in [21]. This behavior is a prediction of the Vargas and Torres model [11,12]. An important advantage of this model is that it allows the calculation of the linear charge density in a simple way, without having to simulate every time the charge density is required. The principal results have shown that when the currents are calculated using a homogeneous charge density for both, the straight and the tortuous channel, they are present time delays and peak
29
current attenuations in the currents for the tortuous channel with respect to the straight channel. When the linear charge density model is included and considered together with the channel geometry, the time delays and peak current attenuations are much more notable than in the case of considering only the channel tortuosity. It was found that the effect of tortuosity is to cause a time delay in the currents and to reduce the return stroke propagation velocity. The effect of the linear charge density was found to produce a larger upward connecting leader, thus reducing also the return stroke front velocity and causing a major peak current attenuation. Finally, and according to the previous conclusions, this work has shown that when modeling the return stroke currents, the model for the linear charge density is more important than the channel tortuosity, because the latter produces major time delays and peak current attenuations. References [1] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromagn. Compat. 40 (4) (1998) 403–426. [2] G. Lupò, C. Petrarca, V. Tucci, M. Vitelli, EM fields generated by lightning channels with arbitrary location and slope, IEEE Trans. Electromagn. Compat. 42 (1) (2000) 39–53. [3] Z. kuo Zhao, Q. lin Zhang, Influence of channel tortuosity on the lightning return stroke electromagnetic field in the time domain, Atmos. Res. 91 (2-4) (2009) 404–409. [4] D.K.L. Chia, A.C. Liew, Effect of tortuosity of lightning stroke path on lightning electromagnetic fields, in: 2008 Asia-Pacific Symposium on Electromagnetic Compatibility & 19th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore, 2008. [5] F. Guo, B. Zhou, Influence of branch geometry on radiation field characteristics from lightning return strokes in the time domain, in: CEEM 2009, 5th AsiaPacific Conference on Environmental Electromagnetics, 2009. [6] M. Saito, M. Ishii, N. Itamoto, Influence of geometrical shape of return stroke channel on associated electromagnetic fields, in: 2011 7th Asia-Pacific International Conference on Lightning, November 1–4, 2011, Chengdu, China, 2011. [7] M. Saito, M. Ishii, N. Itamoto, Influence of geometry of – GC strokes on associated electromagnetic waveforms, in: 2011 International Symposium on Lightning Protection (XI SIPDA), Fortaleza, Brazil, October 3–7, 2011, 2011. [8] A. Andreotti, U.D. Martinis, C. Petrarca, V.A. Rakov, L. Verolino, Lightning electromagnetic fields and induced voltages: influence of channel tortuosity, in: 2011 XXXth URSI General Assembly and Scientific Symposium, 2011. [9] D. Amarasinghe, U. Sonnadara, M. Berg, V. Cooray, Channel tortuosity of long laboratory sparks, J. Electrostat. 65 (8) (2007) 521–526. [10] A. Gulyás, N. Szedenik, 3D simulation of the lightning path using a mixed physical-probabilistic model – the open source lightning model, J. Electrostat. 67 (2-3) (2009) 518–523. [11] M. Vargas, H. Torres, On the development of a lightning leader model for tortuous or branched channels – Part I: Model description, J. Electrostat. 66 (2008) 482–488. [12] M. Vargas, H. Torres, On the development of a lightning leader model for tortuous or branched channels – Part II: Model results, J. Electrostat. 66 (2008) 489–495. [13] R. Hill, Tortuosity of lightning, Atmos. Res. 23 (3) (1988) 217–233. [14] R.D. Hill, Analysis of irregular paths of lightning channels, J. Geophys. Res. 73 (6) (1968) 1897–1906. [15] V.P. Idone, R.E. Orville, Channel tortuosity variation in Florida triggered lightning, Geophys. Res. Lett. 15 (7) (1988) 645–648. [16] V. Cooray, R. Montano, V. Rakov, A model to represent negative and positive lightning first strokes with connecting leaders, J. Electrostat. 60 (2004) 97–109. [17] D.M. Bates, D.G. Watts, in: Nonlinear Regression Analysis and Its Applications, John Wiley and Sons, Inc., Hoboken, New Jersey, 1988. [18] V. Cooray (Ed.), The Lightning Flash, The Institution of Electrical Engineers, IEE, London, United Kingdom, 2003. [19] V. Cooray, Predicting the spatial and temporal variaton of the electromagnetc fields, currents and speeds of subsequent return strokes, IEEE Trans. Electromagn. Compat. 40 (4) (1998) 427–435. [20] D.M. LeVine, R. Meneghini, Simulation of radiation from lightning return strokes: the effects of tortuosity, Radio Sci. 13 (5) (1978) 801–809. [21] V. Cooray, V. Rakov, N. Theethayi, The lightning striking distance-revisited, J. Electrostat. 65 (2007) 296–306.