A New Estimation Model of the Lightning Shielding ... - IEEE Xplore

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach Jianbiao Li, Qing Yang, Wenxia Sima, Caixin Sun, Tao Yuan State Key Laboratory of Power Transmission Equipment & System Security and New Technology Chongqing University, Shapingba District, Chongqing, 400044, P. R. China and Markus Zahn Department of Electrical Engineering and Computer Science, Research Laboratory of Electronics, Laboratory for Electromagnetic and Electronic Systems High Voltage Research Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

ABSTRACT The path of the lightning discharge shows characteristics of branching and tortuosity, and the lightning shielding failure of the transmission line also has statistical features. In this paper, an improved leader progression model using a fractal approach is introduced, which considers both the deterministic and stochastic features of the downward lightning leader. Based on the proposed model, the statistical features of lightning shielding failure on transmission lines will be analyzed. First, the fractal dimensions of the downward lightning trajectory in simulation are calculated and the value of breakdown probability constant η will be determined based on the fractal dimensions of natural lightning. Second, using the proposed model the shielding failure probability contour around the transmission line space is also obtained and used to describe the shielding failure zone and then the contour curve is applied to analyzing the characteristics of transmission lines that suffer from direct lightning strokes. Moreover, a method to estimate the lightning shielding failure rate of transmission lines is put forward, and the influence of the line height and the shielding angle of ground wires on the shielding failure rate of transmission lines are also investigated. Index Terms — LPM, fractal approach, estimation method, shielding failure, UHV transmission line.

1 INTRODUCTION LIGHTNING protection plays a crucial role in the safe operation of transmission lines. Circuit breaker trips on transmission lines caused by direct lightning strikes can lead to economic loss. Such trips can be mainly classified as backflashover trips and lightning shielding failure trips. According to statistical records in Russia [1], with the increase of line voltage the proportion of lightning shielding failures in total trip number also increase. UHV and EHV transmission line trips are mainly caused by lightning shielding failures [1]. Therefore, lightning shielding protection has become an important issue in the protection of transmission lines. Currently, the lightning protection design against direct lightning strikes is generally based on the accurate estimation of the transmission line’s lightning shielding performance. Manuscript received on 5 November 2010, in final form 28 March 2011.

Therefore, a precise estimation model for lightning shielding performance plays an important role in the lightning protection design of transmission lines. Since the 20th century, scholars have conducted research on lightning shielding models [2-8]. Currently, the models frequently used in lightning shielding research are: the Electric Geometry Model (EGM) [2-4], the improved EGM [5], and the Leader Progression Model (LPM) [6, 7]. Young et al [2] proposed the initial concept of EGM. Based on the earlier research and field experiments on lightning shielding failures of transmission lines, Armstrong and Whitehead [3] connected the lightning discharges feature with the structural size of transmission lines, and thus developed EGM. The model assumed that the striking distance had some relation to the lightning current amplitude. In terms of this relationship, Anderson [4], Love [8] and others made their respective studies. Afterwards, with the elevation of transmission line voltage level and the increase of tower height, Eriksson [5]

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and others believed that the striking distance was not only related to the lightning current, but also related to the height of grounded objects, and they proposed the improved EGM. EGM and Eriksson’s improved EGM can calculate the choice of shielding angle of the transmission line under different line parameters and topographies, so they are widely used in lightning protection design of transmission lines. However, in [9-14], it is shown that the actual shielding failure number of large-sized transmission lines is larger than the calculated results of EGM and Eriksson’s improved EGM. In the 1990s, based on the observation and research on long air gap discharges in the laboratory, Dallera and Garbagnati [6], Rizk [7] and others put forward the LPM for lightning shielding failure analysis. LPM gives an initial description of the whole direct lightning striking process on a transmission line from a physical perspective. The model uses a mathematical approach to describe the initial upward leaders, the leader progression and the final jump. It also introduces the lateral distance and shielding failure width, which provide an effective method to estimate the lightning shielding performance of transmission lines and to analyze lightning shielding failures. However, LPM assumes the leader propagates in the direction of maximum electric field intensity. In other words, it only considers the deterministic factors in the lightning leader progression and ignores the influence of leader branches on the spatial distribution of the electric field intensity. This simplification influences the determination of the stroke object. If LPM can consider the stochastic features of lightning leader progression, its simulation process will be closer to the natural lightning strike process. Recently, many researchers use the fractal approach to model the observed paths of lightning discharges [15-20]. The photographed data of natural lightning discharge reveals that the lightning leader has an obvious effect of branching and tortuosity, which can be described by fractal mathematics. Kawasaki and Matsuura [15], and Tsonis and Elsner [16] were the first to use the fractal approach to simulate the lightning trajectory. Then Petrov and Petrova [17] use the fractal approach to model the lightning channel and to research the probability of direct lightning strikes to buildings. However, the lightning shielding model of transmission lines is not a simple lightning trajectory simulation. It is also different from lightning protection of buildings. The occurrence of a lightning shielding failure is also affected by the competition of upward leaders from the ground wires and the phase conductors. The upward initial process of a horizontal conductor is different from the upward initial process of buildings. Therefore, the above models using the fractal approach cannot be used directly in the research on lightning shielding performance of transmission lines. The leader progression is not only deterministic, but also stochastic, with the phenomena of branching or tortuosity in the progressing process. The influences of these stochastic factors should not be ignored in the estimation of lightning shielding performance of transmission lines. Therefore, in the present study based on the fractal features of the lightning discharge, the basic theory of LPM is combined with fractal theory, in which the deterministic factors and stochastic factors in downward lightning leaders are considered. A method based on the

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proposed model is also put forward to estimate the lightning shielding performance of transmission lines.

2 BASIC PRINCIPLES OF THE ESTIMATION MODEL The proposed model is developed on the basis of the conventional LPM. According to LPM theory, the direct lightning striking process on transmission lines can be divided into four stages, as shown in Figure 1. First, the downward lightning leader propagates vertically from the cloud bottom. At the same time, since there are many charges in the lightning channel, the electric field intensity is also increased around the lines and the earth. As the downward lightning leader propagates towards the earth, the induced electric field intensity around the lines and the earth also continuously increases. While the downward lightning leader propagates to a critical height, due to the effect of the increased electric field, continuous upward leaders are produced from the surface of the phase conductors and the ground wires. The upward and the downward leaders continue to propagate in the space. At this time, the electric field intensity between the upward and the downward leaders, and the electric field intensity between the downward lightning leader and the ground continuously grow. When the electric field intensity in the gap between leaders is high enough, a final jump will occur. There are thus several factors influencing the direct lightning strike process: the distribution of charge in downward leaders, the initial criterion of continuous upward leaders, the progressing rules of leaders and the criterion of the final jump. In this section the setting of these criteria in the proposed model are discussed.

Figure 1. The direct lightning strike process on transmission lines.

The proposed model simulates the downward lightning leader using the fractal approach from the initial start of an upward leader. That is, the proposed model assumes that when

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

an upward leader has not yet started on ground objects, the lightning leader propagates vertically from a cloud towards the ground, and the leader channel has no fractal features. Once an upward leader has started on ground objects, the fractal approach can be used to simulate the upward and the downward leaders, and to study the direct lightning strike process to transmission lines. Such simplification is based on the following two reasons: (a) The simulation of the proposed model does not start from the lightning leader's initial progression from cloud bottom, because when the lightning leader tip propagates from the cloud bottom to the height Hi ,which is the height of the lightning leader tip at the upward leader initial moment, the stochastic feature of its position will increase the stochastic feature of the lightning shielding failure probability, which is unfavorable for the research on factors influencing the transmission line’s lightning shielding performance. (b) The existing lightning shielding models generally assume that when an upward leader has not been generated, the ground and the earthed objects have very small influence on the progression of the downward lightning leader. This influence is so small that it can be ignored, and the downward lightning leader propagates downward randomly. So, on any horizontal level above the height Hi, the distributions of lightning density are the same. Therefore this simplification will not affect the estimation result of shielding failure rate. 2.1 THE CHARGE DISTRIBUTION IN DOWNWARD LEADERS When lightning leader propagates to ground in discrete steps, the charges in lightning channel will influence the electric field spatial distribution around the lightning channel. Therefore, in order to calculate the variation of spatial electric field during the progress of lightning, the proposed model uses a line charge varied linearly with channel height to simulate the main channel of lightning [21]. When the lightning leader reaches the certain height Hi, the downward leader channel propagates towards ground in a fractal form [17]. Meanwhile, since the lightning leaders will branch and bend, it is difficult to continue to use a linear distribution to describe the distribution of charges in downward lightning leader branches. Thus, the charge density of the downward lightning leader branches, i.e. the part with fractal features, is set as a uniform charge distribution. Moreover, by observing natural lightning discharge process, it is known that during the progressing process of lightning leader in space, the leader tip is brightest [22]. Therefore, a constant charge Q0 is used to simulate the tip of lightning [21]. Dellera [6] believes that higher values of charge in the channel will result in higher amplitudes of first stroke current; and according to the observation data, the relation between the total quantity of charge in the lightning leader channel QT and the lightning current amplitude I can be expressed by the following formula [21, 23, 24]. 1

 I  0.7 QT    (1)  25  where, QT is the total charge in the lightning leader, C; I is the magnitude of lightning current, kA.

The height of the cloud HC is set at 2.5 km in this paper [21]. For the main lightning channel, when the height of the leader is HC, the line charge density τ=0; when the height of the leader tip is H0, the line charge density τ=τmax, with τmax as the largest line charge density. Then according to the linear distribution, the total volume of charge will be [21]  (H  H0 ) QT  max C  Q0 (2) 2 where Q0 (C) is the charge inside the leader tip, which can be obtained by [21]: (3) Q0  2      r0 2  EB where ε is the dielectric constant of air; r0 is the radius (m) of the lightning leader tip, can be set as 6m [25]; and EB (kV/m) is the critical electric field intensity for propagation of the leader. For positive lightning, EB can be set as 500 kV/m, and for negative lightning, EB is set as 1000 kV/m [17]. Therefore, the line charge density inside the main channel of lightning will be [21] m 2(QT  Q0 )  ( h)  ( H C  h)  ( H C  h) (4) HC  H0 ( H C  H 0 )2 where τ(h) (C/m) is the charge intensity inside the lightning leader. For the fractal lightning leader, the line charge density inside the lightning branch is taken to be uniform distribution and is given by:

F  

QT  Q0  

HC

Hi

Lf l



Hi

H0

 (h)  dh

 (h)  dh (5)

Lf l

where τ(h) can be obtained from equation (4); Lfl is the total length (m) of the fractal lightning leader branch. 2.2 THE INITIAL CRITERION OF THE UPWARD LEADER At present, there are mainly three continuous upward leader initial criteria used in the study of lightning shielding failure of transmission lines [7, 21, 26-28]: a) the upward leader induced voltage Uic of the horizontal conductor proposed by Rizk [7]; b) Peek criterion: when the electric field intensity at the conductor surface is higher than the critical field intensity EC, the upward leader will be generated [21, 26]; c) Carrara criterion [27, 28]: the upward leader will start from the conductor surface in a long gap when the voltage applied on a conductor exceeds a critical value. The Peek criterion can consider the influence factors adequately, such as the phase conductor’s working voltage, the induced electric field of the downward leader and the coupling electric field between conductors. From the superposition principle of electric field, the Peek criterion can also explain the actual operating experience that for a direct negative lightning strike, the upward leader is easier to be generated from positive phase conductors. Thus the proposed model uses the Peek criterion to judge the initiation of the continuous upward leader. As for horizontal conductor, Eriksson [5], Dellera et al [6], Rizk [7] employed the critical radius concept to determine the

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ambient field necessary for the upward leader continuous inception, of which the critical radius is the one required to initiate a direct corona-to-leader transition [6]. Dellera et al [6] believe that for the conductor-plane configuration, when the conductor radius is larger than the critical radius, the criterion for the initiation of the continuous upward leader is the same as the criterion for corona onset. When the conductor radius is smaller than the critical radius, the upward leader continuous inception criterion is obtained by calculating the conductor corona inception criterion under critical radius [6]. Therefore, the critical initial electric intensity can be obtained from the formula [21, 26]: 0.03 EC  3000  m    (1  ) (6)  r where EC (kV/m) is the critical initial electric field intensity; m is the roughness coefficient of the conductor surface; δ is relative air density; and r (m) is conductor critical corona radius. For the ground wires, the radius of ground wire is usually smaller than critical radius, so r is set at the critical radius, r= 0.1 m [6]; For the phase conductor, the radius of conductor bundles is usually larger than the critical radius, and r is set as the equivalent radius of the conductor bundles. 2.3 THE PROGRESSION RULES OF LEADERS LPM assumes that the leader’s progression direction always goes towards the position around the leader tip, where the electric field intensity is largest. But in fact, the observation of direct lightning strikes on earthed objects show that the path of lightning discharge can be branching and tortuous, and the effect of branching and tortuosity of the lightning channel would also influence the electric field intensity in space and thus further influence the progression of the leader. Otherwise, the effect of branching and tortuosity of the lightning discharge indicates that the progression of the leader has fractal features. The fractal approach can be used in analyzing the progression patterns of leaders. Therefore, in this paper, considering the deterministic and stochastic factors in downward lightning leader progression fully, a fractal approach is used to model the trajectory of the leaders, and the details will be introduced in the following. 2.4 THE FINAL BREAKDOWN CRITERION In the proposed model, when the electric field intensity between the ground and any one of the leader branches or lightning branches exceeds the critical breakdown electric field 500 kV/m [29], the final jump will happen in the proposed model, and the stroke object is also determined.

3 THE REALIZATION OF THE LEADER’S FRACTAL PROGRESSION The fractal approach is widely used in many kinds of progression, among which, Dielectric Breakdown Model (DBM) [30] is generally used in the computer simulation of the progression path of discharges in dielectrics. According to that simulation, in the process of dielectric breakdown, it is more probable to have breakdown in a position where the electric field intensity in the dielectric is most intense than that in other areas. In the area shielded by other discharge paths,

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the probability to have a discharge is very low and even zero. This phenomenon confirms that in the progression of lightning leaders, the ionization in air has stochastic features, and it is also controlled by the distribution of electric field in space. The proposed model uses the DBM to simulate the trajectory of the leader. The calculation process of the fractal leader progression is shown in Figure 2 and the procedure is described as follows: Step:1 Discretize the space and set the initial fixed particles, which are also called seeds, at the tips of the downward leader, the conductors and the ground wires, as shown by the solid points in Figure 3; Step:2 Set the moving particles around the tip of the leader, as shown by the moving particles in point B on Figure 3. Moving particles have random Brownian motion in the discrete space, which is shown as the trace of the dashed line in Figure 3. The upward and downward leader progresses most quickly at their tips respectively, and in the calculation of lightning shielding, the progression in leader tips plays a decisive role in the issue whether lightning will strike conductors. Thus it is reasonable to set the initial position of moving particles at the tips of the leader. Step:3 In each leader branch, when the moving particles move to the possible breakdown point around any seed, for example shown in Figure 3 at the semisolid points around point P as possible breakdown points. When a particle moves to point P′, its Brownian movement will stop, and the probability to have breakdown at point P′ is calculated by [30]:  p ( P, P ')  0, if E ( P, P ')  EB    ( E ( P, P ')  EB )  p ( P, P ')  N ,if E ( P, P ')  EB (7)   ( E ( P, Pi ')  EB )   i 1

where, p(P, P′) is the probability for breakdown to occur from P to P′ and EB is the critical electric field intensity for propagation of the leader. For positive lightning, EB can be set at 0.5 MV/m, and for negative lightning, EB is set as 1 MV/m [17]. E(P, P′) is the electric field intensity in the space from P to P′; E(P, Pi′) is the ith possible breakdown point which satisfies the formula E ( P , P ')  E B and goes from P to the point around P′, and N is the number of possible breakdown points of this kind. Constant η is an adjustable number that is used to present the relation between local electric field intensity and the breakdown probability in the dielectric. It directly determines the stochastic features of the leader’s progression direction. The selection of the breakdown probability constant η will be discussed in Section 4. Step:4 Make a breakdown judgment from the obtained probability: compute a random number m which satisfies a [0, 1] uniform distribution. Compare the random number m with the obtained probability p; if m is smaller than p, then it can be judged that breakdown will happen at point P′, if m is larger than p, then breakdown will not happen.

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

Step:5 If point P′ meets the condition to have breakdown, and breakdown happens, then the moving particle becomes a new seed. It will form a “seed cluster” together with other seeds and hence a new leader discharging channel comes into being. If point P′ does not meet the breakdown condition, then the moving particle will be stopped. The program returns to Step 2 and keeps on operating, until it determines the occurrence of lightning strikes.

electric field spatial distribution determines that each leader branch progresses most probably towards the point with most intense electric field at their tips. This feature makes the proposed model maintain the deterministic factor of the leader’s progression. This method not only considers the deterministic feature that breakdown happens more probably when the electric field at the leader tip is more intense, but also stochastic features in the process of leader progression.

4 THE SELECTION OF KEY PARAMETERS OF THE FRACTALCHARACTERIZED LPM The fractal dimension reflects how effectively the complex graph occupies space. It is a measure of the orderliness and randomness of the stochastic and deterministic features of fractal graphs. From the above description of the proposed model, it can be found that the value of the breakdown probability constant η in (7) influences the progressing probabilities at every point of the fractal leader channel, and it determines the stochastic and deterministic features of the lightning leader progression. In order to simulate the whole lightning path accurately, it is very important to select a proper η value. In this paper, the lightning leader will be simulated with different η values, hoping to determine how the constant η will influence the dimensions of the lightning path in simulation, and to determine the proper value of the constant η for the proposed model. The fractal dimensions of lightning progression based on the box-counting method is given by [31-33]: Db  lim

lb  0

Figure 2. Calculation process of the fractal leader progression

ln N (lb ) ln(1 / lb )

(8)

where Db is the fractal dimension, lb is the measurement scale, and N(lb) is the least number of squares that can cover the fractal graph, with lb as the side length. Based on the box-counting method, the fractal dimensions of the simulated lightning path with different values of η are calculated and are shown in Table 1. The lightning paths in simulation with three different η values are shown in Figure 4. Table 1. Calculated fractal dimensions with different η values. Η

0.1

0.5

1

5

10

Db

1.46±0.07

1.31±0.05

1.20±0.04

1.09±0.05

1.04±0.02

Figure 3. Branch picture of leader.

In the model calculation process, the progression of every new seed that constitutes the fractal leader will influence the electric field spatial distribution around the lightning channel, change the electric field intensity E(P, P′) in every direction, and thus influence the progressing probability p(P, P′) of the next moving particle. By repeating the above calculation and judgment procedure, the leader progressing path will be determined. In the proposed model, the charges in the downward lightning channel directly determines the electric field spatial distribution around the lightning channel, and the

Figure 4. Simulated figures of the lightning leader for various fractal dimensions

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Input the parameters of the transmission line and the lightning Calculate the electric field intensity in space

Lightning strike or not?

Yes

No

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range of natural lightning discharge paths. Therefore, in the proposed model, the ŋ value is set as 1. The calculation process of the proposed model is shown in Figure 5, and the simulation figure of a lightning strike on transmission lines is shown in Figure 6. The line is a ±800 kV dc transmission line, with the shielding angle at zero degrees, the conductor height at 44 m, and the ground wires height at 55 m. The lightning current is 30 kA, and the lateral distance between lightning and the line is 100 m.

Calculate the electric field intensity on the conductor surface

The upward leader is generated ?

No

Yes The fractal upward leader progresses The fractal lightning leader progresses Calculate the electric field intensity in space

No

Lightning strike or not? Yes Record the struck object

Figure 5. Calculation process of the proposed lightning model.

It can be found from (7) that when the η value is smaller than the proper value, in the criterion of leader progression the effect of the electric field distribution will be reduced, and the progressing probabilities in every direction of the leader tip tend to be the same, which leads to excessive branches in the leader channel, as shown in Figure 4a. In such a condition, the stochastic feature plays the main role in the leader progression, and the corresponding fractal dimension is relatively larger than that of natural lightning. Conversely, when the η value is larger than the proper value, the direction with most intense field in the leader tip will gain more probability to progress, while the probabilities for leaders to progress in other directions are smaller. The leader progresses along the direction with most intense field, but less probably along the branches, as shown in Figure 4c. In such a condition, the deterministic feature plays the main role in the lightning leader progression, and the corresponding fractal dimension is relatively smaller than that of natural lightning. Therefore, to some extent, the selection of ŋ value decides whether the lightning model can reflect the characteristics of branching and tortuosity of lightning path properly. Kawasaki, Matsuura [15] and Tsonis and Elsner [16] measure the fractal dimension of still pictures of natural lightning channels. They find that the empirical fractal dimensions of the natural lightning discharge paths exist in the range from 1.1 to 1.4; and as shown in Table 1, when the ŋ value is set between 0.5 and 5, the fractal dimension of the simulated lightning path Db exists within the fractal dimension

Figure 6. Direct lightning strike on the simulated transmission line.

5 ESTIMATION METHOD 5.1 THE IMPROVED ESTIMATION METHOD Conventional LPM defines a shielding failure width (SFW) to estimate the shielding failure number (SFN) of transmission lines [34]: 

SFN  N L  Aeq  T  N L  L  T   SFW ( I )  PL ( I )  dI 0

(9)

where SFN is the shielding failure number, strokes; NL is the ground flash density, flashes/km2-year; Aeq is the shielding failure areas, km2; I is the lightning current amplitude, kA; SFW is the shielding failure width, km; L is the length of the transmission line, km; T is the observation time, years; and PL is the probability density distribution of lightning currents, %/kA. Conventional LPM believes that when lightning occurs in the shielding failure area of transmission lines, shielding failure will inevitably happen. In the proposed model, it is believed that the lightning discharge has stochastic features. Even though lightning reaches into the shielding failure area of transmission lines, because of the attraction from the upward leader of ground wires and from the earth, shielding failure is not inevitable. There exists a probability to have shielding failures, as shown in Figure 7. Therefore, for a certain lightning current amplitude I, weight SFW as in (9) and use the equivalent SFW, SFWeq(I) to replace it. The weighting value is set to be the lightning probability PS so that SFWeq(I) is: 

SFW ( I )  SFWeq ( I )   PS ( I , D)  dD 0

(10)

where PS (%) is the probability of lightning strikes on the phase conductor, and D (km) is the lateral distance between lightning leader and the transmission line.

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

PS can be obtained by a large number of repeated calculations in the proposed model. By the model, the shielding failure probability PS at all levels of lightning currents can be calculated. For example, the 500 kV transmission line tower in Guangdong province of China is shown as Figure 8 and PS can be shown as the distribution curve in Figure 9:

has its peak value decrease as the lightning current amplitude I grows. Therefore, under the same line parameters, PS is a function of lightning current I and lateral distance D. In order to obtain SFR, calculate the shielding failure probability distribution under every lightning current level I for different lateral distance D, and by data fitting, a shielding failure probability function is fitted with lateral distance D and lightning current I as its variables [35]: PS  PS ( I , D) (11) Based on (9), (10) and (11), the SFN and SFR of the transmission line can be calculated as follows:

SFN  N L  T  L   PS ( I , D)  PL ( I )  dI  dD SFR 

SFN 100  L  T

(12) (13)

 0.01  N L   PS ( I , D )  PL ( I )  dI  dD

Figure 7. The shielding failure probability PS as a function of lateral distance.

Figure 8. Structural figure of 500 kV EHV transmission line in Guangdong province of China. 100

80

60

40

20

0

0

50

100

150

200

Figure 9. Probability of lightning strike on the conductor versus lateral distance D.

It can be found from Figure 9 that under a certain lightning current I, as the lateral distance D increases, PS will increase first and then decrease. PS under different lightning currents I

where SFN is shielding failure number of the transmission line, strokes; SFR is shielding failure rate of the transmission line, strokes/100 km-year; NL is the ground flash density, flashes/km2-year; PL is the probability density distribution of lightning currents, %/kA; T is the observation time, years; L (km) is the length of the transmission line; I is the current amplitude, kA; and D (km) is the lateral distance between the lightning leader and the transmission line. If the parameters of different parts of the line are different, they can be calculated one by one, and then by adding those up the total SFN will be calculated. 5.2 VALIDATION OF THE ESTIMATION METHOD As for the EHV transmission line, the China Southern Power Grid uses the lightning location system (LLS) to record the lightning trip numbers of 2550 km 500kV AC transmission line in Guangdong Province of China [36], and they also measured the lightning current amplitude of every lightning stroke on the line. From 1999 to 2003, there have been 40 lightning trip accidents on the line [36]. The 500 kV transmission line in Guangdong province of China [28] has its structure model in Figure 8. The SFR of the 500 kV transmission line is estimated by Eriksson’s improved EGM, LPM and the proposed model. The thunderstorm days and the ground flash density are 40 days/year and 2.9 flashes/km2-year, respectively [28], and the probability density function of the lightning crest current is distributed as [3, 9]: I I PL ( I )  0.0475  exp( )  0.0010  exp( ) (14) 20 50 The observation results of direct lightning strikes and the calculation results of the three estimation methods are shown in Figure 10 [36]. The data in Figure 10 show that, compared with Eriksson’s improved EGM, the calculation results of LPM and the proposed model are closer to observation data. According to Eriksson’s improved EGM, the largest lightning shielding failure current amplitude is 9.5 kA, that is, that lightning current higher than 10 kA will not cause lightning shielding failure. This result is obviously different from the observation data. Moreover, Eriksson’s improved EGM believes that with the increase of lightning current, the lightning shielding failure

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arc of transmission lines will decrease gradually, as shown in Figure 11. Therefore, with the increase of lightning current, the corresponding SFR will continue to decrease to zero. LPM and the proposed model show that as the lightning current I increases, SFR shows a tendency of growing at first and then decreasing. As shown in Figure 9, the calculation of the proposed model reveals that for the 5 kA lightning current, though the peak value of lightning shielding failure probability PS is larger than that of the 25 kA lightning current, SFW (I=5kA, PS>10%) is smaller than the SFW (I=25kA, PS>10%). Through the calculations of (10) and (13), it is revealed that from 5 kA to 25 kA, SFWeq rises as the lightning current I increases, so the SFR also increases. Correspondingly, SFWeq (I=5kA) and SFR (I=5kA) are smaller than SFWeq (I=25kA) and SFR (I=25kA) respectively, and the increasing tendency from 5 kA to 25 kA can be calculated by (13). For the 45 kA lightning current, though the SFW (I=45kA, PS>10%) is larger and the SFW (I=25kA, PS>10%), its peak value of PS is less than 20%, and the corresponding SFWeq(I=45kA) is smaller than that of 25 kA lightning current. Therefore, based on the calculation of (10) and (13), the SFR under lightning current higher than 25 kA shows a decreasing tendency. This result is closer to the observation data. 0.14

the improved EGM LPM proposed model observation

0.12 0.1 0.08 0.06 0.04 0.02 0

among which there were 79 negative lightning strikes. The number of lightning strikes on the upper, middle and lower phase conductors are 34, 27 and 18 strokes respectively. The proposed model estimates the ratio of strokes reaching each phase conductor, and also compares the calculation results of the conventional EGM, the improved EGM by Sakae Taniguchi [9] and the 3-D LPM [37]. The calculation and observations are shown in Figure 12. Conventional EGM and the Sakae Taniguchi’s improved EGM show that the lightning shielding failure arc length of the lower phase conductor is the largest, so the percentage of strikes on the lower phase conductor is higher than that of the upper and middle phase conductors. However, observation data show that the SFN of upper phase conductor and middle phase conductor are close to each other, but the SFN of the lower phase conductor is obviously lower than the other two conductors. Therefore, Conventional EGM and improved EGM by Sakae Taniguchi still give different results from the observations [9]. The proposed model believes that when the downward lightning progresses, the phase conductor in the higher position is easier to generate an upward leader. Meanwhile, the shielding angle also has obvious influence on the inception of upward leaders. For this transmission line in Japan, the shielding angles of upper, middle and lower phase conductor are respectively 6.24, -3.37 and -2.05 [11, 12]. Compared with the other two phase conductors, the upward leader inception of the upper phase conductor is restrained because of smaller shielding angle. These two factors determine the probability distribution of SFN of every conductor. Therefore, the rates of the upper phase conductor and middle phase conductor’s SFN are obviously higher than that of the lower phase conductors. The calculation result of the proposed model is close to 3D-LPM; and compared with EGM, it is closer to the observation data.

Figure 10. Distribution of lightning shielding failure rate under different lightning current amplitude.

Stroke percent (%)

B3 B2

Rgw2

B1

Rgw3 Ground wire Rgw1 Rl1

C2 Rl3

Rl2 Phase conductor

O

Observation 3-D LPM The proposed model Sakae Taniguchi’s improved EGM EGM

50 45 40 35 30 25 20 15 10 5 0 Upper Phase

C1

1719

Middle Phase

Lower Phase

Figure 12. Percentage of lightning strokes reaching each phase conductor.

D2

D1

Rg1

Rg2

Rg3

Ground

Figure 11. Calculation representation of the lightning shielding failure arc in EGM.

As for the UHV transmission line, Tokyo Electric Power Company [9, 11] has recorded the SFN of 1000kV UHV AC double-circuit transmission line in Japan. From 1998 to 2004, there have been 81 lightning shielding failure accidents,

6 LIGHTNING SHIELDING FAILURE ZONE The lightning shielding failure zone is used by EGM to determine whether a direct lightning strike will strike to a phase conductor. According to EGM, as shown in Figure 11, when direct lightning strikes happen around transmission lines, there exists a lightning shielding failure zone around the transmission lines so that it is possible for lightning to strike the phase conductor. It is possible to determine the lightning shielding failure zone by calculating the shielding failure arcs, but the results ignore the stochastic feature of lightning

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

discharges. In fact, the dividing border between the lightning shielding failure zone and the effective protection zone is not so obvious. Moreover, when downward lightning enters into the lightning shielding failure zone, due to the stochastic feature of lightning discharges, the phenomenon of lightning strikes on line conductors occurs in the form of a statistical probability. So, based on the proposed model, the lightning shielding failure zone is described in the form of lightning shielding failure probability in this paper. Through research on the distribution of lightning shielding failure probability around the transmission line space, the probability for different direct lightning strikes to cause lightning shielding failure can be better analyzed, which provides theoretical support to the analysis of lightning shielding failures and the lightning shielding design of transmission lines. In LPM, it is believed that before the leader progresses to the height of Hi, the lightning leader tip height at the upward leader initial moment, the lightning’s progression is not influenced by the grounded objects. When the leader reaches Hi, the lightning leader will be attracted by the upward leader generated from phase conductors and ground wires and its progressing direction will be changed. For the condition when the line parameters are determinate, Hi is determined by the lightning current amplitude I and the lateral distance D, which can be described as H i  h( D, I ) (15)

500

C 400 300

40 kA

B

200

A 100 0

50 kA

30 kA

10 kA 0

50

20 kA 100 D (m)

150

200

Figure 14. Calculated distribution of Hi(D, I).

Based on the proposed model and combined with equation (16), the lightning current I at each point of the transmission line space is calculated, so that the lightning shielding failure probability at each point in the transmission line space is obtained. The data of the lightning shielding failure probability is processed by mathematical fitting, and the contour line of the shielding failure probability is drawn. The calculation process is shown in Figure 15, and the shielding failure probability contour is shown in Figure 16.

By altering the position of Hi and I, (15) can be changed into: I  y ( D, H i ) (16) In order to attain the functional relation of (16), in this paper the charge simulation method is used to simulate the lightning leader channel, and Peek criterion is used as the initial criterion of upward leaders. The calculation process is shown in Figure 13. The calculation objective is a ±800 kV dc transmission line, with the shielding angle at zero degrees, the conductor height at 44 m, the ground wires height at 55 m, and the ground slope at zero degrees. Through the flow calculation of Figure 13, the functional relation in equation (16) is shown in Figure 14.

Figure 15. Calculation process of the shielding failure probability contour.

Figure 13. Calculation process of the lightning strike current amplitude I.

As Figure 16 shows, the curves No.1 to No.7 are the contour lines of the transmission line’s lightning shielding failure probability that ranges from 10-70%. The largest probability of lightning shielding failure occurs in the area near observation point A. it can be found from Figure 14 that the lightning current amplitude I at point A is 20 kA. On the phase conductors and ground wires of the transmission line, the induced voltage caused by the lightning channel is relatively small. The high operating voltage on the conductors makes the conductors have a stronger ability to trigger lightning than the

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ground wires. Thus, when lightning strokes with current amplitude around 20 kA happen around the transmission line, lightning shielding failure will occur most easily.

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In the above analysis, based on the proposed model, the lightning shielding failure zone is described by the shielding failure probability contour. The result shows statistical characteristics and reflects the deterministic and stochastic features of the lightning discharge. It is also in accord with the data distribution rules of the laboratory lightning simulation experiment [38, 39]. The proposed model can reflect the natural lightning progression rules better than LPM. Compared with the lightning shielding failure zone described in EGM, the shielding failure probability contour can better reflect the lightning shielding failure performance of transmission lines. Therefore, the transmission line estimation method based on the proposed model is more accurate and well-founded.

7 APPLICATION OF THE ESTIMATION METHOD ON THE TRANSMISSION LINES

Figure 16. Contour lines of the calculated lightning shielding failure probability in space.

Under the same lateral distance D, for example in Figure 16, when the lateral distance D is set at 50 m, as Hi grows from zero m, the lightning shielding failure probability of the transmission line tends to increase first, then to decrease. When Hi is less than 50 m, Ps is less than 30%, and it can be found from Figure 16 that the lightning current amplitude I in this zone is smaller than 10 kA; the lightning in this area mainly strikes the earth. When Hi grows gradually, for example, when Hi grows to 150 m, it is shown from Figure 14 that the lightning current amplitude I will increase to 20 kA, then the conductors’ and the ground wires’ ability to trigger lightning also increases. Meanwhile, because of its high operating voltage, the conductor’s upward leader predominates over the ground wire’s upward leader in space. The lightning shielding failure probability of the transmission line tends to increase. When Hi continues to grow, for example, when Hi grows to 300 m, the lightning current amplitude I in the area increases to 30-40 kA. The operating voltage on the conductor no longer exerts great influence on its upward initial leader, which makes the ground wire’s upward leader take predominance in the competition with the conductor’s upward leader. The lightning shielding failure probability of the transmission line tends to decrease. When the observation point (D,Hi) changes from near to far, for example in Figure 16, when observation point (D,Hi) moves from point A to point B then to point C, the lightning shielding failure probability keeps on decreasing in the process. Figure 14 shows that the current at point A is about 20 kA, which is smaller than the current of 30 kA in point B and that of 50 kA in point C. So when lightning strikes happen around the transmission line, lightning shielding failures will happen more easily when the lightning current amplitude has smaller values. As the observation point (D,Hi) keeps on moving further away from the transmission line, the lightning current I will keep on increasing, and the lightning shielding failure probability continues to decrease until zero. This conclusion is in accordance with the views of EGM and LPM that there exists a largest lightning shielding failure current.

The existing data show that the increase of the transmission line’s height will lead to the increase of the line’s ability to trigger lightning, and the lightning shielding performance of transmission lines will be influenced. In this paper, based on the proposed model and its estimation method to evaluate the SFR of transmission lines, the lightning shielding performance of transmission lines with different nominal heights are analyzed. The structural parameter of a ±800 kV UHVDC transmission line is used as the analyzing parameter, in which the ground wire height is higher than the conductor height by 11 m and the shielding angle is zero degrees. The structural size of the tower head being constant, and when the tower height is changed, the calculation results are shown in Table 2. Table 2. The shielding failure rate (strokes/100km-year) of transmission lines with different nominal heights. nominal heights (m) 50 55 60 65 70 SFR

0.107

0.119

0.133

0.152

0.174

From Table 2 it is shown that, as the nominal height is increased, SFR increases gradually. When the nominal height grows to 70 m, the SFR increases by almost 65% over the SFR when the nominal height is 50 m. This is because the increase of nominal height makes the ground wire’s and the conductor’s upward leaders easier to be initiated, and to have more ability to trigger lightning. Thus the lightning shielding failure area will be enlarged, and the SFR of the transmission line will increase correspondingly. Thus, in order to lower the SFR of transmission lines, the nominal height should be lowered as much as possible. For the high tower and long span transmission line sections, enhanced lightning shielding measures should be taken to ensure the lines meet the lightning shielding requirements. The setting of the shielding angle of transmission lines will influence the field distribution on the surface of ground wires and conductors, then influence the initiation of the ground wire’s and the conductor’s upward leaders, and finally to influence the lightning shielding failure probability of transmission lines. In this paper, based on the proposed model, the lightning shielding performance of transmission lines with different shielding angles was researched. When the conductor height is 44 m, the ground wire height is 55 m, the lightning shielding failure probability with different shielding angles is calculated and shown in Table 3.

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

The data in Table 3 shows that decreasing the shielding angle of the transmission line will lower its SFR. When the lightning leader progresses downward, the decrease of shielding angle suppresses the conductor’s upward leader, then the ground wire’s upward leader will predominate over the conductor’s upward leader. The protecting function of the ground wire is enhanced, so the SFR of the line will decrease correspondingly. Therefore, to reduce the shielding angle is the main method to improve the lightning shielding performance of UHVDC transmission lines. Table 3. The shielding failure rate (strokes/100km-year) of transmission lines with different shielding angles. shielding angles (o) -5 0 5 10 SFR

0.086

0.107

0.122

line and the choice of 3-D lightning leader path, would influence the lightning shielding performance of transmission line, so as to improve the current estimation method.

ACKNOWLEDGEMENT This work was supported by the National Basic Research Program of China (973 Program) (2009CB724504), the National “111” Project of China (B08036) and the National Natural Science Foundation of China (NSFC) (50707036).

REFERENCES [1]

0.141 [2]

In the UHVDC transmission project design in YunnanGuangdong province of China, the China Southern Power Grid Corporation uses the calculation estimation of the proposed model. In mountainous regions, the shielding angle of ±800 kV dc transmission lines is set at -9.6 degree; in plain regions, the shielding angle is set at -1.8 degree.

8 CONCLUSIONS In order to make the model closer to the discharge features of natural lightning, the fractal approach is applied to the LMP in this paper. By comparing the fractal dimension of natural lightning, the fractal parameters are adjusted, so as to control the deterministic factor and stochastic factor of lightning progression within a reasonable range. Based on the proposed model, the lightning shielding failure probability distribution of the transmission line is proposed and the method to calculate the SFR is improved. These factors are used to analyze the lightning shielding performance of transmission lines, and lead to the following conclusions: 1) The lightning stroke simulation figure of the proposed model describes the effects of tortuosity and branching of the lightning channel, and reflects the stochastic and deterministic features shown in the progression process of lightning. 2) When lightning occurs within the shielding failure width, lightning shielding failure is not inevitable, but appears with a probability. The lightning shielding failure probability PS is related to the lightning current amplitude I and the lateral distance D. 3) The lightning shielding failures are most likely when lightning strikes with current amplitude under 20 kA and occur in a position that is close to the transmission line, and the peak value of the lightning shielding failure probability under small lightning current amplitude is larger than that of large lightning current amplitude. The estimation method on the lightning shielding performance of transmission lines in the proposed model has been used on the ±800 UHVDC transmission project design in Yunnan- Guangdong province of China, by China Southern Power Grid Corporation. Besides, the authors intend to develop a 3-D fractal LPM in the future, study the 3-D lightning shielding failure probability space of transmission lines, and analyze how the factors such as line sag, the corridor terrain of

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[21] J. He, Y. Tu, R. Zeng, J.B. Lee, S.H. Chang, and Z. Guan, “Numerical analysis model for shielding failure of transmission line under lightning stroke”, IEEE Trans. Power Delivery, Vol. 20, pp. 815- 822, 2005. [22] G. A. Wurden and D. O. Whiteson, “High-speed Plasma Imaging: A Lightning Bolt”, IEEE Trans. Plasma Sci., Vol. 24, pp. 83- 84, 1996. [23] R. H. Golde, Lightning, New York, USA, Academic Press, 1977. [24] B. Vahidi, M. Yahyaabadi, M.R.B. Tavakoli and S.M. Ahadi, “Leader Progression Analysis Model for Shielding Failure Computation by Using the Charge Simulation Method”, IEEE Trans. Power Delivery, Vol. 23, pp. 2201- 2206, 2008. [25] M. A. Uman, The Art and Science of Lightning Protection, Cambridge University Press, New York, USA, 2008. [26] G. N. Alexandrov, V. L. Ivanov, and V. E. Kizevetter, Electric Strength of External High Voltage Insulation, Leningrad: Energija, 1969. (In Russian) [27] G. Carrara and L. Thione, “Switching surge strength of large air gaps: A physical approach”, IEEE Trans. Power App. Syst., Vol. 95, pp. 512524, 1976. [28] B. Wei, Z. Fu, and H. Yuan, “Analysis of lightning shielding failure for 500-kV overhead transmission lines based on an improved leader progression model”, IEEE Trans. Power Delivery, Vol. 24, pp. 14331440, 2009. [29] R H Golde, Lightning Protection, London, UK, Edward Arnold, 1973. [30] L. Niemeyer, L. Pietronero, and H. J. Wiesmann, “Fractal dimension of dielectric breakdown”, Phys. Rev. Letts., Vol. 52, pp. 1033-1036, 1984. [31] A.L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, New York, USA, 1995. [32] V. Rajini and U. K. Kumar, “Quantification of Damage due to Surface Tracking”, IEEE Trans. Dielectr. Electr. Insul., Vol. 14, pp. 1224- 1231, 2007. [33] A. Krivda, E. Gulski, L. Satish and W.S. Zaengl, “The use of fractal features for recognition of 3-D discharge patterns”, IEEE Trans. Dielectr. Electr. Insul., Vol. 2, pp. 889- 892, 1995. [34] J. Fu, W. Sima, J. Li, Q. Yang, and C. Sun, “Estimation on lightning shielding performance of the transmission lines based on the fractal theory”, High Voltage Eng., Vol. 35, pp. 1274- 1278, 2009. (In Chinese) [35] W. Sima, J. Li, Q. Yang, T. Yuan, and J. Fu, “Fractal characteristics of lightning channels and its application to the analysis of the lightning shielding performance of UHVDC transmission lines”, High Voltage Eng., Vol. 36, pp. 86- 91, 2010. (In Chinese) [36] D. Gu and L. Fan, “New consideration of 500 kV transmission line lightning protection in China”, Electric Power, Vol. 37, pp.18-21, 2004 (In Chinese). [37] M.R.B. Tavakoli and B. Vahidi, “Transmission-Lines Shielding FailureRate Calculation by Means of 3-D Leader Progression Models”, IEEE Trans. Power Delivery, Vol. 26, pp. 507-516, 2011. [38] G. Qian, X. Wang, X. Xu, and Z. Ding, “Study of scatter of shielding failure on transmission line”, High Voltage Eng., Vol. 24, pp. 17- 20, 1998. (In Chinese) [39] H. He, J. He, Z. Jiang, C. Wang, H. Ye, X. Wang, K. Bian, and S. Xie, “Experimental study on lightning shielding performance of ±500 kV HVDC transmission line”, Proceedings of the Chinese Society for Electr. Eng., Vol. 29, pp. 20-26, 2009. (In Chinese) Jianbiao Li was born in Guangxi province, China, on 5 October 1984. He received the B.S degree in electrical engineering in 2007 from Chongqing University, China. He is now a Ph.D. candidate in the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University. His major research interest is lightning protection for transmission lines.

Qing Yang received the B.S and Ph.D. degrees in electrical engineering, respectively, in 2002 from North China Electrical Power University and in 2006 from Chongqing University, China. He is now an Associate Professor in the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University. His research interests include outdoor insulation in complex ambient conditions and electric-field calculations. He is the author and coauthor of more than 30 journal and international conference papers.

1723 Wenxia Sima was born in Henan province, China, on 13 July 1965. She graduated from Chongqing University in 1988 and obtained the Ph.D. degree in 1994 from Chongqing University. Her employment experience includes the college of Electrical Engineering of Chonging University. Her fields of interest include high voltage outdoor insulation and overvoltage protection.

Caixin Sun received the B.S. degree in electrical engineering in 1969 from Chongqing University. Currently, he is Professor of Chongqing University, leading the research group on high voltage engineering. He is a member of the Chinese Academy of Engineering and Director of Electrical Power Engineering Committee of the National Science Foundation of China. His research interest is in high voltage engineering, especially online detection of insulation condition and insulation fault diagnosis for high voltage equipment, discharge mechanism of outdoor insulation in complicated environments, and high voltage techniques applied in biomedicine. He is an author and coauthor of over 200 publications. Tao Yuan received the B.S degree in 1999 from Sichuan Polytechnic Institute, China, and received the Master’s and Ph.D. degrees, respectively, in 2001 and 2010 from Chongqing University, China. He is now a lecturer in Chongqing University, and his research interests include overvoltage protection and grounding technology in power systems.

Markus Zahn (S'68-M'71-SM'78-F'93) received the B.S.E.E., M.S.E.E., Electrical Engineer, and Sc.D. degrees from the Department of Electrical Engineering at the Massachusetts Institute of Technology (MIT), Cambridge, MA, in 1968, 1968, 1969, and 1970 respectively. He then joined the Department of Electrical Engineering at the University of Florida, Gainesville until 1980 when he returned to MIT where he is now Professor of Electrical Engineering working in the Research Laboratory of Electronics, Laboratory for Electromagnetic and Electronic Systems and the High Voltage Research Laboratory. He is also the Director of the MIT Course VI-A EECS Internship Program, a cooperative work/study program with industry. He is the author of Electromagnetic Field Theory: A Problem Solving Approach (now out of print) but is working on a new reference book with a complete collection of solved electromagnetism problems. He has also co-developed a set of educational videotapes on demonstrations of electromagnetic fields and energy for the enriched teaching of electromagnetism. He has received numerous excellence in teaching awards at the University of Florida and at MIT. His primary research areas are ferrohydrodynamics and electrohydrodynamics for microfluidic and biomedical applications; nanoparticle technology for improved high voltage performance of electric power apparatus; modeling of electrical streamer initiation and propagation leading to electrical breakdown; Kerr electrooptic field and space charge mapping measurements in high voltage stressed materials, and for the development of model based interdigital dielectrometry and magnetometry sensors for measurement of dielectric permittivity, electrical conductivity, and magnetic permeability with applications to non-destructive testing and evaluation measurements and for identification of metal and low-metal content dielectric landmines and unexploded ordnance. Professor Zahn is co-inventor on 19 patents. He has contributed to about 10 book and encyclopedia chapters, about 115 journal publications, and about 175 conference papers. He was a Distinguished Visiting Fellow of the Royal Academy of Engineering at the University of Manchester, England; was the 1998 J.B. Whitehead Memorial Lecturer, and was the first James R. Melcher Memorial Lecturer in 2003; is an Associate Editor of the IEEE Transactions on Dielectrics and Electrical Insulation; is on the International Scientific Committee on Magnetic Fluids; and has received a Certificate of Achievement for completion of the “Deminers Orientation Course” at the Night Vision and Electronic Sensors Directorate Countermine Division at Ft. Belvoir, VA, USA.