Jianbiao - New Model Lightninig Trans Lines Fractal (2011)

7/31/2019 Jianbiao - New Model Lightninig Trans Lines Fractal (2011)

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1712 J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach

1070-9878/11/$25.00 2011 IEEE

A New Estimation Model of the Lightning ShieldingPerformance 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 TechnologyChongqing 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 SystemsHigh Voltage Research Laboratory,

Massachusetts Institute of Technology, Cambridge, MA 02139 USA

ABSTRACTThe 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 isintroduced, which considers both the deterministic and stochastic features of thedownward lightning leader. Based on the proposed model, the statistical features of lightning shielding failure on transmission lines will be analyzed. First, the fractaldimensions of the downward lightning trajectory in simulation are calculated and thevalue of breakdown probability constant will be determined based on the fractaldimensions of natural lightning. Second, using the proposed model the shielding failureprobability contour around the transmission line space is also obtained and used todescribe the shielding failure zone and then the contour curve is applied to analyzingthe characteristics of transmission lines that suffer from direct lightning strokes.

Moreover, a method to estimate the lightning shielding failure rate of transmission linesis put forward, and the influence of the line height and the shielding angle of groundwires on the shielding failure rate of transmission lines are also investigated.

Index Terms LPM, fractal approach, estimation method, shielding failure, UHVtransmission line.

1 INTRODUCTION LIGHTNING protection plays a crucial role in the safe

operation of transmission lines. Circuit breaker trips ontransmission lines caused by direct lightning strikes can leadto economic loss. Such trips can be mainly classified as back-flashover trips and lightning shielding failure trips. According

to statistical records in Russia [1], with the increase of linevoltage the proportion of lightning shielding failures in totaltrip number also increase. UHV and EHV transmission linetrips are mainly caused by lightning shielding failures [1].Therefore, lightning shielding protection has become animportant issue in the protection of transmission lines.Currently, the lightning protection design against directlightning strikes is generally based on the accurate estimationof the transmission lines lightning shielding performance.

Therefore, a precise estimation model for lightning shielding performance plays an important role in the lightning protection design of transmission lines.

Since the 20 th century, scholars have conducted research onlightning shielding models [2-8]. Currently, the modelsfrequently used in lightning shielding research are: the ElectricGeometry Model (EGM) [2-4], the improved EGM [5], andthe 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 failuresof transmission lines, Armstrong and Whitehead [3] connectedthe lightning discharges feature with the structural size of transmission lines, and thus developed EGM. The modelassumed that the striking distance had some relation to thelightning current amplitude. In terms of this relationship,Anderson [4], Love [8] and others made their respectivestudies. Afterwards, with the elevation of transmission linevoltage level and the increase of tower height, Eriksson [5] Manuscript received on 5 November 2010, in final form 28 March 2011.


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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 18, No. 5; October 2011 1713

and others believed that the striking distance was not onlyrelated to the lightning current, but also related to the height of grounded objects, and they proposed the improved EGM.EGM and Erikssons improved EGM can calculate the choiceof shielding angle of the transmission line under different line

parameters and topographies, so they are widely used inlightning 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 calculatedresults of EGM and Erikssons improved EGM. In the 1990s,

based on the observation and research on long air gapdischarges in the laboratory, Dallera and Garbagnati [6], Rizk [7] and others put forward the LPM for lightning shieldingfailure analysis. LPM gives an initial description of the wholedirect lightning striking process on a transmission line from a

physical perspective. The model uses a mathematical approachto describe the initial upward leaders, the leader progressionand the final jump. It also introduces the lateral distance andshielding failure width, which provide an effective method toestimate the lightning shielding performance of transmissionlines and to analyze lightning shielding failures. However,

LPM assumes the leader propagates in the direction of maximum electric field intensity. In other words, it onlyconsiders the deterministic factors in the lightning leader

progression and ignores the influence of leader branches onthe spatial distribution of the electric field intensity. Thissimplification 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 naturallightning strike process.

Recently, many researchers use the fractal approach tomodel the observed paths of lightning discharges [15-20]. The

photographed data of natural lightning discharge reveals thatthe lightning leader has an obvious effect of branching andtortuosity, which can be described by fractal mathematics.Kawasaki and Matsuura [15], and Tsonis and Elsner [16] werethe first to use the fractal approach to simulate the lightningtrajectory. Then Petrov and Petrova [17] use the fractalapproach 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 asimple lightning trajectory simulation. It is also different fromlightning protection of buildings. The occurrence of alightning shielding failure is also affected by the competitionof upward leaders from the ground wires and the phaseconductors. The upward initial process of a horizontalconductor 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 lightningshielding performance of transmission lines.

The leader progression is not only deterministic, but alsostochastic, with the phenomena of branching or tortuosity in the

progressing process. The influences of these stochastic factorsshould not be ignored in the estimation of lightning shielding

performance of transmission lines. Therefore, in the presentstudy based on the fractal features of the lightning discharge, the

basic theory of LPM is combined with fractal theory, in whichthe deterministic factors and stochastic factors in downwardlightning leaders are considered. A method based on the

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

2 BASIC PRINCIPLES OF THEESTIMATION MODEL

The proposed model is developed on the basis of theconventional LPM. According to LPM theory, the direct

lightning striking process on transmission lines can be dividedinto four stages, as shown in Figure 1. First, the downwardlightning leader propagates vertically from the cloud bottom. Atthe same time, since there are many charges in the lightningchannel, the electric field intensity is also increased around thelines and the earth. As the downward lightning leader

propagates towards the earth, the induced electric field intensityaround the lines and the earth also continuously increases.While the downward lightning leader propagates to a criticalheight, 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 thedownward leaders continue to propagate in the space. At thistime, the electric field intensity between the upward and thedownward leaders, and the electric field intensity between thedownward lightning leader and the ground continuously grow.When the electric field intensity in the gap between leaders ishigh enough, a final jump will occur. There are thus severalfactors influencing the direct lightning strike process: thedistribution of charge in downward leaders, the initial criterionof continuous upward leaders, the progressing rules of leadersand 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 lightningleader using the fractal approach from the initial start of anupward leader. That is, the proposed model assumes that when


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an upward leader has not yet started on ground objects, thelightning leader propagates vertically from a cloud towards theground, and the leader channel has no fractal features. Once anupward leader has started on ground objects, the fractalapproach can be used to simulate the upward and thedownward leaders, and to study the direct lightning strike

process to transmission lines. Such simplification is based onthe 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 lightningleader tip propagates from the cloud bottom to the height

H i ,which is the height of the lightning leader tip at the upwardleader initial moment, the stochastic feature of its position willincrease the stochastic feature of the lightning shielding failure

probability, which is unfavorable for the research on factorsinfluencing the transmission lines lightning shielding

performance. (b) The existing lightning shielding modelsgenerally assume that when an upward leader has not beengenerated, the ground and the earthed objects have very smallinfluence 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 H i, thedistributions of lightning density are the same. Therefore thissimplification will not affect the estimation result of shieldingfailure rate.

2.1 THE CHARGE DISTRIBUTION IN DOWNWARDLEADERS

When lightning leader propagates to ground in discretesteps, the charges in lightning channel will influence theelectric field spatial distribution around the lightning channel.Therefore, in order to calculate the variation of spatial electricfield during the progress of lightning, the proposed model usesa line charge varied linearly with channel height to simulatethe main channel of lightning [21]. When the lightning leader reaches the certain height H i, the downward leader channel

propagates towards ground in a fractal form [17]. Meanwhile,since the lightning leaders will branch and bend, it is difficultto continue to use a linear distribution to describe thedistribution 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 uniformcharge distribution. Moreover, by observing natural lightningdischarge 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 thetip of lightning [21].

Dellera [6] believes that higher values of charge in thechannel will result in higher amplitudes of first stroke current;and according to the observation data, the relation between thetotal quantity of charge in the lightning leader channel QT andthe lightning current amplitude I can be expressed by thefollowing formula [21, 23, 24].

10.7

25T I

Q

(1)

where, QT is the total charge in the lightning leader, C; I is themagnitude of lightning current, kA.

The height of the cloud H C is set at 2.5 km in this paper [21].For the main lightning channel, when the height of the leader is H C, the line charge density =0; when the height of theleader tip is H 0, the line charge density = max , with max as thelargest line charge density. Then according to the linear distribution, the total volume of charge will be [21]

max 00

( )2C

T

H H Q Q

(2)

where Q0 (C) is the charge inside the leader tip, which can beobtained by [21]:

20 02 BQ r E (3)

where is the dielectric constant of air; r 0 is the radius (m) of the lightning leader tip, can be set as 6m [25]; and E B (kV/m)is the critical electric field intensity for propagation of theleader. For positive lightning, E B can be set as 500 kV/m, andfor negative lightning, E B is set as 1000 kV/m [17].

Therefore, the line charge density inside the main channelof lightning will be [21]

02

0 0

2( )( ) ( ) ( )

( )m T

C C C C

Q Qh H h H h

H H H H

(4)

where (h) (C/m) is the charge intensity inside the lightningleader.

For the fractal lightning leader, the line charge densityinside the lightning branch is taken to be uniform distributionand is given by:

0

0 ( )

( )

C

i

i

H

T H F

f l

H

H

f l

Q Q h dh

L

h dh

L

(5)

where (h) can be obtained from equation (4); Lfl is the totallength (m) of the fractal lightning leader branch.

2.2 THE INITIAL CRITERION OF THE UPWARDLEADER

At present, there are mainly three continuous upward leader initial criteria used in the study of lightning shielding failureof transmission lines [7, 21, 26-28]: a) the upward leader induced voltage U ic of the horizontal conductor proposed byRizk [7]; b) Peek criterion: when the electric field intensity atthe conductor surface is higher than the critical field intensity

E C, the upward leader will be generated [21, 26]; c) Carraracriterion [27, 28]: the upward leader will start from theconductor surface in a long gap when the voltage applied on aconductor exceeds a critical value. The Peek criterion can

consider the influence factors adequately, such as the phaseconductors working voltage, the induced electric field of thedownward leader and the coupling electric field betweenconductors. From the superposition principle of electric field,the Peek criterion can also explain the actual operatingexperience that for a direct negative lightning strike, theupward leader is easier to be generated from positive phaseconductors. Thus the proposed model uses the Peek criterionto 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 continuousinception, of which the critical radius is the one required toinitiate a direct corona-to-leader transition [6]. Dellera et al [6]

believe that for the conductor-plane configuration, when theconductor radius is larger than the critical radius, the criterionfor the initiation of the continuous upward leader is the sameas the criterion for corona onset. When the conductor radius issmaller 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 theformula [21, 26]:

0.033000 (1 )C E m

r

(6)

where E C (kV/m) is the critical initial electric field intensity; m is the roughness coefficient of the conductor surface; isrelative air density; and r (m) is conductor critical coronaradius. For the ground wires, the radius of ground wire isusually smaller than critical radius, so r is set at the criticalradius, r = 0.1 m [6]; For the phase conductor, the radius of conductor bundles is usually larger than the critical radius, andr is set as the equivalent radius of the conductor bundles.

2.3 THE PROGRESSION RULES OF LEADERS

LPM assumes that the leaders progression direction alwaysgoes towards the position around the leader tip, where theelectric 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 theeffect of branching and tortuosity of the lightning channelwould also influence the electric field intensity in space andthus further influence the progression of the leader. Otherwise,the effect of branching and tortuosity of the lightningdischarge indicates that the progression of the leader has

fractal features. The fractal approach can be used in analyzingthe progression patterns of leaders. Therefore, in this paper,considering the deterministic and stochastic factors indownward lightning leader progression fully, a fractalapproach is used to model the trajectory of the leaders, and thedetails 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 electricfield 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 LEADERSFRACTAL 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 tothat simulation, in the process of dielectric breakdown, it ismore probable to have breakdown in a position where theelectric field intensity in the dielectric is most intense than thatin other areas. In the area shielded by other discharge paths,

the probability to have a discharge is very low and even zero.This phenomenon confirms that in the progression of lightningleaders, the ionization in air has stochastic features, and it isalso controlled by the distribution of electric field in space.The proposed model uses the DBM to simulate the trajectoryof the leader. The calculation process of the fractal leader

progression is shown in Figure 2 and the procedure isdescribed as follows:

Step:1 Discretize the space and set the initial fixed particles,which are also called seeds, at the tips of the downwardleader, the conductors and the ground wires, as shown bythe solid points in Figure 3;

Step:2 Set the moving particles around the tip of the leader, asshown by the moving particles in point B on Figure 3.Moving particles have random Brownian motion in thediscrete space, which is shown as the trace of the dashedline in Figure 3. The upward and downward leader

progresses most quickly at their tips respectively, and inthe calculation of lightning shielding, the progression inleader 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 pointsaround point P as possible breakdown points. When a

particle moves to point P , its Brownian movement willstop, and the probability to have breakdown at point P iscalculated by [30]:

,

1

( , ') 0, if ( , ')

( ( , ') )( , ') if ( , ')

( ( , ') )

B

B B N

i Bi

p P P E P P E

E P P E p P P E P P E

E P P E

(7)

where, p( P , P ) is the probability for breakdown to occur from P to P and E B is the critical electric field intensityfor propagation of the leader. For positive lightning, E B can be set at 0.5 MV/m, and for negative lightning, E B isset as 1 MV/m [17]. E ( P , P ) is the electric field intensityin the space from P to P ; E ( P , P i

) is the ith possible breakdown point which satisfies the formula

( , ') B E P P E and goes from P to the point around P ,

and N is the number of possible breakdown points of thiskind. Constant is an adjustable number that is used to

present the relation between local electric field intensity

and the breakdown probability in the dielectric. Itdirectly determines the stochastic features of the leaders

progression direction. The selection of the breakdown probability constant will be discussed in Section 4.

Step:4 Make a breakdown judgment from the obtained probability: compute a random number m which satisfiesa [0, 1] uniform distribution. Compare the randomnumber m with the obtained probability p; if m is smaller than p, then it can be judged that breakdown will happenat point P , if m is larger than p, then breakdown will nothappen.


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Step:5 If point P meets the condition to have breakdown, and breakdown happens, then the moving particle becomes anew seed. It will form a seed cluster together withother seeds and hence a new leader discharging channelcomes into being. If point P does not meet the

breakdown condition, then the moving particle will bestopped. The program returns to Step 2 and keeps onoperating, until it determines the occurrence of lightning

strikes.

Figure 2. Calculation process of the fractal leader progression

Figure 3. Branch picture of leader.

In the model calculation process, the progression of everynew seed that constitutes the fractal leader will influence theelectric 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 thenext moving particle. By repeating the above calculation and

judgment procedure, the leader progressing path will bedetermined. In the proposed model, the charges in thedownward lightning channel directly determines the electricfield spatial distribution around the lightning channel, and the

electric field spatial distribution determines that each leader branch progresses most probably towards the point with mostintense electric field at their tips. This feature makes the

proposed model maintain the deterministic factor of theleaders progression. This method not only considers thedeterministic feature that breakdown happens more probablywhen the electric field at the leader tip is more intense, butalso stochastic features in the process of leader progression.

4 THE SELECTION OF KEYPARAMETERS OF THE FRACTAL-

CHARACTERIZED LPM The fractal dimension reflects how effectively the complex

graph occupies space. It is a measure of the orderliness andrandomness of the stochastic and deterministic features of fractal graphs. From the above description of the proposedmodel, it can be found that the value of the breakdown

probability constant in (7) influences the progressing probabilities at every point of the fractal leader channel, and itdetermines the stochastic and deterministic features of the

lightning leader progression. In order to simulate the wholelightning path accurately, it is very important to select a proper value. In this paper, the lightning leader will be simulatedwith different values, hoping to determine how the constant will influence the dimensions of the lightning path insimulation, and to determine the proper value of the constant for the proposed model.

The fractal dimensions of lightning progression based onthe box-counting method is given by [31-33]:

0

ln ( )lim

ln(1/ )bb

b l b

N l D

l (8)

where D b is the fractal dimension, l b is the measurement scale,and N (l b) is the least number of squares that can cover thefractal graph, with l b as the side length.

Based on the box-counting method, the fractal dimensionsof the simulated lightning path with different values of arecalculated and are shown in Table 1. The lightning paths insimulation with three different values are shown in Figure 4.

Table 1 . Calculated fractal dimensions with different values.

0.1 0.5 1 5 10

D b 1.460.07 1.310.05 1.200.04 1.090.05 1.040.02

Figure 4. Simulated figures of the lightning leader for various fractaldimensions


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Input the parameters of the transmissionline and the lightning

Calculate the electric fieldintensity in space

Lightning strikeor not?

The upward leader is generated ?

The fractal upward leader progresses

The fractal lightning leader progresses

Calculate the electric fieldintensity in space

Lightning strikeor not?

Record the struck object

Yes

Yes

Yes

No

No

No

Calculate the electric field intensityon the conductor surface

Figure 5 . Calculation process of the proposed lightning model.

It can be found from (7) that when the value is smaller thanthe proper value, in the criterion of leader progression the effectof the electric field distribution will be reduced, and the

progressing probabilities in every direction of the leader tip tendto be the same, which leads to excessive branches in the leader channel, as shown in Figure 4a. In such a condition, thestochastic feature plays the main role in the leader progression,and the corresponding fractal dimension is relatively larger thanthat of natural lightning. Conversely, when the value is larger than the proper value, the direction with most intense field inthe leader tip will gain more probability to progress, while the

probabilities for leaders to progress in other directions aresmaller. The leader progresses along the direction with mostintense field, but less probably along the branches, as shown inFigure 4c. In such a condition, the deterministic feature playsthe main role in the lightning leader progression, and thecorresponding fractal dimension is relatively smaller than thatof natural lightning. Therefore, to some extent, the selection of value decides whether the lightning model can reflect thecharacteristics of branching and tortuosity of lightning path

properly. Kawasaki, Matsuura [15] and Tsonis and Elsner [16]measure the fractal dimension of still pictures of naturallightning channels. They find that the empirical fractaldimensions of the natural lightning discharge paths exist in therange from 1.1 to 1.4; and as shown in Table 1, when the value is set between 0.5 and 5, the fractal dimension of thesimulated lightning path D b exists within the fractal dimension

range of natural lightning discharge paths. Therefore, in the proposed model, the value is set as 1.

The calculation process of the proposed model is shown inFigure 5, and the simulation figure of a lightning strike ontransmission lines is shown in Figure 6. The line is a 800 kVdc transmission line, with the shielding angle at zero degrees,the conductor height at 44 m, and the ground wires height at55 m. The lightning current is 30 kA, and the lateral distance

between lightning and the line is 100 m.

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 transmissionlines [34]:

0( ) ( ) L eq L LSFN N A T N L T SFW I P I dI (9)

where SFN is the shielding failure number, strokes; N L is theground flash density, flashes/km 2-year; Aeq is the shieldingfailure areas, km 2; I is the lightning current amplitude, kA;SFW is the shielding failure width, km; L is the length of thetransmission line, km; T is the observation time, years; and P L is the probability density distribution of lightningcurrents, %/kA.

Conventional LPM believes that when lightning occurs in theshielding failure area of transmission lines, shielding failure willinevitably happen. In the proposed model, it is believed that thelightning discharge has stochastic features. Even thoughlightning reaches into the shielding failure area of transmissionlines, because of the attraction from the upward leader of ground wires and from the earth, shielding failure is notinevitable. There exists a probability to have shielding failures,as shown in Figure 7. Therefore, for a certain lightning currentamplitude I , weight SFW as in (9) and use the equivalent SFW ,SFW eq( I ) to replace it. The weighting value is set to be thelightning probability P S so that SFW eq( I ) is:

0( ) ( ) ( , )eq S SFW I SFW I P I D dD (10)

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


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P S can be obtained by a large number of repeatedcalculations in the proposed model. By the model, theshielding failure probability P S at all levels of lightningcurrents can be calculated. For example, the 500 kVtransmission line tower in Guangdong province of China isshown as Figure 8 and P S can be shown as the distributioncurve in Figure 9:

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

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

0 50 100 150 2000

20

40

60

80

100

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

It can be found from Figure 9 that under a certain lightningcurrent I , as the lateral distance D increases, P S will increasefirst and then decrease. P S under different lightning currents I

has its peak value decrease as the lightning current amplitude I grows. Therefore, under the same line parameters, P S is afunction of lightning current I and lateral distance D. In order to obtain SFR, calculate the shielding failure probabilitydistribution under every lightning current level I for differentlateral distance D, and by data fitting, a shielding failure

probability function is fitted with lateral distance D andlightning current I as its variables [35]:

( , )S S P P I D (11)Based on (9), (10) and (11), the SFN and SFR of the

transmission line can be calculated as follows:

( , ) ( ) L S LSFN N T L P I D P I dI dD (12)

100

0.01 ( , ) ( ) L S L

SFN SFR

L T

N P I D P I dI dD

(13)

where SFN is shielding failure number of the transmission line,strokes; SFR is shielding failure rate of the transmission line,strokes/100 km-year; N L is the ground flash density,flashes/km 2-year; P L 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 currentamplitude, kA; and D (km) is the lateral distance between thelightning 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 thoseup the total SFN will be calculated.

5.2 VALIDATION OF THE ESTIMATION METHOD

As for the EHV transmission line, the China SouthernPower Grid uses the lightning location system (LLS) to recordthe lightning trip numbers of 2550 km 500kV AC transmissionline in Guangdong Province of China [36], and they alsomeasured the lightning current amplitude of every lightningstroke on the line. From 1999 to 2003, there have been 40lightning trip accidents on the line [36]. The 500 kVtransmission line in Guangdong province of China [28] has itsstructure model in Figure 8. The SFR of the 500 kVtransmission line is estimated by Erikssons improved EGM,LPM and the proposed model.

The thunderstorm days and the ground flash density are 40days/year and 2.9 flashes/km 2-year, respectively [28], and the

probability density function of the lightning crest current isdistributed as [3, 9]:

( ) 0.0475 exp( ) 0.0010 exp( )20 50 L I I

P I (14)

The observation results of direct lightning strikes and thecalculation results of the three estimation methods are shownin Figure 10 [36].

The data in Figure 10 show that, compared with Erikssonsimproved EGM, the calculation results of LPM and the

proposed model are closer to observation data. According toErikssons improved EGM, the largest lightning shieldingfailure current amplitude is 9.5 kA, that is, that lightningcurrent higher than 10 kA will not cause lightning shieldingfailure. This result is obviously different from the observationdata. Moreover, Erikssons improved EGM believes that withthe increase of lightning current, the lightning shielding failure


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arc of transmission lines will decrease gradually, as shown inFigure 11. Therefore, with the increase of lightning current,the corresponding SFR will continue to decrease to zero. LPMand the proposed model show that as the lightning current I increases, SFR shows a tendency of growing at first and thendecreasing. 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

P S is larger than that of the 25 kA lightning current, SFW ( I =5kA, P S>10%) is smaller than the SFW ( I =25kA, P S>10%).Through the calculations of (10) and (13), it is revealed thatfrom 5 kA to 25 kA, SFW eq rises as the lightning current I increases, so the SFR also increases. Correspondingly, SFW eq( I =5kA) and SFR ( I =5kA) are smaller than SFW eq ( I =25kA)and SFR ( I =25kA) respectively, and the increasing tendencyfrom 5 kA to 25 kA can be calculated by (13). For the 45 kAlightning current, though the SFW ( I =45kA, P S>10%) is larger and the SFW ( I =25kA, P S>10%), its peak value of P S is lessthan 20%, and the corresponding SFW eq( I =45kA) is smaller than that of 25 kA lightning current. Therefore, based on thecalculation of (10) and (13), the SFR under lightning current

higher than 25 kA shows a decreasing tendency. This result iscloser to the observation data.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14the improved EGM

LPM

proposed model

observation

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

Ground wire

Phase conductor

Rl1

Rl2

Rl3

Rgw3

Rgw2

Rgw1

R g 1 R g 2 R g3

B1

B2

B3

C1C2

O

D1

D2

Ground

Figure 11. Calculation representation of the lightning shielding failure arc inEGM.

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

among which there were 79 negative lightning strikes. Thenumber 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 SakaeTaniguchi [9] and the 3-D LPM [37]. The calculation andobservations are shown in Figure 12. Conventional EGM and

the Sakae Taniguchis improved EGM show that the lightningshielding failure arc length of the lower phase conductor is thelargest, so the percentage of strikes on the lower phaseconductor is higher than that of the upper and middle phaseconductors. However, observation data show that the SFN of upper phase conductor and middle phase conductor are closeto each other, but the SFN of the lower phase conductor isobviously lower than the other two conductors. Therefore,Conventional EGM and improved EGM by Sakae Taniguchistill 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 shieldingangle. These two factors determine the probability distributionof SFN of every conductor. Therefore, the rates of the upper

phase conductor and middle phase conductors SFN areobviously higher than that of the lower phase conductors. Thecalculation result of the proposed model is close to 3D-LPM;and compared with EGM, it is closer to the observation data.

05

101520253035404550

Upper Phase Middle Phase Lower Phase

Observation3-D LPM

The proposed modelSakae Taniguchis improved EGMEGM

S t r o k e p e r c e n

t ( % )

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

6 LIGHTNING SHIELDING FAILUREZONE

The lightning shielding failure zone is used by EGM todetermine 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 thetransmission lines so that it is possible for lightning to strikethe phase conductor. It is possible to determine the lightningshielding failure zone by calculating the shielding failure arcs,

but the results ignore the stochastic feature of lightning


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discharges. In fact, the dividing border between the lightningshielding failure zone and the effective protection zone is notso obvious. Moreover, when downward lightning enters intothe lightning shielding failure zone, due to the stochasticfeature of lightning discharges, the phenomenon of lightningstrikes on line conductors occurs in the form of a statistical

probability. So, based on the proposed model, the lightningshielding failure zone is described in the form of lightning

shielding failure probability in this paper. Through research onthe distribution of lightning shielding failure probabilityaround the transmission line space, the probability for different direct lightning strikes to cause lightning shieldingfailure can be better analyzed, which provides theoreticalsupport to the analysis of lightning shielding failures and thelightning shielding design of transmission lines.

In LPM, it is believed that before the leader progresses tothe height of H i, the lightning leader tip height at the upwardleader initial moment, the lightnings progression is notinfluenced by the grounded objects. When the leader reaches

H i, 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 whenthe line parameters are determinate, H i is determined by thelightning current amplitude I and the lateral distance D, whichcan be described as

( , )i H h D I (15)

By altering the position of H i and I , (15) can be changedinto:

( , )i I y D H (16)

In order to attain the functional relation of (16), in this paper the charge simulation method is used to simulate thelightning leader channel, and Peek criterion is used as theinitial criterion of upward leaders. The calculation process is

shown in Figure 13. The calculation objective is a 800 kV dctransmission line, with the shielding angle at zero degrees, theconductor height at 44 m, the ground wires height at 55 m, andthe ground slope at zero degrees. Through the flow calculationof Figure 13, the functional relation in equation (16) is shownin Figure 14.

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

0 50 100 150 2000

100

200

300

400

500

10 kA 20 kA

30 kA

40 kA

50 kA

D (m)

A

B

C

Figure 14 . Calculated distribution of H i( D, I ).

Based on the proposed model and combined with equation(16), the lightning current I at each point of the transmissionline space is calculated, so that the lightning shielding failure

probability at each point in the transmission line space isobtained. The data of the lightning shielding failure

probability is processed by mathematical fitting, and thecontour line of the shielding failure probability is drawn. Thecalculation process is shown in Figure 15, and the shieldingfailure probability contour is shown in Figure 16.

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

As Figure 16 shows, the curves No.1 to No.7 are the contour lines of the transmission lines 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 lightningcurrent amplitude I at point A is 20 kA. On the phaseconductors and ground wires of the transmission line, theinduced voltage caused by the lightning channel is relativelysmall. The high operating voltage on the conductors makes theconductors have a stronger ability to trigger lightning than the


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

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 H i grows from zerom, the lightning shielding failure probability of the transmissionline tends to increase first, then to decrease. When H i is less than50 m, P s is less than 30%, and it can be found from Figure 16that the lightning current amplitude I in this zone is smaller than10 kA; the lightning in this area mainly strikes the earth. When

H i grows gradually, for example, when H i grows to 150 m, it isshown from Figure 14 that the lightning current amplitude I willincrease to 20 kA, then the conductors and the ground wiresability to trigger lightning also increases. Meanwhile, becauseof its high operating voltage, the conductors upward leader

predominates over the ground wires upward leader in space.The lightning shielding failure probability of the transmissionline tends to increase. When H i continues to grow, for example,when H i grows to 300 m, the lightning current amplitude I in thearea increases to 30-40 kA. The operating voltage on theconductor no longer exerts great influence on its upward initialleader, which makes the ground wires upward leader take

predominance in the competition with the conductors upwardleader. The lightning shielding failure probability of thetransmission line tends to decrease.

When the observation point ( D, H i) changes from near to far,for example in Figure 16, when observation point ( D, H i)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 about20 kA, which is smaller than the current of 30 kA in point Band that of 50 kA in point C. So when lightning strikes happenaround the transmission line, lightning shielding failures willhappen more easily when the lightning current amplitude hassmaller values. As the observation point ( D, H i) keeps onmoving further away from the transmission line, the lightningcurrent I will keep on increasing, and the lightning shieldingfailure probability continues to decrease until zero. Thisconclusion is in accordance with the views of EGM and LPMthat there exists a largest lightning shielding failure current.

In the above analysis, based on the proposed model, thelightning shielding failure zone is described by the shieldingfailure probability contour. The result shows statisticalcharacteristics and reflects the deterministic and stochasticfeatures of the lightning discharge. It is also in accord with thedata distribution rules of the laboratory lightning simulationexperiment [38, 39]. The proposed model can reflect the naturallightning progression rules better than LPM. Compared with the

lightning shielding failure zone described in EGM, the shieldingfailure probability contour can better reflect the lightningshielding failure performance of transmission lines. Therefore,the transmission line estimation method based on the proposedmodel is more accurate and well-founded.

7 APPLICATION OF THE ESTIMATIONMETHOD ON THE TRANSMISSION LINES The existing data show that the increase of the transmission

lines height will lead to the increase of the lines ability totrigger lightning, and the lightning shielding performance of transmission lines will be influenced. In this paper, based onthe proposed model and its estimation method to evaluate theSFR of transmission lines, the lightning shielding performanceof transmission lines with different nominal heights areanalyzed. The structural parameter of a 800 kV UHVDCtransmission line is used as the analyzing parameter, in whichthe ground wire height is higher than the conductor height by11 m and the shielding angle is zero degrees. The structuralsize 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 lineswith 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 isincreased, SFR increases gradually. When the nominal heightgrows 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 wires and the conductorsupward leaders easier to be initiated, and to have more ability totrigger lightning. Thus the lightning shielding failure area will beenlarged, and the SFR of the transmission line will increasecorrespondingly. Thus, in order to lower the SFR of transmissionlines, 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 ensurethe lines meet the lightning shielding requirements.

The setting of the shielding angle of transmission lines willinfluence the field distribution on the surface of ground wires andconductors, then influence the initiation of the ground wires andthe conductors upward leaders, and finally to influence thelightning shielding failure probability of transmission lines. In this

paper, based on the proposed model, the lightning shielding performance of transmission lines with different shielding angleswas researched. When the conductor height is 44 m, the groundwire height is 55 m, the lightning shielding failure probabilitywith different shielding angles is calculated and shown in Table 3.


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The data in Table 3 shows that decreasing the shielding angle of the transmission line will lower its SFR. When the lightningleader progresses downward, the decrease of shielding anglesuppresses the conductors upward leader, then the ground wiresupward leader will predominate over the conductors upwardleader. The protecting function of the ground wire is enhanced, sothe SFR of the line will decrease correspondingly. Therefore, toreduce 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 lineswith different shielding angles.

shielding angles ( o) -5 0 5 10

SFR 0.086 0.107 0.122 0.141

In the UHVDC transmission project design in Yunnan-Guangdong province of China, the China Southern Power Grid Corporation uses the calculation estimation of the

proposed model. In mountainous regions, the shielding angleof 800 kV dc transmission lines is set at -9.6 degree; in plainregions, 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 inthis paper. By comparing the fractal dimension of naturallightning, the fractal parameters are adjusted, so as to control thedeterministic factor and stochastic factor of lightning progressionwithin a reasonable range. Based on the proposed model, thelightning shielding failure probability distribution of thetransmission line is proposed and the method to calculate the SFR is improved. These factors are used to analyze the lightningshielding performance of transmission lines, and lead to the

following conclusions:

1) The lightning stroke simulation figure of the proposedmodel describes the effects of tortuosity and branching of the lightning channel, and reflects the stochastic anddeterministic features shown in the progression processof lightning.

2) When lightning occurs within the shielding failure width,lightning shielding failure is not inevitable, but appearswith a probability. The lightning shielding failure

probability P S is related to the lightning current amplitude I and the lateral distance D.

3) The lightning shielding failures are most likely whenlightning strikes with current amplitude under 20 kA andoccur 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 islarger 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 inYunnan- Guangdong province of China, by China SouthernPower Grid Corporation. Besides, the authors intend to developa 3-D fractal LPM in the future, study the 3-D lightningshielding failure probability space of transmission lines, andanalyze how the factors such as line sag, the corridor terrain of

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

ACKNOWLEDGEMENT

This work was supported by the National Basic ResearchProgram of China (973 Program) (2009CB724504), the

National 111 Project of China (B08036) and the National Natural Science Foundation of China (NSFC) (50707036).

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Jianbiao Li was born in Guangxi province, China, on5 October 1984. He received the B.S degree inelectrical engineering in 2007 from ChongqingUniversity, China. He is now a Ph.D. candidate in theState Key Laboratory of Power TransmissionEquipment & 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 inelectrical engineering, respectively, in 2002 from

North China Electrical Power University and in 2006from Chongqing University, China. He is now anAssociate Professor in the State Key Laboratory of Power Transmission Equipment & System Securityand New Technology, Chongqing University. Hisresearch interests include outdoor insulation incomplex ambient conditions and electric-fieldcalculations. He is the author and coauthor of more

than 30 journal and international conference papers.

Wenxia Sima was born in Henan province, China, on13 July 1965. She graduated from ChongqingUniversity in 1988 and obtained the Ph.D. degree in1994 from Chongqing University. Her employmentexperience includes the college of ElectricalEngineering of Chonging University. Her fields of interest include high voltage outdoor insulation andovervoltage protection.

Caixin Sun received the B.S. degree in electricalengineering in 1969 from Chongqing University.Currently, he is Professor of Chongqing University,leading the research group on high voltageengineering. He is a member of the Chinese Academyof Engineering and Director of Electrical Power Engineering Committee of the National ScienceFoundation of China. His research interest is in highvoltage engineering, especially online detection of insulation condition and insulation fault diagnosis for

high voltage equipment, discharge mechanism of outdoor insulation incomplicated 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 theMasters and Ph.D. degrees, respectively, in 2001 and2010 from Chongqing University, China. He is now alecturer in Chongqing University, and his researchinterests include overvoltage protection and groundingtechnology in power systems.

Markus Zahn (S'68-M'71-SM'78-F'93) received theB.S.E.E., M.S.E.E., Electrical Engineer, and Sc.D.degrees from the Department of Electrical Engineering atthe Massachusetts Institute of Technology (MIT),Cambridge, MA, in 1968, 1968, 1969, and 1970respectively. He then joined the Department of ElectricalEngineering at the University of Florida, Gainesvilleuntil 1980 when he returned to MIT where he is nowProfessor of Electrical Engineering working in theResearch Laboratory of Electronics, Laboratory for

Electromagnetic and Electronic Systems and the High Voltage ResearchLaboratory. He is also the Director of the MIT Course VI-A EECS InternshipProgram, a cooperative work/study program with industry. He is the author of Electromagnetic Field Theory: A Problem Solving Approach (now out of print) butis working on a new reference book with a complete collection of solvedelectromagnetism problems. He has also co-developed a set of educationalvideotapes on demonstrations of electromagnetic fields and energy for the enrichedteaching of electromagnetism. He has received numerous excellence in teachingawards at the University of Florida and at MIT. His primary research areas areferrohydrodynamics and electrohydrodynamics for microfluidic and biomedicalapplications; nanoparticle technology for improved high voltage performance of electric power apparatus; modeling of electrical streamer initiation and propagationleading 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 withapplications to non-destructive testing and evaluation measurements and for identification of metal and low-metal content dielectric landmines and unexplodedordnance. Professor Zahn is co-inventor on 19 patents. He has contributed to about10 book and encyclopedia chapters, about 115 journal publications, and about 175conference papers. He was a Distinguished Visiting Fellow of the Royal Academyof Engineering at the University of Manchester, England; was the 1998 J.B.Whitehead Memorial Lecturer, and was the first James R. Melcher MemorialLecturer in 2003; is an Associate Editor of the IEEE Transactions on Dielectricsand Electrical Insulation; is on the International Scientific Committee on MagneticFluids; and has received a Certificate of Achievement for completion of theDeminers Orientation Course at the Night Vision and Electronic SensorsDirectorate Countermine Division at Ft. Belvoir, VA, USA.

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Jianbiao - New Model Lightninig Trans Lines Fractal (2011)