petrov - quantification probability lightning fractal - petrov

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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 10, No. 4; August 2003 641

Quantification of the Probability of Lightning Strikes toStructures Using a Fractal Approach

N. I. Petrov, G. N. PetrovaLenina Str., 19-39

Istra, Moscow region, Russia

and F. DAlessandroERICO Lightning Technologies

Hobart, GPO 536Tasmania, 7001 Australia

ABSTRACTRecently fractal theory has been used to model the observed paths of lightning dis-

charges. This paper extends previous work by using a fractal approach to describethe effects of tortuosity and branching of the lightning channel. In particular, weuse the model to make predictions of the probability of lightning strikes to practi -cal structures. Some of the specific estimates include the probability of strikes as afunction of interception angle, predictions of the strike points on structures, andthe probability of side strikes to tall structures. Significant polarity and geomet-ric effects are shown. The shielding effects of nearby taller structures, shieldingfailure or breakthrough probabilities of so-called protected objects, and the ef-fects of ground topography are also examined. The results are discussed in termsof the implications for the protection of structures against lightning.

Index Terms Fractals, lightning, lightning protection, strike probability, po-larity effects, shielding angle, tall structures, side strikes, topography.

1 INTRODUCTION

NE of the main properties of a lightning dis-Ocharge is the random behavior of its propagation ortrajectory in space. The trajectory of a lightning leader

has a complex structure in which the tortuous main chan-

nel and branches deviate from the electrical force lines.

This inherent random characteristic of propagation leads

to a spread in the breakdown voltage and orientation of

the leader channel in space. The approach of a lightning

leader from the thundercloud does not depend on the ex-

istence of ground objects until it enters within a strikingw xdistance of the ground structure 1 . Therefore, the de-

velopment of the lightning leader up to this point is usu-

ally ignored in any analysis of the strike mechanism. How-

ever, the striking distance for a lightning discharge of agiven charge per unit length or prospective peak stroke

.current may differ from case to case. This is due to varia-

tions in the electric field intensification in the vicinity of

the structure caused by the random characteristics of the

lightning channel.

Manuscript receied on 27 May 2002 , in final form 7 April 2003 .

Up to now, experimental results have been used to de-

velop empirical methods for the determination of the

protection zone of a structure. Many of the results are

based on HV laboratory tests involving relatively short

sparks. In order to translate some of these results to the

ultra-long sparks comprising natural lightning, it is impor-

tant to investigate the phenomenon by means of theoreti-

cal models. Such modeling enables a rapid, loss cost anal-

ysis across a wide range of parameters.

One of the more advanced models for the computation

of the protection zones of structures is the Leader Pro-

w xgression Model 2 . It makes the assumption that the light-ning leader progression will be in the direction of maxi-

mum electric field. However, it is also necessary to take

into account the random tortuosities of the lightning

leader and the corresponding upward discharge from the .structure hereafter called the upward leader . This is

the fundamental basis of the approach outlined in the

present paper.

There is considerable support in published research

findings for assuming the basic characteristic of a light-

ning channel has fractal geometry with a reproducible di-

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Petro et al.: Quantification of the Probability of Lightning Strikes to Structures Using a Fractal Approach642

w xmension 38 . Hence, any model which incorporates the

fractal structure of a breakdown channel in air is of prac-

tical interest in the field of lightning protection.

A fractal model for a streamer discharge in gases wasw xfirst used by Niemeyer et al. 9 . Further development of

w x w xthis model was presented in 10 . In 1114 , this approach

was first used for the modeling of lightning discharges. Inw x12,13 , the fractal approach was first used to determinew xthe strike probabilities for isolated, earthed objects. In 15 ,

it was used for the modeling of lightning, taking into ac-w xcount intra-cloud discharges. In 16 , the model was used

to investigate lightning discharges between the top of the

thundercloud and the ionosphere.

A number of practical issues of relevance to lightning

protection studies fall within the domain of fractal model-

ing. For example, the possibility of side strikes to tall

structures and the selection of the striking point on an

earthed structure can be attributed to the random behav-

ior of the lightning channel. As part of such a study, an

investigation of the probability of upward leader inceptionand propagation from earthed structures is also of practi-

cal interest. In general, it is assumed that the develop-

ment of an upward leader determines the striking dis-

tance of lightning to the structure. However, some obser-w xvations show that this criterion cannot always be used 17 .

The latter results follow from an analysis using fractalw xmodeling 18,19 .

The method considered in this paper allows for the

characteristic bending and branching of the lightning

channel between the thundercloud and the ground. The

degree of bending and branching depends on the parame-

ters of the thundercloud and the lightning leader. Themultiple termination phenomenon that has been observedw xin multi-stroke lightning flashes 20,21 is also seen as an

output of the fractal model. The simulated discharge tra-

jectories also reproduce characteristic features such as theoccurrence of the discharge stopping phenomenon in-

.complete discharges , the possibility of side strikes to tall

structures and the main geometry of the breakdown chan-

nel.

To date, studies have addressed the problem from a

two-dimensional perspective. This modeling is limited

since natural lightning is a three-dimensional phe-

nomenon. Hence, direct comparisons cannot be made with

observational data. Ideally, it is necessary to carry out .three-dimensional 3-D modeling, which takes into ac-

count the true geometry of the lightning channel and the

earthed structures involved in the lightning attachment

process.

In this paper, the fractal model is used to investigate

the effects of tortuosity and branching of the lightning

channel on the probability of lightning strikes to earthed

structures. The modeling is performed in three dimen-

sions. Both slender and extended structures are consid-

ered. The model proposed in the paper takes into account

the polarity and potential of the lightning leader, geome-

try of the discharge gap, random tortuosity and branching

of lightning channel and the upward leader from the

structure.

Section 2 reviews the model and calculation method. In

Section 3, the fractal approach is used to quantify the

probability of strikes as a function of interception angle,strike points on structures, and side strikes to tall struc-

tures. The significance of polarity and geometric effects

are given detailed consideration. The shielding effects

of n e ar by t al le r s tr uc tu r es , s hiel ding f ai lu re or

breakthrough probabilities of so-called protected ob-

jects, and the effects of ground topography are also exam-

ined. Section 4 discusses the results and conclusions are

drawn in Section 5.

2 OVERVIEW OF THE CALCULATION

METHOD

The basic fractal model has been presented in previousw xpapers 12,13,18,19 . The main parameters describing this

model are: EU, the critical electric field intensity for prop-

agation of a leader; E , the electric field intensity in thechleader channel; U , the initial potential of the lightning

0

.leader tip or thundercloud potential ; H, the distance be-

tween the leader tip and the earthed structure. The elec-

tric field intensity is determined from the solution of the

Poisson equation, i.e., taking space charge into account.

The development of the upward discharge from the struc-

ture is determined by a condition of leader propagationw x22 . In this criterion, the critical distance is taken to be

0.7 metres and the critical electric field intensity for posi-

tive and negative leaders is taken to be E

U

s5 kVrcmqand EU s10 kVrcm respectively. Unlike methods whichy

assume the leader propagates in the direction of maxi-

mum electric field strength, the present model allows the

leader to propagate in a direction which may deviate from

the electric field lines. The probabilistic nature of the

leader orientation is applied up until the final jump phase

of its propagation.

Although the breakdown model is described in the

aforementioned papers, it is reviewed here for complete-

ness. The propagation of a leader discharge is considered

as a discrete process. The discharge gap is divided into a

coordinate mesh with square sides, the size of which is

equal to the step length l . The points of the coordinatestmesh passed by the discharge have potentials U , appro-

0

priately decreased by an amount equal to the voltage drop

along the leader channel. The voltage drop is determined

by the channel length and electric field intensity in the

channel, E . The probability of breakdown is propor-chtional to a power, , of the local electric field in the re-

gion in front of the leader, viz.

p;E 1 .

where )0.

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At every step of the development of the discharge, the

potentials U at specified points are determined via thei jkuse of the iteration method for the solution of Laplace s

equation 2Us0. This procedure is repeated until the

probabilities in all directions become zero.

The fractal dimension, D, may be determined from the

relation between the total length, L, of the structural ele-ments of the discharge inside a sphere of radius, R, and

the radius R itself, namely

L R ; r rdy1.dr;RD 2 . . .H

where d is the space dimension. It follows that the dis-

tribution of the space charge density satisfies the relation

R ;rydyD . 3 . .

The influence of the initial conditions on the dischargew x

trajectories was investigated in 13,18 . These studiesshowed that the degree of tortuosity and branching of the

discharge channel and its fractal dimension increase with

leader tip potential.

3 ORIENTATION OF LIGHTNING TO

STRUCTURES

3.1 BASIS

The number of strikes to a given structure is a functionw xof the lightning current 23 and the structure geometry

w x22, 24 . Observations of lightning strikes show that there

is a scatter in the number of strikes to structures of thew xsame height in a given region 24,25 . Usually, the scatter

in the number of strikes is a result of different lightning

currents, i.e., due to the well-known distribution of peakw xcurrents 26 .

The determination of the probability of lightning strikes

to particular points on a practical structure is of interest

from a risk management perspective. Although a lightning

leader propagates predominantly in the direction of high-

est electric field intensification, the selection of the at-

tachment point on an earthed structure is partly a randomw xprocess. Bazelyan and Raizer 27 have discussed this pro-

cess in terms of the probability of orientation to a struc-

ture, and once this is inevitable, the probability of selec-

tion of a particular point. These probabilities depend onthe lightning parameters, such as the potential and polar-

ity of the leader, and the structure height.

In the modeling results which follow, the probability P

of a strike to a given point is simply defined as the ratio of

the number of strikes to that point and the total number

of trial discharges or strikes that were applied. Generally,

this probability will be presented as a percentage. The

number of discharges applied in these modeling trials

range from 30 to 100, depending on the computational

demands of the scenarios considered.

The value of the initial potential U of a leader channel0

was chosen with the recognition that an over-voltage

does not exist in the discharge gap between the leader

and the structure, i.e., the value U corresponds to the0

minimum breakdown voltage. The leader potential is

computed from the nominated leader channel charge,w x

which in turn gives the prospective lightning current 22 .Finally, the initial parameters for the lightning leader

were chosen on the basis that they should correspond to .low lightning currents in the range 310 kA. The reason

for this choice is that lightning discharges with such low

currents have the largest probability of breakthrough to

protected objects, i.e., bypass of the protection system.

Hence, these results are of most practical interest in the

field of lightning protection. Hereafter, all references to

strikes relate to these low-intensity discharges.

3.2 DETERMINATION OF INTERCEPTION

ANGLE

Consider a lightning leader approaching a structure at

different lateral distances or different angles to the verti-

cal axis. In this case, the probability of strikes as function .of interception angle sarcsin rr can be determined,

where r is the distance between the structure top and

lightning leader tip, and is the lateral displacement of

the leader.

The results of these modeling trials are shown in Table

1. The results show that the strike probability decreases

with increasing lateral displacement of the downward

leader. A similar analysis can be performed in the case of

an earthed rod located on top of a practical, 3-D struc-

ture. An example of this type of approach is shown in

Figure 1.

Simulations of strikes to unprotected, 3-D structures

show that negative lightning discharges terminate at the

points of highest field intensification, such as corners,

more often than positive lightning. The latter selects the

strike point in a more random manner.

Finally, Table 2 shows the portion of the total number

of strikes, P, which are deflected through an angle greater

than , where ranges from 1560. It is seen from the

table that an estimated 95% of all strikes to a structure

have trajectories that fit within a cone of half-angle ;60.

This result is consistent with field observations of light-w xning strikes to the Ostankino TV tower in Moscow 28 , in

Table 1. Percentage probability of negative and positive strikes to a . .slender structure SS and the ground GND for different intercep-

tion angles. In this analysis, the structure height was 60 m.

Angle, 0 30 45 60

Location SS GND SS GND SS GND SS GND

.P % 100 0 76 24 45 55 36 64y .P % 64 36 28 72 15 85 q

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Figure 1. Downward negative lightning orientation to a structurefrom a given lateral distance. The height of the building is 60 m and

the width is 40 m. The lightning rod on the structure has a heightof 20 m.

Table 2. Percentage probability of negative and positive strikes aris-ing from a deflection angle of the lightning leader trajectory up to .

15 30 45 60

.P % 92 31 13 5y .P % 83 21 3.6 q

which 95% of all strikes were deviated by an angle less

than 55.

3.3 INTERCEPTION EFFICIENCY

It is common practice to strategically place lightning

rods on structures in order to provide protection against

lightning strikes. This is called the lightning protection .system LPS . A quantitative measure used to determine

the effectiveness of the LPS is the interception effi-

ciency.This is commonly defined as the probability of the

minimum value of the lightning current that the LPS can

intercept.

To date, most models of lightning interception have

been highly simplified and only account for the de-terministic aspects of the process. However, a lightning

leader approaching an earthed structure may bypass the

structure due to the tortuosity of the leader trajectory.

Furthermore, the probability of a strike to a lightning rod

depends on the polarity of the lightning leader.

Table 3 presents the results of some simulations and

calculations that quantify the latter effect. The height of

building in this study was 60 m, the width was 40 m and

the lightning rod on the building is 20 m high. The table

shows the probability of a strike to the lightning rod and

Table 3. Percentage probability of negative and positive strikes to alightning rod on a structure, the structure and the ground.

Prob. Rod Structure Ground

.P % 76 12 12y .P % 42 16 42q

Figure 2. Lightning strike to the horizontal edge of a three-dimen-sional structure.

Figure 3. Negative lightning strike to the corner of a structure witha lightning rod.

the structure. Figures 25 are examples of the simulations

that were performed. The results in Table 3 show that the

interception efficiency of a lightning rod is higher for neg-

ative polarity lightning.

Table 4 presents the same type of scenario, except this

time the lightning rod is absent and the structure height is

varied. The results show that the strike probability in-

creases with increasing structure height, consistent with

field observations.

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Figure 4. Strikes to a lightning rod located on top of a structure.

Figure 5. Positive lightning strike to the ground adjacent to a struc-ture.

Table 4. Percentage probability of negative and positive strikes to . .an unprotected structure US and the adjacent ground GND .

.h m 30 50 100

Location US GND US GND US GND

.P % 65 35 85 15 88 12y .P % 41 59 58 42 66 34q

In another study of strike probabilities, we investigated

the effect of structure width, i.e., practical, extended

structures versus slender ones, such as communication

towers. The probability of strikes to a slender structure is

less than the probability of strikes to an extended struc-

ture, provided its height is greater than its width. This

suggests that extended structures have larger striking dis-

tances than slender masts. The parametric simulations for

structures, with heights and widths in the range 5 400 and

2100 m respectively, revealed significant differences in

the strike probabilities. The difference in strike probabil-

ity between extended and slender structures was found to

be in the range 1540%, depending on the height-to-width

ratio of the structure. Larger differences were found for

structures of larger width.

3.3.1 SELECTION OF STRIKING POINTS

w xIn one photographic survey 29 , it was found that light-

ning strikes to corners and the nearby vertical and hori-

zontal edges of structures account for more than 90% of

all incident flashes. This indicates that the points on the

structure with the highest degree of electric field en-

hancement are much more likely to be struck. However,

our fractal modeling studies show that the probability of a

strike to a given point on a structure also depends on the

polarity of the lightning strike.

The simulations show that negative lightning does in-

deed strike, with high probability, the points of the struc-

ture where the field intensification is highest. For thestructure presented in Figures 15, over many trial simu-

lations, we found that negative polarity discharges struck

the corners, upper horizontal edges, flat horizontal sur-

face, vertical edges and flat vertical surfaces with a per-

centage probability of 49, 38, 3, 7 and 3 respectively. The

vast majority of strike points on the upper horizontal edges

were very close to the corners, so the first two probabili-

ties can effectively be combined.

On the other hand, positive lightning strikes these points

in a more random fashion. From the simulations we con-

ducted with this polarity, the corresponding percentage

probabilities were 18, 32, 18, 23 and 9 respectively. One

reason for this difference is that upward, negative leadersrequire approximately twice the electric field strength ;1

. w xMVrm to sustain propagation 22,27 . This means that

their probability of inception or physical length will bemuch less for a given ambient field or, conversely, the

.downward positive leader must approach more closely .

Hence, the influence of the upward leader in determining

the attachment point is greatly diminished, magnifying the

influence of random effects such as channel tortuosity.

In reality, structures are often composite in nature.

Hence, a series of strike simulations was carried out onthe stepped surfaces of these structures Figures 6 and

.7 . The geometric composition of the structure was alsofound to have an influence on the distribution of strike

points. In this study, a building 100 m high is adjoined to a

20, 40, 60 and 80 m building, respectively. Since the ad-

joined building has a lower height than the main building,

it is either partly or fully shielded by the main building,

i.e., situated in the protection zone of the taller structure.

The simulations show that the probability of a strike to

the adjoined building with heights of 20 and 40 m is less

than 3%. Hence, for this structure, a protection zone de-

fined by a protection angle of ;45 seems appropriate.

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.Figure 6. Orientation of lightning to an extended 3-D structure.The central building has a height of 60 m, and the adjoining build-ings have heights of 20 and 40 m. The width of each building is 40 m.a, positive polarity strike; b, negative polarity strike.

Figure 7. Orientation of lightning to a composite structure. Thetaller part has a height of 60 m, and the other part is 40 m high. Thewidth of both components is 40 m.

Figure 8. Example of a side strike to a 3-D structure of height 100m and width 50 m.

For the adjoined buildings with heights of 60 and 80 m,

the strike probabilities are 14% and 20% respectively. In-w xternational standards on lightning protection 30 claim

that the collection area of a structure in flat terrain is

determined by a distance from the structure perimeter

which is three times the structure height. This means that

a protection angle of 72 is accepted for structures. It fol-

lows from the simulations that such an angle correspondsw xonly to protection level IV in 30 , i.e., an interception

efficiency of;80%.

3.4 SIDE STRIKES

As the height of a structure increases, so does the likeli-

hood that lightning will strike its upper vertical surfaces.

An example of a so-called side strike is shown in Figure

8. This phenomenon indicates that the shielding zone of

a structure begins at some point below the upper surface.

Hence, it is of practical interest to determine the proba-

bility of side strikes and the dependence of these on the

structure height.

Intuitively, it may be postulated that side flashes take

place if the height of the structure exceeds the length of

the streamer zone of the downward leader at the final

jump phase of the lightning attachment process. Also, it isreasonable to expect that the probability of a side strike

increases slightly with structure height and is more preva-

lent for positive lightning than negative lightning.

The simulations we carried out confirm these postu-

lates. Firstly, the overall probability of negative side strikes

is F2% whilst the probability of positive side strikes can

be as high as 20%. To keep these results in perspective,

however, negative polarities typically comprise more than

90% of all flashes. Hence, the overall probability of a side

strike is very low.

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Table 5. Distribution of side strikes along the vertical surface of atall structure of height h for different values of hrh, where h isthe distance of the strike point from the top of structure.

hrh 00.1 0.10.2 0.20.3 0.30.5 0.50.7

.P % 31 38 14 7 5y .P % 23 29 16 16 16q

Secondly, negative side strikes were mainly located

around the upper regions of the structure, whilst positive

side strikes were sometimes located near the lower re-

gions of the structure. Table 5 displays the distribution of

side strike probabilities along the vertical surface of tall

structures. The results have a logical explanation. Nega-

tive lightning predominantly strikes the top of the struc-

ture due to the development of a positive upward leader.

For positive lightning, the development of the upward

negative leader is delayed because of the higher electric

field intensification needed for the inception and propa-

gation of such a leader.

The number of side strikes may also be estimated fromw xtheoretical considerations 31

r f2.4 i2r3 4 .ss 0

where r is the striking distance for side strikes in metresssand i is the lightning current in kA. The probability of a

0

side strike is significantly less than for a downward strike

to the top surface of a structure because of the relatively

low field intensification near the sides of the structure.

The striking distance of lightning to the plane earth sur-w xface is given by 31

r f 4.8 i2r3 , 5 .se 0

again with r in metres and i in kA. These striking dis-se 0tance approximations can then be used to determine the

attractive area for side flashes A , namelyss

A (2r hyr , 6 . .ss ss se

where h is the structure height.

The contribution of side strikes to the total lightning

incidence is quite low and they display a low sensitivity to

the structure height due to the increase of total strikes

with increasing structure height.

3.5 BREAKTHROUGH PROBABILITIES

As shown in Section 3.3, the interception efficiency of a

lightning rod is higher for negative lightning. Hence, the

breakthrough or bypass probability of positive lightning

to protected objects should be higher than for negative

lightning.

To quantify this phenomenon, consider the arrange-

ment shown in Figure 9. It shows an object B of height h,

situated between two lightning rods, A, of height H. Since

H)h, object B is protected by A to some extent. Table

Figure 9. Configuration used to investigate the concept ofbreakthrough probability to a protected object.

Table 6. Percentage probability of negative and positive strikes toan object B of height h located between two taller rods A of heightHs120 m.

. .h m S m Strikes to A Strikes to B Strikes to ground

. . . . . .P % P % P % P % P % P %y q y q y q

90 83 0 1720 60 92 76 0 1.4 8 22.6

30 90 0 10

40 30 77 0 2360 83 72 0 4 17 24

60 60 90 3 7

6 displays the results of simulated strikes to this arrange-

ment. Results are presented for different distances S be-

tween the object and lightning rods, for both negative and

positive polarity lightning.

This analysis shows that the shielding zone between

two lightning rods is larger than the shielding zones of two

individual rods. Also, it can be seen that the breakthrough

probability of positive lightning to protected objects is

higher than for negative lightning. With respect to the de-

pendence on height and distance between the object and

lightning rods, the probability of breakthrough for positive

lightning is 35 times higher than for negative lightning.

Hence, despite the fact that only ;10% of lightning

flashes are of positive polarity, this result indicates that

the contribution of positive polarity strikes to the break-

through probability of protected objects cannot be ig-

nored. Assuming that a strike has bypassed the LPS and

terminated on the protected object, the probability that

this event is due to positive lightning can be determined

from the expression

P q =P br 0.1=0.25 . .qP q s s s0.36 7 . .br

P br 0.07 .

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Figure 10. Orientation of lightning to a gable structure. a, negative polarity strike; b, positive polarity strike.

.where P br is the conditional probability of break-q

.through of positive lightning, P q is the fraction of posi- .tive lightning in all flashes, and P br is the probability of

lightning breakthrough. Thus, the probability that the by-

pass is of positive polarity is 36%. Hence, polarity effects

should be taken into account in an assessment of lightning

rod efficiency.

Note that the geometrical form of the top of a structure

also has an influence on the breakthrough probability of

lightning, particularly for negative polarities. Figure 10

shows the trajectories of lightning orientation to a gable

structure protected with a lightning rod. The break-

through probability for this type of structure is higher and

so a taller lightning rod or more rods should be used forthe protection of gable structures than for structures with

flat roofs.

3.6 DETERMINATION OF THE

SHIELDING ANGLE

w xAccording to international standards 30 , a protection

angle can be defined for structures not exceeding 20 m in

height. This angle defines a cone of protectionaroundsthe structure, where is the apex angle of the cone. Forstaller structures, the rolling sphere method is usually used

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Table 7. Percentage probability of negative and positive strikes to an object B of varying height h located a distance S from a slender structureA of height 120 m.

Strikes to A Strikes to B Strikes to ground

. . . . . . . . .h m S m P % P % P % P % P % P %s y q y q y q

90 30 45 76 60 7 17 17 23

60 30 27 85 70 5 13 10 17

30 18 83 80 0 3 17 1730 60 34 80 73 3 7 17 20

90 45 85 70 5 3 10 27

w xfor the determination of protection zone 30 . Hence, a

determination of the protection angle and the sphere ra-

dius for a given structure is of practical interest.

Table 7 presents the calculated probability of negative

and positive strikes to a protected object B of varying

height h, situated a distance S from a slender structure of

height 120 m. Once again, the results show that the prob-

ability of strikes entering the shielding or protectiony1w .xzone

defined by a cone angle s tan Sr Hyh is

shigher for lightning of positive polarity. Hence, the shield-

ing angle for positive lightning is less than the angle for

negative lightning. Furthermore, the probability of strikes

to a protected object B situated in the shielding zone de-

creases with decreasing angle .s

3.7 EFFECT OF SHIELDING BY NEARBY

STRUCTURES

In practical scenarios, earthed structures cannot be

considered as isolated objects, as dense city blocks are

commonplace. Hence, the effect of surrounding structures

must be taken into account. Surrounding structures may

cause a difference in the number of lightning strikes to agiven structure because of the smoothing of the electric

field distribution around the structures. This effect would

be expected to reduce the number of strikes to a struc-

ture. The influence of surrounding buildings on the num-w xber of observed lightning strikes was shown in 24 .

In this investigation, the breakthrough probability of

lightning to a lower structure is determined for negative

and positive polarities. This particular analysis considered

only 2-D structures, since the large number of trial strikes

required to obtain suitable statistics using multiple 3-D

models requires an extremely large amount of computa-

tion time. This approximation may be used if the lightning

.channel has a low degree of tortuosity fractal dimension .w xSuch a case is presented in 32 , where the 3-D image of

the lightning trajectory was constructed using two stan-

dard video cameras and an image processing system. The

fractal dimensions of the 2-D and 3-D discharge trajecto-

ries determined from this experiment were Ds1.03 and

1.06, respectively, i.e., they were very similar.

Examples of the simulations are shown in Figure 11.

The simulations confirm a decrease in the interception

probability of lightning by structures in built-up situa-

tions. Even though lightning interception probability in-

Figure 11. Examples of the shielding effects of nearby structures. a,negative polarity strike; b, positive polarity strike; c, trajectories of 20negative strikes; d, trajectories of 20 positive strikes.

creases with structure height, the taller structures de-

crease the probability of strikes to surrounding, lower

structures. In Figure 11c, the taller structures cause a

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fourfold decrease in the probability of strikes to the cen-

tral, 50 m-high building. For positive lightning, the influ-

ence of the height of surrounding structures is not signifi-

cant. In Figure 11d, the probability of strikes to the cen-

tral structure is the same as the neighboring, taller struc-

tures.

3.8 EFFECT OF GROUND TOPOGRAPHY

The topography of the ground surface is known to in-

fluence the orientation of lightning to earthed structures,

e.g., increased incidence of strikes in mountainous re-

gions. Some investigations of this kind were carried out inw x w x2,33 . Rizk 33 found that the attractive radius of a struc-

ture on a mountain top is more sensitive to the back-

ground electric field than the same structure on flat

ground. The critical, ambient, ground field for upward

lightning initiation is much lower in mountainous than in

flat regions, resulting in a much higher incidence of up-w xward flashes. In a separate analysis 34 supporting these

observations, vertical and near-vertical ground planes werefound to have a strong influence on the lightning striking

distance.

The fractal approach implemented in this paper is ideal

for investigating these effects. For example, it can be used

to compute the breakthrough probability of lightning to

an object located near a mountain slope. The protection

zone and shielding angle of a lightning rod in this case is

less than the corresponding values for a rod located on

flat terrain. For example, the probability of breakthrough

to a structure, situated near the base of a mountain with a

45 slope, exceeds by 5060% the equivalent probabilityw xfor flat terrain 31 .

The probability of strikes to a structure in different lo-

cations between two mountains was also calculated. Some

examples of the simulations are shown in Figure 12. The

calculations show that the number of strikes to the struc-

ture in this case is less than the number of strikes in a flat

region. However, the number of side strikes increased sig-

nificantly.

Structures of different height, located at the foot, on

the side and at the top of a mountain were also consid-

ered. The results show that the dependence of the num-

ber of strikes on structure height is greater for structures

situated at the foot or on the side of a mountain than for

those located on top of a mountain. For a fourfold in-crease in structure height, the probability of strikes to

structures situated at the foot and on the side of a moun-

tain increased by a factor of 2.5. The corresponding in-

crease for structures on top of the mountain was by a fac-

tor of 1.7.

4 DISCUSSION

4.1 POLARITY EFFECTS

One of the most important findings of this study is the

model-predicted effect of the polarity. Hence, it is impor-

Figure 12. Simulation of the effect of different ground topogra-

phies. a, negative polarity strikes; b, positive polarity strikes.

tant to attempt to relate these predictions to field obser-

vations. Unfortunately, an analysis of lightning polarity ef-

fects from the observational data is not straightforward.

Firstly, most instrumented observations have been made

for negative flashes. Negative flashes are the most com-

mon, typically accounting for more than 90% of all cloud-

ground flashes. These negative-polarity observations have

generally been made for flashes to tall or slender mastsw xand towers, e.g., Bergers measurements 23 , and not to

practical, extended structures. Hence, there are very few

photographs of positive lightning strikes to structures.

Those that do exist usually show trajectories far from the

structures. For example, only one photograph was secured

of a positive downward stroke at Bergers research station

on Mount San Salvatore during the period 1955 to 1965w x23 .

Secondly, the complexity of the lightning trajectory de-

pends on many parameters, such as over-voltage in the

gap between the thundercloud and the earths surface, as

well as the leader potential. In other words, in lightning

observations, it is difficult to distinguish true polarity ef-

fects from these other influences. Hence, to a certain ex-

tent, models must rely upon laboratory observations of

long sparks. There are not enough instrumented observa-

tions of real lightning which would lead us to conclude

that the observations are substantially different to labora-

tory long-spark polarity effects. The polarity effect which

is seen in our fractal model of lightning attachment is due

to the well-documented difference in the electric field re-

quired for the development of positive and negativewstreamer-leader systems in laboratory experiments 27,

x3638 .

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w xThirdly, it is worthwhile considering the Berger 23

classification for cloud-ground lightning flashes and where

the present modelling results lie in relation to these desig-

nations. Bergers classification covers two main scenarios,

namely flashes over open country and flashes to tall,

earthed objects. The former includes the classical down-

ward-upward leader scenarios for both polarities, and the

latter involves so-called upward flashes of both polari-

ties. The present study considers only flashes of the for-

mer type, but with common, everyday structures on theground in a sense, this is an intermediate case, not con-

.sidered in Bergers classifications . In this scheme, at some

distance from the ground or the structure, a negative

downward leader induces a positive upward leader from

the grounded object and, likewise, a positive downward

leader induces a negative upward leader.

Fourthly, the main polarity effect considered in our

study relates to the difference in the electric field re-

quired to initiate and propagate the different upward

leaders, as discussed above, and the ultimate effect thisw xhas on strike probabilities. It was shown in 39 that, ex-

cept for very tall masts, the striking distance for positive

lightning has only a low sensitivity to mast height. How-

ever, for structures of height above 200 m, large striking

distances were predicted. In addition, our present model

also predicts long negative connecting leaders from very

tall structures. This type of discharge was first observed byw xBerger on Monte San Salvatore 40 . A similar type of dis-

charge, where a negative upward leader progresses in the

form of a very long connecting leader into an existingw xintra-cloud flash, was also reported in 40,41 . These types

of discharges take place in mountainous areas or near tall

structures.

Looking specifically at the case of strikes to Bergers

towerwhich is 70-80 m high and physically located on

top of Mount San Salvatore, some 650 m above the sur-

rounding terrainthe resultant data in respect of flash

incidence is somewhat inconsistent with that for struc-

tures of similar height. In Bergers observations, 84% of

flashes were upward in nature, i.e., an upward leader pro-

gressed uninterruptedly from tower top towards the cloud.

Electrically, this mountain-top setting is equivalent to a

tall, slender structure having an effective height of aboutw x350 m 26 . In most practical situations involving buildings,

lightning flashes are due to a downward leader progress-ing towards a structure of moderate height typically less

.than 100 m situated in relatively flat terrain. In the dataw xobtained by Eriksson et al. 42,43 , there were no flashes

of positive polarity recorded on the 60 m research mast

over an 11-year period. Likewise, only one positive flash

was identified in 40 or so records of direct flashes to a 10w xkm length of 11 kV test-line over five years 43 . This indi-

cates that long, negative, connecting leaders generally do

not develop from structures of moderate height in rela-

tively flat terrain. On the other hand, measurements on

tall masts or equivalent scenarios, such as Bergers, show

an increasing tendency to positive flashes with increasingw xstructure height 44 .

w xTo sum up, the Berger data 41 showed that negative

connecting leaders are much longer than positive connect- .ing leaders. These observations are inconsistent with i

our simulation results for practical structure heights in .relatively flat terrain and ii the field data obtained from

lightning incidence to masts of similar height in flat ter-

rain. It is important to note also that the apparent differ-

ence may be due to the interpretation of the experimental

data. The parameters of positive flashes were originallyw xanalysed by Berger et al. 45 in 1975on the assumption

that these were downward flashes. The connecting up-

ward leaders in these cases are expected to be negative.w xIn Bergers later analysis 46 , he has classified all these

records as upward.

4.2 IMPLICATIONS OF THE RESULTS

From the preceding sections, an interception angle

may be introduced for the determination of the collec-

tion volume of earthed structures. The calculations show

that this angle is larger for negative polarity lightning.

Hence, the interception efficiency of lightning rods is

higher for this polarity of downward lightning. This also

indicates that the capture radius of structures is larger for

negative lightning. Whilst negative lightning strikes the top .in particular the corners of a structure with higher prob-

ability, positive lightning results in a larger scatter of strike

points on the structure. This result can be explained in

terms of the inception of an upward leader from the

structure. As discussed previously, a lower electric fieldstrength is needed for positive upward leader inception.

As long as the upward leader starts from the highest points

of the structure, then basically the top of the structure is

struck. For positive downward lightning, the selection of

striking points takes place in a more random manner due

to the higher field required for upward leader inception.

These polarity effects have also been observed in labora-w xtory experiments 47 .

Lightning strikes to the sides of tall structures are wellw xknown 35 , although such strikes are rare in occurrence.

For example, side strikes comprised only 7% of all ob-

w xserved strikes to the 540 m Ostankino TV tower 28 . Al-most all of these side strikes were due to downward light-

.ning as opposed to upward initiated flashes . Given that

typically up to 10% of lightning flashes have a positive

polarity, it may therefore be postulated that these side

strikes were predominantly of positive discharges.

The structure geometry also has an influence on the

probability of strikes. The simulations show that there are

differences in the orientation of lightning to slender and .extended 3-D structures. Although an upward leader may

be initiated earlier from a slender structure, the intercep-

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tion probability of a 3-D structure is higher due to the

larger electric field intensification it creates over a greater

distance. If the conditions for upward leader inception are

fulfilled, then the upward leader propagates more easily

from a 3-D structure than from the slender object. It fol-

lows that the protection zone of a slender structure is less

than the protection zone of 3-D structure of the same

height. Also, it may be postulated that the efficiency of

lightning rods placed on top of buildings is higher than

the efficiency of a slender structure of the same overall

height. The calculations also show that upward leader in-

ception from an earthed structure does not always result

in a strike to the structure, e.g., see Figure 1. Hence, the

striking distance determined from the criterion for up-

ward leader inception cannot, in isolation, be used to ac-

curately estimate the strike probability.

The degree of the tortuosity and branching of a light-

ning channel, determined by the fractal dimension, is sen-

sitive to the potential of the leader tip, which in turn de-

termines the return stroke peak current. The fractal di-mension increases with an increase in leader tip potentialw x13,18 . The fractal dimension of a lightning channel also

determines the spectral density of the electromagnetic ra-w xdiation. In 48 , a method for the determination of light-

ning parameters from measurements of the statistical

properties of the electromagnetic field was proposed. It

was shown that the characteristic scale of tortuosity of a

channel, which is related to the streamer zone length of a

leader or the charge per unit channel length, may be de-

termined from the measurement of the correlation func-

tions of the lightning radiation field.

Finally, the protection area of tall structures subjected

to positive lightning is substantially less than the area pro-

tected against negative-polarity strikes. It follows that the

interception efficiency and hence protection level of

lightning rods placed on structures is lower for positive

lightning.

5 CONCLUSIONS

HE fractal approach and results presented in theTpaper demonstrate that the probability of strikes topractical structures may be determined for negative and

positive polarity lightning. The following conclusions may

be drawn from the analysis:

.a The strike probability decreases with an increase inthe approach angle of downward lightning. The majority

of discharges striking the structure fit within a cone of

half-angle -60.

.b For any given structure, the interception probability

is higher for negative lightning strikes than those of posi-

tive polarity.

.c The structure geometry is an important para-

metersimulations show that the interception probability .of an extended 3-D structure is up to 40% higher than

that of a slender structure of the same height.

.d The selection of strike points on structures is strongly

influenced by the structural features and the polarity of

lightning. Negative lightning overwhelmingly strikes the

points of highest electric field enhancement, whilst posi-

tive lightning strikes more randomly. For negative light-

ning, corners, horizontal and vertical edges accounted for

;

94% of all strikes, whilst for positive lightning the prob-ability for the same points was ;75%.

.e The shielding zone of two neighboring lightning rods

is larger than the combined shielding zones of two individ-

ual rods.

.f For negative lightning, tall structures cause a signifi-

cant amount of shielding of nearby, lower structures. The

resulting decrease in strike probability may be as high as a

factor of four.

.g The incidence of side strikes to tall structures is very

low, F2%, and increases only slightly with structure

height. Most side strikes appear to be due to positive dis-

charges which, in contrast to negative lightning, may ter-minate on the structure at points significantly below the

top.

.h The breakthrough probability of positive lightning is

35 times higher than for negative lightning.

.i The ground topography has an influence on the prob-

ability of strikes to structures. The number of strikes to

objects situated near the base of mountains is less than

for the same objects located on flat terrain. This is due to

increased probability of the orientation of lightning to

mountains.

.j The influence of structure height on the number of

strikes received is lower for structures on mountain topsthan structures in flat regions.

.k The protection angle guidelines for structures, as

proposed in international standards on lightning protec-

tion, need to be revised to take into account stochastic

effects and the polarity of the lightning discharge, as these

have a significant effect on the interception efficiency of a

lightning protection system.

For many years, lightning protection principles have re-

lied solely on electrogeometric models, without any ac-

count for random effects. On the other hand, the fractal

approach demonstrated in this paper can be used to de-

termine an interception efficiency or protection level forany practical scenario. This approach takes into account

all of the important physical criteria, such as the lightning

parameters and conditions for upward leader inception.

Most importantly, it allows for the random effects seen in

lightning discharges, such as channel tortuosity and leader

branching.

ACKNOWLEDGMENT

N. I. Petrov and G. N. Petrova thank ERICO Lightning

Technologies for financial support of this study. All the

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authors wish to thank the anonymous referee, whose com-

ments helped improve the clarity of the paper.

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