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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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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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Petro et al.: Quantification of the Probability of Lightning Strikes to Structures Using a Fractal Approach650
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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Petro et al.: Quantification of the Probability of Lightning Strikes to Structures Using a Fractal Approach652
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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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 10, No. 4; August 2003 653
authors wish to thank the anonymous referee, whose com-
ments helped improve the clarity of the paper.
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