Download Li, J. Q. Yang, W. Sima, C. Sun, T. Yuan, and M. Zahn, A New Estimation Model on the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach, accepted for publication, IEEE Transactions on Dielectrics and Electrical Insulation

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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
A New Estimation Model of the Lightning Shielding
Performance of Transmission Lines
Using a Fractal Approach
Jianbiao Li, Qing Yang, Wenxia Sima, Caixin Sun, Tao Yuan
State Key Laboratory of Power Transmission Equipment & System Security and New Technology
Chongqing University, Shapingba District, Chongqing, 400044, P. R. China
and Markus Zahn
Department of Electrical Engineering and Computer Science,
Research Laboratory of Electronics, Laboratory for Electromagnetic and Electronic Systems
High Voltage Research Laboratory,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
ABSTRACT
The path of the lightning discharge shows characteristics of branching and tortuosity,
and the lightning shielding failure of the transmission line also has statistical features.
In this paper, an improved leader progression model using a fractal approach is
introduced, which considers both the deterministic and stochastic features of the
downward lightning leader. Based on the proposed model, the statistical features of
lightning shielding failure on transmission lines will be analyzed. First, the fractal
dimensions of the downward lightning trajectory in simulation are calculated and the
value of breakdown probability constant η will be determined based on the fractal
dimensions of natural lightning. Second, using the proposed model the shielding failure
probability contour around the transmission line space is also obtained and used to
describe the shielding failure zone and then the contour curve is applied to analyzing
the characteristics of transmission lines that suffer from direct lightning strokes.
Moreover, a method to estimate the lightning shielding failure rate of transmission lines
is put forward, and the influence of the line height and the shielding angle of ground
wires on the shielding failure rate of transmission lines are also investigated.
Index Terms — LPM, fractal approach, estimation method, shielding failure, UHV
transmission line.
1 INTRODUCTION
LIGHTNING protection plays a crucial role in the safe
operation of transmission lines. Circuit breaker trips on
transmission lines caused by direct lightning strikes can lead
to economic loss. Such trips can be mainly classified as backflashover trips and lightning shielding failure trips. According
to statistical records in Russia [1], with the increase of line
voltage the proportion of lightning shielding failures in total
trip number also increase. UHV and EHV transmission line
trips are mainly caused by lightning shielding failures [1].
Therefore, lightning shielding protection has become an
important issue in the protection of transmission lines.
Currently, the lightning protection design against direct
lightning strikes is generally based on the accurate estimation
of the transmission line’s lightning shielding performance.
Manuscript received on 5 November 2010, in final form 28 March 2011.
Therefore, a precise estimation model for lightning shielding
performance plays an important role in the lightning
protection design of transmission lines.
Since the 20th century, scholars have conducted research on
lightning shielding models [2-8]. Currently, the models
frequently used in lightning shielding research are: the Electric
Geometry Model (EGM) [2-4], the improved EGM [5], and
the Leader Progression Model (LPM) [6, 7]. Young et al [2]
proposed the initial concept of EGM. Based on the earlier
research and field experiments on lightning shielding failures
of transmission lines, Armstrong and Whitehead [3] connected
the lightning discharges feature with the structural size of
transmission lines, and thus developed EGM. The model
assumed that the striking distance had some relation to the
lightning current amplitude. In terms of this relationship,
Anderson [4], Love [8] and others made their respective
studies. Afterwards, with the elevation of transmission line
voltage level and the increase of tower height, Eriksson [5]
1070-9878/11/$25.00 © 2011 IEEE
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 18, No. 5; October 2011
and others believed that the striking distance was not only
related to the lightning current, but also related to the height of
grounded objects, and they proposed the improved EGM.
EGM and Eriksson’s improved EGM can calculate the choice
of shielding angle of the transmission line under different line
parameters and topographies, so they are widely used in
lightning protection design of transmission lines. However, in
[9-14], it is shown that the actual shielding failure number of
large-sized transmission lines is larger than the calculated
results of EGM and Eriksson’s improved EGM. In the 1990s,
based on the observation and research on long air gap
discharges in the laboratory, Dallera and Garbagnati [6], Rizk
[7] and others put forward the LPM for lightning shielding
failure analysis. LPM gives an initial description of the whole
direct lightning striking process on a transmission line from a
physical perspective. The model uses a mathematical approach
to describe the initial upward leaders, the leader progression
and the final jump. It also introduces the lateral distance and
shielding failure width, which provide an effective method to
estimate the lightning shielding performance of transmission
lines and to analyze lightning shielding failures. However,
LPM assumes the leader propagates in the direction of
maximum electric field intensity. In other words, it only
considers the deterministic factors in the lightning leader
progression and ignores the influence of leader branches on
the spatial distribution of the electric field intensity. This
simplification influences the determination of the stroke object.
If LPM can consider the stochastic features of lightning leader
progression, its simulation process will be closer to the natural
lightning strike process.
Recently, many researchers use the fractal approach to
model the observed paths of lightning discharges [15-20]. The
photographed data of natural lightning discharge reveals that
the lightning leader has an obvious effect of branching and
tortuosity, which can be described by fractal mathematics.
Kawasaki and Matsuura [15], and Tsonis and Elsner [16] were
the first to use the fractal approach to simulate the lightning
trajectory. Then Petrov and Petrova [17] use the fractal
approach to model the lightning channel and to research the
probability of direct lightning strikes to buildings. However,
the lightning shielding model of transmission lines is not a
simple lightning trajectory simulation. It is also different from
lightning protection of buildings. The occurrence of a
lightning shielding failure is also affected by the competition
of upward leaders from the ground wires and the phase
conductors. The upward initial process of a horizontal
conductor is different from the upward initial process of
buildings. Therefore, the above models using the fractal
approach cannot be used directly in the research on lightning
shielding performance of transmission lines.
The leader progression is not only deterministic, but also
stochastic, with the phenomena of branching or tortuosity in the
progressing process. The influences of these stochastic factors
should not be ignored in the estimation of lightning shielding
performance of transmission lines. Therefore, in the present
study based on the fractal features of the lightning discharge, the
basic theory of LPM is combined with fractal theory, in which
the deterministic factors and stochastic factors in downward
lightning leaders are considered. A method based on the
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proposed model is also put forward to estimate the lightning
shielding performance of transmission lines.
2 BASIC PRINCIPLES OF THE
ESTIMATION MODEL
The proposed model is developed on the basis of the
conventional LPM. According to LPM theory, the direct
lightning striking process on transmission lines can be divided
into four stages, as shown in Figure 1. First, the downward
lightning leader propagates vertically from the cloud bottom. At
the same time, since there are many charges in the lightning
channel, the electric field intensity is also increased around the
lines and the earth. As the downward lightning leader
propagates towards the earth, the induced electric field intensity
around the lines and the earth also continuously increases.
While the downward lightning leader propagates to a critical
height, due to the effect of the increased electric field,
continuous upward leaders are produced from the surface of the
phase conductors and the ground wires. The upward and the
downward leaders continue to propagate in the space. At this
time, the electric field intensity between the upward and the
downward leaders, and the electric field intensity between the
downward lightning leader and the ground continuously grow.
When the electric field intensity in the gap between leaders is
high enough, a final jump will occur. There are thus several
factors influencing the direct lightning strike process: the
distribution of charge in downward leaders, the initial criterion
of continuous upward leaders, the progressing rules of leaders
and the criterion of the final jump. In this section the setting of
these criteria in the proposed model are discussed.
Figure 1. The direct lightning strike process on transmission lines.
The proposed model simulates the downward lightning
leader using the fractal approach from the initial start of an
upward leader. That is, the proposed model assumes that when
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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
an upward leader has not yet started on ground objects, the
lightning leader propagates vertically from a cloud towards the
ground, and the leader channel has no fractal features. Once an
upward leader has started on ground objects, the fractal
approach can be used to simulate the upward and the
downward leaders, and to study the direct lightning strike
process to transmission lines. Such simplification is based on
the following two reasons: (a) The simulation of the proposed
model does not start from the lightning leader's initial
progression from cloud bottom, because when the lightning
leader tip propagates from the cloud bottom to the height
Hi ,which is the height of the lightning leader tip at the upward
leader initial moment, the stochastic feature of its position will
increase the stochastic feature of the lightning shielding failure
probability, which is unfavorable for the research on factors
influencing the transmission line’s lightning shielding
performance. (b) The existing lightning shielding models
generally assume that when an upward leader has not been
generated, the ground and the earthed objects have very small
influence on the progression of the downward lightning leader.
This influence is so small that it can be ignored, and the
downward lightning leader propagates downward randomly.
So, on any horizontal level above the height Hi, the
distributions of lightning density are the same. Therefore this
simplification will not affect the estimation result of shielding
failure rate.
2.1 THE CHARGE DISTRIBUTION IN DOWNWARD
LEADERS
When lightning leader propagates to ground in discrete
steps, the charges in lightning channel will influence the
electric field spatial distribution around the lightning channel.
Therefore, in order to calculate the variation of spatial electric
field during the progress of lightning, the proposed model uses
a line charge varied linearly with channel height to simulate
the main channel of lightning [21]. When the lightning leader
reaches the certain height Hi, the downward leader channel
propagates towards ground in a fractal form [17]. Meanwhile,
since the lightning leaders will branch and bend, it is difficult
to continue to use a linear distribution to describe the
distribution of charges in downward lightning leader branches.
Thus, the charge density of the downward lightning leader
branches, i.e. the part with fractal features, is set as a uniform
charge distribution. Moreover, by observing natural lightning
discharge process, it is known that during the progressing
process of lightning leader in space, the leader tip is brightest
[22]. Therefore, a constant charge Q0 is used to simulate the
tip of lightning [21].
Dellera [6] believes that higher values of charge in the
channel will result in higher amplitudes of first stroke current;
and according to the observation data, the relation between the
total quantity of charge in the lightning leader channel QT and
the lightning current amplitude I can be expressed by the
following formula [21, 23, 24].
1
 I  0.7
QT   
(1)
 25 
where, QT is the total charge in the lightning leader, C; I is the
magnitude of lightning current, kA.
The height of the cloud HC is set at 2.5 km in this paper [21].
For the main lightning channel, when the height of the leader
is HC, the line charge density τ=0; when the height of the
leader tip is H0, the line charge density τ=τmax, with τmax as the
largest line charge density. Then according to the linear
distribution, the total volume of charge will be [21]
 (H  H0 )
QT  max C
 Q0
(2)
2
where Q0 (C) is the charge inside the leader tip, which can be
obtained by [21]:
(3)
Q0  2      r0 2  EB
where ε is the dielectric constant of air; r0 is the radius (m) of
the lightning leader tip, can be set as 6m [25]; and EB (kV/m)
is the critical electric field intensity for propagation of the
leader. For positive lightning, EB can be set as 500 kV/m, and
for negative lightning, EB is set as 1000 kV/m [17].
Therefore, the line charge density inside the main channel
of lightning will be [21]
m
2(QT  Q0 )
 ( h)  ( H C  h)
 ( H C  h)
(4)
HC  H0
( H C  H 0 )2
where τ(h) (C/m) is the charge intensity inside the lightning
leader.
For the fractal lightning leader, the line charge density
inside the lightning branch is taken to be uniform distribution
and is given by:
F 

QT  Q0  
HC
Hi
Lf l

Hi
H0
 (h)  dh
 (h)  dh
(5)
Lf l
where τ(h) can be obtained from equation (4); Lfl is the total
length (m) of the fractal lightning leader branch.
2.2 THE INITIAL CRITERION OF THE UPWARD
LEADER
At present, there are mainly three continuous upward leader
initial criteria used in the study of lightning shielding failure
of transmission lines [7, 21, 26-28]: a) the upward leader
induced voltage Uic of the horizontal conductor proposed by
Rizk [7]; b) Peek criterion: when the electric field intensity at
the conductor surface is higher than the critical field intensity
EC, the upward leader will be generated [21, 26]; c) Carrara
criterion [27, 28]: the upward leader will start from the
conductor surface in a long gap when the voltage applied on a
conductor exceeds a critical value. The Peek criterion can
consider the influence factors adequately, such as the phase
conductor’s working voltage, the induced electric field of the
downward leader and the coupling electric field between
conductors. From the superposition principle of electric field,
the Peek criterion can also explain the actual operating
experience that for a direct negative lightning strike, the
upward leader is easier to be generated from positive phase
conductors. Thus the proposed model uses the Peek criterion
to judge the initiation of the continuous upward leader.
As for horizontal conductor, Eriksson [5], Dellera et al [6],
Rizk [7] employed the critical radius concept to determine the
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 18, No. 5; October 2011
ambient field necessary for the upward leader continuous
inception, of which the critical radius is the one required to
initiate a direct corona-to-leader transition [6]. Dellera et al [6]
believe that for the conductor-plane configuration, when the
conductor radius is larger than the critical radius, the criterion
for the initiation of the continuous upward leader is the same
as the criterion for corona onset. When the conductor radius is
smaller than the critical radius, the upward leader continuous
inception criterion is obtained by calculating the conductor
corona inception criterion under critical radius [6]. Therefore,
the critical initial electric intensity can be obtained from the
formula [21, 26]:
0.03
EC  3000  m    (1 
)
(6)
 r
where EC (kV/m) is the critical initial electric field intensity; m
is the roughness coefficient of the conductor surface; δ is
relative air density; and r (m) is conductor critical corona
radius. For the ground wires, the radius of ground wire is
usually smaller than critical radius, so r is set at the critical
radius, r= 0.1 m [6]; For the phase conductor, the radius of
conductor bundles is usually larger than the critical radius, and
r is set as the equivalent radius of the conductor bundles.
2.3 THE PROGRESSION RULES OF LEADERS
LPM assumes that the leader’s progression direction always
goes towards the position around the leader tip, where the
electric field intensity is largest. But in fact, the observation of
direct lightning strikes on earthed objects show that the path of
lightning discharge can be branching and tortuous, and the
effect of branching and tortuosity of the lightning channel
would also influence the electric field intensity in space and
thus further influence the progression of the leader. Otherwise,
the effect of branching and tortuosity of the lightning
discharge indicates that the progression of the leader has
fractal features. The fractal approach can be used in analyzing
the progression patterns of leaders. Therefore, in this paper,
considering the deterministic and stochastic factors in
downward lightning leader progression fully, a fractal
approach is used to model the trajectory of the leaders, and the
details will be introduced in the following.
2.4 THE FINAL BREAKDOWN CRITERION
In the proposed model, when the electric field intensity
between the ground and any one of the leader branches or
lightning branches exceeds the critical breakdown electric
field 500 kV/m [29], the final jump will happen in the
proposed model, and the stroke object is also determined.
3 THE REALIZATION OF THE LEADER’S
FRACTAL PROGRESSION
The fractal approach is widely used in many kinds of
progression, among which, Dielectric Breakdown Model
(DBM) [30] is generally used in the computer simulation of
the progression path of discharges in dielectrics. According to
that simulation, in the process of dielectric breakdown, it is
more probable to have breakdown in a position where the
electric field intensity in the dielectric is most intense than that
in other areas. In the area shielded by other discharge paths,
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the probability to have a discharge is very low and even zero.
This phenomenon confirms that in the progression of lightning
leaders, the ionization in air has stochastic features, and it is
also controlled by the distribution of electric field in space.
The proposed model uses the DBM to simulate the trajectory
of the leader. The calculation process of the fractal leader
progression is shown in Figure 2 and the procedure is
described as follows:
Step:1 Discretize the space and set the initial fixed particles,
which are also called seeds, at the tips of the downward
leader, the conductors and the ground wires, as shown by
the solid points in Figure 3;
Step:2 Set the moving particles around the tip of the leader, as
shown by the moving particles in point B on Figure 3.
Moving particles have random Brownian motion in the
discrete space, which is shown as the trace of the dashed
line in Figure 3. The upward and downward leader
progresses most quickly at their tips respectively, and in
the calculation of lightning shielding, the progression in
leader tips plays a decisive role in the issue whether
lightning will strike conductors. Thus it is reasonable to
set the initial position of moving particles at the tips of
the leader.
Step:3 In each leader branch, when the moving particles move
to the possible breakdown point around any seed, for
example shown in Figure 3 at the semisolid points
around point P as possible breakdown points. When a
particle moves to point P′, its Brownian movement will
stop, and the probability to have breakdown at point P′ is
calculated by [30]:
 p ( P, P ')  0,
if E ( P, P ')  EB



( E ( P, P ')  EB )
 p ( P, P ')  N
,if E ( P, P ')  EB (7)


( E ( P, Pi ')  EB )


i 1
where, p(P, P′) is the probability for breakdown to occur
from P to P′ and EB is the critical electric field intensity
for propagation of the leader. For positive lightning, EB
can be set at 0.5 MV/m, and for negative lightning, EB is
set as 1 MV/m [17]. E(P, P′) is the electric field intensity
in the space from P to P′; E(P, Pi′) is the ith possible
breakdown point which satisfies the formula
E ( P , P ')  E B and goes from P to the point around P′,
and N is the number of possible breakdown points of this
kind. Constant η is an adjustable number that is used to
present the relation between local electric field intensity
and the breakdown probability in the dielectric. It
directly determines the stochastic features of the leader’s
progression direction. The selection of the breakdown
probability constant η will be discussed in Section 4.
Step:4 Make a breakdown judgment from the obtained
probability: compute a random number m which satisfies
a [0, 1] uniform distribution. Compare the random
number m with the obtained probability p; if m is smaller
than p, then it can be judged that breakdown will happen
at point P′, if m is larger than p, then breakdown will not
happen.
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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
Step:5 If point P′ meets the condition to have breakdown, and
breakdown happens, then the moving particle becomes a
new seed. It will form a “seed cluster” together with
other seeds and hence a new leader discharging channel
comes into being. If point P′ does not meet the
breakdown condition, then the moving particle will be
stopped. The program returns to Step 2 and keeps on
operating, until it determines the occurrence of lightning
strikes.
electric field spatial distribution determines that each leader
branch progresses most probably towards the point with most
intense electric field at their tips. This feature makes the
proposed model maintain the deterministic factor of the
leader’s progression. This method not only considers the
deterministic feature that breakdown happens more probably
when the electric field at the leader tip is more intense, but
also stochastic features in the process of leader progression.
4 THE SELECTION OF KEY
PARAMETERS OF THE FRACTALCHARACTERIZED LPM
The fractal dimension reflects how effectively the complex
graph occupies space. It is a measure of the orderliness and
randomness of the stochastic and deterministic features of
fractal graphs. From the above description of the proposed
model, it can be found that the value of the breakdown
probability constant η in (7) influences the progressing
probabilities at every point of the fractal leader channel, and it
determines the stochastic and deterministic features of the
lightning leader progression. In order to simulate the whole
lightning path accurately, it is very important to select a proper
η value. In this paper, the lightning leader will be simulated
with different η values, hoping to determine how the constant
η will influence the dimensions of the lightning path in
simulation, and to determine the proper value of the constant η
for the proposed model.
The fractal dimensions of lightning progression based on
the box-counting method is given by [31-33]:
Db  lim
lb  0
Figure 2. Calculation process of the fractal leader progression
ln N (lb )
ln(1 / lb )
(8)
where Db is the fractal dimension, lb is the measurement scale,
and N(lb) is the least number of squares that can cover the
fractal graph, with lb as the side length.
Based on the box-counting method, the fractal dimensions
of the simulated lightning path with different values of η are
calculated and are shown in Table 1. The lightning paths in
simulation with three different η values are shown in Figure 4.
Table 1. Calculated fractal dimensions with different η values.
Η
0.1
0.5
1
5
10
Db
1.46±0.07
1.31±0.05
1.20±0.04
1.09±0.05
1.04±0.02
Figure 3. Branch picture of leader.
In the model calculation process, the progression of every
new seed that constitutes the fractal leader will influence the
electric field spatial distribution around the lightning channel,
change the electric field intensity E(P, P′) in every direction,
and thus influence the progressing probability p(P, P′) of the
next moving particle. By repeating the above calculation and
judgment procedure, the leader progressing path will be
determined. In the proposed model, the charges in the
downward lightning channel directly determines the electric
field spatial distribution around the lightning channel, and the
Figure 4. Simulated figures of the lightning leader for various fractal
dimensions
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 18, No. 5; October 2011
Input the parameters of the transmission
line and the lightning
Calculate the electric field
intensity in space
Lightning strike
or not?
Yes
No
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range of natural lightning discharge paths. Therefore, in the
proposed model, the ŋ value is set as 1.
The calculation process of the proposed model is shown in
Figure 5, and the simulation figure of a lightning strike on
transmission lines is shown in Figure 6. The line is a ±800 kV
dc transmission line, with the shielding angle at zero degrees,
the conductor height at 44 m, and the ground wires height at
55 m. The lightning current is 30 kA, and the lateral distance
between lightning and the line is 100 m.
Calculate the electric field intensity
on the conductor surface
The upward leader
is generated ?
No
Yes
The fractal upward leader
progresses
The fractal lightning leader
progresses
Calculate the electric field
intensity in space
No
Lightning strike
or not?
Yes
Record the struck object
Figure 5. Calculation process of the proposed lightning model.
It can be found from (7) that when the η value is smaller than
the proper value, in the criterion of leader progression the effect
of the electric field distribution will be reduced, and the
progressing probabilities in every direction of the leader tip tend
to be the same, which leads to excessive branches in the leader
channel, as shown in Figure 4a. In such a condition, the
stochastic feature plays the main role in the leader progression,
and the corresponding fractal dimension is relatively larger than
that of natural lightning. Conversely, when the η value is larger
than the proper value, the direction with most intense field in
the leader tip will gain more probability to progress, while the
probabilities for leaders to progress in other directions are
smaller. The leader progresses along the direction with most
intense field, but less probably along the branches, as shown in
Figure 4c. In such a condition, the deterministic feature plays
the main role in the lightning leader progression, and the
corresponding fractal dimension is relatively smaller than that
of natural lightning. Therefore, to some extent, the selection of ŋ
value decides whether the lightning model can reflect the
characteristics of branching and tortuosity of lightning path
properly. Kawasaki, Matsuura [15] and Tsonis and Elsner [16]
measure the fractal dimension of still pictures of natural
lightning channels. They find that the empirical fractal
dimensions of the natural lightning discharge paths exist in the
range from 1.1 to 1.4; and as shown in Table 1, when the ŋ
value is set between 0.5 and 5, the fractal dimension of the
simulated lightning path Db exists within the fractal dimension
Figure 6. Direct lightning strike on the simulated transmission line.
5 ESTIMATION METHOD
5.1 THE IMPROVED ESTIMATION METHOD
Conventional LPM defines a shielding failure width (SFW)
to estimate the shielding failure number (SFN) of transmission
lines [34]:

SFN  N L  Aeq  T  N L  L  T   SFW ( I )  PL ( I )  dI
0
(9)
where SFN is the shielding failure number, strokes; NL is the
ground flash density, flashes/km2-year; Aeq is the shielding
failure areas, km2; I is the lightning current amplitude, kA;
SFW is the shielding failure width, km; L is the length of the
transmission line, km; T is the observation time, years; and PL
is the probability density distribution of lightning
currents, %/kA.
Conventional LPM believes that when lightning occurs in the
shielding failure area of transmission lines, shielding failure will
inevitably happen. In the proposed model, it is believed that the
lightning discharge has stochastic features. Even though
lightning reaches into the shielding failure area of transmission
lines, because of the attraction from the upward leader of
ground wires and from the earth, shielding failure is not
inevitable. There exists a probability to have shielding failures,
as shown in Figure 7. Therefore, for a certain lightning current
amplitude I, weight SFW as in (9) and use the equivalent SFW,
SFWeq(I) to replace it. The weighting value is set to be the
lightning probability PS so that SFWeq(I) is:

SFW ( I )  SFWeq ( I )   PS ( I , D)  dD
0
(10)
where PS (%) is the probability of lightning strikes on the
phase conductor, and D (km) is the lateral distance between
lightning leader and the transmission line.
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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
PS can be obtained by a large number of repeated
calculations in the proposed model. By the model, the
shielding failure probability PS at all levels of lightning
currents can be calculated. For example, the 500 kV
transmission line tower in Guangdong province of China is
shown as Figure 8 and PS can be shown as the distribution
curve in Figure 9:
has its peak value decrease as the lightning current amplitude I
grows. Therefore, under the same line parameters, PS is a
function of lightning current I and lateral distance D. In order
to obtain SFR, calculate the shielding failure probability
distribution under every lightning current level I for different
lateral distance D, and by data fitting, a shielding failure
probability function is fitted with lateral distance D and
lightning current I as its variables [35]:
PS  PS ( I , D)
(11)
Based on (9), (10) and (11), the SFN and SFR of the
transmission line can be calculated as follows:
SFN  N L  T  L   PS ( I , D)  PL ( I )  dI  dD
SFR 
SFN
100  L  T
(12)
(13)
 0.01  N L   PS ( I , D )  PL ( I )  dI  dD
Figure 7. The shielding failure probability PS as a function of lateral distance.
Figure 8. Structural figure of 500 kV EHV transmission line in Guangdong
province of China.
100
80
60
40
20
0
0
50
100
150
200
Figure 9. Probability of lightning strike on the conductor versus lateral
distance D.
It can be found from Figure 9 that under a certain lightning
current I, as the lateral distance D increases, PS will increase
first and then decrease. PS under different lightning currents I
where SFN is shielding failure number of the transmission line,
strokes; SFR is shielding failure rate of the transmission line,
strokes/100 km-year; NL is the ground flash density,
flashes/km2-year; PL is the probability density distribution of
lightning currents, %/kA; T is the observation time, years; L
(km) is the length of the transmission line; I is the current
amplitude, kA; and D (km) is the lateral distance between the
lightning leader and the transmission line.
If the parameters of different parts of the line are different,
they can be calculated one by one, and then by adding those
up the total SFN will be calculated.
5.2 VALIDATION OF THE ESTIMATION METHOD
As for the EHV transmission line, the China Southern
Power Grid uses the lightning location system (LLS) to record
the lightning trip numbers of 2550 km 500kV AC transmission
line in Guangdong Province of China [36], and they also
measured the lightning current amplitude of every lightning
stroke on the line. From 1999 to 2003, there have been 40
lightning trip accidents on the line [36]. The 500 kV
transmission line in Guangdong province of China [28] has its
structure model in Figure 8. The SFR of the 500 kV
transmission line is estimated by Eriksson’s improved EGM,
LPM and the proposed model.
The thunderstorm days and the ground flash density are 40
days/year and 2.9 flashes/km2-year, respectively [28], and the
probability density function of the lightning crest current is
distributed as [3, 9]:
I
I
PL ( I )  0.0475  exp( )  0.0010  exp( ) (14)
20
50
The observation results of direct lightning strikes and the
calculation results of the three estimation methods are shown
in Figure 10 [36].
The data in Figure 10 show that, compared with Eriksson’s
improved EGM, the calculation results of LPM and the
proposed model are closer to observation data. According to
Eriksson’s improved EGM, the largest lightning shielding
failure current amplitude is 9.5 kA, that is, that lightning
current higher than 10 kA will not cause lightning shielding
failure. This result is obviously different from the observation
data. Moreover, Eriksson’s improved EGM believes that with
the increase of lightning current, the lightning shielding failure
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 18, No. 5; October 2011
arc of transmission lines will decrease gradually, as shown in
Figure 11. Therefore, with the increase of lightning current,
the corresponding SFR will continue to decrease to zero. LPM
and the proposed model show that as the lightning current I
increases, SFR shows a tendency of growing at first and then
decreasing. As shown in Figure 9, the calculation of the
proposed model reveals that for the 5 kA lightning current,
though the peak value of lightning shielding failure probability
PS is larger than that of the 25 kA lightning current, SFW
(I=5kA, PS>10%) is smaller than the SFW (I=25kA, PS>10%).
Through the calculations of (10) and (13), it is revealed that
from 5 kA to 25 kA, SFWeq rises as the lightning current I
increases, so the SFR also increases. Correspondingly, SFWeq
(I=5kA) and SFR (I=5kA) are smaller than SFWeq (I=25kA)
and SFR (I=25kA) respectively, and the increasing tendency
from 5 kA to 25 kA can be calculated by (13). For the 45 kA
lightning current, though the SFW (I=45kA, PS>10%) is larger
and the SFW (I=25kA, PS>10%), its peak value of PS is less
than 20%, and the corresponding SFWeq(I=45kA) is smaller
than that of 25 kA lightning current. Therefore, based on the
calculation of (10) and (13), the SFR under lightning current
higher than 25 kA shows a decreasing tendency. This result is
closer to the observation data.
0.14
the improved EGM
LPM
proposed model
observation
0.12
0.1
0.08
0.06
0.04
0.02
0
among which there were 79 negative lightning strikes. The
number of lightning strikes on the upper, middle and lower
phase conductors are 34, 27 and 18 strokes respectively. The
proposed model estimates the ratio of strokes reaching each
phase conductor, and also compares the calculation results of
the conventional EGM, the improved EGM by Sakae
Taniguchi [9] and the 3-D LPM [37]. The calculation and
observations are shown in Figure 12. Conventional EGM and
the Sakae Taniguchi’s improved EGM show that the lightning
shielding failure arc length of the lower phase conductor is the
largest, so the percentage of strikes on the lower phase
conductor is higher than that of the upper and middle phase
conductors. However, observation data show that the SFN of
upper phase conductor and middle phase conductor are close
to each other, but the SFN of the lower phase conductor is
obviously lower than the other two conductors. Therefore,
Conventional EGM and improved EGM by Sakae Taniguchi
still give different results from the observations [9]. The
proposed model believes that when the downward lightning
progresses, the phase conductor in the higher position is easier
to generate an upward leader. Meanwhile, the shielding angle
also has obvious influence on the inception of upward leaders.
For this transmission line in Japan, the shielding angles of
upper, middle and lower phase conductor are respectively 6.24, -3.37 and -2.05 [11, 12]. Compared with the other two
phase conductors, the upward leader inception of the upper
phase conductor is restrained because of smaller shielding
angle. These two factors determine the probability distribution
of SFN of every conductor. Therefore, the rates of the upper
phase conductor and middle phase conductor’s SFN are
obviously higher than that of the lower phase conductors. The
calculation result of the proposed model is close to 3D-LPM;
and compared with EGM, it is closer to the observation data.
Figure 10. Distribution of lightning shielding failure rate under different
lightning current amplitude.
Stroke percent (%)
B3
B2
Rgw2
B1
Rgw3
Ground wire
Rgw1
Rl1
C2 Rl3
Rl2
Phase conductor
O
Observation
3-D LPM
The proposed model
Sakae Taniguchi’s improved EGM
EGM
50
45
40
35
30
25
20
15
10
5
0
Upper Phase
C1
1719
Middle Phase
Lower Phase
Figure 12. Percentage of lightning strokes reaching each phase conductor.
D2
D1
Rg1
Rg2
Rg3
Ground
Figure 11. Calculation representation of the lightning shielding failure arc in
EGM.
As for the UHV transmission line, Tokyo Electric Power
Company [9, 11] has recorded the SFN of 1000kV UHV AC
double-circuit transmission line in Japan. From 1998 to 2004,
there have been 81 lightning shielding failure accidents,
6 LIGHTNING SHIELDING FAILURE
ZONE
The lightning shielding failure zone is used by EGM to
determine whether a direct lightning strike will strike to a
phase conductor. According to EGM, as shown in Figure 11,
when direct lightning strikes happen around transmission lines,
there exists a lightning shielding failure zone around the
transmission lines so that it is possible for lightning to strike
the phase conductor. It is possible to determine the lightning
shielding failure zone by calculating the shielding failure arcs,
but the results ignore the stochastic feature of lightning
1720
J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
discharges. In fact, the dividing border between the lightning
shielding failure zone and the effective protection zone is not
so obvious. Moreover, when downward lightning enters into
the lightning shielding failure zone, due to the stochastic
feature of lightning discharges, the phenomenon of lightning
strikes on line conductors occurs in the form of a statistical
probability. So, based on the proposed model, the lightning
shielding failure zone is described in the form of lightning
shielding failure probability in this paper. Through research on
the distribution of lightning shielding failure probability
around the transmission line space, the probability for
different direct lightning strikes to cause lightning shielding
failure can be better analyzed, which provides theoretical
support to the analysis of lightning shielding failures and the
lightning shielding design of transmission lines.
In LPM, it is believed that before the leader progresses to
the height of Hi, the lightning leader tip height at the upward
leader initial moment, the lightning’s progression is not
influenced by the grounded objects. When the leader reaches
Hi, the lightning leader will be attracted by the upward leader
generated from phase conductors and ground wires and its
progressing direction will be changed. For the condition when
the line parameters are determinate, Hi is determined by the
lightning current amplitude I and the lateral distance D, which
can be described as
H i  h( D, I )
(15)
500
C
400
300
40 kA
B
200
A
100
0
50 kA
30 kA
10 kA
0
50
20 kA
100
D (m)
150
200
Figure 14. Calculated distribution of Hi(D, I).
Based on the proposed model and combined with equation
(16), the lightning current I at each point of the transmission
line space is calculated, so that the lightning shielding failure
probability at each point in the transmission line space is
obtained. The data of the lightning shielding failure
probability is processed by mathematical fitting, and the
contour line of the shielding failure probability is drawn. The
calculation process is shown in Figure 15, and the shielding
failure probability contour is shown in Figure 16.
By altering the position of Hi and I, (15) can be changed
into:
I  y ( D, H i )
(16)
In order to attain the functional relation of (16), in this
paper the charge simulation method is used to simulate the
lightning leader channel, and Peek criterion is used as the
initial criterion of upward leaders. The calculation process is
shown in Figure 13. The calculation objective is a ±800 kV dc
transmission line, with the shielding angle at zero degrees, the
conductor height at 44 m, the ground wires height at 55 m, and
the ground slope at zero degrees. Through the flow calculation
of Figure 13, the functional relation in equation (16) is shown
in Figure 14.
Figure 15. Calculation process of the shielding failure probability contour.
Figure 13. Calculation process of the lightning strike current amplitude I.
As Figure 16 shows, the curves No.1 to No.7 are the contour
lines of the transmission line’s lightning shielding failure
probability that ranges from 10-70%. The largest probability of
lightning shielding failure occurs in the area near observation
point A. it can be found from Figure 14 that the lightning
current amplitude I at point A is 20 kA. On the phase
conductors and ground wires of the transmission line, the
induced voltage caused by the lightning channel is relatively
small. The high operating voltage on the conductors makes the
conductors have a stronger ability to trigger lightning than the
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 18, No. 5; October 2011
ground wires. Thus, when lightning strokes with current
amplitude around 20 kA happen around the transmission line,
lightning shielding failure will occur most easily.
1721
In the above analysis, based on the proposed model, the
lightning shielding failure zone is described by the shielding
failure probability contour. The result shows statistical
characteristics and reflects the deterministic and stochastic
features of the lightning discharge. It is also in accord with the
data distribution rules of the laboratory lightning simulation
experiment [38, 39]. The proposed model can reflect the natural
lightning progression rules better than LPM. Compared with the
lightning shielding failure zone described in EGM, the shielding
failure probability contour can better reflect the lightning
shielding failure performance of transmission lines. Therefore,
the transmission line estimation method based on the proposed
model is more accurate and well-founded.
7 APPLICATION OF THE ESTIMATION
METHOD ON THE TRANSMISSION LINES
Figure 16. Contour lines of the calculated lightning shielding failure
probability in space.
Under the same lateral distance D, for example in Figure 16,
when the lateral distance D is set at 50 m, as Hi grows from zero
m, the lightning shielding failure probability of the transmission
line tends to increase first, then to decrease. When Hi is less than
50 m, Ps is less than 30%, and it can be found from Figure 16
that the lightning current amplitude I in this zone is smaller than
10 kA; the lightning in this area mainly strikes the earth. When
Hi grows gradually, for example, when Hi grows to 150 m, it is
shown from Figure 14 that the lightning current amplitude I will
increase to 20 kA, then the conductors’ and the ground wires’
ability to trigger lightning also increases. Meanwhile, because
of its high operating voltage, the conductor’s upward leader
predominates over the ground wire’s upward leader in space.
The lightning shielding failure probability of the transmission
line tends to increase. When Hi continues to grow, for example,
when Hi grows to 300 m, the lightning current amplitude I in the
area increases to 30-40 kA. The operating voltage on the
conductor no longer exerts great influence on its upward initial
leader, which makes the ground wire’s upward leader take
predominance in the competition with the conductor’s upward
leader. The lightning shielding failure probability of the
transmission line tends to decrease.
When the observation point (D,Hi) changes from near to far,
for example in Figure 16, when observation point (D,Hi)
moves from point A to point B then to point C, the lightning
shielding failure probability keeps on decreasing in the
process. Figure 14 shows that the current at point A is about
20 kA, which is smaller than the current of 30 kA in point B
and that of 50 kA in point C. So when lightning strikes happen
around the transmission line, lightning shielding failures will
happen more easily when the lightning current amplitude has
smaller values. As the observation point (D,Hi) keeps on
moving further away from the transmission line, the lightning
current I will keep on increasing, and the lightning shielding
failure probability continues to decrease until zero. This
conclusion is in accordance with the views of EGM and LPM
that there exists a largest lightning shielding failure current.
The existing data show that the increase of the transmission
line’s height will lead to the increase of the line’s ability to
trigger lightning, and the lightning shielding performance of
transmission lines will be influenced. In this paper, based on
the proposed model and its estimation method to evaluate the
SFR of transmission lines, the lightning shielding performance
of transmission lines with different nominal heights are
analyzed. The structural parameter of a ±800 kV UHVDC
transmission line is used as the analyzing parameter, in which
the ground wire height is higher than the conductor height by
11 m and the shielding angle is zero degrees. The structural
size of the tower head being constant, and when the tower
height is changed, the calculation results are shown in Table 2.
Table 2. The shielding failure rate (strokes/100km-year) of transmission lines
with different nominal heights.
nominal heights (m)
50
55
60
65
70
SFR
0.107
0.119
0.133
0.152
0.174
From Table 2 it is shown that, as the nominal height is
increased, SFR increases gradually. When the nominal height
grows to 70 m, the SFR increases by almost 65% over the SFR
when the nominal height is 50 m. This is because the increase of
nominal height makes the ground wire’s and the conductor’s
upward leaders easier to be initiated, and to have more ability to
trigger lightning. Thus the lightning shielding failure area will be
enlarged, and the SFR of the transmission line will increase
correspondingly. Thus, in order to lower the SFR of transmission
lines, the nominal height should be lowered as much as possible.
For the high tower and long span transmission line sections,
enhanced lightning shielding measures should be taken to ensure
the lines meet the lightning shielding requirements.
The setting of the shielding angle of transmission lines will
influence the field distribution on the surface of ground wires and
conductors, then influence the initiation of the ground wire’s and
the conductor’s upward leaders, and finally to influence the
lightning shielding failure probability of transmission lines. In this
paper, based on the proposed model, the lightning shielding
performance of transmission lines with different shielding angles
was researched. When the conductor height is 44 m, the ground
wire height is 55 m, the lightning shielding failure probability
with different shielding angles is calculated and shown in Table 3.
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J. Li et al.: A New Estimation Model of the Lightning Shielding Performance of Transmission Lines Using a Fractal Approach
The data in Table 3 shows that decreasing the shielding angle of
the transmission line will lower its SFR. When the lightning
leader progresses downward, the decrease of shielding angle
suppresses the conductor’s upward leader, then the ground wire’s
upward leader will predominate over the conductor’s upward
leader. The protecting function of the ground wire is enhanced, so
the SFR of the line will decrease correspondingly. Therefore, to
reduce the shielding angle is the main method to improve the
lightning shielding performance of UHVDC transmission lines.
Table 3. The shielding failure rate (strokes/100km-year) of transmission lines
with different shielding angles.
shielding angles (o)
-5
0
5
10
SFR
0.086
0.107
0.122
line and the choice of 3-D lightning leader path, would
influence the lightning shielding performance of transmission
line, so as to improve the current estimation method.
ACKNOWLEDGEMENT
This work was supported by the National Basic Research
Program of China (973 Program) (2009CB724504), the
National “111” Project of China (B08036) and the National
Natural Science Foundation of China (NSFC) (50707036).
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[1]
0.141
[2]
In the UHVDC transmission project design in YunnanGuangdong province of China, the China Southern Power
Grid Corporation uses the calculation estimation of the
proposed model. In mountainous regions, the shielding angle
of ±800 kV dc transmission lines is set at -9.6 degree; in plain
regions, the shielding angle is set at -1.8 degree.
8 CONCLUSIONS
In order to make the model closer to the discharge features of
natural lightning, the fractal approach is applied to the LMP in
this paper. By comparing the fractal dimension of natural
lightning, the fractal parameters are adjusted, so as to control the
deterministic factor and stochastic factor of lightning progression
within a reasonable range. Based on the proposed model, the
lightning shielding failure probability distribution of the
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1) The lightning stroke simulation figure of the proposed
model describes the effects of tortuosity and branching of
the lightning channel, and reflects the stochastic and
deterministic features shown in the progression process
of lightning.
2) When lightning occurs within the shielding failure width,
lightning shielding failure is not inevitable, but appears
with a probability. The lightning shielding failure
probability PS is related to the lightning current amplitude
I and the lateral distance D.
3) The lightning shielding failures are most likely when
lightning strikes with current amplitude under 20 kA and
occur in a position that is close to the transmission line,
and the peak value of the lightning shielding failure
probability under small lightning current amplitude is
larger than that of large lightning current amplitude.
The estimation method on the lightning shielding
performance of transmission lines in the proposed model has
been used on the ±800 UHVDC transmission project design in
Yunnan- Guangdong province of China, by China Southern
Power Grid Corporation. Besides, the authors intend to develop
a 3-D fractal LPM in the future, study the 3-D lightning
shielding failure probability space of transmission lines, and
analyze how the factors such as line sag, the corridor terrain of
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Jianbiao Li was born in Guangxi province, China, on
5 October 1984. He received the B.S degree in
electrical engineering in 2007 from Chongqing
University, China. He is now a Ph.D. candidate in the
State Key Laboratory of Power Transmission
Equipment & System Security and New Technology,
Chongqing University. His major research interest is
lightning protection for transmission lines.
Qing Yang received the B.S and Ph.D. degrees in
electrical engineering, respectively, in 2002 from
North China Electrical Power University and in 2006
from Chongqing University, China. He is now an
Associate Professor in the State Key Laboratory of
Power Transmission Equipment & System Security
and New Technology, Chongqing University. His
research interests include outdoor insulation in
complex ambient conditions and electric-field
calculations. He is the author and coauthor of more
than 30 journal and international conference papers.
1723
Wenxia Sima was born in Henan province, China, on
13 July 1965. She graduated from Chongqing
University in 1988 and obtained the Ph.D. degree in
1994 from Chongqing University. Her employment
experience includes the college of Electrical
Engineering of Chonging University. Her fields of
interest include high voltage outdoor insulation and
overvoltage protection.
Caixin Sun received the B.S. degree in electrical
engineering in 1969 from Chongqing University.
Currently, he is Professor of Chongqing University,
leading the research group on high voltage
engineering. He is a member of the Chinese Academy
of Engineering and Director of Electrical Power
Engineering Committee of the National Science
Foundation of China. His research interest is in high
voltage engineering, especially online detection of
insulation condition and insulation fault diagnosis for
high voltage equipment, discharge mechanism of outdoor insulation in
complicated environments, and high voltage techniques applied in
biomedicine. He is an author and coauthor of over 200 publications.
Tao Yuan received the B.S degree in 1999 from
Sichuan Polytechnic Institute, China, and received the
Master’s and Ph.D. degrees, respectively, in 2001 and
2010 from Chongqing University, China. He is now a
lecturer in Chongqing University, and his research
interests include overvoltage protection and grounding
technology in power systems.
Markus Zahn (S'68-M'71-SM'78-F'93) received the
B.S.E.E., M.S.E.E., Electrical Engineer, and Sc.D.
degrees from the Department of Electrical Engineering at
the Massachusetts Institute of Technology (MIT),
Cambridge, MA, in 1968, 1968, 1969, and 1970
respectively. He then joined the Department of Electrical
Engineering at the University of Florida, Gainesville
until 1980 when he returned to MIT where he is now
Professor of Electrical Engineering working in the
Research Laboratory of Electronics, Laboratory for
Electromagnetic and Electronic Systems and the High Voltage Research
Laboratory. He is also the Director of the MIT Course VI-A EECS Internship
Program, a cooperative work/study program with industry. He is the author of
Electromagnetic Field Theory: A Problem Solving Approach (now out of print) but
is working on a new reference book with a complete collection of solved
electromagnetism problems. He has also co-developed a set of educational
videotapes on demonstrations of electromagnetic fields and energy for the enriched
teaching of electromagnetism. He has received numerous excellence in teaching
awards at the University of Florida and at MIT. His primary research areas are
ferrohydrodynamics and electrohydrodynamics for microfluidic and biomedical
applications; nanoparticle technology for improved high voltage performance of
electric power apparatus; modeling of electrical streamer initiation and propagation
leading to electrical breakdown; Kerr electrooptic field and space charge mapping
measurements in high voltage stressed materials, and for the development of model
based interdigital dielectrometry and magnetometry sensors for measurement of
dielectric permittivity, electrical conductivity, and magnetic permeability with
applications to non-destructive testing and evaluation measurements and for
identification of metal and low-metal content dielectric landmines and unexploded
ordnance. Professor Zahn is co-inventor on 19 patents. He has contributed to about
10 book and encyclopedia chapters, about 115 journal publications, and about 175
conference papers. He was a Distinguished Visiting Fellow of the Royal Academy
of Engineering at the University of Manchester, England; was the 1998 J.B.
Whitehead Memorial Lecturer, and was the first James R. Melcher Memorial
Lecturer in 2003; is an Associate Editor of the IEEE Transactions on Dielectrics
and Electrical Insulation; is on the International Scientific Committee on Magnetic
Fluids; and has received a Certificate of Achievement for completion of the
“Deminers Orientation Course” at the Night Vision and Electronic Sensors
Directorate Countermine Division at Ft. Belvoir, VA, USA.