![distribution line may experience, as a function of its insulation. The attention is
mostly focused on the number and intensity of lightning-induced voltages, either
because distribution lines are surrounded by elevated objects or because direct
strikes protection is uneconomical. The issue has been the object of several studies
in the past and it is still of interest due to the stringent power quality requirements
of modern distribution networks, especially nowadays, given the increasing amount
of distributed generation that is connected to them [1–5].
In [1,2], the frequency of overvoltages exceeding a given insulation level is
evaluated by means of analytical methods for the case of an infinite long line over a
perfect conducting plane. The amplitude of the lightning current at the channel base
is considered as a random variable considering its probability distribution, while
the front time of the lightning current and the return stroke velocity are assumed
fixed.
In [3], a statistical method is employed. Both the probability distribution of
amplitude and that of front time of the lightning current are considered, along with
the correlation coefficient between the two above-mentioned parameters. The
positive value of that coefficient indicates that the higher the current amplitude, the
longer the front time of the impulse. The striking distance of the indirect stroke
from the line (lightning strokes occurring within a certain critical distance from the
line will directly strike the line) is evaluated as a function of the return stroke peak
current (while in [1], it is considered as independent of the current). The return
stroke velocity is fixed and the ground is assumed as a perfectly conductive plane.
In [4], the Monte Carlo (MC) method has been employed to solve the problem.
The induced voltages are calculated at the termination of a 2 km line matched at
both ends. The MC simulation involves 10,000 events taking place over a surface
covering the line and 1 km away from it. The same striking distance equation from
the line as in [3] is adopted. The correlation between peak value and front time of
lightning current distributions is disregarded.
Lightning originated overvoltages in overhead lines are due to both direct
strikes and the coupling between the conductors and the electromagnetic pulse
generated by nearby strikes to ground (see, e.g., [6,7]). For power distribution
overhead lines, characterized by lower insulation and height with respect to trans-
mission lines, most of the lightning related flashovers are caused by strikes to the
ground or structures located in proximity of the line [8]. As a consequence, the
influence of these events on the lightning performance of the line needs to be
appropriately assessed to grant the adequate protection. It is also worth adding that
while for distribution lines all direct strikes are expected to cause flashovers (unless
a large amount of surge arresters and shield wires are installed, ideally one per
phase and per pole), only a fraction of nearby strikes to ground are expected to
induce voltages larger than the line withstand insulation level. Different is the case
for those lightning strokes that, although not hitting directly the line conductor, hit
those elevated objects that are located close to the line and therefore can be able to
cause flashovers.
As described in [5], the standard MC approach for the lightning performance
assessment of power distribution lines consists in generating a large number of
2 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-39-320.jpg)
![events, each characterized by different values of lightning current waveshape
parameters and different coordinates of the perspective stroke location (i.e., the
stroke location in absence of the line). The values of the lightning current para-
meters are generated in agreement with the relevant probability distributions
available in the literature or provided by lightning location systems (see, e.g.,
[9,10]). The coordinates of the stroke location are assumed uniformly distributed
within a region around the line wide enough to include all the events that may cause
a flashover.
In several papers (e.g., [11,12]), the calculation of the lightning-induced
overvoltages is performed by using simplified formulas in order to reduce the
computational effort. However, as the accurate appraisal of the induced voltages
can be achieved only by a time-domain electromagnetic transient simulation (in
this chapter, performed by using the LIOV–EMTP-RV code described in [6,13,14]
and validated by the comparison of the results with several experimental results
[15,16]), the lightning performance of distribution networks by means of standard
MC simulations involves a significant amount of computational resources.
Some recent papers have dealt with the lightning performance assessment by
means of electromagnetic transients simulations (e.g., [17–19]), which are quite
time consuming. Nearby strikes to ground are often referred as indirect strokes in
the relevant literature; this term is also used to indicate other indirect lightning
events that are not considered in this chapter, such as side flashes and the line
interaction with a ground current (e.g., [20,21]). In [22], a method to reduce the
computational effort is presented. It is based on the application of the so-called
Mean Square Pure Error (MSPE) algorithm to determine the optimal number of
MC extractions and on a 3D interpolation that bypasses the necessity of the time-
domain simulation of each MC event.
In order to reduce the number of LIOV–EMTP-RV simulations, in [23] a
heuristic technique has been proposed for the case of an unprotected network,
which has been adapted in [24] to the case of distribution networks with surge
arresters. The heuristic technique is conceived to avoid the time-domain compu-
tation of events expected to be less harmful than the previously calculated ones.
This requires to fix a priori conditions so to discard the upcoming events which, in
general, have characteristics (e.g., amplitude) that depend on the particular network
configuration, especially in presence of nonlinearities [25].
An improved estimation accuracy can be achieved by two approaches: by
increasing the number of replications or reducing the variance of the estimator. A
typical variance reduction technique adopted in MC methods is the stratified
sampling (e.g., [26,27]) and the application of this technique for the lightning
performance assessment of distribution lines has been presented in [28].
The structure of the chapter is the following. Section 1.2 is devoted to the
description of the MC approach with particular reference to the random generation
of the values of the lightning parameters from a multivariate probability distribu-
tion. Section 1.3 describes the identification of the functions that describes the
lightning current waveforms. Section 1.4 describes the stratified sampling techni-
que. Section 1.5 illustrates the application of the MC method to the case of a
Application of the Monte Carlo method to lightning protection 3](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-40-320.jpg)
![medium voltage (MV) overheard line in open terrain. As mentioned in the con-
clusions of the chapter (Section 1.6), the application of the MC method for the
lightning performance assessment of lines and networks with realistic configuration
will be presented in Chapter 4 of this volume.
1.2 Description of the MC-based procedure
As described in detail in what follows, the proposed procedure is based on the
application of the MC method and on the calculation of the induced voltages by
using the LIOV code. From now on, this procedure will be called LIOV-MC.
Such a procedure is defined by the following steps.
1. A large number of lightning events ntot is randomly generated. Each event is
characterized by the parameters that describe the current waveform at the
channel base and the coordinates of the stroke location. Only negative down-
ward first strokes are taken into account; the effects of the presence of positive
flashes and subsequent strokes in negative flashes on the line lightning per-
formance are assumed negligible.*
The stroke locations are assumed to be
uniformly distributed within striking area A, having a size large enough to
contain the entire line and all the lightning events that could cause voltages
larger than the minimum voltage value of interest for the analysis. Typical
lightning channel base current waveforms are defined by the following para-
meters: peak Ip, front time tf, maximum front steepness Sm and wavetail time to
half value th. The relevant probability distributions are provided in [29,30].†
The correlation between these parameters has been recently reviewed by
Rakov et al. in [9]. In particular, in direct current measurements relatively
strong correlation is observed between the current rate-of-rise characteristics
and current peak. The MC random generation of lightning current parameters
is described hereafter.
2. From the total set of events, those relevant to indirect lightning are selected by
adopting a lightning incidence model for the line. In [5], it is analyzed the
influence of the adoption of different incidence models. For the calculations of
this chapter, we have adopted the electro-geometric model suggested in [8].
*
Further investigations are needed to include the effects of subsequent return strokes, solving several
open issues, such as (i) relationship between the subsequent stroke’s path and that of the first stroke,
(ii) correlation between first and subsequent stroke current parameters, and (iii) number of subsequent
strokes.
†
Note that the simpler Anderson equation for the peak current distribution adopted by IEEE Std. [8],
follows the trend of the Cigré two-line distributions comparatively well [29]. These statistical distribu-
tions have been inferred mostly from measurements obtained by using instrumented towers. The mea-
surements at the towers are affected by reflections [56]. Moreover, the current amplitude distributions of
the lightning events collected by the towers are biased toward values higher than those of the distribu-
tions of the flashes to ground, as analyzed in, e.g., [31,32] and references therein. These aspects will be
deliberately disregarded in this chapter.
4 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-41-320.jpg)
![3. For each lightning event, the maximum induced voltage value on the line is
calculated – as earlier mentioned – by means of the using the LIOV–EMTP-
RV code [6,13] that is described in Chapter 12 of this volume.
4. For the entire line, the expected annual numbers of events Fp that causes
overvoltages with amplitude larger than a given value V is:
Fp ¼
n
ntot
ANg (1.1)
where n ¼ ni þ nd, being ni and nd the number of indirect events and direct
events, respectively, that generate overvoltages larger than V and Ng is the
annual ground flash density. In this study, we assume Ng ¼ 1 flash/km2
/yr.
The estimation of the mean time between failures (MTBF) expected at
specific poles of the line, generally of interest for the protection of the trans-
formers connected to those poles, is calculated as the inverse of the Fp value
given by (1.1) with ni and nd evaluated by comparing the overvoltage at the
pole with the withstand voltage of the connected transformer.
The application of the MC method requires the knowledge of the multivariate
distribution of the lightning current parameters. We reasonably assume that every
parameter follows the log-normal probability distribution, as generally done in the
literature on the subject.
Let x1; . . .; xn be n jointly Gaussian random variable. In our case, they are the
four natural logarithms of peak amplitude Ip, equivalent front times tf ¼ t30=0:6 (t30
is the interval from 30% to 90% amplitude intercepts on the wavefront), maximum
front steepness Sm, and wavetail time to half value th. The multivariate normal
distribution is said to be non-degenerate when the symmetric covariance matrix K
is positive definite. In this case, the probability density function is
f x1; . . .; xn
ð Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2p
ð Þn
jKj
p exp
1
2
x m
ð ÞT
K1
x m
ð Þ
(1.2)
where x is a real n-dimensional column vector, m is the corresponding mean vector,
and jKj is the determinant of K.
The ijth off-diagonal element of K is given by correlation coefficient rij
between xi and xj multiplied by the product of their two corresponding standard
deviations (i.e., sxi and sxj), while the iith diagonal element is equal to variance s2
xi
of random variable xi.
Let be Q ¼ K1
, the conditional variance of xn is
ðs
xn
Þ2
¼ Var xnjx1; . . .; xn1
ð Þ ¼
1
Qnn
(1.3)
and the conditional mean of xn is
m
xn
¼ E xnjx1; . . .; xn1
ð Þ ¼ mn
1
Qnn
X
n1
j¼1
Qnjðxj mjÞ (1.4)
where Qnj is the njth element of matrix Q.
Application of the Monte Carlo method to lightning protection 5](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-42-320.jpg)
![The MC random generation of a quadruple of lightning current parameters
values is obtained by applying the following steps, where Zk; Zkþ1; Zkþ2; Zkþ3 are
four standard normal variates.
Step (1) for the calculation of an Ip value:
(1.1) Ip ¼ expðmln Ip
þ sln Ip
ZkÞ;
Step (2) for the calculation of a tf value:
(2.1) s
ln tf
, s
ln Sm
, s
ln th
are calculated by using (1.3);
(2.2) m
ln tf
is calculated by using (1.4);
(2.3) tf ¼ expðm
ln tf
þ s
ln tf
Zkþ1Þ;
Step (3) for the calculation of a Sm value:
(3.1) m
ln Sm
is calculated by using (1.4);
(3.2) Sm ¼ expðm
ln Sm
þ s
ln Sm
Zkþ2Þ;
Step (4) for the calculation of a th value:
(4.1) m
ln th
is calculated by using (1.4);
(4.2) th ¼ expðm
ln th
þ s
ln th
Zkþ3Þ.
A complete set of the data required for the MC generation procedure is pro-
vided by Berger and Garbagnati in [30] and is reported in Tables 1.1 and 1.2. For
each parameter y, Table 1.1 provides median value
yðmln y ¼ ln
yÞ and sln y.
Table 1.2 provides correlation coefficients rln y1ln y2
between parameters y1 and y2.
Since these statistical distributions have been inferred mostly from measure-
ments obtained by using instrumented towers, the current amplitude distributions of
the lightning events collected by the towers are biased toward values higher than
those of the distributions of the flashes to ground, as analyzed in, for example,
[31,32] and references therein. This aspect will be deliberately disregarded in this
analysis, as earlier mentioned at note.†
Table 1.1 Statistical parameters of the log-normal distributions
for negative downward first strokes [30]
Parameter Median value Standard deviation of the
parameter logarithm (base 10)
Ip 30 kA 0.26
Tcr 5.5 ms 0.31
Sm 12 kA/ms 0.26
th 75 ms 0.26
Table 1.2 Correlation coefficients between parameters [30]
Parameter Ip Tcr Sm th
Tcr 0.37 1
Sm 0.36 –0.21 1
th 0.56 0.33 0.1 1
6 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-43-320.jpg)
![The parameters of the distribution relevant to the front duration Tcr is given instead
of the ones relevant to tf. Since [29] provides both the parameters of Tcr (the same as
Table 1.1) and of tf (
tf ¼ 3:8 ms, sln tf
¼ 0:55) obtained from almost the same set of
experimental measurements used in [30], the implemented MC procedure generates
the values of Tcr according to step (2). The values relevant to tf are obtained by mul-
tiplying each value of Tcr by the ratio between
tf and
Tcr provided by [29].
If a simple current function, that is, a step waveform, a linearly rising current,
linearly rising with flat top (trapezoidal) or with drooping tail, is used for the cal-
culation of the maximum overvoltages along the line, the values generated for Ip, tf,
and th are directly used. However, for more complex current functions, a specific
identification procedure is needed as described in Section 1.3.
1.3 Identification of the lightning current functions
As known, the waveform of the return stroke current at the channel base as well at
its peak value have a significant influence on the lightning originated overvoltages
along the lines. Different functions have been proposed in order to represent the
typical lightning current waveform. The most commonly used are: the one adopted
by CIGRE WG [29] and the one proposed by Heidler in [33]. Other functions that
can be found in the literature are the classical double exponential [34], others
derived from it (e.g., [35]), the combination of multiple Heidler functions (e.g.,
[36]) and more recent ones (e.g., [37]). Functions for multi-peaked waveforms have
been also proposed in [38,39].
For a given quadruple of values for Ip, tf, Sm, and th, we describe here the
procedures to identify the parameters of the Cigré function and of the Heidler
function, as presented in [40] that analyzes the effects of different current wave-
forms on the lightning performance of distribution lines for both direct and indirect
strokes.
Cigré function
The current waveform is [29]:
i t
ð Þ ¼ At þ Btn
; t tn
i t
ð Þ ¼ I1e ttn
ð Þ=t1
I2e ttn
ð Þ=t2
; t tn
(1.5)
where
SN ¼ Smtf =Ip
n ¼ 1 þ 2 SN 1
ð Þ 2 þ 1=SN
ð Þ
tn ¼ 0:6tf 3SN
2
= 1 þ SN
2
A ¼
1
n 1
0:9
Ip
tn
n Sm B ¼
1
tn
n n 1
ð Þ
Smtn 0:9Ip
t1 ¼ th tn
ð Þ=ln 2 t2 ¼ 0:1Ip=Sm
I1 ¼
t1t2
t1 t2
Sm þ 0:9
Ip
t2
I2 ¼
t1t2
t1 t2
Sm þ 0:9
Ip
t1
(1.6)
Application of the Monte Carlo method to lightning protection 7](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-44-320.jpg)
![This formulation presents some numerical issues if n 1 or n 55. In case a
MC event presents a value of n out of these bounds, the value of Sm is adjusted as
Sm ¼ 1:01Ip=tf if n 1
Sm ¼ 12Ip=tf if n 55
(1.7)
As this procedure can lead to small errors on the resulting current peak, the current
is normalized to the desired peak value.
Heidler function
The Heidler function is [33]:
i t
ð Þ ¼
I0
h
t=t1
ð ÞN
1 þ t=t1
ð ÞN
exp t=t2
ð Þ
h ¼ exp
t1
t2
t2N
t1
1=N
# (1.8)
It is completely defined by four parameters, that is, I0; t1; t2 and N, which cannot
be fully obtained from the values Ip
, tf
, Sm
, and th
by analytical equations. In [41], an
iterative graphical method is presented to identify the parameters as a function of a
waveform, while the useof a genetic algorithm (GA) is adopted in, for example, [42,43].
The following procedure based on MATLAB
GA function has been devel-
oped. The objective of the algorithm determines a set of values I0; t1; t2 and N such
to minimize the following fitness function
f ¼ c1
Ipc Ip
Ip
þ c2
tfc tf
tf
þ c3
thc th
th
(1.9)
where Ipc, tfc, and thc are the peak value, the equivalent front time, and the time to
half value of the current calculated at every iteration of the algorithm, respectively.
Parameters c1, c2, and c3 are the weights ascribed to the relative errors of the three
parameters Ip
, tf
, and th
, respectively. The algorithm is stopped if the relative
errors on the three parameters satisfy all the three following conditions
Ipc Ip
Ip
0:5%;
tfc tf
tf
0:5%;
thc th
th
1% (1.10)
The possible values of N are limited to the integer values 2, 3, or 4. At first, the
values of c1, c2, and c3 are equal to each other. The initial population size and the
maximum number of generations are set to 50 and 100, respectively, and they are
subsequently enlarged in case any of the conditions of (1.10) is not satisfied. If after
some tens of attempts, conditions (1.10) are still not satisfied, the time to half value
is penalized by means of a reduction of c3 with respect to c1 and c2. At the end of
this procedure, only for 10 out of 20,000 events the constraint on th is not satisfied,
while the constraints relevant to peak and equivalent front time are always fulfilled.
Table 1.3 compares the expected median value of the parameters with those
obtained by 20,000 current waveforms calculated by using Cigré function (1.5).
8 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-45-320.jpg)
![The small deviations in the Cigré model are due to the corrections previously
mentioned. Table 1.3 also compares the expected median value of the parameters
with those obtained by 20,000 current waveforms calculated by using Heidler
function (1.8) with the parameter given by the GA. The mean errors resulting on
20,000 Heidler waveforms are: 0.004% for Ip, 0.17% for tf, and 0.21% for th.
Although the Sm is not taken directly into account by the GA, also the relevant median
value is in close agreement with the expected one.
1.4 Stratified sampling technique
As mentioned in the Introduction, the calculation of the lightning performance of dis-
tribution lines considering also indirect strokes is a typical rare event calculation.
Indeed, the endpoints of the 95% confidence interval of the estimateb
p ¼ n=ntot of (2.1)
are b
p Cp b
p where Cp is the relative error that can be evaluated as [26]:
Cp ¼ 1:96
ffiffiffiffiffiffiffiffiffiffiffi
1 p
ntotp
s
(1.11)
under the assumption that the estimator b
p is a normal random variable with mean
value p and variance equal to p 1 p
ð Þ=ntot.
As area A needs to be quite wide and the density of events that cause a flash-
over decreases with the distance from the feeder, p is generally small. Therefore, in
a standard MC approach, ntot needs to be large so to achieve the desired level of
accuracy (i.e., a predefined value of Cp).
As shown in [28], several advantages with respect to the standard MC method
are obtained by the use of the stratified sampling technique:
● it allows a significant computational time reduction (up to more than 75% for
the cases analyzed in [28]) while maintaining the accuracy of the solution;
● it reduces the importance of the choice of the smallest area A that includes the
perspective stroke locations of all the events inducing voltages exceeding the
insulation withstand capability;
● it is directly applicable to the case of networks with complex configuration,
with surge arresters and with the adoption of detailed flashover models, while
heuristic rules need to be adapted for each specific case.
Table 1.3 Median values of the parameters obtained from (1.5) and from the GA
compared to the expected values given in [30]
Ip (kA) tf (ms) Sm (kA/ms) th (ms)
Expected 30.0 3.80 12.0 75.0
Cigré 30.0 3.89 13.1 74.7
Heidler 30.0 3.81 12.0 75.4
Application of the Monte Carlo method to lightning protection 9](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-46-320.jpg)
![According to this technique, area A is divided in subdomains. The number of
events generated in each subdomain is proportional to the variance of the local
estimator that is recursively updated. Therefore, the larger the distance to the lines
or the shorter the distance to surge arresters, the lower the number of events in the
specific subdomain.
Let us define X as a random variable such that Xk ¼ 1 if MC event k causes an
overvoltage greater than W, 0 if not. As in standard MC methods, the probability p
of observing an overvoltage greater than W is estimated by
b
p ¼
n
ntot
¼
1
ntot
X
ntot
k¼1
Xk (1.12)
and the perspective stroke locations of the events in the absence of the power lines
and other structures are assumed to be uniformly distributed over area A.
As already mentioned, for the application of stratified sampling, total area A is
divided in m subdomains and probability p is estimated by
b
J ¼
X
m
j¼1
aj
A
1
Nj
X
Nj
k¼1
Xjk
!
(1.13)
where aj is the area of subdomain j, Nj is the number of MC events allocated in
subdomain j such that
Pm
j¼1 Nj ¼ ntot, and Xjk is the k-th observation of X in domain j.
As shown in [27], the variance of estimator b
J is given by
Varðb
JÞ ¼
X
m
j¼1
aj
A
2 s2
j
Nj
(1.14)
where s2
j is conditional variance of X in domain j. Varðb
JÞ is always smaller or
equal to Var X
ð Þ=ntot.
Since the values of s2
j are not known a priori, they are initially estimated by a
certain number of pilot runs, that is, by the simulation of some MC events. For this
purpose, the same number Ns
j of starting events is generated in each sub-domain,
with Ns
j chosen large enough to estimate s2
j even with a very small probability p,
for example, in presence of surge arresters (SAs).
In order to obtain a near-random sample distribution for both the parameters of
the lightning current waveshape and the stroke location with a limited Ns
j , the
starting events in each subdomain j are generated by using the Latin hypercube
sampling (LHS) [44]. The use of LHS allows an improved accuracy on the initial
estimation of s2
j with respect to the usual random sampling.
The endpoints of the 95% confidence interval of the estimate b
J are b
J CJ b
J
where CJ is the relative error provided by
CJ ¼
1:96
b
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
m
j¼1
aj
A
2 s2
j
Nj
v
u
u
t (1.15)
10 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-47-320.jpg)
![Starting from the initial value of s2
j , the procedure adds new MC events until
CJ becomes lower than the desired estimation error.
Each new MC event is allocated in the area A according to a weighted uniform
distribution with different weights for each subdomain. The weight of each sub-
domain j is proportional to the corresponding conditional variance s2
j and the sum
of the weights of all the m subdomains is equal to 1. Indeed, the minimum variance
value given by (1.14) is obtained when the events are allocated proportionally to
the conditional variance s2
j of each subdomain, provided that all subdomains have
the same area [27].
If a too small Ns
j is adopted, a null value can be obtained in the first estimate of
s2
j , in this case the initial guess of the weight of the subdomain can be set according
to the mean values of the variances of the adjacent subdomains.
The values of s2
j are updated recursively after each MC run, so that their
estimation is progressively improved as the number of simulated events increases.
1.5 Application results for a MV overhead line
in open terrain
1.5.1 Influence of the return stroke current waveform
In order to illustrate the application of the MC method, we consider here a simple
three-phase overhead line, straight in shape. The conductors are assumed horizon-
tally placed at 9.3 m above ground, with diameter equal to 1 cm. The distances
between the lateral conductors and the central one are 1.5 and 0.7 m. In all the
calculations of this section, we have assumed the soil conductivity equal to 1 mS/m.
The striking area A is chosen as a 1-km band from the line. The number of
lightning events n is 20,000. Direct events nd are 1,208. Indirect events nd are
18,792.
Figure 1.1 shows the annual number of overvoltages of the line with length
equal to 2 km for the three different current waveforms adopted caused by indirect
events only. Such a result is here denoted as the perspective lightning performance
of the distribution line, as the calculations are run in absence of surge arresters and
by neglecting the flashovers along the line. The steady state voltage at the utility
frequency is not taken into account in the calculations. Figure 1.1 shows that the
choice of different current waveforms has a limited impact on the estimation of the
perspective lightning performance. It is worth mentioning that the lowest curve is
obtained by using the Heidler function.
Without surge arresters and flashovers, all direct events result in overvoltages
larger than the maximum value in abscissa (i.e., 0.24 events/year for the case of the
2-km long line), as expected, since all the Ip of the MC generated events are greater
than 2 kA. Indeed, the peak current values included in the Berger-Garbagnati dis-
tribution are all above 2 kA (i.e., 2 or 3 kA was the triggering level of the Italian
measuring stations, while the minimum peak current value included in Berger’s
distribution is 2 kA).
Application of the Monte Carlo method to lightning protection 11](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-48-320.jpg)
![Let us now consider the same three-conductor line but with sets of three surge
arresters installed at different distance intervals. The voltage–current characteristic
of the adopted 15-kV class surge arresters is the one reported in [45]. The con-
sidered rated voltage at the utility frequency is 13.8 kV.
The distance between subsequent poles is 50 m. The parameters of the dis-
ruptive effect criterion (adopted for the representation of the flashovers in the line
insulators) are reported in Table 1.4. These parameters have been obtained in [46]
from the results of laboratory tests performed on a 15 kV pin-type ceramic insulator.
We assume the presence of a transformer at the middle point of the line. The
withstand voltage of the transformer is assumed to be constant and equal to 110 kV,
as suggested in [47].‡
In [48], a procedure able to take into account the withstand
probability distribution of transformer insulation is described.
Figure 1.2 shows the top view of the line; the position of the transformer is
denoted by the cross in the middle of the line while the circles indicate the surge
arrester locations. Distance d defines the interval between consecutive surge
arresters. The length of the line is 2 km and the number of generated events in the
MC procedure is again 20,000. Tables 1.5 and 1.6 show the MTBF values relevant
to a transformer connected to the middle of the line for the two different insulators
described in Table 1.4, corresponding to a critical flashover voltage (CFO) of
100
10–1
100 150 200 250
Voltage (kV)
300 350
Trapz
Heidler
Cigré
Number
of
events
having
amplitude
larger
than
the
abscissa/yr
Figure 1.1 Comparison of the perspective indirect lightning performances
calculated by using three different current waveforms.
Length of the line equal to 2 km. Adapted from [40]
‡
In the computation of the MTBF values, the transformer failure is expected to occur if the voltage
amplitude exceeds the withstand voltage of the transformers even for a very short time interval.
12 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-49-320.jpg)
![165 and 100 kV, respectively. The results are reported for the three different current
waveforms and for different distances d between consecutive SAs, namely 100, 200,
300, and 400 m.
The comparison between Tables 1.5 and 1.6 shows that with SAs at d ¼ 200 m
and above, the MTBF values calculated for insulator CFO ¼ 165 kV are higher
than those calculated for CFO ¼ 100 kV, while the opposite happens without SAs
or with d ¼ 100 m. Such a difference is due to additional flashovers near the
transformer for the case of 100 kV CFO insulators with respect to the case with
165 kV CFO insulators. The oscillatory transients originated by these flashovers
and by the associated reflections at the surrounding SAs may cause voltages, at the
transformer location, with peak value higher than without flashovers. These over-
voltages are not limited by the nearby SAs if the distance between them is sig-
nificant, namely d ¼ 200 m and above. A similar unfavorable effect of large
separating distances between SAs has been already observed and discussed in [49].
The adoption of a different current waveform generally results in slight dif-
ferences between the MTBF values. The larger differences – concerning the direct
events only – appear to be those relevant to the case with d ¼ 100 m and
CFO ¼ 165 kV, for which the adoption of the Heidler and the Cigré turn out in an
8% increase of the MTBF. By increasing the distance between SAs, these differ-
ences tend to be negligible. Concerning the indirect events only, the differences are
negligible for the cases of d ¼ 100 m and d ¼ 200 m due to the very low probability
of exceeding the withstand voltage of the transformer. The differences turn out to
be appreciable, instead, by increasing d, while again they are negligible if the line is
unprotected. These variations in the results are ascribed to the effect of nonlinearity
introduced by SAs and flashover model that enhance the effect of the difference
among the chosen current waveforms and in particular of their front.
The computational cost due to the assumption of more realistic current
waveforms rather than the trapezoidal one is quite heavy. The time required to
obtain the results of both Tables 1.5 and 1.6 of this volume relevant to indirect
Table 1.4 Parameters assumed for the disruptive effect model
DE model parameters
CFO (kV) V0 (kV) k DE (kVms)
100 90 1 60.9
165 132 1 255
2,000 m
d
Figure 1.2 Top view of the line with the indication of the observation point in the
middle of the line and the position of the SAs
Application of the Monte Carlo method to lightning protection 13](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-50-320.jpg)
![strikes for the case of trapezoidal current waveform is about 8 h and about twice the
time for the two other currents waveforms (almost all the computational effort is
required by the calculation of the lightning electromagnetic pulse (LEMP), which is
performed only once for each MC event and then used for all the different SA
configurations and insulator types).
1.5.2 Application of the recursive stratified
sampling technique
In this section, the earlier illustrated recursive stratified sampling procedure is
applied to the case of a single-conductor straight line, assuming a withstand voltage
value W equal to 150 kV. The line is 2 km long, 10 m high, and is matched at both
terminations with the matrix of surge impedances to render more straightforward
the interpretation of the results.
In the simulations of this section, a linearly rising current with flat top is assumed
for the representation of the channel base lightning-current waveform, with peak
amplitude Ip and equivalent front time tf. The return-stroke propagation speed is set to
1.5 108
m/s. The lightning performance is evaluated for overhead lines above a soil
with conductivity equal to 0.001 S/m and relative permittivity equal to 10.
In Figure 1.3, the top view of one half of the considered area A is reported
(the line is located on the x-axis and the distribution of the stroke locations is
2,000
1,800
1,600
1,200
1,400
1,000
800
400
200
600
–1,000 –600 –400
–800 –200 200
m
m
600 800
400 1,000
0
0
Figure 1.3 Position of the events generated by the stratified sampling procedure
for the case of an unprotected line (direct strikes in red, nearby strikes
to ground in blue). Adapted from [28]
16 Lightning interaction with power systems, volume 2](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-53-320.jpg)
![obviously symmetric with respect to the line). The figure shows also the stroke
locations of all the simulated events, that is, the initial pilot events and those gen-
erated by the stratified sampling procedure. The red dots represent direct strikes to
the line while the nearby strikes to ground are indicated in blue. For the considered
case, the semi-area is a 2 2 km2
, divided in m ¼ 400 subdomains of area
0.1 0.1 km.
The number of pilot events simulated before starting the stratification proce-
dure is 5,200. The stratified sampling calculation is stopped when relative error CJ
reaches the same value of Cp calculated with the standard MC method with 100,000
events for W ¼ 150 kV.
The total number of events generated by the stratified sampling procedure are
3,464 direct strikes and 23,093 nearby strikes, as reported in Table 1.7 of this
volume. Since only the evaluation of the voltages induced by nearby strikes needs
time domain LIOV–EMTP-RV simulations, the computational time reduction
indicated in Table 1.7 of this volume is estimated as 100 nind;p nind;J
=nind;p,
where nind,J is the number of events required by the stratified sampling and nind,p is
the corresponding number calculated in the standard MC.
As shown by Figure 1.3, the procedure allocates the majority of the events in
the subdomains closest to the line. A very few events are allocated farther than
1.2 km from the line, since the initial variance in those subdomains is null.
In order to limit the computational time of the standard MC procedure, the
smallest area A that includes all the dangerous events is to be chosen, as discussed
in [5]. The capability of the stratified sampling to recursively allocate the events in
the subdomains with the largest variance value reduces the importance of an accurate
choice of the smallest area A.
Figure 1.3 also shows that fewer events are allocated near the matched termi-
nations with respect to the subdomains close to the internal part of the line, due to
the risers effect that reduce the induced overvoltages [50].
The above-described assessment has been repeated for the case of a line pro-
tected with SAs. Two cases are considered with SAs placed every d ¼ 500 m and
d ¼ 200 m, respectively, starting from the line terminations. The line terminations
are open. The voltage–current characteristic of the considered SAs is the same used
in [51].
Table 1.7 Comparison between number of direct and nearby strikes in the
standard MC and stratified sampling. Computational time reduction
due to stratified sampling
Relative
error %
Direct strikes Nearby strikes Time
saved %
Standard/
stratified
Standard/
stratified
Unprotected 2.4 3,103/3,464 96,897/23,093 76
With SAs d ¼ 500 m 2.8 12,166/12,095 187,834/42,455 77
With SAs d ¼ 200 m 10.7 24,665/18,687 175,335/50,163 71
Application of the Monte Carlo method to lightning protection 17](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-54-320.jpg)
![[156]
[157]
Eight hours after she had crept into the luxurious bed in
the guest room of the strange lodge, Marian stirred,
then half awake, felt the drowsy warmth of wolf-skin
rugs. For a moment she lay there and inhaled the drug-
like perfume of balsam and listened to the steady
breathing of the Eskimo girl beside her. She was about
to turn over for another sleep, when, from some cell of
her brain where it had been stowed the night before,
there came the urge that told her she must make haste.
“Haste! Haste! Haste!” came beating in upon her drowsy
senses. It was as if her brain were a radio, and the
message was coming from the air.
Suddenly she sat bolt upright. At the same instant she
found herself wide awake, fully alert and conscious of
the problems she must face that day—the passing of
the rapids and covering a long span of that trail which
still lay between them and their goal.
She did not waken Attatak. That might not be necessary
for another hour. She sprang out upon the heavy bear
skin rug, and there went through a set of wild, whirling
gestures that limbered every muscle in her body and
sent the red blood racing through her veins. After that
she quickly slipped into her blouse, knickers, stockings
and deerskin boots, to at last go tiptoeing down the
corridor toward the large living-room where she heard
the roar of the open fire as it raced up the chimney.
She found her host sitting by the fire. In the uncertain
light he appeared haggard and worn, as if quite done in
from some great exertion. Of course Marian could not
so much as guess how he had spent the night. She had
slept through it all.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-56-320.jpg)
![[158]
With a smile of greeting the old man motioned her to a
seat beside him.
“You’ll not begrudge an old man a half hour’s
company?” he said.
“Indeed not.”
“You’ll wish to ask me things. Everyone who passes this
way wants to. Mostly they ask and I don’t tell. A fair
lady, though,” there was something of ancient gallantry
in his tone, “fair ladies usually ask what they will and
get it, too.”
For a moment he sat staring silently into the fire.
“This house,” he said at last, “is a bit unusual. That pipe
organ, for instance—you wouldn’t expect it here. It
came here as if by accident; Providence, I call it. A rich
young man had more things than he knew what to do
with. The Creator sent some of them to me.
“As for me, I came here voluntarily. You have probably
taken me for a prospector. I have never bought pick nor
pan. There are things that lure me, but gold is not one
of them.
“I had troubles before I came here. Troubles are the
heritage of the aged. I sometimes think that it is not
well to live too long.
“And yet,” he shook himself free of the mood; his face
lighting up as he exclaimed, “And yet, life is very
wonderful! Wonderful, even up here in the frozen north.
I might almost say, especially here in the north.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-57-320.jpg)
![[159]
[160]
“I came here to be alone. I brought in food with a dog
team. I built a cabin of logs, and here I lived for a year.
“One day a young man came up the river in a wonderful
pleasure yacht and anchored at the foot of the rapids.
Being a lover of music, he had built a pipe organ into
his yacht; the one you heard last night.”
“And did—did he die?” Marian asked, a little break
coming in her voice.
“No,” the old man smiled, “he tarried too long. Being a
lover of nature—a hunter and an expert angler—and
having found the most ideal spot in the world as long as
summer lasted, he stayed on after the frosts and the
first snow. I was away at the time, else I would have
warned him. I returned the day after it happened. There
had been a heavy freeze far up the river, then a storm
came that broke the ice away. The ice came racing
down over the rapids like mad and wrecked his
wonderful yacht beyond all repair.
“We did as much as we could about getting the parts on
shore; saved almost all but the hull. He stayed with me
for a few days; then, becoming restless, traded me all
there was left of his boat for my dog team.
“That winter, with the help of three Indians and their
dogs, I brought the wreckage up here. Gradually, little
by little, I have arranged it into the form of a home that
is as much like a boat as a house. The organ was
unimpaired, and here it sings to me every day of the
great white winter.”
He ceased speaking and for a long time was silent.
When he spoke again his tones were mellow with
kindness and a strange joy.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-58-320.jpg)
![[161]
“I am seldom lonely now. The woods and waters are full
of interesting secrets. Travellers, like you, come this way
now and again. I try to be prepared to serve them; to
be their friend.”
“May—may I ask one question?” Marian suggested
timidly.
“As many as you like.”
“How did you know I was at the door last night when
you were playing? You did not see me. You couldn’t
have heard me.”
“That,” he smiled, “is a question I should like to ask
someone myself; someone much wiser than I am. I
knew you were there. I had been feeling your presence
for more than an hour before you came. I knew I had
an audience. I was playing for them. How did I know? I
cannot tell. It has often been so before. Perhaps all
human presence can be felt by some specially endowed
persons. It may be that in the throngs of great cities the
message of soul to soul is lost, just as a radio message
is lost in a jumble of many messages sent at once.
“But then,” he laughed, “why speculate? Life’s too short.
Some things we must accept as they are. What’s more
important to you is that your sleds are beyond the
rapids. When breakfast is over, you can strap your
sleeping bags on your deer and I will guide you over the
trail around the rapids to the point where I left your
sleds.”
A look of consternation flashed over Marian’s face. She
was thinking of the ancient dishes and how fragile they
were. “I have some fragile articles in the sleeping bags,”
she said. “They—they might break!”](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-59-320.jpg)
![[162]
[163]
“Break?” He wore a puzzled look.
For a second she hesitated; then, reassured by the
kindly face of the gentle old man, decided to tell him
the story of their adventure in the cave. Then she
launched into the story with all the eagerness of a
discoverer.
“I see,” he said, when she had finished the story. “I
know just how you feel. However, there is now only one
safe thing to do. Leave these treasures with me. If the
rapids are frozen over when the time comes for the
return trip, you can pass here and get them. You’ll
always be welcome. Better leave an address to which
they may be sent in case you should not pass this way.
The rapids freeze over every winter. I will surely be able
to get them off on the first river boat. They can be sent
to any spot in the world. To attempt to pack them over
on your deer would mean certain destruction.”
Reluctant as Marian was to leave the treasure behind,
she saw the wisdom of his advice. So, feeling a perfect
confidence in him, she decided to leave her treasure in
his care. Then she gave him her address at Nome, with
instructions for shipping should she fail to return this
way.
“One thing more I wanted to ask you,” she said. “How
many men are there at the Station?”
“One man; the trader. He stays there the year ’round.”
“One man!” she exclaimed.
“One is all. Time was when there were twenty.
Prospectors, traders, Indians, trappers. Two years ago
forest fires destroyed the timber. The game sought](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-60-320.jpg)
![[164]
[165]
other feeding grounds and the trappers, traders and
Indians went with them. Gold doesn’t seem to exist in
the streams hereabouts, so the prospectors have left,
too. Now one man keeps the post; sort of holding on, I
guess, just to see if the old days won’t return.”
“Do you suppose he could—could leave for a week or
two?” Marian faltered.
“Guess not. Company wouldn’t permit it.”
“Then—then—” Marian set her lips tight. She would not
worry this kind old man with her troubles. The fact
remained, however, that if there was but one man at
the Station, and he could not leave, there was no one
who could be delegated by the Government Agent to go
back with her to help fight her battles against Scarberry.
Suddenly, as she thought of the weary miles they had
travelled, of the hardships they had endured, and of the
probability that they would, after all, fail in fulfilling their
mission, she felt very weak and as one who has
suddenly grown old.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-61-320.jpg)
![[166]
CHAPTER XX
A MESSAGE FROM THE AIR
A cup of perfect coffee, followed by a dash into the
bracing Arctic morning, completely revived Marian’s
spirits. Casting one longing look backward at the
mysterious treasure of ancient dishes and old ivory,
throwing doubt and discouragement to the winds, with
energy and courage she set herself to face the problems
of the day.
The passing of the rapids by the overland trail was all
that their host had promised. Struggling over rocky,
snow-packed slopes; slipping, sliding, buffeted by strong
winds, beaten back by swinging overhanging branches
of ancient spruce and firs, they made their way
pantingly forward until at last, with a little cry of joy,
Marian saw their own sleds in the trail ahead.
“That’s over,” she breathed. “How thankful I am that we
did not attempt to make it with the sleds, or with our
treasure on the backs of the deer. There would not have
been left a fragment of our dishes as big as a dime. As
for the sleds, well it simply couldn’t be done.”
“No-me,” sighed Attatak.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-62-320.jpg)
![[167]
“I wonder how he could have brought them by the
rapids?” Marian mused as she examined the sleds.
There were flakes of ice frozen to the runners. She
could only guess at the method he had used, only dimly
picture the struggle it must have taken. Even as she
attempted to picture the night battle, a great wave of
admiration and trust swept over her.
“The treasure is safer in his hands than in ours,” she
told herself.
“But, after it has left his hands?” questioned her
doubting self.
“Oh well,” she sighed at last, “what must be, will be.
The important thing after all is to reach the station
before the Agent has started on his way.”
Again her brow clouded. What if there was no one to go
back with her?
To dispel this doubt, she hastened to hitch her deer to
her sled. Soon they were racing away over the trail,
causing the last miles of their long journey to melt away
like ice in the river before a spring thaw.
In the meantime a third startling revelation had come to
Patsy. First she had discovered that at least one of the
persons connected with the strange purple flame was a
girl. Next she had found the red trail of blood that
apparently was made by one of Marian’s slain deer, and
which led to the door of their tent. The third discovery
had nothing to do with the first two, nor with the purple
flame. It was of a totally different nature, and was most
encouraging.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-63-320.jpg)
![[168]
[169]
“If only Marian were here!” she said to herself as she
paced the floor after receiving the important message.
This message came to her over the radiophone. It was
not meant particularly for her, nor for Marian. It was
just news; not much more than a rumor, at that. Yet
such news as it was, if only it were true!
Faint and far away, it came drifting in upon the air from
some powerful sending station. Perhaps that station was
Fairbanks, Dawson or Nome. She missed that part of
the message.
Only this much came to her that night as she sat at
their compact, powerful receiving set, beguiling the
lonesome hours by catching snatches of messages from
near and far:
“Rumor has it that the Canadian Government plans the
purchase of reindeer to be given to her Eskimo people
on the north coast of the Arctic. Five or six hundred will
be purchased as an experiment, if the plan carries. It
seems probable that the deer purchased will be
procured in Alaska. It is thought possible to drive herds
across the intervening space and over the line from
Alaska, and that in this way they may be purchased by
the Canadian Agent on Canadian soil. A call for such
herds may be issued later over the radio, as it is well
known that many owners of herds have their camps
equipped with radio-phones.”
There the message ended. It had left Patsy in a fever of
excitement. Marian and her father wished to sell the
herd. It was absolutely necessary to sell it if Marian’s
hopes of continuing her education were not to be
blasted. There was no market now for a herd in Alaska.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-64-320.jpg)
![[170]
In the future, as pastures grew scarcer, and as herds
increased in numbers, there would be still less
opportunity for a sale.
“What a wonderful opportunity!” Patsy exclaimed. “To
sell the whole herd to a Government that would pay fair
prices and cash! And what a glorious adventure! To
drive a reindeer herd over hundreds of miles of rivers,
forests, tundra, hills and mountains; to camp each night
in some spot where perhaps no man has been before;
surely that would be wonderful! Wonderful!”
Just at that moment there entered her mind a startling
thought. Scarberry’s camp, too, was equipped with a
radio-phone. Probably he, too, at this very moment, was
smiling at the prospect of selling six hundred of his deer.
He wanted to sell. Of course he did. Everyone did. He
would make the drive. Certainly he would.
“And then,” she breathed, pressing her hands to her
fluttering heart, “then it will be a race; a race between
two reindeer herd; a race over hundreds of miles of
wilderness for a grand prize. What a glorious
adventure!”
“If only Marian were here,” she sighed again. “The
message announcing the plans may come while she is
gone. Then—”
She sat in a study for a long time. Finally she whispered
to herself:
“If the message comes while she is gone; if the
opportunity is sure to be lost unless the herd starts as
soon as the message comes, I wonder if I’d dare to
start on the race with the herd, with Terogloona and
without Marian and Attatak. I wonder if I would?”](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-65-320.jpg)
![[171]
[172]
For a long time she sat staring at the fire. Perhaps she
was attempting to read the answer in the flames.
At last, with cheeks a trifle flushed, she sprang to her
feet, did three or four leaps across the floor, and
throwing off her clothing, crept between the deer-skins
in the strange little sleeping compartment.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-66-320.jpg)
![[173]
CHAPTER XXI
FADING HOPES
Just at dawn of a wonderfully crisp morning, Marian
found herself following her reindeer over a trail that had
recently been travelled by a dog team. She was just
approaching the Trading Station where the questions
that haunted her tired brain would be answered.
Since leaving the cabin in the forest above the rapids,
she and Attatak had travelled almost day and night. A
half hour for a hasty lunch here and there, an hour or
two for sleep and for permitting the deer to feed; that
was all they had allowed themselves.
An hour earlier, Marian had felt that she could not travel
another mile. Then they had come upon the trail of the
dog team, and realizing that they were nearing their
goal, her blood had quickened like a marathon racer’s at
the end of his long race. No longer feeling fatigue, she
urged her weary reindeer forward. Contrary to her
usually cautious nature, she even cast discretion to the
winds and drove her deer straight toward the
settlement. That there were dogs which might attack
her deer she knew right well. That they were not of the](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-67-320.jpg)
![[174]
species that attacked deer, or that they were chained,
was her hope.
So, with her heart throbbing, she rounded a sudden
turn to find herself within sight of a group of low-lying
cabins that at one time had been a small town.
Now, as her aged host had said, it was a town in name
only. She knew this at a glance. One look at the
chimneys told her the place was all but deserted.
“No smoke,” she murmured.
“Yes, one smoke,” Attatak said, pointing.
It was true. From one long cabin there curled a white
wreath of smoke.
For a moment Marian hesitated. No dogs had come out
to bark, yet they might be there.
“You stay with the deer,” she said to Attatak. “Tether
them strongly to the sleds. If dogs come, beat them
off.”
She was away like an arrow. Straight to that cabin of
the one smoke she hurried. She caught her breath as
she saw a splendid team of dogs standing at the door.
Someone was going on a trip. The sled was loaded for
the journey. Was it the Agent’s sled? Had she arrived in
time?
She did not have long to wait before knowing. She had
come within ten feet of the cabin when a tall, deep-
chested man opened the door and stepped out. She
caught her breath. Instantly she knew him. It was the
Agent.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-68-320.jpg)
![[175]
[176]
He, in turn, recognized her, and with cap in hand and
astonishment showing in his eyes, he advanced to meet
her.
“You here!” he exclaimed. “Why Marian Norton, you
belong in Nome.”
“Once I did,” she smiled, “but now I belong on the
tundra with our herd. It is the herd that has brought me
here. May I speak to you about it?”
“Certainly you may. But you look tired and hungry. The
Trader has a piping Mulligan stew on the stove. It will
do you good. Come inside.”
An Indian boy, who made his home with the Trader, was
dispatched to relieve Attatak of her watch, and Marian
sat down to enjoy a delicious repast.
There are some disappointments that come to us so
gradually that, though the matters they effect are of the
utmost importance, we are not greatly shocked when at
last their full meaning is unfolded to us. It was so with
Marian. She had dared and endured much to reach this
spot. She had arrived at the critical moment. An hour
later the Agent would have been gone. The Agent was
her friend. Ready to do anything he could to help her,
he would gladly have gone back with her to assist in
defending her rights. But duty called him over another
trail. He had no one, absolutely no one to send from
this post to execute his orders.
“Of course,” he said after hearing her story, “I can give
you a note to that outlaw, Scarberry, but he’d pay no
attention to it.”](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-69-320.jpg)
![[177]
“He’d tear it up and throw it in my face,” asserted
Marian stoutly.
“I’ll tell you what I’ll do,” said the Agent, rising and
walking the floor. “There is Ben Neighbor over at the
foot of Sugar Loaf Mountain. His cabin is only three
days travel from your camp. He’s a good man, and a
brave one. He is a Deputy Marshal. If I give you a note
to him, he will serve you as well as I could.”
“Would we need take a different trail home?”
“Why? Which way did you come?”
Marian described their course. The Agent whistled. “It’s
a wonder you didn’t perish!”
“Here,” he said, “is a rough map of the country. I will
mark out the course to Ben’s cabin. You’ll find it a much
safer way.”
“Oh, all right,” she said slowly. “Thanks. That’s surely
the best way.”
She was thinking of the treasure left at the cabin. She
had hoped to return by that route and claim it. Now that
hope was gone.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-70-320.jpg)
![[178]
[179]
This called for courage. Born with a brave soul, Marian
was equal to any emergency. Sheer weariness and lack
of sleep added to this a touch of daring.
Without pausing, she drove straight up to the door.
Reassured by the snow banked up against it, she hastily
scooped away the bank with her snow-shoe, and having
shoved the door open, boldly entered.
It was a cheerless place, black and empty. The wind
whistled through the cracks where the planks had rotted
away. Yet it was a shelter. Passing through another door,
she found herself in an inner room that housed the
boiler of the engine that had furnished power to the
dredge. The boiler, a great red drum of rust, stood
directly in front of her.
“Here’s where we camp,” she said to Attatak. “We can
build a fire in the fire-box of the boiler and broil some
steak. That will be splendid!”
“Eh-eh,” grinned Attatak.
“And Attatak, bring the deer through the outer door,
then close it. They were fed two hours ago. That will do
until morning.”
She lighted a candle, gathered up some bits of wood
that lay strewn about the narrow room, and began to
kindle a fire while Attatak went out after the deer.
For the moment, being alone, she began to think of the
herd. How was the herd faring? What had happened to
Patsy during those many days of her absence? Were Bill
Scarberry’s deer rapidly destroying her herd ground.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-72-320.jpg)
![[180]
[181]
“Well, if they are, we are powerless to prevent it,” she
told herself with a sigh.
As she looked back upon it now, she felt that her whole
journey had been a colossal failure. They had
discovered the mountain cave treasure, only to be
obliged to leave the treasure behind. They had reached
the Station in time to talk with the Government Agent,
but he had not been able to come with her. Only
twenty-four hours before they had reached the cabin of
Ben Neighbor, only to find it dark and deserted. He had
gone somewhere, as people in the Arctic have a way of
doing; and where that might be she could not even
hazard a guess. At last, in despair, she had headed her
deer toward her own camp. In thirty-six hours she
would be there.
“Well, at any rate,” she sighed, “it will be a pleasure to
see Patsy and to sleep the clock round in our own sweet
little deerskin bedroom.”
She was indeed to see Patsy, but the privilege of
sleeping the clock round was not to be hers for many a
day. She was destined to find the immediate future far
too stirring for that.
Twenty-four hours later saw Marian well on her way
home. Ten hours more, she felt sure, would bring her to
camp. And then what? She could not even guess. Had
she been able to even so much as suspect what was
going on at camp, she would have urged her reindeer to
do their utmost.
Patsy was right in the middle of a peck of trouble.
Because of the fact that for the last few days she had
been living in a realm of exciting dreams, the troubles](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-73-320.jpg)
![[182]
that had come down upon her seemed all the more
grievous. Since that most welcome radio message
regarding the proposed purchase of reindeer by the
Canadian Government had come drifting in over the air,
she had, during every available moment, hovered over
the radio-phone in the momentary expectation of
receiving the confirmation of that rumor which might
send the herd over mountains and tundra in a wild race
for a prize, a prize worth thousands of dollars to her
uncle and cousin—the sale of the herd.
Perhaps it was because of her too close application to
the radio-phone that she failed to note the approach of
Scarberry’s herd as it returned to ravish their feeding
ground. Certain it was that the first of the deer, with the
entire herd close upon their heels, were already over
the hills before she knew of their coming.
It was night when Terogloona brought this bit of
disquieting news.
“And this time,” Patsy wailed, “we have not so much as
one hungry Eskimo with his dog to send against them.”
As if in answer to the complaint, the aged herder
plucked at her sleeve, then led her out beneath the
open sky.
With an impressive gesture, he waved his arm toward
the distant hills that lay in the opposite direction of
Scarberry’s herd. To her great surprise and
mystification, she saw gleaming there the lights of
twenty or more campfires.
“U-bogok,” (see there) he said.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-74-320.jpg)
![[183]
[184]
“What—what does it mean?” Patsy stammered, grasping
at her dry throat.
“It is that I fear,” said Terogloona. “They come. To-
morrow they are here. You gave food for a week for a
few; flour, sugar, bacon. They like him. Now come whole
village of Sitne-zok. Want food. You gave them food.
What you think? No food for herders, no herders. No
herders, no herd. What you think?”
Patsy did not know what to think. Gone was all her little
burst of pride over the way she had handled the other
situation that had confronted her. Now she felt that she
was but a girl, a very small girl, and very, very much
alone. She wished Marian would come. Oh, how she did
wish that she would come!
“In the morning we will see what can be done,” was all
she could say to the faithful old herder as she turned to
re-enter the igloo.
That night she did not undress. She sat up for hours,
trying to think of some way out. She sat long with the
radio head-set over her ears. She entertained some wild
notion of fleeing with the herd toward the Canadian
border, providing the message confirming the offer for
the deer came. But the message did not come.
At last, in utter exhaustion, she threw herself among the
deerskins and fell into a troubled sleep.
She was roused from this sleep by a loud: “Hello there!”
followed by a cheery: “Where are you? Are you asleep?”
It was Marian. The next moment poor, tired, worried
Patsy threw herself sobbing into her cousin’s strong
arms.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-75-320.jpg)
![[185]
“There now,” said Marian, soothingly, as Patsy’s sobbing
ceased, “sit down and tell me all about it. You’re safe;
that’s something. Your experiences can’t have been
worse than ours.”
“The Eskimo! Bill Scarberry’s herd!” burst out Patsy,
“They’re here. All of them!”
“Tell me all about it,” encouraged Marian.
“Wait till I get my head-set on,” said Patsy, more
hopefully. “It’s been due for days; may come at any
time.”
“What’s due?” asked Marian, mystified.
“Wait! I’ll tell you. One thing at a time. Let’s get it all
straight.”
She began at the beginning and recited all that had
transpired since Marian had left camp. When she came
to tell of her discovery that one of the mysterious
occupants of the tent of the purple flame was a girl,
Marian’s astonishment knew no bounds. When told of
the bloody trail, Marian was up in arms. The camp of
the purple flame must be raided at once. They would
put a stop to that sort of thing. They would take their
armed herders and raid that camp this very night.
“But wait!” Patsy held up a warning finger, “I am not
half through yet. There is more. Too much more!”
She was in the midst of recounting her experiences with
the band of wandering Eskimo and Scarberry’s herd,
when suddenly she clapped the radio receiver tightly to
her ears and stopped talking. Then she murmured:](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-76-320.jpg)
![[186]
“It’s coming! At last, it is coming!”
“For goodness sake!” exclaimed Marian, out of all
patience, “Will you kindly tell me what is coming?”
But Patsy only held the receiver to her ears and listened
the more intently as she whispered:
“Shush! Wait!”](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-77-320.jpg)
![[187]
CHAPTER XXIII
PLANNING THE LONG DRIVE
The message that was holding Patsy’s attention was one
from the Canadian Government. It was a bonafide offer
from that Government to purchase the first herd of from
four to six hundred reindeer that should reach Fort
Jarvis.
When Patsy had imparted the exciting news to her,
Marian sat long in silent thought. Fort Jarvis, as she well
knew, lay some five hundred miles away over hills and
tundra. She had just returned from one such wearisome
journey. Should she start again? And would this second
great endeavor prove more successful than the first? Of
all the herds in Alaska, two were closest to Fort Jarvis;
Scarberry’s and her own. She had not the slightest
doubt that Scarberry would start driving a section of his
herd toward that goal. It would be a race; a race that
would be won by the bravest, strongest and most
skillful. Marian believed in her herders. She believed in
herself and Patsy. She believed as strongly in her herd,
her sled-deer and her dogs. It was the grand
opportunity; the way out of all troubles. That the band
of begging natives would not follow, she knew right](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-78-320.jpg)
![[188]
well. Nor would the mysterious persons of the purple
flame camp; at least, she hoped not. As for their little
herd range, if they sold their deer, Scarberry might have
it, and welcome; if they did not sell, they could
doubtless find pasture in some far away Canadian
valley.
“Yes,” she said in a tone of decision, “we will go. We will
waken the herders at once. Come on, let’s go.”
As they burst breathlessly into the cabin of their Eskimo
herders, they received something of a shock. Since all
the work of the day had long since been done, they had
expected to find the entire group of four assembled in
the cabin, or asleep in their bunks. But here was only
old Terogloona and Attatak.
“Where’s Oatinna? Where’s Azazruk?” demanded
Marian.
“Gone,” said Terogloona solemnly.
“Where? Go call them, quick!”
Terogloona did not move. He merely shrugged his
shoulders and mumbled:
“No good. Gone long way. Bill Scarberry’s camp. No
come back, say that one.”
“What!” exclaimed Marian in consternation. “Gone?
Deserted us?”
“Eh-eh,” Terogloona nodded his head. “Say Bill
Scarberry pay more money; more deer; say that one
Oatinna, that one Azazruk. No good, that one Bill
Scarberry, me think.” He shook his head solemnly. “Not](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-79-320.jpg)
![[189]
listen that one Oatinna, that one Azazruk. Say wanna
go. Go, that’s all.”
“Then we can’t start the herd,” murmured Marian,
sinking down upon a rolled up sleeping-bag. “Yes, we
will!” she exclaimed resolutely. “Terogloona, where are
the rifles?”
“Gone,” he repeated like a parrot. “Mebby you forget.
That one rifle b’long herder boys.”
“And your rifle?” questioned Marian, “where is your
rifle?”
“Broke-tuk. Hammer not want come down hard. Not
want shoot, that one rifle, mine.”
Marian was stunned with surprise and chagrin. She and
Patsy returned silently to their igloo.
“Oh, that treacherous Bill Scarberry!” she exploded. “He
has known this was coming. He knew our herders were
energetic and capable. He thought if they remained with
us, we might beat him to the prize; so he sent some spy
over here to buy them away from us with promises of
more pay.”
“And now?” asked Patsy.
“Now he will drive his herd to Fort Jarvis and sell it, and
our grand chance is gone forever.”
“No!” exclaimed Patsy, “He won’t! He shall not! We will
beat him yet. We are strong. Terogloona and Attatak are
faithful. We have our three collies. We can do it. We will
beat him yet. Our herd is better than his. It will travel](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-80-320.jpg)
![[190]
[191]
faster. Oh, Marian! Somehow, somehow we must do it.
It’s your chance! Your one big, wonderful opportunity.”
“Yes,” exclaimed Marian, suddenly fired by her cousin’s
hot blooded southern enthusiasm, “we will do it or
perish in the attempt. It’s to be a race,” she exclaimed,
“a race for a wonderful prize, a race between two large
herds of reindeer over five hundred miles of hills, tundra
and forest. There may be wolves in the forests. In
Alaska dangers lurk at every turn; rivers too rapid to
freeze over and blizzards and wild beasts. We will be
terribly handicapped from the very start. But for father’s
sake we must try it.”
“For your father’s and for your own sake,” murmured
Patsy. “And, Marian, I have always believed that our
great Creator was on the side of those who are kind and
just. Bill Scarberry played us a mean trick. Perhaps God
will somehow even the score.”
An hour was spent in consultation with old Terogloona.
His face became very sober at the situation, but in the
end, with the blood of youth coursing eternally in his
veins, he sprang to his feet and exclaimed:
“Eh-eh!” (Yes-yes) “We will go. Before it is day we will
be away. You go sleep. You must be very strong. In the
morning Terogloona will have reindeer and sleds ready.
We will call to the dogs. We will be away before the sun.
We will shout ‘Kul-le-a-muck, Kul-le-a-muck’ (Hurry!
Hurry!) to dogs and reindeer. We will beat that one Bill
yet.
“You know what?” he exclaimed, his face darkening like
a thundercloud, “You know that mean man, that one Bill
Scarberry. Want my boy, So-queena, work for him. Want](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-81-320.jpg)
![[192]
[193]
pay him reindeer. Give him bad rifle, very bad rifle.
Want shoot, my boy So-queena. Shot at carabou, So-
queena. Rifle go flash. Crooch! Just like that. Shoot
back powder, that rifle. Came in So-queena’s eyes, that
powder. Can’t see, that one. Almost lost to freeze, that
one, So-queena. Bye’m bye find camp. Stay camp
mebby five days. Can see, not very good. Bill, he say:
‘Go herd reindeer,’ So-queena, he say: ‘Can’t see. Mebby
get lost. Mebby freeze’.
“He say Bill very mad. ‘Get out! No good, you! Go
freeze. Who cares?’
“So-queena come my house—long way. Plenty starve.
Plenty freeze. No give reindeer that one So-queena, that
one Bill. Bad one, that Bill. So me think; beat Bill. Sell
reindeer herd white man. Think very good. Work hard.
Mebby beat that one Bill Scarberry.”
There came a look of determination to Patsy’s face such
as Marian had never seen there.
“If that’s the kind of man he is; if he would send an
Eskimo boy, half-blinded by his own worthless rifle, out
into the snow and the cold, then we must beat him. We
must! We must!” said Patsy vehemently.
“That’s exactly the kind of man he is,” said Marian
soberly. “We must beat him if we can. But it will be a
long, hard journey.”
They had hardly crept between their deerskins when
Patsy was fast asleep. Not so Marian. The full
responsibility of this perilous journey rested upon her
shoulders. She knew too well the hardships and dangers
they must face. They must pass through broad
stretches of forest where food for the deer was scarce,](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-82-320.jpg)
![[194]
and where lurking wolves, worn down to mere skeletons
by the scarcity of food, might attack and scatter their
herd beyond recovery.
They must cross high hills, from whose summits the
snow at times poured like smoke from volcanoes in
circling sweeps hundreds of feet in extent. Here there
would be danger of losing their deer in some wild
blizzard, or having them buried beneath the snows of
some thundering avalanche.
“It’s not for myself alone that I’m afraid,” she told
herself. “It’s for Patsy, Patsy from Kentucky. Who would
have thought a girl from the sunny south could be so
brave, such a good sport.”
As she thought of the courageous, carefree manner in
which Patsy had insisted on the journey, a lump rose in
her throat, and she brushed a hand hastily over her
eyes.
“And yet,” she asked herself, “ought I to allow her to do
it? She’s younger than I, and not so strong. Can she
stand the strain?”
Again her mind took up the thought of the perils they
must face.
There were wandering tribes of Indians in the territory
they must cross; the skulking and oft-times treacherous
Indians of the Little Sticks. What if they were to cross
the path of these? What if a great band of caribou
should come pouring down some mountain pass and,
having swallowed up their little herd, go sweeping on,
leaving them in the midst of a great wilderness with
only their sled-deer to stand between them and
starvation.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-83-320.jpg)
![[195]
[196]
As if dreaming of Marian’s thoughts, Patsy suddenly
turned over with a little sobbing cry, and wound her
arms about Marian.
“What is it?” Marian whispered.
Patsy did not answer. She was still asleep. The dream
soon passed, her muscles relaxed, and with a deep sigh
she sank back into her place.
This little drama left Marian in an exceedingly troubled
state of mind.
“We ought not to go,” she told herself. “We will not.”
Then, from sheer exhaustion, she too, fell asleep.
Three hours before the tardy Arctic sunrise, she heard
Terogloona pounding at their door. She found that sleep
had banished fear, and that every muscle in her body
and every cell of her brain was ready for action, eager
to be away.
As for Patsy, she could not dress half fast enough, so
great was her desire for the wonderful adventure.](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-84-320.jpg)
![[197]
CHAPTER XXIV
CAMP FOLLOWERS
It was just as Marian was tightening the ropes to the
pack on her sled that, happening to glance away at a
distant hill, she was reminded of Patsy’s latest story of
the purple flame. From the crest of that hill there came
a purple flare of light. Quickly as it had come, just so
quickly it vanished, leaving the hill a faint outline against
the sky.
“The purple flame,” she breathed. “I wonder if we can
leave those mysterious camp-followers of ours behind?”
On the instant a disturbing thought flashed through her
mind. It caused an indignant flash of color to rise to her
cheek.
“I wonder,” she said slowly, “if those mysterious people
are spies set by Bill Scarberry to dog our tracks?”
“They may start with us,” she smiled to herself, as she
at last dismissed the subject from her mind, “but unless
they really are Bill Scarberry’s spies and set to watch us,
they’ll never finish with us. Camp-followers don’t follow](https://image.slidesharecdn.com/16465759-250525204032-c4b8b15d/85/Lightning-Interaction-With-Power-Systems-Applications-Volume-2-Energy-Engineering-Alexandre-Piantini-Editor-85-320.jpg)
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- 6. IET ENERGY ENGINEERING 172 Lightning Interaction with Power Systems
- 7. Other volumes in this series: Volume 1 Power Circuit Breaker Theory and Design C.H. Flurscheim (Editor) Volume 4 Industrial Microwave Heating A.C. Metaxas and R.J. Meredith Volume 7 Insulators for High Voltages J.S.T. Looms Volume 8 Variable Frequency AC Motor Drive Systems D. Finney Volume 10 SF6 Switchgear H.M. Ryan and G.R. Jones Volume 11 Conduction and Induction Heating E.J. Davies Volume 13 Statistical Techniques for High Voltage Engineering W. Hauschild and W. Mosch Volume 14 Uninterruptible Power Supplies J. Platts and J.D. St Aubyn (Editors) Volume 15 Digital Protection for Power Systems A.T. Johns and S.K. Salman Volume 16 Electricity Economics and Planning T.W. Berrie Volume 18 Vacuum Switchgear A. Greenwood Volume 19 Electrical Safety: A guide to causes and prevention of hazards J. Maxwell Adams Volume 21 Electricity Distribution Network Design, 2nd Edition E. Lakervi and E.J. Holmes Volume 22 Artificial Intelligence Techniques in Power Systems K. Warwick, A.O. Ekwue and R. Aggarwal (Editors) Volume 24 Power System Commissioning and Maintenance Practice K. Harker Volume 25 Engineers’ Handbook of Industrial Microwave Heating R.J. Meredith Volume 26 Small Electric Motors H. Moczala et al. Volume 27 AC–DC Power System Analysis J. Arrillaga and B.C. Smith Volume 29 High Voltage Direct Current Transmission, 2nd Edition J. Arrillaga Volume 30 Flexible AC Transmission Systems (FACTS) Y.-H. Song (Editor) Volume 31 Embedded generation N. Jenkins et al. Volume 32 High Voltage Engineering and Testing, 2nd Edition H.M. Ryan (Editor) Volume 33 Overvoltage Protection of Low-Voltage Systems, Revised Edition P. Hasse Volume 36 Voltage Quality in Electrical Power Systems J. Schlabbach et al. Volume 37 Electrical Steels for Rotating Machines P. Beckley Volume 38 The Electric Car: Development and future of battery, hybrid and fuel-cell cars M. Westbrook Volume 39 Power Systems Electromagnetic Transients Simulation J. Arrillaga and N. Watson Volume 40 Advances in High Voltage Engineering M. Haddad and D. Warne Volume 41 Electrical Operation of Electrostatic Precipitators K. Parker Volume 43 Thermal Power Plant Simulation and Control D. Flynn Volume 44 Economic Evaluation of Projects in the Electricity Supply Industry H. Khatib Volume 45 Propulsion Systems for Hybrid Vehicles J. Miller Volume 46 Distribution Switchgear S. Stewart Volume 47 Protection of Electricity Distribution Networks, 2nd Edition J. Gers and E. Holmes Volume 48 Wood Pole Overhead Lines B. Wareing Volume 49 Electric Fuses, 3rd Edition A. Wright and G. Newbery Volume 50 Wind Power Integration: Connection and system operational aspects B. Fox et al. Volume 51 Short Circuit Currents J. Schlabbach Volume 52 Nuclear Power J. Wood Volume 53 Condition Assessment of High Voltage Insulation in Power System Equipment R.E. James and Q. Su Volume 55 Local Energy: Distributed generation of heat and power J. Wood Volume 56 Condition Monitoring of Rotating Electrical Machines P. Tavner, L. Ran, J. Penman and H. Sedding Volume 57 The Control Techniques Drives and Controls Handbook, 2nd Edition B. Drury Volume 58 Lightning Protection V. Cooray (Editor) Volume 59 Ultracapacitor Applications J.M. Miller
- 8. Volume 62 Lightning Electromagnetics V. Cooray Volume 63 Energy Storage for Power Systems, 2nd Edition A. Ter-Gazarian Volume 65 Protection of Electricity Distribution Networks, 3rd Edition J. Gers Volume 66 High Voltage Engineering Testing, 3rd Edition H. Ryan (Editor) Volume 67 Multicore Simulation of Power System Transients F.M. Uriate Volume 68 Distribution System Analysis and Automation J. Gers Volume 69 The Lightening Flash, 2nd Edition V. Cooray (Editor) Volume 70 Economic Evaluation of Projects in the Electricity Supply Industry, 3rd Edition H. Khatib Volume 72 Control Circuits in Power Electronics: Practical issues in design and implementation M. Castilla (Editor) Volume 73 Wide Area Monitoring, Protection and Control Systems: The enabler for smarter grids A. Vaccaro and A. Zobaa (Editors) Volume 74 Power Electronic Converters and Systems: Frontiers and applications A.M. Trzynadlowski (Editor) Volume 75 Power Distribution Automation B. Das (Editor) Volume 76 Power System Stability: Modelling, analysis and control B. Om P. Malik Volume 78 Numerical Analysis of Power System Transients and Dynamics A. Ametani (Editor) Volume 79 Vehicle-to-Grid: Linking electric vehicles to the smart grid J. Lu and J. Hossain (Editors) Volume 81 Cyber-Physical-Social Systems and Constructs in Electric Power Engineering S. Suryanarayanan, R. Roche and T.M. Hansen (Editors) Volume 82 Periodic Control of Power Electronic Converters F. Blaabjerg, K.Zhou, D. Wang and Y. Yang Volume 86 Advances in Power System Modelling, Control and Stability Analysis F. Milano (Editor) Volume 87 Cogeneration: Technologies, optimisation and implementation C.A. Frangopoulos (Editor) Volume 88 Smarter Energy: From smart metering to the smart grid H. Sun, N. Hatziargyriou, H.V. Poor, L. Carpanini and M.A. Sánchez Fornié (Editors) Volume 89 Hydrogen Production, Separation and Purification for Energy A. Basile, F. Dalena, J. Tong and T.N.Veziroğlu (Editors) Volume 90 Clean Energy Microgrids S. Obara and J. Morel (Editors) Volume 91 Fuzzy Logic Control in Energy Systems with Design Applications in MATLAB‡/Simulink‡ İ.H. Altaş Volume 92 Power Quality in Future Electrical Power Systems A.F. Zobaa and S.H.E.A. Aleem (Editors) Volume 93 Cogeneration and District Energy Systems: Modelling, analysis and optimization M.A. Rosen and S. Koohi-Fayegh Volume 94 Introduction to the Smart Grid: Concepts, technologies and evolution S.K. Salman Volume 95 Communication, Control and Security Challenges for the Smart Grid S.M. Muyeen and S. Rahman (Editors) Volume 96 Industrial Power Systems with Distributed and Embedded Generation R Belu Volume 97 Synchronized Phasor Measurements for Smart Grids M.J.B. Reddy and D.K. Mohanta (Editors) Volume 98 Large Scale Grid Integration of Renewable Energy Sources A. Moreno- Munoz (Editor) Volume 100 Modeling and Dynamic Behaviour of Hydropower Plants N. Kishor and J. Fraile-Ardanuy (Editors) Volume 101 Methane and Hydrogen for Energy Storage R. Carriveau and D.S.-K. Ting Volume 104 Power Transformer Condition Monitoring and Diagnosis A. Abu-Siada (Editor) Volume 106 Surface Passivation of Industrial Crystalline Silicon Solar Cells J. John (Editor) Volume 107 Bifacial Photovoltaics: Technology, applications and economics J. Libal and R. Kopecek (Editors)
- 9. Volume 108 Fault Diagnosis of Induction Motors J. Faiz, V. Ghorbanian and G. Joksimović Volume 110 High Voltage Power Network Construction K. Harker Volume 111 Energy Storage at Different Voltage Levels: Technology, integration, and market aspects A.F. Zobaa, P.F. Ribeiro, S.H.A. Aleem and S.N. Afifi (Editors) Volume 112 Wireless Power Transfer: theory, technology and application N.Shinohara Volume 115 DC Distribution Systems and Microgrids T. Dragičević, F.Blaabjerg and P. Wheeler Volume 117 Structural Control and Fault Detection of Wind Turbine Systems H.R. Karimi Volume 119 Thermal Power Plant Control and Instrumentation: The control of boilers and HRSGs, 2nd Edition D. Lindsley, J. Grist and D. Parker Volume 120 Fault Diagnosis for Robust Inverter Power Drives A. Ginart (Editor) Volume 123 Power Systems Electromagnetic Transients Simulation, 2nd Edition N. Watson and J. Arrillaga Volume 124 Power Market Transformation B. Murray Volume 125 Wind Energy Modeling and Simulation, Volume 1: Atmosphere and plant P.Veers (Editor) Volume 126 Diagnosis and Fault Tolerance of Electrical Machines, Power Electronics and Drives A.J.M. Cardoso Volume 128 Characterization of Wide Bandgap Power Semiconductor Devices F. Wang, Z. Zhang and E.A. Jones Volume 129 Renewable Energy from the Oceans: From wave, tidal and gradient systems to offshore wind and solar D. Coiro and T. Sant (Editors) Volume 130 Wind and Solar Based Energy Systems for Communities R. Carriveau and D.S.-K. Ting (Editors) Volume 131 Metaheuristic Optimization in Power Engineering J. Radosavljević Volume 132 Power Line Communication Systems for Smart Grids I.R.S. Casella and A. Anpalagan Volume 139 Variability, Scalability and Stability of Microgrids S.M. Muyeen, S.M. Islam and F. Blaabjerg (Editors) Volume 155 Energy Generation and Efficiency Technologies for Green Residential Buildings D. Ting and R. Carriveau (Editors) Volume 157 Electrical Steels, 2 Volumes A. Moses, K. Jenkins, Philip Anderson and H. Stanbury Volume 905 Power System Protection, 4 Volumes
- 10. Lightning Interaction with Power Systems Volume 2: Applications Edited by Alexandre Piantini The Institution of Engineering and Technology
- 11. Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2020 First published 2020 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the authors to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN 978-1-83953-092-0 (Hardback Volume 2) ISBN 978-1-83953-093-7 (PDF Volume 2) ISBN 978-1-83953-090-6 (Hardback Volume 1) ISBN 978-1-83953-091-3 (PDF Volume 1) ISBN 978-1-83953-094-4 (Hardback Volumes 1 and 2) Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon
- 12. Contents About the editor xv Preface xvii Acknowledgments xxi About the authors xxiii 1 Application of the Monte Carlo method to lightning protection and insulation coordination practices 1 Alberto Borghetti, Fabio Napolitano, Carlo Alberto Nucci and Fabio Tossani 1.1 Introduction 1 1.2 Description of the MC-based procedure 4 1.3 Identification of the lightning current functions 7 1.4 Stratified sampling technique 9 1.5 Application results for a MV overhead line in open terrain 11 1.5.1 Influence of the return stroke current waveform 11 1.5.2 Application of the recursive stratified sampling technique 16 1.6 Conclusions 21 References 21 2 Lightning interaction with power substations 27 Shigemitsu Okabe 2.1 Fundamental concepts 27 2.1.1 Definition and procedure of insulation coordination 27 2.1.2 Lightning overvoltage in insulation coordination 28 2.1.3 Lightning surge analysis 31 2.2 Simplified statistical approach of lightning surge analysis 33 2.2.1 Basics 33 2.2.2 Calculation of the limit distance 33 2.2.3 Estimation of the lightning overvoltage amplitude 35 2.2.4 Simplified method 38 2.2.5 Assumed maximum value of the representative lightning overvoltage 40 2.3 Detailed deterministic approach of lightning surge analysis 40 2.3.1 Basic analysis conditions 40 2.3.2 Analysis conditions and models 44
- 13. 2.4 Failure rate evaluation considering front time of lightning current 46 2.4.1 Crest value and wavefront time of lightning stroke current 46 2.4.2 Wavefront time of lightning stroke current and amplitude of lightning surge 48 2.4.3 Lightning failure rates in substations in consideration of lightning current waveforms 50 References 53 3 Lightning interaction with power transmission lines 57 William A. Chisholm 3.1 Lightning attachment to overhead transmission lines 59 3.1.1 Overhead line attachment rates using ground flash density and typical dimensions 60 3.1.2 Local voltage rise from lightning attachment to transmission line phase conductor 64 3.1.3 Role of span length and nearby arresters on peak insulator voltage 70 3.1.4 Shielding of transmission line phase conductors using overhead groundwires 72 3.2 Lightning impulse flashover of power transmission line insulation 73 3.2.1 Lightning impulse voltage test waveshapes 74 3.2.2 Single gap full-wave flashover strength for dry arc distance of 0.5 to 10 m 76 3.2.3 Strength of multiple air gaps in parallel under shielding failure conditions 80 3.2.4 Strength of multiple air gaps in series 81 3.2.5 Evolution of surge protective devices for insulation coordination 81 3.2.6 Design and performance of unshielded power transmission lines 82 3.3 Bonding, earthing and equalisation of potential differences on transmission lines 85 3.3.1 Analysis of transient voltage rise on connections from OHGW to earthing electrodes 85 3.3.2 Analysis of transient voltage rise on earthing electrodes 89 3.3.3 Analysis of transient voltage reduction from adjacent phases 98 3.3.4 Analysis of transient voltage rise on insulated phases from surge impedance coupling 99 3.3.5 The backflashover from OHGW to phase 102 3.3.6 Design and performance of shielded power transmission lines 103 3.3.7 Methods for increasing the backflashover critical current 104 3.3.8 Methods for improving the equalisation of potential differences 106 viii Lightning interaction with power systems, volume 2
- 14. 3.4 Considerations in the design trade-off: arresters versus earthing 107 References 108 4 Lightning interaction with medium-voltage overhead power distribution systems 113 Alexandre Piantini, Alberto Borghetti and Carlo Alberto Nucci 4.1 Flash collection rate 114 4.2 Effects of various parameters on lightning overvoltages 115 4.2.1 Direct strokes 115 4.2.2 Indirect strokes 117 4.3 Lightning protection of MV systems 132 4.3.1 Increase of the line withstand capability 132 4.3.2 Use of shield wires 133 4.3.3 Application of surge arresters 146 4.4 Lightning performance of overhead distribution lines 153 4.4.1 Influence of the environment around the line 153 4.4.2 Lines located above open ground 154 4.4.3 Lines surrounded by buildings 158 4.4.4 Hybrid configuration (MV and HV lines mounted on the same poles) 160 4.5 Concluding remarks 164 References 164 5 Lightning interaction with low-voltage overhead power distribution networks 173 Alexandre Piantini 5.1 Typical configurations of LV networks 174 5.2 Lightning surges on LV power systems 176 5.2.1 Cloud discharges 176 5.2.2 Direct strikes 177 5.2.3 Indirect strikes 178 5.2.4 Transference from the MV line 201 5.3 Lightning protection of LV networks 207 5.3.1 Distribution transformers 208 5.3.2 LV power installations 212 5.4 Concluding remarks 217 References 218 6 Lightning protection of structures and electrical systems inside of buildings 227 Fridolin Heidler 6.1 Lightning currents 228 6.1.1 Current components 230 Contents ix
- 15. 6.1.2 Lightning protection level 230 6.1.3 Simulation of the lightning currents for analytical purpose 233 6.2 Lightning protection of buildings 233 6.2.1 Lightning protection zone 234 6.2.2 Lightning protection system 236 6.2.3 Surge protection measure (SPM) system 238 6.3 Volume protected against direct lightning strike 241 6.3.1 Striking distance 241 6.3.2 Rolling sphere method 241 6.3.3 Simplifications of the rolling sphere method 242 6.4 Air-termination and down-conductor system 244 6.4.1 Air-termination system 244 6.4.2 Down-conductor system 245 6.4.3 Materials and dimensions 245 6.5 Earth-termination system 247 6.5.1 Earth-termination system for the lightning protection system (LPS) 247 6.5.2 Improved earth-termination system for the surge protection measure (SPM) system 249 6.6 Lightning equipotential bonding 250 6.6.1 Lightning equipotential bonding required for LPS 250 6.6.2 Lightning equipotential bonding according to the surge protection measure (SPM) system 251 6.7 Separation distance 255 6.7.1 Material coefficient km 256 6.7.2 Current steepness coefficient ki 256 6.7.3 Configuration coefficient kc 256 6.8 Currents and voltages on lines 260 6.8.1 Protection of connection lines at the entrance into LPZ 260 6.8.2 Shielded connection lines 261 6.8.3 Lines in reinforced concrete cable duct 262 6.8.4 Current share on lines in case of direct lightning 263 6.8.5 Reduction of the induced over-voltage on internal lines by line routing 264 6.9 Grid-like spatial shield 265 6.9.1 Magnetic field inside LPZ 1 in the case of a direct lightning strike 265 6.9.2 Magnetic field inside LPZ 1 in the case of a nearby lightning strike 266 6.9.3 Magnetic field inside LPZ 2 and higher 267 References 268 x Lightning interaction with power systems, volume 2
- 16. 7 Lightning protection of Smart Grids 271 William A. Chisholm and Kenneth L. Cummins 7.1 Introduction: history of power system technologies 272 7.1.1 Electric power systems and mathematics 272 7.1.2 Electric power systems and communication 273 7.1.3 Electric power systems and lightning measurements 276 7.2 Smart Grid functions and technologies 278 7.2.1 Wide-area monitoring and visualization 278 7.2.2 Flow control 281 7.2.3 Enhanced fault identification 282 7.2.4 Adaptive protection and automated feeder switching 284 7.2.5 Automated islanding and reconnection 286 7.2.6 Diagnosis and notification of equipment condition 286 7.2.7 Dynamic thermal rating capabilities 287 7.3 Lightning and digital recording technology 288 7.3.1 Digital recording systems for lightning overvoltages 288 7.3.2 Voltage sensors for lightning overvoltages 291 7.3.3 Combined current and voltage sensor for lightning measurements 292 7.3.4 Non-contact sensors for impulse voltage and current 293 7.3.5 Commercial current sensors for equipment monitoring 294 7.4 Lightning protection of Smart Grid sensors 295 7.4.1 Reliability requirements for Smart Grid sensors and systems 295 7.4.2 Candidate wiring configurations for Smart Grid sensors 297 7.4.3 Industry standards for lightning protection of Smart Grid sensors 298 7.4.4 Industry standards for lightning protection of Smart Grid communication systems 298 7.4.5 Case study: EMC and residential Smart Grid interoperability 301 7.5 Conclusions 302 References 303 8 Lightning protection of wind power systems 309 Masaru Ishii and Joan Montanyà 8.1 Wind turbine components and overview of the lightning protection system 310 8.2 Lightning phenomenology and wind turbines 311 8.2.1 Interaction with downward lightning 311 8.2.2 Upward lightning 312 Contents xi
- 17. 8.3 Lightning damage to wind turbines due to direct impacts 317 8.3.1 Lightning damage mechanisms 317 8.3.2 Overview of types of lightning damage to wind turbines 318 8.3.3 Statistics on lightning damage to wind turbines 324 8.4 Lightning protection of wind turbine components 324 8.4.1 Blades 324 8.4.2 Consideration of CFRP in blades and other components 327 8.4.3 Other components: hub, bearings and nacelle 329 8.4.4 Overvoltages caused by direct lightning 330 8.5 Overvoltages in wind farms 335 8.5.1 Structure of wind farms 336 8.5.2 Sources of lightning overvoltages in wind farms: the back-flow surge 336 References 339 9 Renewable energy systems—photovoltaic systems 343 Kazuo Yamamoto and Yarú Méndez 9.1 Solar energy: solar radiation, parameters, hourly and daily parameters 343 9.1.1 Daily parameters 344 9.1.2 Second, minute or hourly based parameters 345 9.1.3 Extraterrestrial and terrestrial solar radiation 346 9.2 Photovoltaics: PV cells, PV modules, partial shading and its effects 346 9.3 PV systems: off-grid and grid-connected, considerations of the grid connection 349 9.4 Earthing (grounding) of PV-systems 351 9.4.1 The importance of earthing characteristics 351 9.4.2 The earthing characteristics of photovoltaic systems 354 9.4.3 The earthing characteristics for optimal selection of SPDs 359 9.5 Internal and overvoltage lightning protection 363 9.5.1 Protection at the PV generator’s side or DC side 363 9.5.2 Protection at the AC side 364 9.6 External lightning protection 364 9.6.1 Internal lightning protection 366 9.7 Mounting (racking) systems as air-termination systems 368 9.8 External dedicated mounting systems (non-isolated, isolated) 369 9.9 Concluding remarks 369 References 369 10 Measurement of lightning currents and voltages 371 Ruy Alberto Corrêa Altafim, Wenxia Sima and Qing Yang 10.1 Historical introduction 371 10.2 Lightning current measurements 372 xii Lightning interaction with power systems, volume 2
- 18. 10.2.1 Lightning current measurement methodology on transmission lines 372 10.2.2 Lightning current measurement methodology on high towers 378 10.3 Measurement method of lightning voltage 380 10.3.1 Voltage divider 380 10.3.2 Capacitive sensor connected to bushings 383 10.3.3 Noncontact capacitive voltage divider 384 10.3.4 Integrated optical waveguide voltage (electric field) sensor 385 10.3.5 Crystal-based batteryless and contactless optical transient overvoltage sensor 386 10.3.6 Optical voltage (electric field) sensor based on converse piezoelectric effect 388 10.4 Application of various lightning overvoltage sensors in power systems 389 References 390 11 Application of the FDTD method to lightning studies 393 Yoshihiro Baba and Vladimir A. Rakov 11.1 Introduction 393 11.2 FDTD method 396 11.2.1 Fundamentals 396 11.2.2 Advantages and disadvantages 398 11.3 Representations of lightning source 399 11.3.1 Lightning return-stroke channel 399 11.3.2 Excitation methods 402 11.4 Applications 403 11.4.1 Surges on grounding electrodes 403 11.4.2 Lightning surges on overhead power transmission lines and towers 404 11.4.3 Lightning surges on overhead power distribution and telecommunication lines 405 11.4.4 Lightning electromagnetic environment and surges in power substation 407 11.4.5 Lightning surges on underground distribution and telecommunication lines 408 11.4.6 Lightning surges in wind-turbine-generator towers 408 11.4.7 Lightning surges in photovoltaic arrays 409 11.4.8 Lightning surges and electromagnetic environment in buildings 409 11.4.9 Lightning electromagnetic fields at close and far distances 410 11.4.10 Other applications 412 Contents xiii
- 19. 11.5 Summary 413 References 413 12 Software tools for the lightning performance assessment 425 Alberto Borghetti, William A. Chisholm, Fabio Napolitano, Carlo Alberto Nucci, Farhad Rachidi and Fabio Tossani 12.1 Introduction 425 12.2 FLASH program 426 12.2.1 Simplified modelling of shielding and backflashover calculations 427 12.2.2 Adoption of ‘red book’ method by IEEE, 1982–85 428 12.2.3 Adjustments of IEEE FLASH program, 1985–93 429 12.2.4 Standardizing the IEEE FLASH program, 1993–97 430 12.2.5 Maintaining the IEEE FLASH program, 1997–2007 430 12.2.6 Developing the IEEE FLASH V2.0 program, 2007–19 430 12.3 Lightning-induced overvoltages–electromagnetic transients program 431 12.3.1 Interfacing LIOV with EMTP 432 12.3.2 LIOV–EMTP input parameters 436 12.3.3 Application examples 442 12.4 Application to a real distribution network 444 References 448 Index 453 xiv Lightning interaction with power systems, volume 2
- 20. About the editor Alexandre Piantini graduated in electrical engi- neering from the Federal University of Paraná in 1985 and got his masters and doctoral degrees from the Polytechnic School of the University of São Paulo in 1991 and 1997, respectively. He joined the University of São Paulo in 1986 and served as the director of Technological Development of the Institute of Energy and Environment (1998–2011), where he is Associate Professor and the head of the Lightning and High Voltage Research Centre. He has participated in 26 research projects related mainly to lightning and electromagnetic compatibility (EMC). He coordinated 21 of these projects, of which 15 funded mainly by power com- panies and national agencies for research support. IEEE Senior Member since 2004, he was the Convener of the CIGRÉ WG C4.408 “Lightning Protection of Low-Voltage Networks” and member of various IEEE and CIGRÉ working groups. He is Associate Editor of the IEEE Trans. Electromagnetic Compatibility, High Voltage (IET), Electrical Engineering (Springer), and member of the Editorial Advisory Panel of the Electric Power Systems Research (Elsevier). He is member of the Steering Committee of the Int. Project on Electromagnetic Radiation from Lightning to Tall Structures. He was deputy editor-in-chief of the Journal of Lightning Research (2005–15) and associate editor of The Open Atmospheric Science Journal (2008–13). He has given various invited lectures and courses related to lightning in universities and international conferences organized in Brazil, Sweden, Spain, Colombia, Russia, and China. Prof. Piantini is the chairman of the Int. Symposium on Lightning Protection (SIPDA), vice-chairman of the Int. Conf. Grounding and Earthing & Int. Conf. Lightning Physics and Effects, and member of scientific committees of various conferences such as the Int. Conf. Lightning Protection (ICLP). He is a founder member of the Institute for Lightning Protection and Safety (ILPS), guest professor of the Chongqing University, China, and member of the IEEE Award Committee of the Sun & Grzybowski Award. In 2018, he was the recipient of the ICLP Rudolf Heinrich Golde Award, “for extraordinary theoretical and experimental achievements in lightning protection of power systems.” He is author or coauthor of four book chapters and over 150 scientific papers published in prestigious peer-reviewed jour- nals or presented at international conferences with review board. He has given over 190 interviews to national and regional TV stations, radios, newspapers, etc. in topics related mainly to lightning.
- 22. Preface The importance of improving the reliability and robustness of power systems makes protection of transmission and distribution lines against lightning-related effects a primary concern. This situation stems mainly from the increasing emphasis on overall power system efficiency, the continuous proliferation of equipment sensitive to short duration voltage disturbances, the increasing level of consumer demand for power quality, and the high economic losses associated with power-quality issues. Numerous studies have been carried out in this area with a view to a better understanding of the phenomena involved and the identification of technically and economically viable solutions that provide effective improvement of the quality of energy supplied to consumers. Lightning is particularly noteworthy in this context, as it is often responsible for a significant number of unscheduled outages of power transmission lines and distribution networks even in regions with relatively moderate ground flash den- sities. Besides, renewable electricity generation capacity has been increasing rapidly all over the world. Wind turbines are growing not only in number but also in size, leading to an increasing concern for lightning protection of wind power plants. Lightning is a major source of damages to wind turbines and can cause failures either hitting the turbines directly or inducing transients on the control systems that lead to equipment failure, malfunction or degradation. Photovoltaic (PV) systems may be vulnerable to lightning transients associated with both direct and nearby strikes, which can damage sensitive electronics or weaken the dielectric strength of the PV module insulation. Lightning is a multidisciplinary subject and the importance of understanding the physics of the phenomenon and its interaction with various objects and mate- rials, as well as the need to effectively protect structures, systems, people, and animals against its deleterious effects, has led to the existence of several books involving different lightning-related aspects. However, the current literature lacks a comprehensive work with specific focus on the interaction between lightning and electrical power systems that addresses in depth the lightning protection of trans- mission and distribution networks, including smart grids and renewable energy systems. This is the aim of this book, which contains well-established information and includes the most recent advancements in the field. This book is intended primarily for a two-semester course for undergraduate and graduate students in energy and electrical engineering, but it can be used also for a one-semester or even shorter courses. It is also useful as reference for aca- demic scientists, researchers, and engineers in the areas of electrical engineering
- 23. and physics, power systems consultants, and professionals from electric power companies involved in the fields of lightning protection, electromagnetic compat- ibility, renewable energy systems, and smart grids. The secondary readership consists of professionals from telecommunication companies and manufacturers of power equipment. This book is divided into two volumes. The chapters in Volume 1 describe and discuss the main concepts, fundamentals, and models necessary to understand and evaluate the interaction between lightning and electrical systems. The first chapter is concerned with an assessment of how global lightning may respond to global climate change. In Chapter 2, basic lightning terminology is introduced and the main lightning processes are described. The “classical” dis- tributions of lightning parameters needed in engineering applications are reviewed along with the distributions based on more recent direct current measurements. Correlations between the parameters are discussed and mathematical expressions used to represent lightning current waveforms are reviewed. Chapter 3 introduces the reader to the various concepts used to construct engineering return stroke models. After describing the most important models, it provides a review of the basic features of lightning electromagnetic fields and presents methods for their calculation, including the horizontal electric field associated with return stroke over finitely conducting ground. Chapter 4 provides the basis for calculating ground flash densities, details of techniques used by modern lightning location systems (LLSs), examples of well-established LLSs in different parts of the world, and methods used to validate the performance characteristics of LLSs. In Chapter 5, the physical process and engineering models of lightning attachment to overhead power lines are described in detail and a general procedure for the estimation of lightning incidence to overhead power lines is presented. Chapter 6 presents the coupling of lightning electromagnetic fields to overhead and underground lines based on the transmission line approximation, whereas Chapter 7 addresses the lightning response of grounding electrodes. Chapter 8, which deals with surge protective devices, presents the most common definitions, character- istics, operating mechanisms, classifications, and applications of devices used in transmission and distribution networks, including low-voltage (LV) systems. Chapters 9 and 10 present and discuss models of the most important power trans- mission and distribution (medium and LV) system components for simulations of lightning electromagnetic transients. The second volume, devoted to the applications, contains Chapters 1–12, which cover lightning protection of various systems, including structures and buildings, transmission and distribution networks, renewable energy systems, and smart grids. Chapter 1 is devoted to the application of the Monte Carlo method to lightning protection and insulation coordination practices and describes also the application of the stratified sampling technique to reduce the computational effort usually required. The effect of lightning on the insulation performance of substation equipment is dealt with in Chapter 2, which includes also the evaluation of the failure rates of gas-insulated switchgear and transformers. Chapter 3 organizes the lightning interactions with power transmission lines from the simple consequences of a direct stroke attachment to an unshielded line to the complex consequences of a xviii Lightning interaction with power systems, volume 2
- 24. stroke attachment to a shielded line with multiple ground wires, including the effects from phases protected with line arresters. It builds on the information in previous chapters to develop important measures in transmission line lightning performance. Chapter 4 deals with the lightning impacts on medium-voltage power distribu- tion systems and discusses the effects of the most important parameters on the overvoltages, as well as the effectivenesses of the main protective measures that can be applied to improve the line lightning performance. A procedure for estimating the mean annual number of line flashovers of overhead lines is presented and the light- ning performances of lines with different protective measures are compared. In Chapter 5, devoted to the lightning interaction with LV power distribution networks, the major mechanisms by which lightning overvoltages can be produced are explained and the general surge characteristics are evaluated. The effectiveness of the installation of secondary arresters along the network in protecting the LV side of transformers and consumers’ entrances is also discussed. Chapter 6 is dedicated to the lightning protection of common structures, including their installations and content, and persons as well. Such protection requires the combination of external and internal countermeasures, which are also discussed in the chapter. A broad view of “lightning protection” finds many smart grid applications of real-time lightning information in proactive protection strategies. After presenting the history of power system technologies and describing the roles that lightning research plays in successful integration of digital technologies into electric power systems, Chapter 7 discusses lightning-related digital recording technologies and addresses the lightning protection of smart grid sensors. Chapters 8 and 9 focus on the lightning protection of renewable energy systems. Chapter 8 gives an introduction to wind power generators and their components from the perspective of lightning protection, as well as an overview of lightning occurrence in relation to wind turbines. It presents the mechanisms of lightning damage to wind turbines, their classification and sta- tistics, discusses the protection of the most sensitive components, and describes the mechanisms whereby lightning surges invade a wind farm through a lightning-struck wind turbine. Chapter 9 deals with PV systems and gives a brief introduction to solar radiation, PV cells, modules, and the associated effects of shading. Off-grid and grid- connected PV systems are described and the common configurations of external and internal lightning protection systems are discussed. Chapter 10, which is about measurements of lightning currents and voltages, describes various types of sensors and discusses their application in power systems. Chapter 11 presents the fundamentals of the finite-difference time-domain method and reviews the application of the method to the analysis of lightning electro- magnetic fields and lightning-caused surges in various systems. Chapter 12 describes two of the most adopted software tools for the evaluation of the lightning performance of transmission and distribution lines, namely, FLASH and LIOV- EMTP, together with some application examples. The chapters follow a logical order and ideally should be read sequentially by a beginner reader, but they are self-contained and can be read independently, so that a reader interested in a specific topic can go directly to the relevant chapter. Alexandre Piantini Preface xix
- 26. Acknowledgments I would like to express my sincere thanks to my colleagues and friends, authors of the chapters, for their dedication and esteemed contributions. My special thanks go to Prof. Carlo Alberto Nucci, Prof. Farhad Rachidi, Prof. Marcos Rubinstein, Prof. Vernon Cooray, Prof. Vladimir A. Rakov, and Prof. William A. Chisholm, for the valuable discussions and continuous support. I am also grateful to Dr. Christoph von Friedeburg, Senior Books Commissioning Editor at the IET, for the interesting discussions, and to Ms. Olivia Wilkins, assistant editor at the IET, for her kindness, sincerity, and patience to deal with submission delays. Working with her was indeed a great pleasure. I am indebted to all my former and current students, postdocs, and colleagues, and specially thank Miss Michele N.N. Santos, Ph.D. student, for her precious help during the organization of the book. Finally, I would like to thank my parents, Farley and Elza, my 102-year-old grandmother Nair, my sisters Andrea and Barbara, and my nieces Angel, Farly, and Isabella, for their unlimited love and support and for always bringing joy to my life. Alexandre Piantini
- 28. About the authors Ruy Alberto Corrêa Altafim was born in Agudos, Brazil, on January 4, 1957. He received his Ph.D. degree in electrical engineering from the University of São Paulo, Brazil, in 1991. In 1994, He worked as a guest researcher for the National Institute of Standards and Technology-NIST in USA, with liquid dielectrics. In 1995, he was promoted to associate professor and, in 1997, he had got a university posi- tion as full professor of electrical engineering in the University of São Paulo. In 2001, he was nominated as a member of CEIDP/IEEE board and, in 2013, as an AdCom member of DEIS-IEEE society. He is a member of SIPDA scientific board. He is a member of Editorial Board Associate Editor of IEEE Transactions on Dielectric and Electrical Insulation, and until 2018, at IEEE Electrical Insulation Magazine— regional editor. In 2010, He worked as a guest professor at the University of Potsdam—Germany, with PROBRAL/CAPES financial support in the piezo- electret research area. He is a senior member of IEEE and his special fields of interest are solid and liquid dielectrics, liquid crystal, electrets, piezoelectric sensors, and high-voltage engineering. Dr. Altafim was also head of the Electrical and Computer Department of EESC-USP for 10 years, vice-dean at Engineering School of São Carlos-USP and Pro-Rector at the University of São Paulo. He has also worked as leader of the Applied Electrical Metrology and High Voltage Group and leaded many research projects in areas such as refor- estation wood cross-arm, lightning-induced voltages on distribution power sys- tems, piezoelectret sensors, and impulse impedance of grounding systems. He is Senior Professor of the Electrical Engineering Department of EESC/USP and Visiting Professor at the Federal University of Paraı́ba. He has continually pub- lished in these areas in many journals such as IEEE Transactions on Dielectric and Electrical Insulation, IEEE Transactions on Industry Application, Molecular Crystals, Applied Physics, Journal of Applied Physics and Liquid Crystals, and on many international conferences.
- 29. Yoshihiro Baba received the B.Sc., M.Sc., and Ph.D. degrees from the University of Tokyo in 1994, 1996, and 1999, respectively. In 1999, he joined Doshisha University, Kyoto, Japan, where since 2012 he has been a professor. From April 2003 to August 2004, he was a visiting scholar at the University of Florida. He received the Technical Achievement Award from the IEEE EMC Society in 2014. He is the Chairperson of Technical Program Committee of the 2015 Asia- Pacific International Conference on Lightning (APL), Nagoya, Japan. He has been the vice chairperson of the APL Steering Committee since 2017. He has been the convener of CIGRÉ C4.37 Working Group since 2014. He had been an editor of the IEEE Trans. Power Delivery from 2009 until 2018. He has been an editor of the IEEE Power Engineering Letters since 2009, a guest associate editor of the IEEE Trans. EMC since 2018, and an associate editor of Electric Engineering (Springer Journal) since 2019. He is a fellow of both IET and IEEE. Alberto Borghetti was born in Cesena, Italy, in 1967. He graduated (with honors) in electrical engineering from the University of Bologna, Italy, in 1992. Since then, he has been working with the power system group of the same university, where he is now a professor of electrical power systems. His research and teaching activities are in the areas of power system analysis, power system restoration after blackout, electromagnetic tran- sients, optimal generation scheduling, and dis- tribution system operation. He is the author or coauthor of over 150 scientific papers published in peer-reviewed journals or pre- sented at international conferences. He has served as Technical Program Committee chairperson of the 2010 30th Int. Conf. on Lightning Protection and chair of the 2016 Bologna CIGRÉ Colloquium on Lightning and Power systems. He was special reporter for the Study Committee C4 (System technical perfor- mance) of CIGRÉ 2018, recipient of the Int. Conf. on Lightning Protection Scientific Committee Award in 2016, and of the 2018 CIGRÉ Technical Council Award. He is a fellow of the Institute of Electrical and Electronics Engineers (class 2015) for contributions to modeling of power distribution systems under transient conditions. From 2010 to 2017, he has served as an editor of IEEE Trans. on Smart Grid. Since 2018, he is serving as an editor of IEEE Trans. on Power Systems and as an associate editor of Journal of Modern Power Systems and Clean Energy (MPCE), SGEPRI Press and Springer. Since 2019, he serves as editor in chief of xxiv Lightning interaction with power systems, volume 2
- 30. Electrical Engineering—Archiv für Elektrotechnik, Springer. Since 2008, he is as editorial board member of Electric Power Systems Research, Elsevier. William A. Chisholm (IEEE M’79–SM’90– F’07) was born in New York, USA, in 1955. He received the B.A.Sc. degree in engineering science and the M.Eng. degree in electrical engi- neering from the University of Toronto, Toronto, ON, Canada, in 1977 and 1979, respectively, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1983. He was with Kinectrics, the former Ontario Hydro Research Division, from 1977 to 2007. He continues to serve as a consultant to industry, professor at UQAC, Chicoutimi, QC, Canada, and the University of Toronto and mentor at METSCO, Mississauga, ON, Canada. He has coauthored reference books on icing for IEEE/Wiley, a textbook for Mc-Graw-Hill, and chapters in the Electric Power Research Institute Red, Blue and Gray books and the CRC/IEEE Electric Power Engineering Handbook. Dr. Chisholm was a chair of the IEEE Power and Energy Society Transmission and Distribution Committee, with many contributions to IEEE and CIGRÉ litera- ture and standards related to the effects of adverse weather, including lightning, earthing, icing, and low wind on overhead lines. In addition to IEEE fellow in 2007, he received an IEEE “Best Standard” award for Std. 1243-1997 (‘99), Masters Swim Canada #1 Rank (‘98, ‘00, ‘01, ‘05, ‘06, ‘10, ‘11) and national record (‘15), the Masoud Farzaneh Prize (2014) and the INMR Claude de Tourreil Award (2017). Kenneth L. Cummins (IEEE S’73–M’78– SM’99) received the B.S. degree in electrical engineering from the University of California, Irvine, in 1973, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1974 and 1978, respectively. Until 1989, he was involved in the field of neurosciences as a Research Scientist at Stanford Medical Center and the Veterans’ Administration, and then, as a Staff Scientist at Nicolet Instrument. From 1989 to 2005, he was the R&D Manager and the chief scientist for the Thunderstorm Business Unit, Vaisala (formally Global Atmospherics, Inc.) in Tucson, AZ. He is currently a research professor in the Department of Hydrology and Atmospheric Sciences at the University of Arizona. He is the author or coauthor of more than 85 scientific papers and holds 9 US patents. About the authors xxv
- 31. Dr. Cummins is a member of NASA’s Science Team for the space-based Geostationary Lightning Mappers on the GOES weather satellites. He has served in various IEEE and CIGRÉ Working Groups related to lightning. Over the last 5 years, he received three NASA awards for his research activities and for his service on NASA’s Lightning Advisory Panel. Fridolin H. Heidler received the B.Eng. and the M.Eng. degrees in electrical engineering with special emphasis on high-voltage engineering from the Technical University Munich, Munich, Germany, in 1978 and 1982, respectively, and the Ph.D. and Dr.-Ing. habilitation degrees in the high-voltage engineering, in 1987 and 1999, respectively. From 1987 to 1991, he was with Industrial Engineering Company (IABG), where he was engaged in the field of electro-dynamic calculations in the frequency and time domains. In 1991, he joined the Institute of High Voltage Engineering, University of the Federal Armed Forces, where he is currently a professor of high- voltage engineering. His current research interests include the fields of lightning research, lightning protection, and electromagnetic compatibility (EMC) with main emphasis on numerical calculations of lightning discharge process, and the mea- surement of the currents and electric or magnetic fields from lightning striking the Peissenberg telecommunication tower nearby Munich, Germany. He has authored or coauthored more than 180 scientific papers on lightning protection, lightning research, and electromagnetic compatibility. Masaru Ishii received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Tokyo, Tokyo, Japan, in 1971, 1973, and 1976, respectively. In 1976, he joined the Institute of Industrial Science, the University of Tokyo, where he was a professor during 1992–2013. He became an emeritus professor of the University of Tokyo and an Advisor of Central Research Institute of Power Industry (CRIEPI) in 2013. He became an honorary research advisor of CRIEPI in 2019. He was the vice president of the Institute of Electrical Engineers of Japan from 2007 to 2008 and is the president of the Institute of Electrical Installation Engineers of Japan since 2015. Prof. Ishii is a fellow of IEEE, a fellow of IEE of Japan, and a distinguished member of CIGRÉ. xxvi Lightning interaction with power systems, volume 2
- 32. Since 2015, Dr. Yarú Méndez works as a lecturer in electrical engineering (EE) at the Universidad Simón Bolı́var (USB) and entrepreneur at the company Murayh in Caracas, Venezuela. Main focus of the professional and academic activity is on power systems and renewable energy-based power generation. Previously, he was “Director of Engineering” at the company Raycap GmbH in Munich, Germany, and “Research Engineer” at the company General Electric Global Research (GEGR) in Munich, Germany, working on topics of renewable energy (wind and solar) and their interaction to the grid in terms of lightning and transients. Concerning education, he owns a degree in electrical engineering in power systems from the Universidad Simón Bolı́var (USB), a Dr.-Ing. degree from the University of Kassel (UNIK) in Germany, and a MBA degree from the University of Applied Sciences Munich (HM) in Germany. Currently, Mr. Méndez owns 17 patents and has published 54 scientific pub- lications as author and coauthor. Joan Montanyà was born in Súria (Barcelona), Spain, in 1975. He received the B.S. degree in industrial engineering and M.S. and Ph.D. degrees in electrical engineering from the Polytechnic University of Catalonia, Barcelona, Spain, in 2000 and 2004, respectively. He joined the Department of Electrical Engineering of the Polytechnic University of Catalonia as adjunct lecturer in 1997. In 2003, he became assistant professor, in 2011 associated professor, and in 2017 obtained a full professor position. He did several short stays at the University of Arizona (Tucson, AZ, US), the Laboratoire d’Aérologie (Toulosue, France), and the Massachusetts Institute of Technology (Cambridge, MA, US). He is author and coauthor of more than 150 publications related to atmospheric electricity including lightning protection, transient luminous events, terrestrial gamma ray flashes (TGF), lightning warning, high energy radiation from lightning and laboratory sparks, thunderstorm electrification, severe weather, and Schumann resonance. He has special interest in lightning protection of wind turbine blades with composite materials. He is currently the head of the UPC Lightning Research Group. He participated in more than 15 research projects related to lightning research being principal investigator of 10 projects. Five of these projects are About the authors xxvii
- 33. related to the Atmosphere-Space Interactions Monitor (ASIM) an ESA mission in order to investigate the origin of the TGF. Since 2014, he is member of the International Commission on Atmospheric Electricity. He is also member of several international standardization groups for lightning protection. He is convener of the EU CENELEC TC81X/WG5 for the standard EN 50536 “Protection against lightning—Thunderstorm warning sys- tems.” He is an active member of the IEC TC 88 PT 24 about lightning protection of wind turbines. He is also active in several CIGRÉ SC C4 committees including the committee WG C4.409 related to lightning protection of wind turbine blades, the WG C4.410 about lightning to very tall objects, and the WG C4.36 related to winter lightning. Fabio Napolitano received the M.S. degree (with honors) in electrical engineering and the Ph.D. degree in electrical engineering from the University of Bologna, Italy, in 2003 and 2009, respectively. He is an assistant professor at the Department of Electrical, Electronic, and Information Engineering of the University of Bologna, Italy. Since his graduation, he collaborated with the Power Systems group of the University of Bologna on the analysis of power systems tran- sients, in particular those due to indirect lightning strokes, and lightning protection systems. He is senior member of IEEE and member of CEI Technical Committee 81. He is cur- rently associate editor of the journal Electrical Engineering. Carlo Alberto Nucci graduated with honors in electrical engineering from the University of Bologna, Bologna, Italy, in 1982. He is a full professor and head of the Power Systems Laboratory of the Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi,” University of Bologna. He has authored or coauthored over 370 scientific papers published in peer-reviewed journals or in proceedings of international conferences. Prof. Nucci is a fellow of the IEEE and of the International Council on Large Electric Systems (CIGRÉ), of which he is also an honorary mem- ber, and has received some of the best paper/technical international awards, including the CIGRÉ Technical Committee Award and the ICLP Golde Award. From January 2006 to September 2012, he served as chairman of the CIGRÉ Study Committee C4 ªSystem Technical Performance. He has served as IEEE PES Region 8 Rep in 2009 and 2010. Since January 2010, he has served as editor-in-chief of the Electric Power Systems Research journal (Elsevier). He has served as the president xxviii Lightning interaction with power systems, volume 2
- 34. of the Italian Group of the University Professors of Electrical Power Systems (GUSEE) from 2012 to 2015. He is an advisor of the Global Resource Management Program of Doshisha University, Kyoto, Japan, supported by the Japanese Ministry of Education and Science and has represented PES in the IEEE Smart City Initiatives Program since 2014. Prof. Nucci is doctor Honoris Causa of the University Politehnica of Bucharest and a member of the Academy of Science of the Institute of Bologna. Shigemitsu Okabe received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Tokyo in 1981, 1983, and 1986, respectively. Since 1986, he has been with Tokyo Electric Power Company and presently is the chief researcher at the R&D Department. He was a visiting scientist at the Technical University of Munich in 1992. He has been an adjunct professor of Doshisha Universitysince2005andofNagoyaUniversitysince 2006. He is also the visiting lecturer of the University of Tokyo. He has served as the secretary and or member of several WG/MT in CIGRÉ and IEC. He is an associate editor of the IEEE Transactions on Dielectrics and Electrical Insulation. Alexandre Piantini graduated in electrical engi- neering from the Federal University of Paraná in 1985 and got his masters and doctoral degrees from the Polytechnic School of the University of São Paulo in 1991 and 1997, respectively. He joined the University of São Paulo in 1986 and served as the director of Technological Development of the Institute of Energy and Environment (1998–2011), where he is Associate Professor and the head of the Lightning and High Voltage Research Centre. He has participated in 26 research projects related mainly to lightning and EMC. He coordi- nated 21 of these projects, of which 15 funded mainly by power companies and national agencies for research support. IEEE Senior Member since 2004, he was the Convener of the CIGRÉ WG C4.408 “Lightning Protection of Low-Voltage Networks” and member of various IEEE and CIGRÉ working groups. He is Associate Editor of the IEEE Trans. Electromagnetic Compatibility, High Voltage (IET), Electrical Engineering (Springer), and member of the Editorial Advisory Panel of the Electric Power Systems Research (Elsevier). He is member of the Steering Committee of the Int. Project on Electromagnetic Radiation from Lightning to Tall Structures. He was Deputy Editor-in-Chief of the Journal of Lightning Research (2005–15) and Associate Editor of The Open Atmospheric Science Journal (2008–13). He has given various invited lectures and courses related to lightning in universities and About the authors xxix
- 35. international conferences organized in Brazil, Sweden, Spain, Colombia, Russia, and China. Prof. Piantini is the chairman of the Int. Symposium on Lightning Protection (SIPDA), vice-chairman of the Int. Conf. Grounding and Earthing & Int. Conf. Lightning Physics and Effects, and member of scientific committees of various conferences such as the Int. Conf. Lightning Protection (ICLP). He is a founder member of the Institute for Lightning Protection and Safety (ILPS), guest professor of the Chongqing University, China, and member of the IEEE Award Committee of the Sun & Grzybowski Award. In 2018, he was the recipient of the ICLP Rudolf Heinrich Golde Award, “for extraordinary theoretical and experimental achieve- ments in lightning protection of power systems.” He is author or coauthor of four book chapters and over 150 scientific papers published in prestigious peer- reviewed journals or presented at international conferences with review board. He has given over 190 interviews to national and regional TV stations, radios, news- papers, etc. in topics related mainly to lightning. Farhad Rachidi (M’93–SM’02–F’10) received the M.S. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1986 and 1991, respectively. He was with the Power Systems Laboratory, Swiss Federal Institute of Technology, until 1996. In 1997, he joined the Lightning Research Laboratory, University of Toronto, Toronto, ON, Canada. From 1998 to 1999, he was with Montena EMC, Rossens, Switzerland. He is currently a Titular Professor and the head of the EMC Laboratory with the Swiss Federal Institute of Technology, Lausanne, Switzerland. He has authored or coauthored over 190 scientific papers published in peer-reviewed journals and over 400 papers presented at international conferences. Dr. Rachidi is currently a member of the Advisory Board of the IEEE Transactions on Electromagnetic Compatibility and the president of the Swiss National Committee of the International Union of Radio Science. He has received numerous awards including the 2005 IEEE EMC Technical Achievement Award, the 2005 CIGRÉ Technical Committee Award, the 2006 Blondel Medal from the French Association of Electrical Engineering, Electronics, Information Technology and Communication (SEE), the 2016 Berger Award from the International Conference on Lightning Protection, the 2016 Best Paper Award of the IEEE Transactions on EMC, and the 2017 Motohisa Kanda Award for the most cited paper of the IEEE Transactions on EMC (2012–16). In 2014, he was conferred the title of honorary professor of the Xi’an Jiaotong University in China. He served as the vice-chair of the European COST Action on the Physics of Lightning Flash and its Effects from 2005 to 2009, the chairman of the 2008 European Electromagnetics International Symposium, the president of the International Conference on Lightning Protection from 2008 to 2014, the editor-in-chief of the Open Atmospheric Science Journal (2010–12), and the editor-in-chief of the IEEE Transactions on xxx Lightning interaction with power systems, volume 2
- 36. Electromagnetic Compatibility from 2013 to 2015. He is a fellow of the IEEE and of the SUMMA Foundation, and a member of the Swiss Academy of Sciences. Vladimir A. Rakov received the M.S. and Ph.D. degrees in electrical engineering from the Tomsk Polytechnical University, Russia, in 1977 and 1983, respectively. He is currently a professor in the Department of Electrical and Computer Engineering, University of Florida, Gainesville, and co-director of the International Center for Lightning Research and Testing (ICLRT). He is the author or coauthor of 4 books and over 700 other publications on various aspects of lightning, with about 300 papers being published in peer- reviewed journals. Dr. Rakov is a fellow of four major professional societies, the IEEE, the American Meteorological Society, the American Geophysical Union, and the Institution of Engineering and Technology (formerly IEE). He is also a recipient of Karl Berger Award for distinguished achievements in lightning research, develop- ing new fields in theory and practice, modeling and measurements (2012), and Toshio Takeuti Award for outstanding contribution to worldwide recognition of winter lightning (2017). In 2015, he was awarded honorary doctoral degree by the Institute of Applied Physics of the Russian Academy of Sciences (IAP RAS). Wenxia Sima received the B.S. and Ph.D. degrees in electrical engineering from the Chongqing University, Chongqing, China, in 1988 and 1994, respectively. In 1988, she was involved in high-voltage research at the High Voltage Research Institute (a division of Chongqing University), where from 1996 to 2014, she held the position of the vice director of High Voltage and Insulation Technology Department. From 1994 to 1997, she was an assistant professor of Electrical Engineering at Chongqing University. From 1997 to 2001, she was an associate professor. In 2001, she became a professor at Chongqing University. She is currently the director of High Voltage and Insulation Technology Department, and a professor of the State Key Laboratory of Power Transmission Equipment & System Security and New Technology. She is the author or coauthor of 1 book, 6 patents, and more than 100 papers. She has been in charge of more than 30 scientific research projects, including 3 projects supported by National Science Foundation of China (NSFC) and 2 National Basic Research Program of China. Her research interests are in mechanism of long air gap discharge, lightning shield model and lightning pro- tection theory, space charge measurement in liquids and discharge mechanism, overvoltage monitoring, grounding, and grounding grid diagnosis. About the authors xxxi
- 37. Fabio Tossani (S’15–M’16) received the B.S. (Hons.), M.S. (Hons.), and Ph.D. degree in elec- trical engineering from the University of Bologna, Italy, in 2010, 2012, and 2016, respectively. He is currently a junior assistant professor of the Electrical Power Systems group of the University of Bologna. His research interests are power system transients, with particular reference to lightning electromagnetic pulse interaction with electrical networks, power systems protection, and the integration of renewables in power distribution networks. He is assistant editor of the journal Electric Power Systems Research. Kazuo Yamamoto was born in Osaka, Japan, in 1974. He received the B.E., M.E., and Ph.D. degrees in engineering from Doshisha University, Kyoto, Japan, in 1997, 2000, and 2007, respec- tively. From 1998 to 1999, he was a visiting researcher with Manitoba HVDC Research Centre, Winnipeg, MB, Canada. From 2000 to 2006, he was with Nara National College of Technology, Nara, Japan, and from 2007 to 2012, he was with Kobe City College of Technology, Kobe, Japan. In 2012, he was a visiting researcher with Electro Magnetic Applications, Inc., Lakewood, CO, USA. He is currently an associate professor with the Department of Electrical and Electronic Engineering, College of Engineering, Chubu University, Kasugai, Japan. His research interests include lightning protection for renewable energy systems and automobiles. 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 a professor in the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University. His research interests include lightning protection, overvoltage protection, electric-field measurement, and space charge dynamics. He is the author and coauthor of more than 60 journal and international conference papers. xxxii Lightning interaction with power systems, volume 2
- 38. Chapter 1 Application of the Monte Carlo method to lightning protection and insulation coordination practices Alberto Borghetti1 , Fabio Napolitano1 , Carlo Alberto Nucci1 and Fabio Tossani1 Lightning insulation coordination is based on statistical approaches. This allows to correlate the electrical stress caused by lightning and the electrical strength of the insulations, both having probabilistic nature. This chapter provides an example of lightning insulation coordination. Specifically, it deals with the statistical appraisal of the so-called lightning performance of distribution systems, carried out by means of Monte Carlo (MC) simulations. The relevant application to both the cases of direct and indirect lightning events, considering the correlation between the prob- ability distributions of the lightning current parameters, is described and discussed. In particular, the application to the indirect events is based on the definition of a surface around the power line and on the calculation of the induced voltages along the line caused by indirect events having stroke location uniformly distributed within such a surface. The result obtained through the MC simulations is finally scaled taking into account the annual number of flashes per square kilometer expected in the region of interest. In order to obtain significant results, two aspects need to be considered: the surface around the power line should be large enough in order to collect all the events that may endanger the insulation, and the density of the stroke locations should be sufficiently high. Therefore, for medium voltage systems, or even more for the case of low voltage ones, the area can reach a large value indeed, and the number of events to be considered can be consequently huge. The chapter also describes the application of the stratified sampling technique able to reduce the computation effort typical to this type of calculation. 1.1 Introduction The basic question for the purposes of insulation and protection coordination of distribution lines is how many lightning originated flashovers per year a certain 1 Department of Electrical, Electronic and Information Engineering, University of Bologna, Italy
- 39. distribution line may experience, as a function of its insulation. The attention is mostly focused on the number and intensity of lightning-induced voltages, either because distribution lines are surrounded by elevated objects or because direct strikes protection is uneconomical. The issue has been the object of several studies in the past and it is still of interest due to the stringent power quality requirements of modern distribution networks, especially nowadays, given the increasing amount of distributed generation that is connected to them [1–5]. In [1,2], the frequency of overvoltages exceeding a given insulation level is evaluated by means of analytical methods for the case of an infinite long line over a perfect conducting plane. The amplitude of the lightning current at the channel base is considered as a random variable considering its probability distribution, while the front time of the lightning current and the return stroke velocity are assumed fixed. In [3], a statistical method is employed. Both the probability distribution of amplitude and that of front time of the lightning current are considered, along with the correlation coefficient between the two above-mentioned parameters. The positive value of that coefficient indicates that the higher the current amplitude, the longer the front time of the impulse. The striking distance of the indirect stroke from the line (lightning strokes occurring within a certain critical distance from the line will directly strike the line) is evaluated as a function of the return stroke peak current (while in [1], it is considered as independent of the current). The return stroke velocity is fixed and the ground is assumed as a perfectly conductive plane. In [4], the Monte Carlo (MC) method has been employed to solve the problem. The induced voltages are calculated at the termination of a 2 km line matched at both ends. The MC simulation involves 10,000 events taking place over a surface covering the line and 1 km away from it. The same striking distance equation from the line as in [3] is adopted. The correlation between peak value and front time of lightning current distributions is disregarded. Lightning originated overvoltages in overhead lines are due to both direct strikes and the coupling between the conductors and the electromagnetic pulse generated by nearby strikes to ground (see, e.g., [6,7]). For power distribution overhead lines, characterized by lower insulation and height with respect to trans- mission lines, most of the lightning related flashovers are caused by strikes to the ground or structures located in proximity of the line [8]. As a consequence, the influence of these events on the lightning performance of the line needs to be appropriately assessed to grant the adequate protection. It is also worth adding that while for distribution lines all direct strikes are expected to cause flashovers (unless a large amount of surge arresters and shield wires are installed, ideally one per phase and per pole), only a fraction of nearby strikes to ground are expected to induce voltages larger than the line withstand insulation level. Different is the case for those lightning strokes that, although not hitting directly the line conductor, hit those elevated objects that are located close to the line and therefore can be able to cause flashovers. As described in [5], the standard MC approach for the lightning performance assessment of power distribution lines consists in generating a large number of 2 Lightning interaction with power systems, volume 2
- 40. events, each characterized by different values of lightning current waveshape parameters and different coordinates of the perspective stroke location (i.e., the stroke location in absence of the line). The values of the lightning current para- meters are generated in agreement with the relevant probability distributions available in the literature or provided by lightning location systems (see, e.g., [9,10]). The coordinates of the stroke location are assumed uniformly distributed within a region around the line wide enough to include all the events that may cause a flashover. In several papers (e.g., [11,12]), the calculation of the lightning-induced overvoltages is performed by using simplified formulas in order to reduce the computational effort. However, as the accurate appraisal of the induced voltages can be achieved only by a time-domain electromagnetic transient simulation (in this chapter, performed by using the LIOV–EMTP-RV code described in [6,13,14] and validated by the comparison of the results with several experimental results [15,16]), the lightning performance of distribution networks by means of standard MC simulations involves a significant amount of computational resources. Some recent papers have dealt with the lightning performance assessment by means of electromagnetic transients simulations (e.g., [17–19]), which are quite time consuming. Nearby strikes to ground are often referred as indirect strokes in the relevant literature; this term is also used to indicate other indirect lightning events that are not considered in this chapter, such as side flashes and the line interaction with a ground current (e.g., [20,21]). In [22], a method to reduce the computational effort is presented. It is based on the application of the so-called Mean Square Pure Error (MSPE) algorithm to determine the optimal number of MC extractions and on a 3D interpolation that bypasses the necessity of the time- domain simulation of each MC event. In order to reduce the number of LIOV–EMTP-RV simulations, in [23] a heuristic technique has been proposed for the case of an unprotected network, which has been adapted in [24] to the case of distribution networks with surge arresters. The heuristic technique is conceived to avoid the time-domain compu- tation of events expected to be less harmful than the previously calculated ones. This requires to fix a priori conditions so to discard the upcoming events which, in general, have characteristics (e.g., amplitude) that depend on the particular network configuration, especially in presence of nonlinearities [25]. An improved estimation accuracy can be achieved by two approaches: by increasing the number of replications or reducing the variance of the estimator. A typical variance reduction technique adopted in MC methods is the stratified sampling (e.g., [26,27]) and the application of this technique for the lightning performance assessment of distribution lines has been presented in [28]. The structure of the chapter is the following. Section 1.2 is devoted to the description of the MC approach with particular reference to the random generation of the values of the lightning parameters from a multivariate probability distribu- tion. Section 1.3 describes the identification of the functions that describes the lightning current waveforms. Section 1.4 describes the stratified sampling techni- que. Section 1.5 illustrates the application of the MC method to the case of a Application of the Monte Carlo method to lightning protection 3
- 41. medium voltage (MV) overheard line in open terrain. As mentioned in the con- clusions of the chapter (Section 1.6), the application of the MC method for the lightning performance assessment of lines and networks with realistic configuration will be presented in Chapter 4 of this volume. 1.2 Description of the MC-based procedure As described in detail in what follows, the proposed procedure is based on the application of the MC method and on the calculation of the induced voltages by using the LIOV code. From now on, this procedure will be called LIOV-MC. Such a procedure is defined by the following steps. 1. A large number of lightning events ntot is randomly generated. Each event is characterized by the parameters that describe the current waveform at the channel base and the coordinates of the stroke location. Only negative down- ward first strokes are taken into account; the effects of the presence of positive flashes and subsequent strokes in negative flashes on the line lightning per- formance are assumed negligible.* The stroke locations are assumed to be uniformly distributed within striking area A, having a size large enough to contain the entire line and all the lightning events that could cause voltages larger than the minimum voltage value of interest for the analysis. Typical lightning channel base current waveforms are defined by the following para- meters: peak Ip, front time tf, maximum front steepness Sm and wavetail time to half value th. The relevant probability distributions are provided in [29,30].† The correlation between these parameters has been recently reviewed by Rakov et al. in [9]. In particular, in direct current measurements relatively strong correlation is observed between the current rate-of-rise characteristics and current peak. The MC random generation of lightning current parameters is described hereafter. 2. From the total set of events, those relevant to indirect lightning are selected by adopting a lightning incidence model for the line. In [5], it is analyzed the influence of the adoption of different incidence models. For the calculations of this chapter, we have adopted the electro-geometric model suggested in [8]. * Further investigations are needed to include the effects of subsequent return strokes, solving several open issues, such as (i) relationship between the subsequent stroke’s path and that of the first stroke, (ii) correlation between first and subsequent stroke current parameters, and (iii) number of subsequent strokes. † Note that the simpler Anderson equation for the peak current distribution adopted by IEEE Std. [8], follows the trend of the Cigré two-line distributions comparatively well [29]. These statistical distribu- tions have been inferred mostly from measurements obtained by using instrumented towers. The mea- surements at the towers are affected by reflections [56]. Moreover, the current amplitude distributions of the lightning events collected by the towers are biased toward values higher than those of the distribu- tions of the flashes to ground, as analyzed in, e.g., [31,32] and references therein. These aspects will be deliberately disregarded in this chapter. 4 Lightning interaction with power systems, volume 2
- 42. 3. For each lightning event, the maximum induced voltage value on the line is calculated – as earlier mentioned – by means of the using the LIOV–EMTP- RV code [6,13] that is described in Chapter 12 of this volume. 4. For the entire line, the expected annual numbers of events Fp that causes overvoltages with amplitude larger than a given value V is: Fp ¼ n ntot ANg (1.1) where n ¼ ni þ nd, being ni and nd the number of indirect events and direct events, respectively, that generate overvoltages larger than V and Ng is the annual ground flash density. In this study, we assume Ng ¼ 1 flash/km2 /yr. The estimation of the mean time between failures (MTBF) expected at specific poles of the line, generally of interest for the protection of the trans- formers connected to those poles, is calculated as the inverse of the Fp value given by (1.1) with ni and nd evaluated by comparing the overvoltage at the pole with the withstand voltage of the connected transformer. The application of the MC method requires the knowledge of the multivariate distribution of the lightning current parameters. We reasonably assume that every parameter follows the log-normal probability distribution, as generally done in the literature on the subject. Let x1; . . .; xn be n jointly Gaussian random variable. In our case, they are the four natural logarithms of peak amplitude Ip, equivalent front times tf ¼ t30=0:6 (t30 is the interval from 30% to 90% amplitude intercepts on the wavefront), maximum front steepness Sm, and wavetail time to half value th. The multivariate normal distribution is said to be non-degenerate when the symmetric covariance matrix K is positive definite. In this case, the probability density function is f x1; . . .; xn ð Þ ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p ð Þn jKj p exp 1 2 x m ð ÞT K1 x m ð Þ (1.2) where x is a real n-dimensional column vector, m is the corresponding mean vector, and jKj is the determinant of K. The ijth off-diagonal element of K is given by correlation coefficient rij between xi and xj multiplied by the product of their two corresponding standard deviations (i.e., sxi and sxj), while the iith diagonal element is equal to variance s2 xi of random variable xi. Let be Q ¼ K1 , the conditional variance of xn is ðs xn Þ2 ¼ Var xnjx1; . . .; xn1 ð Þ ¼ 1 Qnn (1.3) and the conditional mean of xn is m xn ¼ E xnjx1; . . .; xn1 ð Þ ¼ mn 1 Qnn X n1 j¼1 Qnjðxj mjÞ (1.4) where Qnj is the njth element of matrix Q. Application of the Monte Carlo method to lightning protection 5
- 43. The MC random generation of a quadruple of lightning current parameters values is obtained by applying the following steps, where Zk; Zkþ1; Zkþ2; Zkþ3 are four standard normal variates. Step (1) for the calculation of an Ip value: (1.1) Ip ¼ expðmln Ip þ sln Ip ZkÞ; Step (2) for the calculation of a tf value: (2.1) s ln tf , s ln Sm , s ln th are calculated by using (1.3); (2.2) m ln tf is calculated by using (1.4); (2.3) tf ¼ expðm ln tf þ s ln tf Zkþ1Þ; Step (3) for the calculation of a Sm value: (3.1) m ln Sm is calculated by using (1.4); (3.2) Sm ¼ expðm ln Sm þ s ln Sm Zkþ2Þ; Step (4) for the calculation of a th value: (4.1) m ln th is calculated by using (1.4); (4.2) th ¼ expðm ln th þ s ln th Zkþ3Þ. A complete set of the data required for the MC generation procedure is pro- vided by Berger and Garbagnati in [30] and is reported in Tables 1.1 and 1.2. For each parameter y, Table 1.1 provides median value yðmln y ¼ ln yÞ and sln y. Table 1.2 provides correlation coefficients rln y1ln y2 between parameters y1 and y2. Since these statistical distributions have been inferred mostly from measure- ments obtained by using instrumented towers, the current amplitude distributions of the lightning events collected by the towers are biased toward values higher than those of the distributions of the flashes to ground, as analyzed in, for example, [31,32] and references therein. This aspect will be deliberately disregarded in this analysis, as earlier mentioned at note.† Table 1.1 Statistical parameters of the log-normal distributions for negative downward first strokes [30] Parameter Median value Standard deviation of the parameter logarithm (base 10) Ip 30 kA 0.26 Tcr 5.5 ms 0.31 Sm 12 kA/ms 0.26 th 75 ms 0.26 Table 1.2 Correlation coefficients between parameters [30] Parameter Ip Tcr Sm th Tcr 0.37 1 Sm 0.36 –0.21 1 th 0.56 0.33 0.1 1 6 Lightning interaction with power systems, volume 2
- 44. The parameters of the distribution relevant to the front duration Tcr is given instead of the ones relevant to tf. Since [29] provides both the parameters of Tcr (the same as Table 1.1) and of tf ( tf ¼ 3:8 ms, sln tf ¼ 0:55) obtained from almost the same set of experimental measurements used in [30], the implemented MC procedure generates the values of Tcr according to step (2). The values relevant to tf are obtained by mul- tiplying each value of Tcr by the ratio between tf and Tcr provided by [29]. If a simple current function, that is, a step waveform, a linearly rising current, linearly rising with flat top (trapezoidal) or with drooping tail, is used for the cal- culation of the maximum overvoltages along the line, the values generated for Ip, tf, and th are directly used. However, for more complex current functions, a specific identification procedure is needed as described in Section 1.3. 1.3 Identification of the lightning current functions As known, the waveform of the return stroke current at the channel base as well at its peak value have a significant influence on the lightning originated overvoltages along the lines. Different functions have been proposed in order to represent the typical lightning current waveform. The most commonly used are: the one adopted by CIGRE WG [29] and the one proposed by Heidler in [33]. Other functions that can be found in the literature are the classical double exponential [34], others derived from it (e.g., [35]), the combination of multiple Heidler functions (e.g., [36]) and more recent ones (e.g., [37]). Functions for multi-peaked waveforms have been also proposed in [38,39]. For a given quadruple of values for Ip, tf, Sm, and th, we describe here the procedures to identify the parameters of the Cigré function and of the Heidler function, as presented in [40] that analyzes the effects of different current wave- forms on the lightning performance of distribution lines for both direct and indirect strokes. Cigré function The current waveform is [29]: i t ð Þ ¼ At þ Btn ; t tn i t ð Þ ¼ I1e ttn ð Þ=t1 I2e ttn ð Þ=t2 ; t tn (1.5) where SN ¼ Smtf =Ip n ¼ 1 þ 2 SN 1 ð Þ 2 þ 1=SN ð Þ tn ¼ 0:6tf 3SN 2 = 1 þ SN 2 A ¼ 1 n 1 0:9 Ip tn n Sm B ¼ 1 tn n n 1 ð Þ Smtn 0:9Ip t1 ¼ th tn ð Þ=ln 2 t2 ¼ 0:1Ip=Sm I1 ¼ t1t2 t1 t2 Sm þ 0:9 Ip t2 I2 ¼ t1t2 t1 t2 Sm þ 0:9 Ip t1 (1.6) Application of the Monte Carlo method to lightning protection 7
- 45. This formulation presents some numerical issues if n 1 or n 55. In case a MC event presents a value of n out of these bounds, the value of Sm is adjusted as Sm ¼ 1:01Ip=tf if n 1 Sm ¼ 12Ip=tf if n 55 (1.7) As this procedure can lead to small errors on the resulting current peak, the current is normalized to the desired peak value. Heidler function The Heidler function is [33]: i t ð Þ ¼ I0 h t=t1 ð ÞN 1 þ t=t1 ð ÞN exp t=t2 ð Þ h ¼ exp t1 t2 t2N t1 1=N # (1.8) It is completely defined by four parameters, that is, I0; t1; t2 and N, which cannot be fully obtained from the values Ip , tf , Sm , and th by analytical equations. In [41], an iterative graphical method is presented to identify the parameters as a function of a waveform, while the useof a genetic algorithm (GA) is adopted in, for example, [42,43]. The following procedure based on MATLAB GA function has been devel- oped. The objective of the algorithm determines a set of values I0; t1; t2 and N such to minimize the following fitness function f ¼ c1 Ipc Ip Ip þ c2 tfc tf tf þ c3 thc th th (1.9) where Ipc, tfc, and thc are the peak value, the equivalent front time, and the time to half value of the current calculated at every iteration of the algorithm, respectively. Parameters c1, c2, and c3 are the weights ascribed to the relative errors of the three parameters Ip , tf , and th , respectively. The algorithm is stopped if the relative errors on the three parameters satisfy all the three following conditions Ipc Ip Ip 0:5%; tfc tf tf 0:5%; thc th th 1% (1.10) The possible values of N are limited to the integer values 2, 3, or 4. At first, the values of c1, c2, and c3 are equal to each other. The initial population size and the maximum number of generations are set to 50 and 100, respectively, and they are subsequently enlarged in case any of the conditions of (1.10) is not satisfied. If after some tens of attempts, conditions (1.10) are still not satisfied, the time to half value is penalized by means of a reduction of c3 with respect to c1 and c2. At the end of this procedure, only for 10 out of 20,000 events the constraint on th is not satisfied, while the constraints relevant to peak and equivalent front time are always fulfilled. Table 1.3 compares the expected median value of the parameters with those obtained by 20,000 current waveforms calculated by using Cigré function (1.5). 8 Lightning interaction with power systems, volume 2
- 46. The small deviations in the Cigré model are due to the corrections previously mentioned. Table 1.3 also compares the expected median value of the parameters with those obtained by 20,000 current waveforms calculated by using Heidler function (1.8) with the parameter given by the GA. The mean errors resulting on 20,000 Heidler waveforms are: 0.004% for Ip, 0.17% for tf, and 0.21% for th. Although the Sm is not taken directly into account by the GA, also the relevant median value is in close agreement with the expected one. 1.4 Stratified sampling technique As mentioned in the Introduction, the calculation of the lightning performance of dis- tribution lines considering also indirect strokes is a typical rare event calculation. Indeed, the endpoints of the 95% confidence interval of the estimateb p ¼ n=ntot of (2.1) are b p Cp b p where Cp is the relative error that can be evaluated as [26]: Cp ¼ 1:96 ffiffiffiffiffiffiffiffiffiffiffi 1 p ntotp s (1.11) under the assumption that the estimator b p is a normal random variable with mean value p and variance equal to p 1 p ð Þ=ntot. As area A needs to be quite wide and the density of events that cause a flash- over decreases with the distance from the feeder, p is generally small. Therefore, in a standard MC approach, ntot needs to be large so to achieve the desired level of accuracy (i.e., a predefined value of Cp). As shown in [28], several advantages with respect to the standard MC method are obtained by the use of the stratified sampling technique: ● it allows a significant computational time reduction (up to more than 75% for the cases analyzed in [28]) while maintaining the accuracy of the solution; ● it reduces the importance of the choice of the smallest area A that includes the perspective stroke locations of all the events inducing voltages exceeding the insulation withstand capability; ● it is directly applicable to the case of networks with complex configuration, with surge arresters and with the adoption of detailed flashover models, while heuristic rules need to be adapted for each specific case. Table 1.3 Median values of the parameters obtained from (1.5) and from the GA compared to the expected values given in [30] Ip (kA) tf (ms) Sm (kA/ms) th (ms) Expected 30.0 3.80 12.0 75.0 Cigré 30.0 3.89 13.1 74.7 Heidler 30.0 3.81 12.0 75.4 Application of the Monte Carlo method to lightning protection 9
- 47. According to this technique, area A is divided in subdomains. The number of events generated in each subdomain is proportional to the variance of the local estimator that is recursively updated. Therefore, the larger the distance to the lines or the shorter the distance to surge arresters, the lower the number of events in the specific subdomain. Let us define X as a random variable such that Xk ¼ 1 if MC event k causes an overvoltage greater than W, 0 if not. As in standard MC methods, the probability p of observing an overvoltage greater than W is estimated by b p ¼ n ntot ¼ 1 ntot X ntot k¼1 Xk (1.12) and the perspective stroke locations of the events in the absence of the power lines and other structures are assumed to be uniformly distributed over area A. As already mentioned, for the application of stratified sampling, total area A is divided in m subdomains and probability p is estimated by b J ¼ X m j¼1 aj A 1 Nj X Nj k¼1 Xjk ! (1.13) where aj is the area of subdomain j, Nj is the number of MC events allocated in subdomain j such that Pm j¼1 Nj ¼ ntot, and Xjk is the k-th observation of X in domain j. As shown in [27], the variance of estimator b J is given by Varðb JÞ ¼ X m j¼1 aj A 2 s2 j Nj (1.14) where s2 j is conditional variance of X in domain j. Varðb JÞ is always smaller or equal to Var X ð Þ=ntot. Since the values of s2 j are not known a priori, they are initially estimated by a certain number of pilot runs, that is, by the simulation of some MC events. For this purpose, the same number Ns j of starting events is generated in each sub-domain, with Ns j chosen large enough to estimate s2 j even with a very small probability p, for example, in presence of surge arresters (SAs). In order to obtain a near-random sample distribution for both the parameters of the lightning current waveshape and the stroke location with a limited Ns j , the starting events in each subdomain j are generated by using the Latin hypercube sampling (LHS) [44]. The use of LHS allows an improved accuracy on the initial estimation of s2 j with respect to the usual random sampling. The endpoints of the 95% confidence interval of the estimate b J are b J CJ b J where CJ is the relative error provided by CJ ¼ 1:96 b J ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X m j¼1 aj A 2 s2 j Nj v u u t (1.15) 10 Lightning interaction with power systems, volume 2
- 48. Starting from the initial value of s2 j , the procedure adds new MC events until CJ becomes lower than the desired estimation error. Each new MC event is allocated in the area A according to a weighted uniform distribution with different weights for each subdomain. The weight of each sub- domain j is proportional to the corresponding conditional variance s2 j and the sum of the weights of all the m subdomains is equal to 1. Indeed, the minimum variance value given by (1.14) is obtained when the events are allocated proportionally to the conditional variance s2 j of each subdomain, provided that all subdomains have the same area [27]. If a too small Ns j is adopted, a null value can be obtained in the first estimate of s2 j , in this case the initial guess of the weight of the subdomain can be set according to the mean values of the variances of the adjacent subdomains. The values of s2 j are updated recursively after each MC run, so that their estimation is progressively improved as the number of simulated events increases. 1.5 Application results for a MV overhead line in open terrain 1.5.1 Influence of the return stroke current waveform In order to illustrate the application of the MC method, we consider here a simple three-phase overhead line, straight in shape. The conductors are assumed horizon- tally placed at 9.3 m above ground, with diameter equal to 1 cm. The distances between the lateral conductors and the central one are 1.5 and 0.7 m. In all the calculations of this section, we have assumed the soil conductivity equal to 1 mS/m. The striking area A is chosen as a 1-km band from the line. The number of lightning events n is 20,000. Direct events nd are 1,208. Indirect events nd are 18,792. Figure 1.1 shows the annual number of overvoltages of the line with length equal to 2 km for the three different current waveforms adopted caused by indirect events only. Such a result is here denoted as the perspective lightning performance of the distribution line, as the calculations are run in absence of surge arresters and by neglecting the flashovers along the line. The steady state voltage at the utility frequency is not taken into account in the calculations. Figure 1.1 shows that the choice of different current waveforms has a limited impact on the estimation of the perspective lightning performance. It is worth mentioning that the lowest curve is obtained by using the Heidler function. Without surge arresters and flashovers, all direct events result in overvoltages larger than the maximum value in abscissa (i.e., 0.24 events/year for the case of the 2-km long line), as expected, since all the Ip of the MC generated events are greater than 2 kA. Indeed, the peak current values included in the Berger-Garbagnati dis- tribution are all above 2 kA (i.e., 2 or 3 kA was the triggering level of the Italian measuring stations, while the minimum peak current value included in Berger’s distribution is 2 kA). Application of the Monte Carlo method to lightning protection 11
- 49. Let us now consider the same three-conductor line but with sets of three surge arresters installed at different distance intervals. The voltage–current characteristic of the adopted 15-kV class surge arresters is the one reported in [45]. The con- sidered rated voltage at the utility frequency is 13.8 kV. The distance between subsequent poles is 50 m. The parameters of the dis- ruptive effect criterion (adopted for the representation of the flashovers in the line insulators) are reported in Table 1.4. These parameters have been obtained in [46] from the results of laboratory tests performed on a 15 kV pin-type ceramic insulator. We assume the presence of a transformer at the middle point of the line. The withstand voltage of the transformer is assumed to be constant and equal to 110 kV, as suggested in [47].‡ In [48], a procedure able to take into account the withstand probability distribution of transformer insulation is described. Figure 1.2 shows the top view of the line; the position of the transformer is denoted by the cross in the middle of the line while the circles indicate the surge arrester locations. Distance d defines the interval between consecutive surge arresters. The length of the line is 2 km and the number of generated events in the MC procedure is again 20,000. Tables 1.5 and 1.6 show the MTBF values relevant to a transformer connected to the middle of the line for the two different insulators described in Table 1.4, corresponding to a critical flashover voltage (CFO) of 100 10–1 100 150 200 250 Voltage (kV) 300 350 Trapz Heidler Cigré Number of events having amplitude larger than the abscissa/yr Figure 1.1 Comparison of the perspective indirect lightning performances calculated by using three different current waveforms. Length of the line equal to 2 km. Adapted from [40] ‡ In the computation of the MTBF values, the transformer failure is expected to occur if the voltage amplitude exceeds the withstand voltage of the transformers even for a very short time interval. 12 Lightning interaction with power systems, volume 2
- 50. 165 and 100 kV, respectively. The results are reported for the three different current waveforms and for different distances d between consecutive SAs, namely 100, 200, 300, and 400 m. The comparison between Tables 1.5 and 1.6 shows that with SAs at d ¼ 200 m and above, the MTBF values calculated for insulator CFO ¼ 165 kV are higher than those calculated for CFO ¼ 100 kV, while the opposite happens without SAs or with d ¼ 100 m. Such a difference is due to additional flashovers near the transformer for the case of 100 kV CFO insulators with respect to the case with 165 kV CFO insulators. The oscillatory transients originated by these flashovers and by the associated reflections at the surrounding SAs may cause voltages, at the transformer location, with peak value higher than without flashovers. These over- voltages are not limited by the nearby SAs if the distance between them is sig- nificant, namely d ¼ 200 m and above. A similar unfavorable effect of large separating distances between SAs has been already observed and discussed in [49]. The adoption of a different current waveform generally results in slight dif- ferences between the MTBF values. The larger differences – concerning the direct events only – appear to be those relevant to the case with d ¼ 100 m and CFO ¼ 165 kV, for which the adoption of the Heidler and the Cigré turn out in an 8% increase of the MTBF. By increasing the distance between SAs, these differ- ences tend to be negligible. Concerning the indirect events only, the differences are negligible for the cases of d ¼ 100 m and d ¼ 200 m due to the very low probability of exceeding the withstand voltage of the transformer. The differences turn out to be appreciable, instead, by increasing d, while again they are negligible if the line is unprotected. These variations in the results are ascribed to the effect of nonlinearity introduced by SAs and flashover model that enhance the effect of the difference among the chosen current waveforms and in particular of their front. The computational cost due to the assumption of more realistic current waveforms rather than the trapezoidal one is quite heavy. The time required to obtain the results of both Tables 1.5 and 1.6 of this volume relevant to indirect Table 1.4 Parameters assumed for the disruptive effect model DE model parameters CFO (kV) V0 (kV) k DE (kVms) 100 90 1 60.9 165 132 1 255 2,000 m d Figure 1.2 Top view of the line with the indication of the observation point in the middle of the line and the position of the SAs Application of the Monte Carlo method to lightning protection 13
- 53. strikes for the case of trapezoidal current waveform is about 8 h and about twice the time for the two other currents waveforms (almost all the computational effort is required by the calculation of the lightning electromagnetic pulse (LEMP), which is performed only once for each MC event and then used for all the different SA configurations and insulator types). 1.5.2 Application of the recursive stratified sampling technique In this section, the earlier illustrated recursive stratified sampling procedure is applied to the case of a single-conductor straight line, assuming a withstand voltage value W equal to 150 kV. The line is 2 km long, 10 m high, and is matched at both terminations with the matrix of surge impedances to render more straightforward the interpretation of the results. In the simulations of this section, a linearly rising current with flat top is assumed for the representation of the channel base lightning-current waveform, with peak amplitude Ip and equivalent front time tf. The return-stroke propagation speed is set to 1.5 108 m/s. The lightning performance is evaluated for overhead lines above a soil with conductivity equal to 0.001 S/m and relative permittivity equal to 10. In Figure 1.3, the top view of one half of the considered area A is reported (the line is located on the x-axis and the distribution of the stroke locations is 2,000 1,800 1,600 1,200 1,400 1,000 800 400 200 600 –1,000 –600 –400 –800 –200 200 m m 600 800 400 1,000 0 0 Figure 1.3 Position of the events generated by the stratified sampling procedure for the case of an unprotected line (direct strikes in red, nearby strikes to ground in blue). Adapted from [28] 16 Lightning interaction with power systems, volume 2
- 54. obviously symmetric with respect to the line). The figure shows also the stroke locations of all the simulated events, that is, the initial pilot events and those gen- erated by the stratified sampling procedure. The red dots represent direct strikes to the line while the nearby strikes to ground are indicated in blue. For the considered case, the semi-area is a 2 2 km2 , divided in m ¼ 400 subdomains of area 0.1 0.1 km. The number of pilot events simulated before starting the stratification proce- dure is 5,200. The stratified sampling calculation is stopped when relative error CJ reaches the same value of Cp calculated with the standard MC method with 100,000 events for W ¼ 150 kV. The total number of events generated by the stratified sampling procedure are 3,464 direct strikes and 23,093 nearby strikes, as reported in Table 1.7 of this volume. Since only the evaluation of the voltages induced by nearby strikes needs time domain LIOV–EMTP-RV simulations, the computational time reduction indicated in Table 1.7 of this volume is estimated as 100 nind;p nind;J =nind;p, where nind,J is the number of events required by the stratified sampling and nind,p is the corresponding number calculated in the standard MC. As shown by Figure 1.3, the procedure allocates the majority of the events in the subdomains closest to the line. A very few events are allocated farther than 1.2 km from the line, since the initial variance in those subdomains is null. In order to limit the computational time of the standard MC procedure, the smallest area A that includes all the dangerous events is to be chosen, as discussed in [5]. The capability of the stratified sampling to recursively allocate the events in the subdomains with the largest variance value reduces the importance of an accurate choice of the smallest area A. Figure 1.3 also shows that fewer events are allocated near the matched termi- nations with respect to the subdomains close to the internal part of the line, due to the risers effect that reduce the induced overvoltages [50]. The above-described assessment has been repeated for the case of a line pro- tected with SAs. Two cases are considered with SAs placed every d ¼ 500 m and d ¼ 200 m, respectively, starting from the line terminations. The line terminations are open. The voltage–current characteristic of the considered SAs is the same used in [51]. Table 1.7 Comparison between number of direct and nearby strikes in the standard MC and stratified sampling. Computational time reduction due to stratified sampling Relative error % Direct strikes Nearby strikes Time saved % Standard/ stratified Standard/ stratified Unprotected 2.4 3,103/3,464 96,897/23,093 76 With SAs d ¼ 500 m 2.8 12,166/12,095 187,834/42,455 77 With SAs d ¼ 200 m 10.7 24,665/18,687 175,335/50,163 71 Application of the Monte Carlo method to lightning protection 17
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. [156] [157] Eight hours after she had crept into the luxurious bed in the guest room of the strange lodge, Marian stirred, then half awake, felt the drowsy warmth of wolf-skin rugs. For a moment she lay there and inhaled the drug- like perfume of balsam and listened to the steady breathing of the Eskimo girl beside her. She was about to turn over for another sleep, when, from some cell of her brain where it had been stowed the night before, there came the urge that told her she must make haste. “Haste! Haste! Haste!” came beating in upon her drowsy senses. It was as if her brain were a radio, and the message was coming from the air. Suddenly she sat bolt upright. At the same instant she found herself wide awake, fully alert and conscious of the problems she must face that day—the passing of the rapids and covering a long span of that trail which still lay between them and their goal. She did not waken Attatak. That might not be necessary for another hour. She sprang out upon the heavy bear skin rug, and there went through a set of wild, whirling gestures that limbered every muscle in her body and sent the red blood racing through her veins. After that she quickly slipped into her blouse, knickers, stockings and deerskin boots, to at last go tiptoeing down the corridor toward the large living-room where she heard the roar of the open fire as it raced up the chimney. She found her host sitting by the fire. In the uncertain light he appeared haggard and worn, as if quite done in from some great exertion. Of course Marian could not so much as guess how he had spent the night. She had slept through it all.
- 57. [158] With a smile of greeting the old man motioned her to a seat beside him. “You’ll not begrudge an old man a half hour’s company?” he said. “Indeed not.” “You’ll wish to ask me things. Everyone who passes this way wants to. Mostly they ask and I don’t tell. A fair lady, though,” there was something of ancient gallantry in his tone, “fair ladies usually ask what they will and get it, too.” For a moment he sat staring silently into the fire. “This house,” he said at last, “is a bit unusual. That pipe organ, for instance—you wouldn’t expect it here. It came here as if by accident; Providence, I call it. A rich young man had more things than he knew what to do with. The Creator sent some of them to me. “As for me, I came here voluntarily. You have probably taken me for a prospector. I have never bought pick nor pan. There are things that lure me, but gold is not one of them. “I had troubles before I came here. Troubles are the heritage of the aged. I sometimes think that it is not well to live too long. “And yet,” he shook himself free of the mood; his face lighting up as he exclaimed, “And yet, life is very wonderful! Wonderful, even up here in the frozen north. I might almost say, especially here in the north.
- 58. [159] [160] “I came here to be alone. I brought in food with a dog team. I built a cabin of logs, and here I lived for a year. “One day a young man came up the river in a wonderful pleasure yacht and anchored at the foot of the rapids. Being a lover of music, he had built a pipe organ into his yacht; the one you heard last night.” “And did—did he die?” Marian asked, a little break coming in her voice. “No,” the old man smiled, “he tarried too long. Being a lover of nature—a hunter and an expert angler—and having found the most ideal spot in the world as long as summer lasted, he stayed on after the frosts and the first snow. I was away at the time, else I would have warned him. I returned the day after it happened. There had been a heavy freeze far up the river, then a storm came that broke the ice away. The ice came racing down over the rapids like mad and wrecked his wonderful yacht beyond all repair. “We did as much as we could about getting the parts on shore; saved almost all but the hull. He stayed with me for a few days; then, becoming restless, traded me all there was left of his boat for my dog team. “That winter, with the help of three Indians and their dogs, I brought the wreckage up here. Gradually, little by little, I have arranged it into the form of a home that is as much like a boat as a house. The organ was unimpaired, and here it sings to me every day of the great white winter.” He ceased speaking and for a long time was silent. When he spoke again his tones were mellow with kindness and a strange joy.
- 59. [161] “I am seldom lonely now. The woods and waters are full of interesting secrets. Travellers, like you, come this way now and again. I try to be prepared to serve them; to be their friend.” “May—may I ask one question?” Marian suggested timidly. “As many as you like.” “How did you know I was at the door last night when you were playing? You did not see me. You couldn’t have heard me.” “That,” he smiled, “is a question I should like to ask someone myself; someone much wiser than I am. I knew you were there. I had been feeling your presence for more than an hour before you came. I knew I had an audience. I was playing for them. How did I know? I cannot tell. It has often been so before. Perhaps all human presence can be felt by some specially endowed persons. It may be that in the throngs of great cities the message of soul to soul is lost, just as a radio message is lost in a jumble of many messages sent at once. “But then,” he laughed, “why speculate? Life’s too short. Some things we must accept as they are. What’s more important to you is that your sleds are beyond the rapids. When breakfast is over, you can strap your sleeping bags on your deer and I will guide you over the trail around the rapids to the point where I left your sleds.” A look of consternation flashed over Marian’s face. She was thinking of the ancient dishes and how fragile they were. “I have some fragile articles in the sleeping bags,” she said. “They—they might break!”
- 60. [162] [163] “Break?” He wore a puzzled look. For a second she hesitated; then, reassured by the kindly face of the gentle old man, decided to tell him the story of their adventure in the cave. Then she launched into the story with all the eagerness of a discoverer. “I see,” he said, when she had finished the story. “I know just how you feel. However, there is now only one safe thing to do. Leave these treasures with me. If the rapids are frozen over when the time comes for the return trip, you can pass here and get them. You’ll always be welcome. Better leave an address to which they may be sent in case you should not pass this way. The rapids freeze over every winter. I will surely be able to get them off on the first river boat. They can be sent to any spot in the world. To attempt to pack them over on your deer would mean certain destruction.” Reluctant as Marian was to leave the treasure behind, she saw the wisdom of his advice. So, feeling a perfect confidence in him, she decided to leave her treasure in his care. Then she gave him her address at Nome, with instructions for shipping should she fail to return this way. “One thing more I wanted to ask you,” she said. “How many men are there at the Station?” “One man; the trader. He stays there the year ’round.” “One man!” she exclaimed. “One is all. Time was when there were twenty. Prospectors, traders, Indians, trappers. Two years ago forest fires destroyed the timber. The game sought
- 61. [164] [165] other feeding grounds and the trappers, traders and Indians went with them. Gold doesn’t seem to exist in the streams hereabouts, so the prospectors have left, too. Now one man keeps the post; sort of holding on, I guess, just to see if the old days won’t return.” “Do you suppose he could—could leave for a week or two?” Marian faltered. “Guess not. Company wouldn’t permit it.” “Then—then—” Marian set her lips tight. She would not worry this kind old man with her troubles. The fact remained, however, that if there was but one man at the Station, and he could not leave, there was no one who could be delegated by the Government Agent to go back with her to help fight her battles against Scarberry. Suddenly, as she thought of the weary miles they had travelled, of the hardships they had endured, and of the probability that they would, after all, fail in fulfilling their mission, she felt very weak and as one who has suddenly grown old.
- 62. [166] CHAPTER XX A MESSAGE FROM THE AIR A cup of perfect coffee, followed by a dash into the bracing Arctic morning, completely revived Marian’s spirits. Casting one longing look backward at the mysterious treasure of ancient dishes and old ivory, throwing doubt and discouragement to the winds, with energy and courage she set herself to face the problems of the day. The passing of the rapids by the overland trail was all that their host had promised. Struggling over rocky, snow-packed slopes; slipping, sliding, buffeted by strong winds, beaten back by swinging overhanging branches of ancient spruce and firs, they made their way pantingly forward until at last, with a little cry of joy, Marian saw their own sleds in the trail ahead. “That’s over,” she breathed. “How thankful I am that we did not attempt to make it with the sleds, or with our treasure on the backs of the deer. There would not have been left a fragment of our dishes as big as a dime. As for the sleds, well it simply couldn’t be done.” “No-me,” sighed Attatak.
- 63. [167] “I wonder how he could have brought them by the rapids?” Marian mused as she examined the sleds. There were flakes of ice frozen to the runners. She could only guess at the method he had used, only dimly picture the struggle it must have taken. Even as she attempted to picture the night battle, a great wave of admiration and trust swept over her. “The treasure is safer in his hands than in ours,” she told herself. “But, after it has left his hands?” questioned her doubting self. “Oh well,” she sighed at last, “what must be, will be. The important thing after all is to reach the station before the Agent has started on his way.” Again her brow clouded. What if there was no one to go back with her? To dispel this doubt, she hastened to hitch her deer to her sled. Soon they were racing away over the trail, causing the last miles of their long journey to melt away like ice in the river before a spring thaw. In the meantime a third startling revelation had come to Patsy. First she had discovered that at least one of the persons connected with the strange purple flame was a girl. Next she had found the red trail of blood that apparently was made by one of Marian’s slain deer, and which led to the door of their tent. The third discovery had nothing to do with the first two, nor with the purple flame. It was of a totally different nature, and was most encouraging.
- 64. [168] [169] “If only Marian were here!” she said to herself as she paced the floor after receiving the important message. This message came to her over the radiophone. It was not meant particularly for her, nor for Marian. It was just news; not much more than a rumor, at that. Yet such news as it was, if only it were true! Faint and far away, it came drifting in upon the air from some powerful sending station. Perhaps that station was Fairbanks, Dawson or Nome. She missed that part of the message. Only this much came to her that night as she sat at their compact, powerful receiving set, beguiling the lonesome hours by catching snatches of messages from near and far: “Rumor has it that the Canadian Government plans the purchase of reindeer to be given to her Eskimo people on the north coast of the Arctic. Five or six hundred will be purchased as an experiment, if the plan carries. It seems probable that the deer purchased will be procured in Alaska. It is thought possible to drive herds across the intervening space and over the line from Alaska, and that in this way they may be purchased by the Canadian Agent on Canadian soil. A call for such herds may be issued later over the radio, as it is well known that many owners of herds have their camps equipped with radio-phones.” There the message ended. It had left Patsy in a fever of excitement. Marian and her father wished to sell the herd. It was absolutely necessary to sell it if Marian’s hopes of continuing her education were not to be blasted. There was no market now for a herd in Alaska.
- 65. [170] In the future, as pastures grew scarcer, and as herds increased in numbers, there would be still less opportunity for a sale. “What a wonderful opportunity!” Patsy exclaimed. “To sell the whole herd to a Government that would pay fair prices and cash! And what a glorious adventure! To drive a reindeer herd over hundreds of miles of rivers, forests, tundra, hills and mountains; to camp each night in some spot where perhaps no man has been before; surely that would be wonderful! Wonderful!” Just at that moment there entered her mind a startling thought. Scarberry’s camp, too, was equipped with a radio-phone. Probably he, too, at this very moment, was smiling at the prospect of selling six hundred of his deer. He wanted to sell. Of course he did. Everyone did. He would make the drive. Certainly he would. “And then,” she breathed, pressing her hands to her fluttering heart, “then it will be a race; a race between two reindeer herd; a race over hundreds of miles of wilderness for a grand prize. What a glorious adventure!” “If only Marian were here,” she sighed again. “The message announcing the plans may come while she is gone. Then—” She sat in a study for a long time. Finally she whispered to herself: “If the message comes while she is gone; if the opportunity is sure to be lost unless the herd starts as soon as the message comes, I wonder if I’d dare to start on the race with the herd, with Terogloona and without Marian and Attatak. I wonder if I would?”
- 66. [171] [172] For a long time she sat staring at the fire. Perhaps she was attempting to read the answer in the flames. At last, with cheeks a trifle flushed, she sprang to her feet, did three or four leaps across the floor, and throwing off her clothing, crept between the deer-skins in the strange little sleeping compartment.
- 67. [173] CHAPTER XXI FADING HOPES Just at dawn of a wonderfully crisp morning, Marian found herself following her reindeer over a trail that had recently been travelled by a dog team. She was just approaching the Trading Station where the questions that haunted her tired brain would be answered. Since leaving the cabin in the forest above the rapids, she and Attatak had travelled almost day and night. A half hour for a hasty lunch here and there, an hour or two for sleep and for permitting the deer to feed; that was all they had allowed themselves. An hour earlier, Marian had felt that she could not travel another mile. Then they had come upon the trail of the dog team, and realizing that they were nearing their goal, her blood had quickened like a marathon racer’s at the end of his long race. No longer feeling fatigue, she urged her weary reindeer forward. Contrary to her usually cautious nature, she even cast discretion to the winds and drove her deer straight toward the settlement. That there were dogs which might attack her deer she knew right well. That they were not of the
- 68. [174] species that attacked deer, or that they were chained, was her hope. So, with her heart throbbing, she rounded a sudden turn to find herself within sight of a group of low-lying cabins that at one time had been a small town. Now, as her aged host had said, it was a town in name only. She knew this at a glance. One look at the chimneys told her the place was all but deserted. “No smoke,” she murmured. “Yes, one smoke,” Attatak said, pointing. It was true. From one long cabin there curled a white wreath of smoke. For a moment Marian hesitated. No dogs had come out to bark, yet they might be there. “You stay with the deer,” she said to Attatak. “Tether them strongly to the sleds. If dogs come, beat them off.” She was away like an arrow. Straight to that cabin of the one smoke she hurried. She caught her breath as she saw a splendid team of dogs standing at the door. Someone was going on a trip. The sled was loaded for the journey. Was it the Agent’s sled? Had she arrived in time? She did not have long to wait before knowing. She had come within ten feet of the cabin when a tall, deep- chested man opened the door and stepped out. She caught her breath. Instantly she knew him. It was the Agent.
- 69. [175] [176] He, in turn, recognized her, and with cap in hand and astonishment showing in his eyes, he advanced to meet her. “You here!” he exclaimed. “Why Marian Norton, you belong in Nome.” “Once I did,” she smiled, “but now I belong on the tundra with our herd. It is the herd that has brought me here. May I speak to you about it?” “Certainly you may. But you look tired and hungry. The Trader has a piping Mulligan stew on the stove. It will do you good. Come inside.” An Indian boy, who made his home with the Trader, was dispatched to relieve Attatak of her watch, and Marian sat down to enjoy a delicious repast. There are some disappointments that come to us so gradually that, though the matters they effect are of the utmost importance, we are not greatly shocked when at last their full meaning is unfolded to us. It was so with Marian. She had dared and endured much to reach this spot. She had arrived at the critical moment. An hour later the Agent would have been gone. The Agent was her friend. Ready to do anything he could to help her, he would gladly have gone back with her to assist in defending her rights. But duty called him over another trail. He had no one, absolutely no one to send from this post to execute his orders. “Of course,” he said after hearing her story, “I can give you a note to that outlaw, Scarberry, but he’d pay no attention to it.”
- 70. [177] “He’d tear it up and throw it in my face,” asserted Marian stoutly. “I’ll tell you what I’ll do,” said the Agent, rising and walking the floor. “There is Ben Neighbor over at the foot of Sugar Loaf Mountain. His cabin is only three days travel from your camp. He’s a good man, and a brave one. He is a Deputy Marshal. If I give you a note to him, he will serve you as well as I could.” “Would we need take a different trail home?” “Why? Which way did you come?” Marian described their course. The Agent whistled. “It’s a wonder you didn’t perish!” “Here,” he said, “is a rough map of the country. I will mark out the course to Ben’s cabin. You’ll find it a much safer way.” “Oh, all right,” she said slowly. “Thanks. That’s surely the best way.” She was thinking of the treasure left at the cabin. She had hoped to return by that route and claim it. Now that hope was gone.
- 71. CHAPTER XXII A FRUITLESS JOURNEY It was night; such a night as only the Arctic knows. Cold stars, gleaming like bits of burnished silver in the sky, shone down upon vast stretches of glistening snow. Out of that whiteness one object loomed, black as ink against the whiteness of its background. Weary with five days of constant travel, Marian found herself approaching this black bulk. She pushed doggedly forward, expecting at every moment to catch a lightning-like zig-zag flash of purple flame shooting up the side of it. The black bulk was the old dredge in Sinrock River. She had passed that way twice before. Each time she had hoped to find there a haven of rest, and each time she had been frightened away by the flash of the purple flame. Those mysterious people had left this spot at one time. Had they returned? Was the dredge now a place of danger, or a haven for weary travellers? The answer to this question was only to be found by marching boldly up to the dredge.
- 72. [178] [179] This called for courage. Born with a brave soul, Marian was equal to any emergency. Sheer weariness and lack of sleep added to this a touch of daring. Without pausing, she drove straight up to the door. Reassured by the snow banked up against it, she hastily scooped away the bank with her snow-shoe, and having shoved the door open, boldly entered. It was a cheerless place, black and empty. The wind whistled through the cracks where the planks had rotted away. Yet it was a shelter. Passing through another door, she found herself in an inner room that housed the boiler of the engine that had furnished power to the dredge. The boiler, a great red drum of rust, stood directly in front of her. “Here’s where we camp,” she said to Attatak. “We can build a fire in the fire-box of the boiler and broil some steak. That will be splendid!” “Eh-eh,” grinned Attatak. “And Attatak, bring the deer through the outer door, then close it. They were fed two hours ago. That will do until morning.” She lighted a candle, gathered up some bits of wood that lay strewn about the narrow room, and began to kindle a fire while Attatak went out after the deer. For the moment, being alone, she began to think of the herd. How was the herd faring? What had happened to Patsy during those many days of her absence? Were Bill Scarberry’s deer rapidly destroying her herd ground.
- 73. [180] [181] “Well, if they are, we are powerless to prevent it,” she told herself with a sigh. As she looked back upon it now, she felt that her whole journey had been a colossal failure. They had discovered the mountain cave treasure, only to be obliged to leave the treasure behind. They had reached the Station in time to talk with the Government Agent, but he had not been able to come with her. Only twenty-four hours before they had reached the cabin of Ben Neighbor, only to find it dark and deserted. He had gone somewhere, as people in the Arctic have a way of doing; and where that might be she could not even hazard a guess. At last, in despair, she had headed her deer toward her own camp. In thirty-six hours she would be there. “Well, at any rate,” she sighed, “it will be a pleasure to see Patsy and to sleep the clock round in our own sweet little deerskin bedroom.” She was indeed to see Patsy, but the privilege of sleeping the clock round was not to be hers for many a day. She was destined to find the immediate future far too stirring for that. Twenty-four hours later saw Marian well on her way home. Ten hours more, she felt sure, would bring her to camp. And then what? She could not even guess. Had she been able to even so much as suspect what was going on at camp, she would have urged her reindeer to do their utmost. Patsy was right in the middle of a peck of trouble. Because of the fact that for the last few days she had been living in a realm of exciting dreams, the troubles
- 74. [182] that had come down upon her seemed all the more grievous. Since that most welcome radio message regarding the proposed purchase of reindeer by the Canadian Government had come drifting in over the air, she had, during every available moment, hovered over the radio-phone in the momentary expectation of receiving the confirmation of that rumor which might send the herd over mountains and tundra in a wild race for a prize, a prize worth thousands of dollars to her uncle and cousin—the sale of the herd. Perhaps it was because of her too close application to the radio-phone that she failed to note the approach of Scarberry’s herd as it returned to ravish their feeding ground. Certain it was that the first of the deer, with the entire herd close upon their heels, were already over the hills before she knew of their coming. It was night when Terogloona brought this bit of disquieting news. “And this time,” Patsy wailed, “we have not so much as one hungry Eskimo with his dog to send against them.” As if in answer to the complaint, the aged herder plucked at her sleeve, then led her out beneath the open sky. With an impressive gesture, he waved his arm toward the distant hills that lay in the opposite direction of Scarberry’s herd. To her great surprise and mystification, she saw gleaming there the lights of twenty or more campfires. “U-bogok,” (see there) he said.
- 75. [183] [184] “What—what does it mean?” Patsy stammered, grasping at her dry throat. “It is that I fear,” said Terogloona. “They come. To- morrow they are here. You gave food for a week for a few; flour, sugar, bacon. They like him. Now come whole village of Sitne-zok. Want food. You gave them food. What you think? No food for herders, no herders. No herders, no herd. What you think?” Patsy did not know what to think. Gone was all her little burst of pride over the way she had handled the other situation that had confronted her. Now she felt that she was but a girl, a very small girl, and very, very much alone. She wished Marian would come. Oh, how she did wish that she would come! “In the morning we will see what can be done,” was all she could say to the faithful old herder as she turned to re-enter the igloo. That night she did not undress. She sat up for hours, trying to think of some way out. She sat long with the radio head-set over her ears. She entertained some wild notion of fleeing with the herd toward the Canadian border, providing the message confirming the offer for the deer came. But the message did not come. At last, in utter exhaustion, she threw herself among the deerskins and fell into a troubled sleep. She was roused from this sleep by a loud: “Hello there!” followed by a cheery: “Where are you? Are you asleep?” It was Marian. The next moment poor, tired, worried Patsy threw herself sobbing into her cousin’s strong arms.
- 76. [185] “There now,” said Marian, soothingly, as Patsy’s sobbing ceased, “sit down and tell me all about it. You’re safe; that’s something. Your experiences can’t have been worse than ours.” “The Eskimo! Bill Scarberry’s herd!” burst out Patsy, “They’re here. All of them!” “Tell me all about it,” encouraged Marian. “Wait till I get my head-set on,” said Patsy, more hopefully. “It’s been due for days; may come at any time.” “What’s due?” asked Marian, mystified. “Wait! I’ll tell you. One thing at a time. Let’s get it all straight.” She began at the beginning and recited all that had transpired since Marian had left camp. When she came to tell of her discovery that one of the mysterious occupants of the tent of the purple flame was a girl, Marian’s astonishment knew no bounds. When told of the bloody trail, Marian was up in arms. The camp of the purple flame must be raided at once. They would put a stop to that sort of thing. They would take their armed herders and raid that camp this very night. “But wait!” Patsy held up a warning finger, “I am not half through yet. There is more. Too much more!” She was in the midst of recounting her experiences with the band of wandering Eskimo and Scarberry’s herd, when suddenly she clapped the radio receiver tightly to her ears and stopped talking. Then she murmured:
- 77. [186] “It’s coming! At last, it is coming!” “For goodness sake!” exclaimed Marian, out of all patience, “Will you kindly tell me what is coming?” But Patsy only held the receiver to her ears and listened the more intently as she whispered: “Shush! Wait!”
- 78. [187] CHAPTER XXIII PLANNING THE LONG DRIVE The message that was holding Patsy’s attention was one from the Canadian Government. It was a bonafide offer from that Government to purchase the first herd of from four to six hundred reindeer that should reach Fort Jarvis. When Patsy had imparted the exciting news to her, Marian sat long in silent thought. Fort Jarvis, as she well knew, lay some five hundred miles away over hills and tundra. She had just returned from one such wearisome journey. Should she start again? And would this second great endeavor prove more successful than the first? Of all the herds in Alaska, two were closest to Fort Jarvis; Scarberry’s and her own. She had not the slightest doubt that Scarberry would start driving a section of his herd toward that goal. It would be a race; a race that would be won by the bravest, strongest and most skillful. Marian believed in her herders. She believed in herself and Patsy. She believed as strongly in her herd, her sled-deer and her dogs. It was the grand opportunity; the way out of all troubles. That the band of begging natives would not follow, she knew right
- 79. [188] well. Nor would the mysterious persons of the purple flame camp; at least, she hoped not. As for their little herd range, if they sold their deer, Scarberry might have it, and welcome; if they did not sell, they could doubtless find pasture in some far away Canadian valley. “Yes,” she said in a tone of decision, “we will go. We will waken the herders at once. Come on, let’s go.” As they burst breathlessly into the cabin of their Eskimo herders, they received something of a shock. Since all the work of the day had long since been done, they had expected to find the entire group of four assembled in the cabin, or asleep in their bunks. But here was only old Terogloona and Attatak. “Where’s Oatinna? Where’s Azazruk?” demanded Marian. “Gone,” said Terogloona solemnly. “Where? Go call them, quick!” Terogloona did not move. He merely shrugged his shoulders and mumbled: “No good. Gone long way. Bill Scarberry’s camp. No come back, say that one.” “What!” exclaimed Marian in consternation. “Gone? Deserted us?” “Eh-eh,” Terogloona nodded his head. “Say Bill Scarberry pay more money; more deer; say that one Oatinna, that one Azazruk. No good, that one Bill Scarberry, me think.” He shook his head solemnly. “Not
- 80. [189] listen that one Oatinna, that one Azazruk. Say wanna go. Go, that’s all.” “Then we can’t start the herd,” murmured Marian, sinking down upon a rolled up sleeping-bag. “Yes, we will!” she exclaimed resolutely. “Terogloona, where are the rifles?” “Gone,” he repeated like a parrot. “Mebby you forget. That one rifle b’long herder boys.” “And your rifle?” questioned Marian, “where is your rifle?” “Broke-tuk. Hammer not want come down hard. Not want shoot, that one rifle, mine.” Marian was stunned with surprise and chagrin. She and Patsy returned silently to their igloo. “Oh, that treacherous Bill Scarberry!” she exploded. “He has known this was coming. He knew our herders were energetic and capable. He thought if they remained with us, we might beat him to the prize; so he sent some spy over here to buy them away from us with promises of more pay.” “And now?” asked Patsy. “Now he will drive his herd to Fort Jarvis and sell it, and our grand chance is gone forever.” “No!” exclaimed Patsy, “He won’t! He shall not! We will beat him yet. We are strong. Terogloona and Attatak are faithful. We have our three collies. We can do it. We will beat him yet. Our herd is better than his. It will travel
- 81. [190] [191] faster. Oh, Marian! Somehow, somehow we must do it. It’s your chance! Your one big, wonderful opportunity.” “Yes,” exclaimed Marian, suddenly fired by her cousin’s hot blooded southern enthusiasm, “we will do it or perish in the attempt. It’s to be a race,” she exclaimed, “a race for a wonderful prize, a race between two large herds of reindeer over five hundred miles of hills, tundra and forest. There may be wolves in the forests. In Alaska dangers lurk at every turn; rivers too rapid to freeze over and blizzards and wild beasts. We will be terribly handicapped from the very start. But for father’s sake we must try it.” “For your father’s and for your own sake,” murmured Patsy. “And, Marian, I have always believed that our great Creator was on the side of those who are kind and just. Bill Scarberry played us a mean trick. Perhaps God will somehow even the score.” An hour was spent in consultation with old Terogloona. His face became very sober at the situation, but in the end, with the blood of youth coursing eternally in his veins, he sprang to his feet and exclaimed: “Eh-eh!” (Yes-yes) “We will go. Before it is day we will be away. You go sleep. You must be very strong. In the morning Terogloona will have reindeer and sleds ready. We will call to the dogs. We will be away before the sun. We will shout ‘Kul-le-a-muck, Kul-le-a-muck’ (Hurry! Hurry!) to dogs and reindeer. We will beat that one Bill yet. “You know what?” he exclaimed, his face darkening like a thundercloud, “You know that mean man, that one Bill Scarberry. Want my boy, So-queena, work for him. Want
- 82. [192] [193] pay him reindeer. Give him bad rifle, very bad rifle. Want shoot, my boy So-queena. Shot at carabou, So- queena. Rifle go flash. Crooch! Just like that. Shoot back powder, that rifle. Came in So-queena’s eyes, that powder. Can’t see, that one. Almost lost to freeze, that one, So-queena. Bye’m bye find camp. Stay camp mebby five days. Can see, not very good. Bill, he say: ‘Go herd reindeer,’ So-queena, he say: ‘Can’t see. Mebby get lost. Mebby freeze’. “He say Bill very mad. ‘Get out! No good, you! Go freeze. Who cares?’ “So-queena come my house—long way. Plenty starve. Plenty freeze. No give reindeer that one So-queena, that one Bill. Bad one, that Bill. So me think; beat Bill. Sell reindeer herd white man. Think very good. Work hard. Mebby beat that one Bill Scarberry.” There came a look of determination to Patsy’s face such as Marian had never seen there. “If that’s the kind of man he is; if he would send an Eskimo boy, half-blinded by his own worthless rifle, out into the snow and the cold, then we must beat him. We must! We must!” said Patsy vehemently. “That’s exactly the kind of man he is,” said Marian soberly. “We must beat him if we can. But it will be a long, hard journey.” They had hardly crept between their deerskins when Patsy was fast asleep. Not so Marian. The full responsibility of this perilous journey rested upon her shoulders. She knew too well the hardships and dangers they must face. They must pass through broad stretches of forest where food for the deer was scarce,
- 83. [194] and where lurking wolves, worn down to mere skeletons by the scarcity of food, might attack and scatter their herd beyond recovery. They must cross high hills, from whose summits the snow at times poured like smoke from volcanoes in circling sweeps hundreds of feet in extent. Here there would be danger of losing their deer in some wild blizzard, or having them buried beneath the snows of some thundering avalanche. “It’s not for myself alone that I’m afraid,” she told herself. “It’s for Patsy, Patsy from Kentucky. Who would have thought a girl from the sunny south could be so brave, such a good sport.” As she thought of the courageous, carefree manner in which Patsy had insisted on the journey, a lump rose in her throat, and she brushed a hand hastily over her eyes. “And yet,” she asked herself, “ought I to allow her to do it? She’s younger than I, and not so strong. Can she stand the strain?” Again her mind took up the thought of the perils they must face. There were wandering tribes of Indians in the territory they must cross; the skulking and oft-times treacherous Indians of the Little Sticks. What if they were to cross the path of these? What if a great band of caribou should come pouring down some mountain pass and, having swallowed up their little herd, go sweeping on, leaving them in the midst of a great wilderness with only their sled-deer to stand between them and starvation.
- 84. [195] [196] As if dreaming of Marian’s thoughts, Patsy suddenly turned over with a little sobbing cry, and wound her arms about Marian. “What is it?” Marian whispered. Patsy did not answer. She was still asleep. The dream soon passed, her muscles relaxed, and with a deep sigh she sank back into her place. This little drama left Marian in an exceedingly troubled state of mind. “We ought not to go,” she told herself. “We will not.” Then, from sheer exhaustion, she too, fell asleep. Three hours before the tardy Arctic sunrise, she heard Terogloona pounding at their door. She found that sleep had banished fear, and that every muscle in her body and every cell of her brain was ready for action, eager to be away. As for Patsy, she could not dress half fast enough, so great was her desire for the wonderful adventure.
- 85. [197] CHAPTER XXIV CAMP FOLLOWERS It was just as Marian was tightening the ropes to the pack on her sled that, happening to glance away at a distant hill, she was reminded of Patsy’s latest story of the purple flame. From the crest of that hill there came a purple flare of light. Quickly as it had come, just so quickly it vanished, leaving the hill a faint outline against the sky. “The purple flame,” she breathed. “I wonder if we can leave those mysterious camp-followers of ours behind?” On the instant a disturbing thought flashed through her mind. It caused an indignant flash of color to rise to her cheek. “I wonder,” she said slowly, “if those mysterious people are spies set by Bill Scarberry to dog our tracks?” “They may start with us,” she smiled to herself, as she at last dismissed the subject from her mind, “but unless they really are Bill Scarberry’s spies and set to watch us, they’ll never finish with us. Camp-followers don’t follow
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