Phd

University of Bologna


Department of Electrical Engineering

PhD in Electrical Engineering
XIV course




MODELING OF LIGHTNING-INDUCED VOLTAGES
ON DISTRIBUTION NETWORKS FOR THE SOLUTION
OF POWER QUALITY PROBLEMS,
AND RELEVANT IMPLEMENTATION
IN A TRANSIENT PROGRAM



PhD Thesis of
MARIO PAOLONE


Tutor
Prof. CARLO ALBERTO NUCCI
PhD Coordinator
Prof. FRANCESCO NEGRINI

1998-2001


I
Index
Chapter 1. – Introduction…………………………………………………………. pag. 1
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines…. pag. 4
2.1. – Engineering return-stroke current models………………………………… pag. 6
2.2. – Lightning electromagnetic field appraisal………………………………… pag. 12
2.3. – Transmission line coupling models……………………………………….. pag. 14
Chapter 3. – LEMP-to transmission-line coupling models for distribution
systems and their implementation in a transient program………..

pag. 24
3.1. – Lines with transverse discontinuities……………………………………… pag. 24
3.1.1. – Proposed model………………………………….…………………… pag. 25
3.1.2. – Numerical implementation…………………………………………… pag. 26
3.1.3. – Comparison with experimental results and simulations …………….. pag. 36
3.2. – Distribution systems………………………………………………………. pag. 49
3.2.1. – Proposed approach…………………………………………………… pag. 50
3.2.2. – Interface between the developed line model and transient programs... pag. 51
3.2.3. – Comparison with experimental results and simulations...…………… pag. 56
Chapter 4. – Influence of lightning-induced overvoltages on power quality…... pag. 70
4.1. – Statistical evaluation of the lightning performance of distribution lines….. pag. 72
4.1.1. – Procedure based on the Monte Carlo method and on the developed
LEMP-to-transmission-line coupling models for distribution lines….

pag. 72
4.1.2. – Comparison with the IEEE Std 1410-1997…………………………... pag. 74
4.1.3. – Sensitivity analysis…………………………………………………… pag. 78
4.2. – Case of a typical Italian distribution line………………………………….. pag. 84
4.2.1. – Model extension……………………………………………………… pag. 84


II
4.2.2. – Results………………………………………………………………... pag. 87
Chapter 5. – Conclusions………………………………………………………….. pag. 90
References………………………………………………………………………….. pag. 94
Appendix A.1. – Numerical treatment of line transverse discontinuities in 1
st

order FDTD scheme……………………………………………..

pag. 104
Appendix A.2. – Comparison between 1
st
and 2
nd
order FDTD integration
schemes…………………………………………………………..

pag. 107



III
List of symbols

A
e
: incident vector potential;
e
z
A : vertical component of the incident vector potential;
B: total magnetic field;
B
e
: external incident magnetic field;
B
r
: radial component of the LEMP magnetic field;
B
s
: scattered magnetic field;
s
y
B : scattered induction magnetic field;
C': single-conductor line capacitance per-unit-length;
c: vacuum light velocity;
C
dyn
: voltage-dependent dynamic capacitance in the corona phenomenon;
[ ]
ij
C' : multi-conductor line capacitance matrix per unit length;
d: distance of the stroke location from the line;
d
l
: lateral attractive distance;
E: total electric field;
E
e
: external incident electric field;
e
z
E : vertical component of the incident electric field along the z axis;
n
k
Eh : spatial-temporal discretized value of the horizontal component of the incident electric
field along the line;
e
x
E : horizontal component of the incident electric field along the x axis;
r
E : radial component of the LEMP electric field;
r
E : Fourier-transform of the electric field radial component;
rp
E : Fourier-transform of the electric field radial component for an ideal ground;
E
s
: scattered electric field;
E
z
: vertical component of the LEMP electric field;
erfc: complementary error function;
F
p
: number of annual insulation flashovers per km of distribution line;
G' : single-conductor line conductance per-unit-length;
G
1
, G
2
: Bergeron equivalent generators;
H: lightning channel height;


IV
h: single-conductor line height;
p
H
φ
: Fourier-transform of the magnetic field radial component for an ideal ground;
i
0’
(t) current at the EMTP node;
i
g
: current that flow through the connection between a line conductor and ground;
I
p
: lightning return stroke peak value;
I
0
: channel-base current peak value of the Heidler function;
) , ( t z i : spatial-temporal lightning current distribution along the return stroke channel;
) , ( t x i : single-conductor line current induced along the line;
n
k
i : spatial-temporal discretized value of the induced current along the line;
[i
gp
]: vector of the induced currents diverted to ground in correspondence of the pole;
[ ] ) , ( t x i
i
: multi-conductor line induced current vector along the line;
k: spatial discretization index;
kmax: maximum value of the spatial discretization index;
k
1,
k
2
: parameters related to the corona model;
L': single-conductor line external inductance per-unit-length;
[ ]
ij
L' : multi-conductor line external inductance matrix per unit length;
n: exponent of the Heidler function;
n: temporal discretization index;
NC: number of line conductors;
N
g
: ground flash density;
nmax: maximum value of the temporal discretization index;
) O( x ∆ : remainder term, which approaches zero as the first power of the spatial increment;
) O(
3
t ∆ : remainder term, which approaches zero as the third power of the temporal
increment;
P
i
: probability of lightning current peak to be within interval i;
R: distance of the electric dipole from the observation point;
r: projection of R in the plane xy;
R
0
: left line termination resistive load;
r
g
: striking distances to ground;
R
g
: pole grounding resistance;
R
L
: right line termination resistive load;


V
[R
gp
]: diagonal matrix of the phase-to-ground resistances of the poles when a flashover
occurs;
r
s
: striking distances to wire;
v: return stroke wave front velocity;
) , ( t x v
i
: single-conductor line incident voltage along the line;
) , ( t x v
s
: single-conductor line scattered voltage along the line;
) , ( t x v : single-conductor line total voltage along the line;
v
0’
(t): voltage at the EMTP node;
n
k
v : spatial-temporal discretized value of the scattered voltage along the line;
) , ( ' t x v
g
: voltage drop due to the transient ground impedance;
( ) t , x v
th
: corona threshold voltage;
[ ] ) , ( t x v
e
i
: multi-conductor line incident voltage along the line;
[ ] ) , ( t x v
s
i
: multi-conductor line scattered voltage vector for along the line;
[ ] ) , ( t x v
i
: multi-conductor line total voltage vector for along the line;
Z
0
: vacuum impedance;
Z
c
: single-conductor line characteristic impedance;
'
g
Z : ground impedances;

'
w
Z : wire impedances;
∆t: time integration step;
∆x: spatial integration step;
δ: skin depth factor;
ε
0
: vacuum permittivity constant;
ε
rg
: relative ground permittivity;
φ : total induced scalar potential;
φ
i
: incident scalar potential;
Γ
0
(t) is the EMTP termination type;
Γ : integro-differential operator for the representation of a single-conductor line transverse
discontinuities;
[ΓK]: integro-differential matrix operators for the representation of a multi-conductor line
transverse discontinuities;


VI
η: correction factor of the Heidler function;
λ: decay constant of the MTLE model;
µ
0
: vacuum permeability constant;
µ
g
; ground permittivity;
ρ: correlation coefficient between lightning current amplitude and its front duration;
σ
g
: ground conductivity;
τ
1
: time constant of the wave front of the Heidler function;
τ
2
: time constant of the wave decay of the Heidler function;
τ
g
: ground time constant;
ω: angular frequency;
ξ’
g
: inverse Fourier-transform of the ground impedance;
⊗: convolution product.
Chapter 1. – Introduction pag.

1
Chapter 1. – Introduction

One of the main causes of a-periodic disturbances on distribution networks, which
seriously affect the power quality, is certainly the lightning activity
1
, and in particular the
so-called lightning ‘indirect’ activity
2
.
Due to the limited height of distribution lines of medium and low voltage distribution
networks as compared to that of the structures in their vicinity, indirect lightning return
strokes are more frequent events than direct strokes
3
, and for this reason we shall focus on
such a type of lightning event.
The analysis of the distribution networks response against Lightning Electro Magnetic
Pulse (LEMP), requires the availability of accurate models of LEMP-illuminated lines.
These should be able to reproduce the real and complex configuration of distribution
systems including the presence of shielding wires and their groundings, as well as that of
surge arresters and distribution transformers.
In addition to the accurate modeling of the overhead lines, the development of models of
the entire distribution networks is clearly necessary. This should allow, in principle, to
optimize the number and location of protective devices and then to minimize the number of
outages.
The first part of the research activity here reported has been dedicated to the development
of a complex LEMP-illuminated line model based on a modification of the Agrawal et al.
[1980] coupling model, able of predicting LEMP response of a multi-conductor overhead
line with transverse discontinuities represented by groundings of shielding wire, or neutral
conductor, and surge arresters on the phase conductors.
The model has been then implemented in a computer code using a numerical integration
scheme based on the finite difference time domain (FDTD) technique. In particular a 2
nd

order FDTD scheme has been selected to improve the numerical stability of the line model.
The developed line model has been validated by means of a series of results obtained
during an experimental campaign carried out at the Swiss Federal Institute of Technology in
Lausanne during the year 2000 by means of a NEMP simulator.
The natural extension of the developed LEMP-illuminated line model consists in a
LEMP-illuminated distribution system. This can be accomplished, in principle, by
appropriately rewriting, for each type of termination or model component, the boundary
condition of each illuminated line forming the distribution system. An alternative approach
has been used to develop such a complex distribution system model. The model is based on

1
[e.g. Boonseng and Kinnares, 2001].
2
[e.g. Gunther and Mehta, 1995].
3
[Rusck, 1977; IEEE WG on lightning performance of distribution lines, 1997].
Chapter 1. – Introduction pag.

2
the contemporary use of the above mentioned illuminated line model, and the Electro
Magnetic Transient Program (EMTP96). Other similar programs have been presented in the
literature: they present limitations concerning the calculation of the LEMP and the type of
phenomena that can be dealt (e.g. corona), which the present model is aimed at overcoming.
The developed complex system model has also been validated by means of experimental
results provided by the lightning group of the University of Saõ Paulo.
We have then made use of the Monte Carlo method and of the developed models, to
assemble a statistical procedure aimed at assessing the lightning performance of distribution
lines.
So far, several studies on the subject have been carried out without considering the real
configuration of the distribution systems, i.e. without considering the presence of shielding
wires, of their groundings, of the steady-state voltage superimposed to that induced by
lightning and, more important, without any indication on the nature of the real number of
faults. The procedure here proposed is aimed at overcoming the above mentioned
limitations.


The structure of the thesis is the following:

Chapter 2. A résumé on the models proposed and applied in the literature to calculate
lightning-induced overvoltages is given. The chapter is divided in three paragraphs to
illustrate: a) the most used lightning current return stroke models, b) the relevant
electromagnetic field appraisal and c) the coupling between the LEMP and the overhead
multi-conductor line. Section c) also compares the simplified formula of Rusck [1958] with
the Agrawal et al. [1980] coupling model.

Chapter 3. This chapter illustrates the models implementation. It gives a description of
the illuminated-line model proposed to take into account line transverse discontinuities. It
illustrates the numerical implementation of a FDTD 2
nd
order integration scheme, applied to
the complex line model. The experimental validation of such model is then shown by means
of the experimental results obtained with the EMP simulator of the Swiss Federal Institute
of Technology. A sensitivity analysis on the effect of periodical groundings of the shielding
wire, and on the presence of surge arresters on lightning induced overvoltages is illustrated
and discussed.
In the same chapter is also described the illuminated distribution system model based on
an interface of the developed line model with the Electro Magnetic Transient Program
(EMTP96). A similar interface has been realized also concerning the Power System
Blockset in Matlab Environment and a description of it is also given.
Chapter 1. – Introduction pag.

3
A comparison between theoretical and experimental results is presented. It is based on
the experimental results obtained using a LEMP simulator on a reduced scale distribution
system by the lightning research group at the University of Saõ Paulo.

Chapter 4. The statistical procedure based on the developed models and on the Monte
Carlo method is first illustrated and then compared, for simple line configuration, with the
similar one proposed in the IEEE Std 1410-1997. A sensitivity analysis is then given and
discussed which involves the main parameters such as the ground conductivity, the multi-
conductor line configuration, the shielding wire height, its grounding resistance and
grounding spacing and the presence of surge arresters. The possibility to take into account
the steady-state line voltage and the change of the phase-to-ground coupling factor when a
flashover occurs, is finally illustrated and its implications are discussed. The application of
the statistical procedure to a typical Italian distribution line is finally presented in order to
evaluate the number and type of flashovers that occur per 100 km per year.

Chapter 5 is devoted to the conclusions.

This thesis represents one of the results obtained in a framework of an international
collaboration between the University of Bologna, the Swiss Federal Institute of Technology
of Lausanne and the University of Rome ‘La Sapienza’.

I would like to thank chiefly prof. Carlo Alberto Nucci for his fundamental guide all
along the three years of PhD and his constant and helpful support. Acknowledgements are
also due to prof. Dino Zanobetti for precious advises during the layout of this thesis and to
prof. Alberto Borghetti for important discussions on the statistical procedure and on the
implementation of the developed models in the transient program.
I would also like to thank prof. Farhad Rachidi for his fundamental help in the
experimental activity carried out at the Swiss Federal Institute of Technology in Lausanne
and for very useful discussions.
Thanks are also due to prof. Alexandre Piantini for providing the experimental results of
the induced surges on the reduced scale distribution system.
Acknowledgements are due to dr. Alberto Gutierrez who has developed the Mat-LIOV
code and who has contributed in developing the Mat-LIOV interface with the Power system
Blockset in Matlab environment.

Finally I would to thank Luigia for her comprehension and fundamental support all along
the three years of PhD.

Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


4
Chapter 2. – Evaluation of lightning-induced voltages on
transmission lines

The estimation of lightning induced overvoltages has been the object of various studies
since the early years of the past century.
The first studies reported by Wagner K.W. [1908], Bewley [1929], Norinder [1936]
considered the overvoltage as being produced basically via electrostatic induction by a
charged cloud. According to Wagner K.W. [1908], when the lightning discharge occurs, the
charge bound to the line is released in form of traveling waves of voltage and current.
Wagner did not consider the electromagnetic field radiated by the lightning return-stroke
current.
In the early 1940's Wagner C.F and McCann [1942], based on Schonland's [1934]
investigations of the nature of the lightning discharge, published a paper in which the
overvoltage was considered mostly due to the return stroke phase; an assumption that was
accepted in practically all the subsequent studies
1
. Rachidi et al. [1994] have shown that for
lightning particularly close to the distribution line (30 m or so), some important
overvoltages can be produced also by leader phase preceding the return stroke one.
However, in what follows, we shall be interested only in the voltages induced by the
electromagnetic field change produced by the return-stroke phase
2
.
In parallel with the theoretical studies, several measurement campaigns on lightning
electromagnetic fields
3
and many tests with voltages induced on experimental lines
4
have
been performed. For our purposes it is worth reminding that practically all the above

1
[Golde, 1954; Lundholm, 1957; Rusck, 1958; Papet-Lépine, 1960; Chowdhuri, 1969,1989; Fisher and
Uman, 1972; Cinieri and Fumi , 1978; Hamelin et al., 1979; Eriksson et al., 1982; Saint-Privat-d'Allier
Research Group, 1982; Yokoyama, 1984; Master and Uman, 1984; Liew and Mar, 1986; Diendorfer, 1990;
Zeddam and Degauque, 1990; Michishita et al., 1991; Imai et al., 1993; Ishii et al. 1994, Nucci, et al.
1993,2000].
2
All the models dealt with in this thesis can indeed be extended to take into account also the leader field in
relatively simple manner. It is reminding that taking into account the leader field not change significantly the
results reported here.
3
[Lin and Uman, 1973; Tiller et al., 1976; Weidman and Krider, 1978; Lin et al., 1979; McDonald et al.,
1979; Yokoyama, 1980; Cooray and Lundquist, 1982; Krider and Guo, 1983; Cooray and Lundquist, 1985;
Rakov and Uman, 1990; Rakov et al., 1998, Shostak et al., 2000; Willett and Krider, 2000; Rachidi et al.,
2001].
4
[Norinder, 1936; Berger, 1955; Koga et al., 1981; Eriksson et al., 1982; Yokoyama et al., 1983;
Darveniza and Uman, 1984; Master et al., 1984; Cooray and de la Rosa, 1986; Yokoyama et al., 1986; de la
Rosa et al., 1988; Rubinstein et al., 1989; Yokoyama et al., 1989; Yacoub et al. 1991; Georgiadis et al., 1992;
Michishita and Ishii, 1992; Barker et al., 1993; Fernandez et al., 1999a,1999b; Mata et al., 2000; Uman et al.,
2000].
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


5
mentioned studies, both theoretical and experimental, have dealt with single line
configuration.
Nowadays the evaluation of lightning induced overvoltages is generally performed in the
following way:

! the lightning return-stroke electromagnetic field change is calculated at a number of
points along the line employing a lightning return-stroke current model, namely a
model that describes the form of the return stroke current as a function of height and
time along the vertical channel. To this purpose, the return stroke channel is
generally considered as a straight vertical antenna (see Fig. 2.1);
! the electromagnetic field (LEMP – Lightning Electro Magnetic Pulse) is then
evaluated and used to calculate the induced overvoltages making use of a coupling
model which describes the interaction between the field and the line conductors.

z'
H
IMMAGINE
dz'
i(z',t)
R
R'
r
x
y
Z
L
z z'
P (x,y,z)
Z
o
E
h
E
r
z
E
E
E
x
P(x,y,z)
φ
IMAGE
v
i(z’,t): lightning current;
H: lightning channel height;
v: lightning wave front velocity.


Fig. 2.1. – Return Stroke channel.

The first point will be discussed in the next paragraph 2.1, while the second one will be
dealt with in the two subsequent paragraphs devoted to the LEMP radiated by a return
stroke 2.2, and to the coupling models for the evaluation of the induced overvoltages 2.3
respectively.

Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


6
2.1. – Engineering return-stroke current models
A comprehensive review of the various models proposed in the literature to predict some
of the observed properties of lightning return strokes is beyond the scope of this thesis; the
interested reader is referred to two papers by Rakov and Uman [1998] and by Gomes and
Cooray [2000]; here we shall limit the discussion to those models that are generally called
‘engineering return-stroke current models’.
A return stroke current model is a mathematical specification of the spatial-temporal
distribution of the lightning current along the discharge channel (or the channel line charge
density). Such a mathematical specification include the return stroke wavefront velocity,
which is generally one of the model inputs [e.g. Uman and McLain, 1969], the charge
distribution along the channel, and a number of adjustable parameters related, to a certain
extent, to the discharge phenomenon [e.g. Gomes and Cooray, 2000] and which should be
inferred by means of model comparison with experimental results [e.g. Nucci and Rachidi,
1989]. Outputs can be directly used for computation of electromagnetic fields.
In these models the lightning channel is in general assumed to be straight, vertical and
perpendicular to the conducting ground plane, as shown in Fig. 2.1, where the geometry of
the problem is also defined.
Now, for the problem we are dealing with, only those models in which the return stroke
current i(z',t) can be simply related to the specified channel-base current i(0,t) are of interest
from an engineering point of view, since it is only the channel-base current that can be
measured directly and for which experimental data are available. We shall call these models,
‘engineering return stroke current models’, or simply ‘engineering models’ in accordance
with Rakov [2001].
The most popular engineering model is probably still the Transmission Line (TL) model,
proposed about 30 years ago by Uman and McLain [1969]. A number of other models have
been subsequently proposed with the primary aim of calculating the return stroke
electromagnetic field given a certain lightning current or, vice versa, inferring the lightning
current characteristics from remote electromagnetic field measurements
5
. In addition and
although not originally formulated as engineering models, we feel worth mentioning certain
more physically plausible models, such as the models by Lin et al. [1980], by Master et al.
[1981] and by Cooray [1993,1998a].
In the large majority of the return-stroke models above mentioned
6
, the channel-base
current is viewed as the result of the flowing towards ground of charges contained in the
leader channel and in the corona sheath around the channel during the return-stroke phase.

5
[Heidler, 1985b; Rakov and Dulzon, 1987; Nucci et al. ,1988; Diendorfer and Uman, 1990; Thottappillil
et al., 1991; Rakov and Uman, 1998; Gomes and Cooray, 2000].
6
[Lin et al., 1980; Master et al. 1981; Heidler, 1985b; Nucci et al., 1988; Diendorfer and Uman, 1990;
Thottappillil et al., 1991; Cooray, 1993; Cooray, 1998].
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


7
They are the so-called traveling-current-source-type [Rakov and Uman, 1998], or discharge-
type [Gomes and Cooray, 2000] models. These models have been conceived to describe the
field change due to the return-stroke phase, the total return-stroke field being the sum of the
return-stroke field change plus the final value of the field produced by the preceding leader
phase.
The validation of the various return-stroke models is not straightforward since, for
natural lightning, no experimental set of simultaneously-measured currents and fields is
available. Thus, although a direct comparison between model predictions and experimental
results is possible for artificially-initiated lightning
7
only a qualitative comparison is
possible for natural lightning. In particular, a return stroke model is to be considered
adequate if, starting from a typical channel-base current, it reproduces the typical features of
the observed fields at different distances. The characterisation of natural lightning
electromagnetic fields is therefore of importance, and an exhaustive survey can be found in
Uman and Krider [1982]; Uman [1987]; Rakov et al. [1995]; Rakov and Uman [1998].
The validation only with the comparison among each other of some of the most popular
return-stroke models can be found in several papers, [e.g. Uman and McLain, 1969; Lin et
al., 1980; Nucci et al., 1990], who used essentially data from natural lightning. Thottappillil
and Uman [1994] used data from triggered lightning to compare some engineering return-
stroke models.
Not all of the above mentioned engineering return-stroke models have been used to
calculate lightning-induced voltages. To the best of our knowledge, the most commonly
adopted for such a purpose are nowadays:

! the Transmission Line (TL) model [Uman and McLain, 1969];
! the Traveling Current Source (TCS) model [Heidler, 1985];
! the Modified Transmission Line Exponential (MTLE) model [Nucci et al., 1988;
Rachidi and Nucci, 1990];
! the Diendorfer-Uman (DU) model [Diendorfer and Uman, 1990].

From the review papers earlier mentioned (in particular from Nucci et al., [1990];
Thottapphillil et al., [1993b]; Rakov and Uman, [1998]; and Gomes and Cooray, [2000]) we
conclude that all the above models allow the reproduction of overall fields that are
reasonable approximations of measured fields from natural and triggered lightning; Gomes
and Cooray [2000] have shown that also the models by Cooray [1993], can predict with
good approximation all typical features of experimentally observed fields.

7
[Saint-Privat-D'Allier Res. Group, 1982; Willett et al., 1988; Leteinturier et al., 1990; Thottappillil and
Uman, 1993b; Rubinstein et al., 1995; Uman et al., 1995].
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


8
If one considers, however, that for lightning-induced voltage calculation it is the early
time region of the field that plays the major role in the coupling mechanism [Nucci et al.,
1993], it follows that the most adequate models are probably the MTL-type ones (see Tab.
2.1). Additionally, although the TL model does not allow for any net charge removal from
the channel and does not reproduce realistic fields for late time calculations [Nucci et al.,
1990], the early time field prediction of the TL model is very similar to that of the more
physically reasonable MTLL and MTLE models and thus, for the problem of interest, it can
be considered a useful and relatively simple engineering tool.

Absolute Error = (E
calc
- E
meas
)/E
meas

TL MTL TCS DU
Mean 0.17 0.16 0.43 0.23
St.Dev. 0.12 0.11 0.22 0.20
Min. 0.00 0.00 0.14 0.00
Max. 0.51 0.45 0.84 0.63

Tab. 2.1. – Summary of statistics on the absolute error of the model peak fields on the basis of triggered
lightning simultaneously measured currents, velocities and fields (subsequent return strokes). With reasonable
approximation the results of the MTL column apply to both MTLE and MTLL. Adapted from Thottappillil
and Uman [1993b].

For the current at the channel base the following analytical expression [Heidler, 1985a] is
adopted
8
:

2
1
1 0
1
) , 0 (
τ
τ
τ
η
t
n
n
e
t
t
I
t i
−
|
|
.
|


\
|
+
|
|
.
|


\
|
=
(2.1)

where

n
n
e
1
1
2
2
1
|
|
.
|


\
|
|
|
.
|


\
|
−
=
τ
τ
τ
τ
η
(2.2)

8
It is worth noting that so far it has been implicitly assumed that the channel base current is not affected by
reflection of the charges flowing downward the channel during the return stroke phase. Heilder and Hopf
[1994] have addressed this issue by extending the TCS model to account for these reflections. This point is
certainly of importance and stresses the need for additional experimental data useful for further model
validation, but is, however, beyond the scopes of this thesis.
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


9
and
! I
0
is the amplitude of the channel-base current;
! τ
1
is the front-time constant;
! τ
2
is the decay-time constant;
! η is the amplitude correction factor;
! n is an exponent (2÷10).

Function (2.1) has been preferred to the commonly used double-exponential function
since it exhibits, as opposed to the double-exponential function, a time-derivative equal to
zero at t=0, consistent with measured return-stroke current wave shapes. Additionally, it
allows for the adjustment of the current amplitude, the maximum current derivative and the
charge transferred nearly independently by varying I
0
, τ
1
and τ
2
respectively.
In what follows we shall describe only two return stroke current models, in particular we
shall focus on the TL (Transmission line) model [Uman and Mc Lain, 1969] and MTLE
(Modified Transmission Line Exponential decay) model.

The Transmission Line (TL) model. In the TL model, it is assumed that the current wave
at the ground travels undistorted and unattenuated up the lightning channel at a constant
speed v. Mathematically:

|
.
|

\
|
− =
v
z
t i t z i
'
, 0 ) , ' ( z'≤v t
i z t ( ' , ) = 0 z'>v t
(2.3)

where
! v is the lightning return stroke wave-front velocity;
! z is the vertical space variable.

The transfer of charge takes place only from the bottom of the leader channel to the top;
thus no net charge is removed from the channel, this being an unrealistic situation given the
present knowledge of lightning physics [Uman, 1987].

The Modified Transmission Line Exponential decay (MTLE) model. In the MTLE model
[Nucci et al., 1988] the lightning current intensity is supposed to decrease exponentially
while propagating up the channel as expressed by:


Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


10
λ
'
'
) , ' (
z
e
v
z
t i t z i
−
|
.
|

\
|
− = z'≤v t
0 ) , ' ( = t z i z'>v t
(2.4)

where
! v is the return-stroke velocity;
! λ is the decay constant which allows the current to reduce its amplitude with height.

This attenuation is not to be considered as due to losses in the channel or to take into
account the already mentioned decay with height of the initial peak luminosity, but has been
proposed by Nucci et al. [1988] to take into account the effect of the charges stored in the
corona sheath of the leader and subsequently discharged during the return stroke phase. Its
value has been determined to be about 2 km by Nucci and Rachidi [1989], by means of tests
with experimental results published by Lin et al. [1979,1980].
The MTLE model represents a modification of the TL model which allows net charge to
be removed from the leader channel via the divergence of the return stroke current with
height, and thus results in a better agreement with experimental results.

Channel-base current measurements have been performed by means of instrumented
towers in some countries
9
, and statistical elaboration of lightning current data have been
presented [e.g. Berger et al. 1975; Anderson and Eriksson, 1980]. Without loosing
generality only lightning with lower negative charge to ground will be considered, here
since it is generally accepted that, at least in temperate climate, positive flashes occur less
frequently and have a lower peak current-derivative.
In Fig. 2.2, typical channel-base current wave shapes for negative first (Fig. 2.2a) and
subsequent (Fig. 2.2b) return strokes, as reported by Berger et al. [1975], are shown. The
statistics of lightning current parameters which are most significant for the evaluation of
induced overvoltages (peak value and front steepness) are shown in Tab. 2.2,2.3 and in Fig.
2.3
10
.

9
e.g. Switzerland [Berger, 1972; Berger et al., 1975; Montandon and Beyeler, 1994], Italy [Garbagnati
and Lo Piparo, 1970,1973,1982; Garbagnati et al. 1980], Russia [Gorin and Shkilev, 1984], South Africa
[Eriksson, 1974,1978; Geldenhuys et al., 1988], Germany [Beierl, 1991,1992] and Canada [Chang et al. 1992;
Janischewskyj et al. 1992,1993,1994; Bermudez et al. 2001].
10
It is worth observing that since channel-base current is generally measured at the top of the instrumented
tower, the statistical distributions of current parameters are presumably affected by current reflections at both
bottom and top of the tower [Montandon and Beyeler, 1994; Beierl, 1992; Janischewskyj et al., 1992,
Bermudez et al. 2001]. In principle, these reflections might alter both the front steepness and the peak value of
the current depending on several factors, e.g. tower height, current wave shape and value of the reflection
coefficients. To elaborate meaningful statistics on lightning current parameters, one should ‘decontaminate’
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


11

Fig. 2.2. – Typical channel-base current wave shapes: a) for first negative return strokes, b) for subsequent
negative return strokes. Adapted from Berger et al. [1975].


95% 50% 5%
Stroke First Subs First Subs First Subs
Ipeak [kA] 14 4.6 30 12 80 30
Time to crest
[µs]
1.8 0.2 5.5 1.1 18 4.5
( )
max dt
di
[kA/ms]
5.5 12 12 40 32 120

Tab. 2.2. – Statistics of peak amplitude, time to crest and maximum front steepness for first ad subsequent
negative return strokes. Adapted from Berger et al. [1975].


the measured currents from the mentioned ‘tower-effects’ [Guerrieri et al. 1994]. The problem of
‘decontaminating’ the lightning current is, however, beyond the scope of this thesis. In what follows we shall
assume that the current distributions found in the literature [Berger et al., 1975; Anderson and Eriksson, 1980]
are relevant to current measured at ‘ground level’ disregarding the presence of the elevated strike object which
would require an appropriate reprocessing of the statistical data [Sabot, 1995].
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


12
1000.00 10000.00 100000.00 1000000.00
IP in A
0.01
0.01
0.05
0.10
0.50
1.
00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
y
Current amplitude [A]
1000.00 10000.00 100000.00 1000000.00
IP in A
0.01
0.01
0.05
0.10
0.50
1.
00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
y
1000.00 10000.00 100000.00 1000000.00
IP in A
0.01
0.01
0.05
0.10
0.50
1.
1000.00 10000.00 100000.00 1000000.00
IP in A
0.01
0.01
0.05
0.10
0.50
1.
00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
y
00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
y
Current amplitude [A]

1.00E-7 1.00E-6 1.00E-5 1.00E-4
Tempo alla cresta tf in s
0.01
0.01
0.05
0.10
0.50
1.00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
P
r
o
b
a
b
i
l
i
t
y
Time to crest [s]
1.00E-7 1.00E-6 1.00E-5 1.00E-4
Tempo alla cresta tf in s
0.01
0.01
0.05
0.10
0.50
1.00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
P
r
o
b
a
b
i
l
i
t
y
1.00E-7 1.00E-6 1.00E-5 1.00E-4
Tempo alla cresta tf in s
0.01
0.01
0.05
0.10
0.50
1.00
2.00
5.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
95.00
98.00
99.00
99.50
99.90
99.95
99.99
P
r
o
b
a
b
i
l
i
t
P
r
o
b
a
b
i
l
i
t
y
Time to crest [s]

Fig. 2.3. – Current amplitude and rise time statistical distributions. Adapted from Anderson and Eriksson
[1980].

95% 50% 5%
Stroke First Subs First Subs First Subs
Ipeak [kA] 14.1 5.2 31.1 12.3 68.5 29.2
Time to crest
[µs]
1.8 0.1 4.5 0.6 11.3 2.8
( )
max dt
di
[kA/ms]
9.1 9.9 24.3 39.9 65.0 161.5

Tab. 2.3. – Statistics of peak amplitude, time to crest and maximum front steepness for first ad subsequent
negative return strokes. Adapted from Anderson and Eriksson [1980]

2.2. – Lightning electromagnetic field appraisal
Expressions of the electromagnetic field radiated by a vertical dipole of length dz' at a
height z' along the lightning channel, assumed as an antenna over a perfectly conducting
plane, have been derived by Master and Uman [1983] by solving Maxwell's equations in
terms of retarded scalar and vector potentials:

[
]
d d E r z t
dz r z z
R
i z R c
r z z
cR
i z t R c
r z z
c R
i z t R c
t
r
o
t
( , , , )
( )
( , / )
( )
( , / )
( ) ( , / )
φ
πε
τ τ
∂
∂
=
′ − ′
′ − +
+
− ′
′ − +
+
− ′ ′ −
∫
4
3
3
5
0
4
2 3


(2.5)
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


13
[
]
t
c R t z i
R c
r
c R t z i
cR
r z z
c R z i
R
r z z z d
t z r E
t
o
z
∂
∂
τ τ
πε
φ
) / , (
) / , (
) ( 2
d ) / , (
) ( 2
4
) , , , ( d
3 2
2
4
2 2
0
5
2 2
− ′
−
+ − ′
− ′ −
+
+ − ′
− ′ − ′
=
∫


(2.6)
[
]
t
c R t z i
cR
r
c R t z i
R
r z d
t z r B
o
r
∂
∂
π
µ
φ
) / , (
) / , (
4
) , , , ( d
2
3
− ′
− ′
′
=
(2.7)

where
! i(z', t) is the current along the channel obtained from one of the return-stroke current
models summarized above;
! c is the speed of light;
! R is the distance of the electric dipole from the observation point;
! r is the projection of R in the plane xy (see Fig. 2.1);

and the geometrical factors are as given in Fig. 2.1.

By integrating along the channel expressions (2.5)-(2.7), where the current distribution as
a function of height and time is given by the return-stroke models discussed in previous
paragraph, one obtains the electromagnetic exciting field.
For distances not exceeding a few kilometers, the perfect ground conductivity
assumption is a reasonable approximation for the vertical component of the electric field
and for the horizontal component of the magnetic field as shown by several authors [Djébari
et al., 1981; Zeddam and Degauque, 1990; Rubinstein, 1996]. In fact, the contributions of
the source dipole and of its image to these field components add constructively and,
consequently, small variations in the image field due to the finite ground conductivity will
have little effect on the total field.
On the other hand, the horizontal component of the electric field, is appreciably affected
by a finite ground conductivity. Indeed, for such a field component, the effects of the two
contributions subtract, and small changes in the image field may lead to appreciable changes
in the total horizontal field. Although the intensity of the horizontal field component is
generally much smaller than that of the vertical one, within the context of certain coupling
models it plays an important role in the coupling mechanism [Master and Uman, 1984;
Cooray and De la Rosa, 1986; Rubinstein et al., 1989; Diendorfer, 1990; Nucci et al. 1993;
Ishi et al., 1994] and, hence, an accurate calculation method has to be chosen for it. Methods
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


14
for the calculation of the horizontal field using the exact Sommerfeld integrals are
inefficient from the point of view of computer-time. Simplified expressions exist. One of the
approach which appears now most promising is that proposed independently by Cooray and
Rubinstein [Rubinstein 1991,1996; Cooray 1992,1994] and discussed by Wait [1997].
The Cooray-Rubinstein expression is given by

δ σ
ω ω ω
φ
g
p rp r
j
j z r H j z r E j z r E
) 1 (
) , 0 , ( ) , , ( ) , , (
+
⋅ = − = (2.8)

where
! p is the subscript that indicates the fields calculated assuming a perfect ground;
! ε
rg
is the relative ground permittivity;
! σ
g
is the ground conductivity.
! ) , , ( ω j z r E
rp
and ) , 0 , ( ω
φ
j r H
p
are the Fourier-transforms of the horizontal
component of the electric field at height z, and of the azimuthal component of the
magnetic field at ground level respectively, both calculated assuming a perfect
conducting ground;
! δ is the skin depth factor, (
g g
σ ωµ / 2 );
! µ
g
is the ground permittivity.

This approach has been shown to produce satisfactory approximation of the horizontal
electric field for some significant cases: in particular, it reproduces the positive, bipolar and
negative polarities of the field at close (one hundred meters), intermediate (some
kilometers), and far (tens of kilometers) distances respectively, and at all these ranges it
predicts results close to those predicted by more accurate expressions [Cooray, 1994;
Rubinstein, 1996; Rachidi et al., 1996]. Motivated by the discussion by Wait [1997], Cooray
[1998b] has proposed a modified, improved expression of (2.8).

2.3. – Transmission line coupling models
To solve the coupling problem, i.e. the determination of voltages and currents induced by
an external field on a conducting system, use could be made of the antenna theory, the
general and rigorous approach based on Maxwell's equations [Tesche, 1992]. However, due
to the length of distribution lines, the use of such theory for the calculation of lightning-
induced overvoltages implies long computing times. In our case, the use of the simplest
approach namely the quasi-static approximation [Johnk, 1975], according to which
propagation is neglected and coupling between incident fields and the line conductors can
be described by means of lumped elements (e.g. an inductance, or a capacitance), is not
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


15
appropriate. In fact, such an approach requires that the dimensions of the line conductors be
smaller than about one tenth of the minimum wavelength of the electromagnetic field, an
unacceptable assumption for the case of power lines illuminated by LEMP fields (above 1
MHz frequency, that is below 300 m wave length). Another possible approach is the
transmission line theory [Taylor et al., 1965]. The basic assumptions of this approximation
are that the response of the line is quasi-transverse electromagnetic (quasi-TEM) and that
the transverse dimension of the line is smaller than the minimum significant wavelength.
The line is represented by a series of elementary sections to which, by virtue of the above
assumptions, the quasi-static approximation applies. Each section is illuminated
progressively by the incident electromagnetic field so that longitudinal propagation effects
are taken into account. The transmission line approximation appears to be the most
promising approach for the problem of interest: indeed, in the power literature the most used
coupling models are based on it, and in this thesis use will be made of it.
We shall now briefly resume the most popular coupling models based on the
transmission line approximation. To do that, let us start considering a single-conductor
overhead line parallel to the x-axis and contained in the xz-plane terminated on two
resistances R
0
and R
L
(Fig. 2.5). It is important to observe that the incident external
electromagnetic field E
e
, B
e
shown in Fig. 2.5 is the sum of the field radiated by the
lightning stroke and of the ground-reflected field, both considered in absence of the wire.
The total field E, B is given by the sum of the incident field E
e
, B
e
and the scattered field E
s
,
B
s
, which represents the reaction of the wire to the incident field. The incident
electromagnetic field is related to the incident scalar potential φ
i
and to the incident vector
potential A
e
by the following expressions:

|
|
.
|


\
|
+ ∇ − =
t
e
e e
∂
∂
φ
A
E (2.9)
e e
A B × ∇ =
(2.10)

L x
y
z
R
0
R
L
0
x x+dx
h
E
z
e
E x
e
By
e


Fig. 2.5. – Geometry used for the calculation of overvoltages induced on an overhead power line
by an indirect lightning return-stroke
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


16
The Agrawal, Price and Gurbaxani model for a single-conductor line. The most adequate
coupling model to describe the coupling between lightning return-stroke fields and overhead
lines is nowadays considered the model by Agrawal et al. [1980], which is based on the
transmission line approximation [Tesche et al., 1997]. It is indeed the model that allows in a
straightforward way to take into account the ground resistivity in the coupling mechanism,
and it is the only one that has been thoroughly tested and validated using experimental
results, as will be discussed next.
The two transmission line coupling equations of the Agrawal model, expressed in the
time domain, are [Agrawal et al., 1980; Tesche et al., 1997; Rachidi et al., 1999]:

) , , (
) , (

) , (
'

) , (
t h x E
t
t x i
L
t
t x i
x
t x v
e
x
i
g
s
= ′ +
∂
∂
⊗ +
∂
∂
ξ
∂
∂

(2.11)
0

) , (
) , ( ' +

) , (
= ′ +
t
t x v
C t x v G
x
t x i
s
s
∂
∂
∂
∂

(2.12)

where
! ) , , ( t h x E
e
x
is the horizontal component of the incident electric field along the x axis at
the conductor's height;
! ) , ( t x i is the current induced along the line;
!
g
' ξ is the inverse Fourier transform of the ground impedance;
! ⊗ denotes the convolution product;
! L', C', G' are the line inductance (external), capacitance and conductance per-unit-
length respectively.

In equation (2.11) L’ is the external per-unit-length inductance calculated for a loss less
wire above a perfectly conducting ground,
g
' ξ is the inverse Fourier transform of the
ground impedance. In particular
g
' ξ is the inverse Fourier-transform of
ω ω j
Z
j
Z Z
g w g
' ' '
≅
+

where
'
w
Z and Z
g
'
are the wire and the ground impedances respectively. Note, further, that
within the frequency range of interest, the wire impedance can be neglected compared with
the ground impedance [Ramo et al., 1984; Rachidi et al., 1996].
The ground impedance can be viewed as a correction factor to the line longitudinal
impedance when the ground is not a perfect conductor and can be defined as Rachidi etl al.
[1996], Guerrieri [1997], Tesche et al. [1997], Cooray and Schuka [1998]:

Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


17
'
) , (
'
L j
I
dx z x B j
Z
h
s
y
g
ω
ω
− =
∫
∞ −

(2.13)

In this thesis we shall make reference to the expression derived by Carson [1926], valid
for the so-called ‘low frequency approximation’ (
rg g
ε ωε σ
o
>> ), and for ground
conductivities not lower than about 0.001 S/m [Rachidi et al. 1999] recently improved in
[Rachidi et al., 2000]:

¦
)
¦
`
¹
(
(
¸
(


¸

−
|
|
.
|


\
|
+
¦
¹
¦
´
¦
=
4
1
4
1
2
1
2
1
t
erfc e
t
,
h
min ) t ( '
g
t
g
g
o
rg o
o
g
g
τ τ
π πτ
µ
ε ε
µ
π
ξ
τ
(2.14)

in which ε
0
and ε
rg
are the air and ground permittivity respectively, µ
0
is the air
permeability, τ
g
=h
2
µ
0
σ
g
(where σ
g
is the ground conductivity) and erfc is the
complementary error function.
In (2.11) and (2.12) the wire impedance, the ground admittance and the line conductance
have been assumed negligible (an approximation valid for typical overhead lines a few
meters above ground, σ
g
=10
-3
-10
-2
S/m, ε
r
=1-10, Rachidi et al. [1996]).
Equations (2.11) and (2.12) are written in terms of scattered voltage ) , ( t x v
s
11
. The total
voltage ) , ( t x v is given by the sum of the scattered voltage ) , ( t x v
s
and the so-called
incident voltage

h t x E dz t z x E v
e
z
h
e
z
i
⋅ − ≈ − =
∫
) , 0 , ( ) , , (
0
(2.15)

namely,

) , ( ) , ( ) , ( t x v t x v t x v
i s
+ =
(2.16)

The boundary conditions, written for the case of resistive terminations, are


11
The total field is the sum of the incident (or exciting) field and of the scattered field. The first is given by
the field radiated by the lightning channel and by the ground-reflected field in absence of the line conductors;
the second is given by the reaction of the overhead conductors to the incident field. To each of these fields one
can associate a voltage, namely the incident voltage and the scattered one.
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


18
) , 0 ( ) , 0 ( ) , 0 ( t v t i R t v
i
o
s
− ⋅ − =
(2.17)
) , ( ) , ( ) , ( t L v t L i R t L v
i
L
s
− ⋅ =

(2.18)

Eqs. (2.11)-(2.16) have the circuit representation of Fig. 2.6.


0
v(x,t)
i(x,t) L'dx
C'dx
x
x+dx
+ -
i(x+dx,t)
+
-
+
-
L
v
s
(x+dx,t)
-v
i
(0,t)
R
0
R
L
E
x
e
(x,h,t)dx
v
e
(x,t)
-v
i
(L,t)
t
(x,t) i
ξ'
i
g
∂
∂
⊗
Fig. 2.6. – Differential equivalent coupling circuit according to the Agrawal et al. formulation for a lossless
single-wire overhead line.

According to the Agrawal model, the forcing functions explicitly appearing in equations
which produce the scattered voltage are: the horizontal component of the incident electric
field along the line and the incident vertical electric field at the vertical line terminations.
These forcing functions are represented by the voltage sources in Fig. 2.6.
An adequate coupling model for lightning-induced overvoltage calculations is a model
that, given as inputs the lightning electromagnetic fields which illuminate a transmission
line, predicts satisfactorily the line overvoltages induced by that field. Thus, to adequately
test a coupling model, data sets of simultaneously measured fields and voltages are needed.
The Agrawal model has been first applied for the calculation of lightning induced
overvoltages by Master and Uman [1984] and then employed by several other authors
12
.
The Agrawal model is the only one that has been thoroughly tested versus experimental
results in the sense above specified. Although some of the first tests did not provide
satisfactory agreement between theory and measurements [Master and Uman, 1984; Master
et al., 1984], in subsequent experiments the agreement was largely improved [Cooray and
de la Rosa, 1986; Rubinstein et al. 1989]. Some tests of the Agrawal model have been
performed making use of Nuclear Electromagnetic Pulse (NEMP) simulators [Master et al.,
1984; Rubinstein et al. 1989] and results have shown a reasonably good agreement between
measurements and theory. It is worth mentioning that one of the aim of this thesis is the
extension of the Agrawal coupling model to the case of a line with transverse discontinuities

12
[Cooray and De la Rosa, 1986; Rubinstein et al., 1989; Diendorfer, 1990; Iorio et al., 1993; Nucci et al.,
1993,2000; Ishii et al., 1994].
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


19
(see paragraphs 2.4. and 3.1.2.), and that the experimental validation of such a model
extension was also successfully accomplished by means of a NEMP simulator.
The Agrawal model has been shown to be completely equivalent to two other models in
which the forcing functions are in terms of different components of the electromagnetic
field, namely the model by Taylor et al. [1965], and the model by Rachidi [1993]. Indeed,
these last three models (Agrawal et al., Taylor et al., and Rachidi) are different though
equivalent formulations of the same coupling equations set in terms of different
combinations of the various components of the electromagnetic field. This means that it is
misleading to speak of the contribution of a given component of the electromagnetic field to
the induced overvoltage without specifying the formulation of the coupling model one is
using [Nucci and Rachidi, 1995]. Additionally, it has been shown that for the case of a
lightning channel perpendicular to a perfectly conductive ground plane the model proposed
by Rusck [1958] provides the same results as the Agrawal one. Although for different
channel geometry the Rusck model is expected to predict results that might be affected by
some inaccuracy, it is still very popular among power engineers, and for this reason we
briefly summarize it below.

The Rusck model. To derive the transmission-line coupling equations of this model,
Rusck [1958] started from the expression relating the total electric field on the conductor
surface to the scalar and vector potentials. The coupling equations obtained by using the
above procedure for a single-conductor line above perfect conducting ground are:

0
) , ( ) , (
= ′ +
t
t x i
L
x
t x
∂
∂
∂
∂φ

(2.19)
[ ] 0 ) , ( ) , (
) , (
= − ′ + t x t x
t
C
x
t x i
i
φ φ
∂
∂
∂
∂

(2.20)

where
! L’ and C’ are the line inductance and the line capacitance per unit length
respectively;
! i(x,t) is the total line current;
! φ is the total induced scalar potential.
The total induced voltage v(x,t) is given by the following expression

dz
t
t z x A
t x t x v
h e
z
∫
+ =
0
) , , (
) , ( ) , (
∂
∂
φ
(2.21)

Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


20
where h is the conductor’s height and
e
z
A is the vertical component of the incident vector
potential. The boundary conditions for (2.19) and (2.20) are [Cooray, 1994]:

dz
t
t z A
t i R t
h e
z
∫
− − =
0
0
) , , 0 (
) , 0 ( ) , 0 (
∂
∂
φ (2.22)
dz
t
t z L A
t L i R t L
h e
z
L
∫
− =
0
) , , (
) , ( ) , (
∂
∂
φ (2.23)

In the Rusck model the forcing functions that explicitly appear in the coupling equations
are the incident scalar potential along the line and the vertical component of the incident
vector potential at the line terminations.
Yokoyama et al. [1984,1989] used the Rusck model to compute induced voltages starting
from a measured lightning current. The field was calculated from the measured current
assuming a simple return-stroke current model and a satisfactory agreement was found
between measurements and calculations. Also Eriksson et al. [1982] obtained a satisfactory
agreement between theory and calculations. The field they input to the Rusck model was not
the one that produced the measured induced voltage but was calculated using a simple
return-stroke current model starting from a typical lightning channel-base current: thus their
comparison has more a qualitative than a quantitative nature.
It is worth mentioning that Rusck proposed a simplified formula that provides a first
estimation of the peak value V
max
of the overvoltages induced on a infinite long line starting
from the peak value I
p
of the lightning current, the velocity of the return-stroke wavefront
and the height of the line:

( ) ( )
|
|
.
|


\
|
⋅ −
+
⋅
=
2
0 max
5 0 1
1
2
1
1
v/c .
c
v
d
h I
Z V
p
(2.24)

where
! I
p
is the lightning peak current;
! d is the distance of the stroke location from the line (in m);
!
0
0
0
4
1
ε
µ
π
= Z = 30 Ω;
! h is the line height (in m);
! v is the return stroke wave front velocity;
! c is the speed of the light.

Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


21
Discussion of the simplified Rusck formula. As mentioned earlier, the Rusck simplified
analytical formula (2.24) gives the maximum value V
max
of the induced overvoltages on an
infinitely long line at the nearest point to the stroke location. Figures 2.7a, b and c show the
variation of the maximum amplitude of the voltage induced at the point closest to the stroke
location for three different lightning current wave shapes (see Tab. 2.4) and for the different
values of ground conductivity, namely infinite, 0.01 and 0.001 S/m calculated adopting the
MTLE return stroke model and the Agrawal coupling model. Each case is compared with
the prediction of the Rusck formula.

Current types
A1 A2 A3
I
p
[kA]
12 12 12
(di/dt)
max
[kA/µs]
12 40 120

Tab. 2.4a. – Subsequent return-stroke current peak values and maximum time derivatives of the adopted
currents.

A1 A2 A3
I
01
[kA] 10,7 10,7 7,4
τ
11
[µs] 0,95 0,25 0,063
τ
12
[µs] 4,7 2,5 0,5
I
02
[kA] 6,5 6,5 9
τ
21
[µs] 4,6 2,1 0,27
τ
22
[µs] 900 230 66
n 2 2 2

Tab. 2.4b. – Parameters of the Heidler functions reproducing the adopted currents. The channel base current is
approximated by the sum of two Heidler functions.

By observing Fig. 2.7 we can conclude that, in general, when the channel base current
exhibits a steep front (current A3) and when the ground is approximated as a perfectly
conducting plane, the Rusck simplified analytical expression provides an estimation for the
induced voltages close to the one predicted by the more general Agrawal model in which the
forcing functions are calculated by means of the MTLE model. The disagreement between
the voltage predicted by the two approaches (Rusck formula and Agrawal-MTLE models)
increases with the return-stroke velocity, with the ground resistivity, and with the front
duration of the lightning current. As a conclusion, in general, the Rusck simplified formula
should not be applied to the case of overhead lines above a lossy ground.
Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


22
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
a)

0
50
100
150
200
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
a)


0
50
100
150
200
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
b)

0
50
100
150
200
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
b)


0
50
100
150
200
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
c)

0
50
100
150
200
0 20 40 60 80 100 120 140 160
Distance (m)
I
n
d
u
c
e
d

V
o
l
t
a
g
e

U
1

(
k
V
)
infinite
0.01 S/m
0.001 S/m
Rusck formula
c)


Fig. 2.7. – Variation of the induced voltage magnitude at the line center as a function of distance to the stroke
location. Left column: return-stroke velocity equal 1.3⋅10
8
m/s; right column: return-stroke velocity equal
1.9⋅10
8
m/s. a) lightning current A1, b) lightning current A2, c) lightning current A3. In solid line we have
reported the results obtained from the Rusck simplified analytical expression. Adapted from Borghetti et al.
[2000a].



Chapter 2. – Evaluation of lightning-induced voltages on transmission lines pag.


23
The Agrawal, Price and Gurbaxani model for a multi-conductor line. The Agrawal
model for the case of a multi-conductor line above a lossy ground reads [Tesche et al., 1997,
Rachidi et al., 1997]:

[ ] [ ] [ ] [ ] [ ] [ ] ) , , ( ) , ( ' ) , ( ' ) , ( t h x E t x i
t
t x i
t
L t x v
x
i
e
x i gij i ij
s
i
=
∂
∂
⊗ +
∂
∂
+
∂
∂
ξ
(2.25)
[ ] [ ] [ ] 0 ) , ( ' ) , ( =
∂
∂
+
∂
∂
t x v
t
C t x i
x
s
i ij i

(2.26)

where
! [ ] ) , , ( t h x E
i
e
x
is the horizontal component of the incident electric field along the x axis
at the conductor's height;
! [ ]
gij
' ξ is the matrix of transient ground resistance;
! [ ]
ij
L' and [ ]
ij
C' are respectively the external inductance and the capacitance matrices
per unit length of the line;
! [ ] ) , ( t x i
i
is the current vector;
! ⊗ denotes the convolution product;
! [ ] ) , ( t x v
s
i
is the scattered voltage vector, related to the [ ] ) , ( t x v
i
, the total voltage
vector, by the following expression:

[ ] [ ] [ ] [ ]
(
(
¸
(


¸

− = + =
∫
i
h
e
z
s
i
e
i
s
i i
dz t z x E t x v t x v t x v t x v
0
) , , ( ) , ( ) , ( ) , ( ) , ( (2.27)

The boundary conditions for the scattered voltage at both line ends are:

[ ] [ ][ ] [ ] ) , 0 ( ) , 0 ( ) , 0 (
0
t v t i R t v
e
i i
s
i
− − =
(2.28)
[ ] [ ][ ] [ ] ) , ( ) , ( ) , ( t L v t L i R t L v
e
i i L
s
i
− =
(2.29)

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


24
Chapter 3. – LEMP-to transmission-line coupling models
for distribution systems and their implementation in a
transient program

The evaluation of the LEMP response of distribution networks is crucial for the
assessment of outages on customers and distribution utilities caused by lightning. For this
purpose the availability of a model of a LEMP illuminated line having realistic geometrical
configuration is of particular interest. This chapter presents the developed complex
overhead line model illuminated by a LEMP, its implementation, and the relevant
distribution system model developed.
The models here proposed are included in the LIOV (lightning induced overvoltage)
code. The LIOV code has been developed in the framework of an international collaboration
involving the University of Bologna (Department of Electrical Engineering), the Swiss
Federal Institute of Technology (Power Systems Laboratory), and the University of Rome
“La Sapienza” (Department of Electrical Engineering). The LIOV code is based on the
field-to-transmission line coupling formulation of Agrawal et al. [1980], suitably adapted
for the case of an overhead line illuminated by an indirect lightning electromagnetic field;
the return stroke electromagnetic field is calculated by assuming the MTLE engineering
model and using the Cooray-Rubinstein formula for the case of lossy grounds [Nucci et al.,
1993; Rachidi et al., 1995]. It allows for the calculation of lightning-induced voltages along
an overhead line as a function of current wave shape (amplitude, front steepness, duration),
return stroke velocity, line geometry (height, length, number and position of conductors),
stroke location with respect to the line, ground resistivity and relative permittivity, and value
of termination impedances.
The models here proposed are aimed at improving the LIOV code in order to permit for
the treatment of more realistic overhead line and distribution systems.

3.1. – Lines with transverse discontinuities
Concerning medium voltage overhead lines the main protective measures against
lightning induced overvoltages can be identified as follow: I) use of shielding wires and II)
use of surge arresters.
Regarding the first protective measure, several authors have addressed theoretically the
issue by assuming the shielding wire at zero potential at any point of it and at any time
[Rusck, 1958; Chowdhuri, 1969,1990; Yokoyama et al., 1983,1986], an assumption that
appears reasonable only when shielding wire is grounded at short intervals along the line.
Further, such an approach does not allow to find the ‘optimal’ distance between two
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


25
consecutives groundings needed to accomplish the required shielding effect. In Yokoyama
[1984] an improved approach was used in which the shielding wire was considered as one
of the conductors of the multi-conductor line. The coupling model adopted is the same
proposed by Rusck [1958] which has been shown to apply for the case of a lightning
channel perpendicular to the ground plane.

3.1.1. – Proposed model
Line transverse discontinuities are represented by the connections between the line
conductors and the ground. These connections are constituted by the groundings of the
shielding wire and by surge arresters connected to the phase conductors. Then the elements
that represent these discontinuities could be a simple linear resistance or a non-linear
component.
In this thesis, we extend the model already presented in Rachidi et al. [1997,1999] in
which the coupling between the LEMP and the multi-conductor transmission line is
described by the more general coupling model by Agrawal et al. and in which the shielding
wire was treated, similarly to [Yokoyama, 1984], as one of the conductors of the multi-
conductor line. In particular, while in [Rachidi et al., 1997,1999] the shielding wire was
grounded only at the line terminations, we here propose a model modification with allows
for the treatment of multiple groundings along the line. The grounding resistance is also
included in our extended model. Comparisons with experimental data obtained by using a
reduced scale line model illuminated by an EMP simulator and, also, with those obtained by
other authors [Yokoyama, 1984], are presented.
The above-mentioned model has also be extended in order to deal with the presence of
surge arresters along the line. We shall essentially assess the effect of both shielding wires
and surge arresters on voltages induced by a nearby cloud-to-ground lightning.
According to the Agrawal transmission line coupling equations (2.25) and (2.26) the
scattered voltage, at node g of the conductor i at which a given impedance is connected to
ground, can be expressed as follows (see Fig. 3.1):

( ) ( )
∫
+ =
i
h
e
z
i
g
s
i
(x,z,t)dz E t i Γ t x v
0
) , ( (3.1)

where Γ is an integro-differential operator which describes the voltage drop across the
impedance as function of current i
g
(Γ=R
g
⋅i
g
for the simple case of a resistance). Since the
Agrawal model is expressed in terms of the scattered voltage, it is necessary to include a
voltage source in series with the impedance, the so-called incident voltage, which is given
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


26
by the integral from the ground level to the line conductor of the incident vertical electric
field (see Fig. 3.1).

∫
i
h
e
z
(x,z,t)dz E
0
Γ
i
g
i
(t)
) , ( t x v
i
s
conductor i
g

Fig. 3.1. – Insertion of discontinuity point in a generic point along a multi-conductor line.

The developed extended model was tested versus experimental data obtained by means
of a reduced line model illuminated by an EMP simulator. Besides, additional validation
was performed by testing it with the theoretical results published by Yokoyama [1984]. The
entire experimental validation results are presented in the paragraph 3.1.3.

3.1.2. – Numerical implementation
Most studies on lightning-induced voltages on overhead power lines use a direct time
domain analysis because of its straightforwardness in dealing with insulation coordination
problems, and its ability to handle non-linearities, which arise in presence of protective
devices such as surge arresters or corona phenomenon.
One of the most popular approaches to solve the transmission line coupling equations in
time domain is the finite difference time domain (FDTD) technique (e.g. [Tafflove, 1995]).
Such technique was used indeed by Agrawal et al. in [1980] when presenting their field-to-
transmission line coupling equations. In the above publications, partial time and space
derivatives were approximated using the 1
st
order FDTD scheme.
We have proposed, instead, the use of a second order finite difference scheme, it is based
on the Lax-Wendroff algorithm [Lax and Wendroff, 1960]. In [Omick Castillo, 1993] this
algorithm is applied to the classical transmission line equations excited by lumped
excitation sources. We here propose an extension of such an integration scheme to take into
account distributed sources due to the action of an external electromagnetic field, according
to the Agrawal et al. coupling model.
We shall first deal with the case of a single conductor line above an ideal ground with no
discontinuities. Then we shall propose an extension of the Agrawal model to deal with the
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


27
case of a lossy ground, and eventually we shall present the algorithm relevant to a line with
transverse periodical discontinuities. In Appendix A.1 we report also the treatment of line
transverse discontinuities concerning the FDTD 1
st
order scheme.

Case of a single-conductor line above an ideal ground. For convenience we write below
the Agrawal coupling equations (2.11) and (2.12) which, for the case of interest, assume the
following form:

) , , (
) , (
'
) , (
t h x E
t
t x i
L
x
t x v
e
x
s
=
∂
∂
+
∂
∂

(3.2)
0
) , (
'
) , (
=
∂
∂
+
∂
∂
t
t x v
C
x
t x i
s

(3.3)

The meaning of the symbols that appear in (3.2) and (3.3) is the same than in equations
(2.11) and (2.12).
If we differentiate with respect to the x and t variables, the system of equations (3.2) and
(3.3) can be rewritten as

t
t h x E
C
t
t x i
C L
x
t x i
e
x
∂
∂
− =
∂
∂
−
∂
∂ ) , , (
'
) , (
' '
) , (
2
2
2
2

(3.4)
x
t h x E
t
t x v
C L
x
t x v
e
x
s s
∂
∂
=
∂
∂
−
∂
∂ ) , , ( ) , (
' '
) , (
2
2
2
2

(3.5)

Expanding the line current and the scattered voltage using Taylor’s series applied to the
time variable, and truncating after the second order term yields

) O(
2
) , ( ) , (
) , ( ) , (
3
2
2
2
0
t
t
t
t x v
t
t
t x v
t x v t x v
s s
s s
∆ +
∆
∂
∂
+ ∆
∂
∂
+ =
(3.6)
) O(
2
) , ( ) , (
) , ( ) , (
3
2
2
2
0
t
t
t
t x i
t
t
t x i
t x i t x i ∆ +
∆
∂
∂
+ ∆
∂
∂
+ =
(3.7)

where ) O(
3
t ∆ is the remainder term, which approaches zero as the third power of the
temporal increment.
If we substitute the time derivatives in (3.6) and (3.7) with the corresponding expressions
using equations (3.2), (3.3), (3.4), and (3.5), we obtain the following second order
differential equation

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


28
) O(
) , ( ) , , (
' ' 2
) , (
'
) , ( ) , (
3
2
2 2
0
t
x
t x v
x
t h x E
C L
t
x
t x i
C
t
t x v t x v
s e
x s s
∆ +
|
|
.
|


\
|
∂
∂
−
∂
∂ ∆
−
∂
∂ ∆
− =
(3.8)
) O(
) , , (
'
) , (
' ' 2
) , , (
) , (
'
) , ( ) , (
3
2
2 2
0
t
t
t h x E
C
x
t x i
C L
t
t h x E
x
t x v
L
t
t x i t x i
e
x e
x
s
∆ +
+
|
|
.
|


\
|
∂
∂
+
∂
∂ ∆
+
|
|
.
|


\
|
−
∂
∂ ∆
− =
(3.9)

In order to represent equations (3.8) and (3.9) using an FDTD technique, we will proceed
with the discretization of time and space as follows

n
k
s s
v t n x k v t x v = ∆ ∆ = ) , ( ) , (
(3.10)
n
k
i t n x k i t x i = ∆ ∆ = ) , ( ) , (
(3.11)
n
k
e
x
e
x
Eh t n h x k E t h x E = ∆ ∆ = ) , , ( ) , , (
(3.12)

where:
! ∆x: spatial integration step;
! ∆t: time integration step;
! k= 1,2,…, kmax;
! n= 1,2, …, nmax.

In the integration scheme, the scattered voltage and current at time step n are known for
all spatial nodes. Therefore equations (3.8) and (3.9) allow us to compute the scattered
voltage and current at the time step n+1.
The spatial derivatives of the scattered voltage, line current, and horizontal electric field
can be written respectively as

) O(
2
) , (
1 1
x
x
v v
x
t x v
n
k
n
k
t n t
s
∆ +
∆
−
=
∂
∂
− +
∆ =
(3.13)
) O(
2
) , (
1 1
x
x
i i
x
t x i
n
k
n
k
t n t
∆ +
∆
−
=
∂
∂
− +
∆ =
(3.14)
) O(
2
) , , (
1 1
x
x
Eh Eh
x
t h x E
n
k
n
k
t n t
e
x
∆ +
∆
−
=
∂
∂
− +
∆ =
(3.15)

On the other hand, the time derivative of the horizontal electric field reads

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


29
) O(
2
) , , (
1 1
t
t
Eh Eh
t
t h x E
n
k
n
k
t n t
e
x
∆ +
∆
−
=
∂
∂
− +
∆ =
(3.16)

The second order spatial derivatives can be written as
) O(
2 ) , (
2
1 1
2
2
x
x
v v v
x
t x v
n
k
n
k
n
k
t n t
s
∆ +
∆
− +
=
∂
∂
− +
∆ =
(3.17)
) O(
2 ) , (
2
1 1
2
2
x
x
i i i
x
t x i
n
k
n
k
n
k
t n t
∆ +
∆
− +
=
∂
∂
− +
∆ =
(3.18)

Inserting equations (3.10)-(3.16) into (3.8) and (3.9), we obtain the 1
st
order FDTD
scheme. If we insert equations (3.10)-(3.18) into (3.8) and (3.9), we obtain the following 2
nd

order FDTD scheme:

|
|
.
|


\
|
∆
− +
−
∆
− ∆
−
|
|
.
|


\
|
∆
− ∆
− =
− + − + − + +
2
1 1 1 1
2
1 1 1
2
2 ' ' 2 2 ' x
v v v
x
Eh Eh
C L
t
x
i i
C
t
v v
n
k
n
k
n
k
n
k
n
k
n
k
n
k n
k
n
k
(3.19)
|
|
.
|


\
|
∆
−
+
∆
− + ∆
+
|
|
.
|


\
|
−
∆
− ∆
− =
− +
− + − + +
t
Eh Eh
C
x
i i i
C L
t
Eh
x
v v
L
t
i i
n
k
n
k
n
k
n
k
n
k n
k
n
k
n
k n
k
n
k
2
'
2
' ' 2 2 '
1 1
2
1 1
2
1 1 1
(3.20)

It is worth noting that, compared to the 1
st
order point centered scheme adopted in
[Agrawal et al., 1980; Rachidi et al., 1996], the spatial-temporal grid distribution for current
and voltage nodes in the 2
nd
order scheme is quite different. Fig. 3.2 shows a schematic
representation of the spatial-temporal grid for both integration schemes.







Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


30

n+1
n

n- 1
k k+1 k - 1
spatial
discretization
time
discretization


n n
i v
0 0
n
k
n
k
i v
max max
n
k
n
k
i v
R
0
∫
h
e
z
dz t E
0
) , 0 , 0 (
∫
h
e
z
dz t L E
0
) , 0 , (
R
L
) t , h , x ( E
e
x
1
st
order FDTD
point centered


n+1
n

n- 1
k k+1 k - 1
spatial
discretization
time
discretization


n n
i v
0 0
,
n
k
n
k
i v
max max
,
n
k
n
k
i v ,
R
0
∫
h
e
z
dz t E
0
) , 0 , 0 (
∫
h
e
z
dz t L E
0
) , 0 , (
R
L
) t , h , x ( E
e
x
2
nd
order FDTD


Fig. 3.2. – FDTD 1
st
and 2
nd
order integration schemes applied to the case of a single-conductor lossless
overhead line above a perfectly conducting ground illuminated by an external electromagnetic field. Left
column: time and spatial discretization; right column: schematic representation of the spatial discretization
along the line.

The boundary conditions for resistive terminations, can be expressed as follows:

∫
+ − =
h
e
z
n n
dz t E i R v
0
0 0 0
) , 0 , 0 (
(3.21)
∫
+ =
h
e
z
n
k L
n
k
dz t L E i R v
0
max max
) , 0 , ( (3.22)

Case of a single-conductor line above a lossy ground. We now extend the FDTD 2
nd

order integration scheme previously presented in order to take into account the presence of
an uniform lossy ground, characterized by its conductivity σ
g
and its relative permittivity
ε
rg
. In this case, the Agrawal et al. coupling equations become [Rachidi et al., 1996]:

) , , (
) , (
) ( '
) , (
'
) , (
0
t h x E d
x i
t
t
t x i
L
x
t x v
e
x
t
g
s
=
∂
∂
− +
∂
∂
+
∂
∂
∫
τ
τ
τ
τ ξ

(3.23)
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


31
0
) , (
'
) , (
=
∂
∂
+
∂
∂
t
t x v
C
x
t x i
s

(3.24)

in which ξ’
g
(t) is the transient ground resistance [Rachidi et al., 1999], which can be
evaluated using the analytical expression (2.14) [Rachidi et al, 2000].
In equations (3.23) and (3.24), the contributions from the wire impedance and the ground
admittance, which have shown to be negligible for typical power lines [Rachidi et al.,
1996], are deliberately disregarded. Following a procedure similar to the case of a lossless
line, equations (3.23) and (3.24) become

t
t h x E
C
t
t x v
C
t
t x i
C L
x
t x i
e
x
g
∂
∂
− =
∂
∂
−
∂
∂
−
∂
∂ ) , , (
'
) , ( '
'
) , (
' '
) , (
2
2
2
2

(3.25)
x
t h x E
x
t x v
t
t x v
C L
x
t x v
e
x
g
s s
∂
∂
=
∂
∂
+
∂
∂
−
∂
∂ ) , , (
) , ( '
) , (
' '
) , (
2
2
2
2

(3.26)

in which

∫
∂
∂
− =
t
g g
d
i
t t x v
0
) (
) ( ' ) , ( ' τ
τ
τ
τ ξ (3.27)

Expanding the current and the scattered voltage using Taylor’s series, and replacing the
time derivative of the current and of the scattered voltage in equations (3.25) and (3.26), we
obtain the following second order differential equation:

) O(
) , ( '
) , ( ) , , (
' ' 2
) , (
'
) , ( ) , (
3
2
2 2
0
t
x
t x v
x
t x v
x
t h x E
C L
t
x
t x i
C
t
t x v t x v
g
s e
x
s s
∆ +
|
|
.
|


\
|
∂
∂
−
∂
∂
−
∂
∂ ∆
−
+
∂
∂ ∆
− =
(3.28)
) O(
) , ( '
'
) , , (
'
) , (
' ' 2
) , ( ' ) , , (
) , (
'
) , ( ) , (
3
2
2 2
0
t
t
t x v
C
t
t h x E
C
x
t x i
C L
t
t x v t h x E
x
t x v
L
t
t x i t x i
g
e
x
g
e
x
s
∆ +
|
|
.
|


\
|
∂
∂
−
∂
∂
+
∂
∂ ∆
+
+
|
|
.
|


\
|
+ −
∂
∂ ∆
− =
(3.29)

Following a procedure similar as that for the ideal ground case, we obtain the 2
nd
order
FDTD discretized equations:

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


32
|
|
.
|


\
|
∆
−
∆
+
+
|
|
.
|


\
|
∆
− +
−
∆
− ∆
−
|
|
.
|


\
|
∆
− ∆
− =
− +
− + − + − + +
x
v v
C L
t
x
v v v
x
Eh Eh
C L
t
x
i i
C
t
v v
n
k
g
n
k
g
n
k
n
k
n
k
n
k
n
k
n
k
n
k n
k
n
k
2
' '
' ' 2
2
2 ' ' 2 2 '
1 1
2
2
1 1 1 1
2
1 1 1
(3.30)
|
|
.
|


\
|
∆
−
∆
−
|
|
.
|


\
|
∆
−
+
∆
− + ∆
+
+
|
|
.
|


\
|
+ −
∆
− ∆
− =
−
− +
− +
− + +
t
v v
L
t
t
Eh Eh
C
x
i i i
C L
t
v Eh
x
v v
L
t
i i
n
k
g
n
k
g
n
k
n
k
n
k
n
k
n
k
n
k
g
n
k
n
k
n
k n
k
n
k
1
2 1 1
2
1 1
2
1 1 1
' '
' 2 2
'
2
' ' 2
'
2 '
(3.31)

Extension to the case of a multi-conductor line. The Agrawal et al. field-to-transmission
line coupling equations for a multi-conductor line above a lossy ground, reported in
paragraph 2.3 (equations (2.25) and (2.26)), can be numerically integrated by means of the
same FDTD 2
nd
order scheme used for the single-conductor case, obtaining the following
second-order differential equation:

[ ] [ ] [ ]
[ ]
[ ][ ] [ ]
[ ] [ ]
[ ][ ] [ ]
[ ]
|
|
.
|


\
|
∂
∂
∆
+
+
|
|
.
|


\
|
∂
∂
−
∂
∂ ∆
−
+
∂
∂
∆ − =
−
−
−
x
t x v
C L
t
x
t x v
x
t h x E
C L
t
x
t x i
C t t x v t x v
gi
ij ij
s
i i
e
x
ij ij
i
ij
s
i
s
i
) , ( '
' '
2
) , ( ) , , (
' '
2
) , (
' ) , ( ) , (
1
2
2
2
1
2
1
0

(3.32)
[ ] [ ] [ ]
[ ]
[ ] [ ]
[ ][ ] [ ]
[ ]
[ ]
[ ]
[ ][ ] [ ] [ ]
[ ]
|
|
.
|


\
|
∂
∂
∆
−
+
|
|
.
|


\
|
∂
∂
+
∂
∂ ∆
+
+
|
|
.
|


\
|
+ −
∂
∂
∆ − =
−
−
−
t
t x v
C L C
t
t
t h x E
C
x
t x i
L C
t
t x v t h x E
x
t x v
L t t x i t x i
gi
ij ij ij
i
e
x
ij
i
ij ij
gi i
e
x
s
i
ij i i
) , ( '
' ' '
2
) , , (
'
) , (
' '
2
) , ( ' ) , , (
) , (
' ) , ( ) , (
1
2
2
2
1
2
1
0
(3.33)

in which

[ ] [ ] [ ] ) , ( ' ) , ( ' t x i
t
t x v
i gij gi
∂
∂
⊗ = ξ
(3.34)

And finally, the 2
nd
order FDTD representation of (3.32) and (3.33) reads

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


33
[ ] [ ] [ ]
[ ] [ ]
[ ][ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ][ ] [ ]
[ ] [ ]
|
|
.
|


\
|
∆
−
∆
+
+
|
|
.
|


\
|
∆
− +
−
∆
− ∆
−
+
|
|
.
|


\
|
∆
−
∆ − =
− +
−
− + − +
−
− +
− +
x
v v
C L
t
x
v v v
x
Eh Eh
C L
t
x
i i
C t v v
n
k
gi
n
k
gi
ij ij
n
k i
n
k i
n
k i
n
k i
n
k i
ij ij
n
k i
n
k i
ij
n
k i
n
k i
2
' '
' '
2
2
2
' '
2
2
'
1 1
1
2
2
1 1 1 1
1
2
1 1
1 1
(3.35)
[ ] [ ] [ ]
[ ] [ ]
[ ] [ ]
[ ][ ] [ ]
[ ] [ ] [ ]
[ ][ ] [ ] [ ]
[ ] [ ]
[ ][ ] [ ] [ ]
[ ] [ ]
|
|
.
|


\
|
∆
−
∆
−
+
|
|
.
|


\
|
∆
− ∆
+
+
|
|
.
|


\
|
∆
− + ∆
+
+
|
|
.
|


\
|
+ −
∆
−
∆ − =
−
−
− +
−
− +
−
− +
− +
t
v v
C L C
t
t
Eh Eh
C L C
t
x
i i i
L C
t
v Eh
x
v v
L t i i
n
k
gi
n
k
gi
ij ij ij
n
k i
n
k i
ij ij ij
n
k i
n
k i
n
k i
ij ij
n
k
gi
n
k i
n
k i
n
k i
ij
n
k i
n
k i
1
1
2
1 1
1
2
2
1 1
1
2
1 1
1 1
' '
' ' '
2
2
' ' '
2
2
' '
2
'
2
'
(3.36)

Treatment of line transverse discontinuities in FDTD 2
nd
order scheme for a single-
conductor line. In the proposed FDTD 2
nd
order scheme, the voltages and currents nodes are
coincident, which allows simplifying the equations for the treatment of the periodical
groundings. The treatment of a shunt impedance representing one of the conductor
groundings for an overhead line illuminated by an external electromagnetic field is
schematically illustrated in Fig. 3.3.


∫
h
e
z
dz ) t , z , x ( E
0
Γ
Voltage and Current
FDTD 2
nd
order node
v
k
n+1
, i
k
n+1
v
k-1
n+1
, i
k-1
n+1

dx
k spatial discretization
n time discretization
dx spatial discretization
i
k’
n

i
k”
n

i
g
n

Known variables:
• All internal node scattered
voltages and currents at
time step n+1
• Operator Γ (=R
g
in case
of ground resistance)
• Vertical Electric Field E
z


Unknown variables:
• v
k
n+1

• i
k
n+1

v
k+1
n+1
, i
k+1
n+1

g

Fig. 3.3. – Insertion of discontinuity point, in a generic point along a single-conductor line, in the 2
nd
order
Finite Difference integration scheme.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


34
Applying equation (3.1) the node voltage v
k
n+1
(see Fig. 3.3) can be expressed as follows:

∫
+ =
+ +
h
e
z
n
g
n
k
(x,z,t)dz E ) Γ ( i v
0
1 1

(3.37)

where
! i
g
n+1
is the current flowing in the grounding impedance;
! Γ is an integral-differential operator, which describes the voltage drop across the
shunt impedance as a function of current ig. (
g g
i R Γ ⋅ = for the simple case of a
resistance).

As for the FDTD 1
st
order scheme (see appendix A.1) current i
g
n+1
can be expressed as
function of the current i
k’
n+1
, i
k”
n+1
applying Kirchhoff’s law on the currents at the
grounding node

1
' '
1
'
1 + + +
− =
n
k
n
k
n
g
i i i
(3.38)

Currents i
k’
n+1
, i
k”
n+1
can be expressed as a function of the adjacent current nodes
assuming the following linear spatial interpolation:

1
2
1
1
1
'
2
+
−
+
−
+
− =
n
k
n
k
n
k
i i i
(3.39)
1
2
1
1
1
' '
2
+
+
+
+
+
− =
n
k
n
k
n
k
i i i
(3.40)

By introducing (3.38) (3.39) and (3.40) in (3.37) we obtain the equation for the scattered
voltage at the grounding point:

∫
+ + − − =
+
+
+
+
+
−
+
−
+
h
e
z
n
k
n
k
n
k
n
k
n
k
(x,z,t)dz E ) i i i i Γ ( v
0
1
2
1
1
1
2
1
1
1
2 2
(3.41)

The node current i
k
n
, needed in equations (3.19) (3.20) to compute voltages and currents
at nodes k-1 and k+1, must be substituted with (3.39) and (3.40) respectively with equations
written for the node k-1 and k+1.

Treatment of line transverse discontinuities in FDTD 2
nd
order scheme for a multi-
conductor line. The treatment of periodical groundings in the FDTD 2
nd
order integration
scheme for a multi-conductor line, is a simple extension in terms of current and voltage
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


35
vector of equations (3.37)-(3.41). In particular the equation for the scattered voltage at the
grounding of the conductor point reads:

[ ] [ ] [ ] [ ] [ ] [ ] ( )
(
(
¸
(


¸

+ + − − =
∫
+
+
+
+
+
−
+
− +
i
h
e
z
n
k i
n
k i
n
k i
n
k i
n
k i
(x,z,t)dz E i i i i ΓK v
0
1
2
1
1
1
2
1
1 1
2 2 (3.42)

Equations (3.39) and (3.40) become:

[ ] [ ] [ ]
1
2
1
1
1
'
2
+
−
+
−
+
− =
n
k i
n
k i
n
k i
i i i
(3.43)
[ ] [ ] [ ]
1
2
1
1
1
' '
2
+
+
+
+
+
− =
n
k i
n
k i
n
k i
i i i
(3.44)

where

[ ] [ ] ;
:
;
:
3
2
1
3
2
1
|
|
|
|
|
|
.
|






\
|
=
|
|
|
|
|
|
.
|






\
|
=
n
k NC
n
k
n
k
n
k
n
k i
n
k NC
n
k
n
k
n
k
n
k i
v
v
v
v
v
i
i
i
i
i (3.45)
|
|
|
|
|
|
|
|
|
|
|
|
.
|












\
|
=
(
(
¸
(


¸

∫
∫
∫
∫
∫
NC
i
h
e
z
h
e
z
h
e
z
h
e
z
h
e
z
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
0
0
0
0
0
:
3
2
1
(3.46)

and matrix [ ΓK ]:

[ ]
|
|
|
|
|
|
.
|






\
|
   
=
NCNC
K
K
K
K
K
Γ | 0 0 0
|
0 | Γ 0 0
0 | 0 Γ 0
0 | 0 0 Γ
Γ
33
22
11

(3.47)

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


36
3.1.3. – Comparison with experimental results and simulations
The validation of the Agrawal coupling model modified for the treatment of periodical
groundings was performed by means of the experimental results obtained on reduced scale
line models illuminated by an EMP simulator. The reported experimental data was obtained
from two different sessions performed at the Swiss Federal Institute of Technology in
Lausanne on the EMP simulator of their laboratory described in more detail in [Arreghini et
al., 1993]. The simulator is a bounded wave vertically-polarized type. A measurement
record of the waveform of the electric vertical field inside the working volume, performed
in absence of the line during the experimental campaign, is presented in Fig. 3.4.

0
5
10
15
20
25
0 50 100 150 200
Time [ns]
V
e
r
t
i
c
a
l


E
l
e
c
t
r
i
c


F
i
e
l
d


[
k
V
/
m
]

Fig. 3.4. – Vertical electric field in absence of the line measured in the working volume of the SEMIRAMIS
EMP simulator. Adapted from Paolone et al. [2000].

The reduced scale line model reproduces single and multi-conductor lines. The procedure
used for the validation is based on the measurement of the electrical field generated in the
simulator (see Fig. 3.4) and the measurement of the induced current in the reduced scale line
model at line terminations for each conductor. The measured vertical electric field is used as
input in the modified Agrawal coupling model, then the results obtained from the
simulations are compared with the measured currents.
In what follows we report the results concerning both a single conductor with a shielding
wire grounded at the line extremities (Fig. 3.5), and a three conductor configuration with a
shielding wire grounded at line extremities and at the line center (Fig. 3.7).
For the configuration with a single conductor, the shielding wire was placed above and
under the phase conductor at different heights (as shown in Fig. 3.5). The comparison
between the measurement and the simulation, reported in Figs. 3.6, show the current at line
terminations on the phase conductor with and without the presence of the shielding wire.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


37
2m
Shielding
Wire
Phase
conductor
2
0

c
m

1
4
-
1
6
-
1
8
-
2
2
-
2
4
-
2
6

c
m

1.4 mm
0

Ω

6
8
0

Ω

6
8
0

Ω

0

Ω

Return
conductor

Fig. 3.5. – Reduced scale line model used in the SEMIRAMIS EMP Simulator composed by a single
conductor with shielding wire grounded at line extremities.

a)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 14 cm measurement
with SW 14 cm simulation

b)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 16 cm measurement
with SW 16 cm simulation

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


38
c)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 18 cm measurement
with SW 18 cm simulation

d)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 22 cm measurement
with SW 22 cm simulation

e)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 24 cm measurement
with SW 24 cm simulation

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


39
f)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
without SW measurement
without SW simulation
with SW 26 cm measurement
with SW 26 cm simulation

Fig. 3.6. – Comparison between the experimental results and simulations relevant to line configuration of Fig.
3.5: induced current on the phase conductor; a) height of shielding 14 cm; b) height of shielding 16 cm; c)
height of shielding 18 cm; d) height of shielding 22 cm; e) height of shielding 24 cm; f) height of shielding 26
cm.

For the line configuration with three conductors, the shielding wire was placed above the
highest phase conductor (as indicated in Fig. 3.7). The comparison between the
measurement and the simulation, reported in Figs. 3.8, show the current at line terminations
on all the line conductors (phase conductors and shielding wire) with and without the
presence of the shielding wire.

2m
Shielding
Wire
Phase
conductors
2
0

c
m

2
7
.
4

c
m

1.4 mm
0

Ω

6
8
0

Ω

0

Ω

6
8
0

Ω

6
8
0

Ω

6
8
0

Ω

0

Ω

6
8
0

Ω

6
8
0

Ω

Return
conductor
3
4
.
8

c
m

4
0
.
9

c
m


Fig. 3.7. – Reduced scale line model used in the SEMIRAMIS EMP Simulator composed by three conductors
with shielding wire grounded at line extremities.


Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


40
a)
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
conductor 1 without SW measurement
conductor 1 without SW simulation
conductor 1 with SW measurement
conductor 1 with SW simulation

b)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
conductor 2 without SW measurement
conductor 2 without SW simulation
conductor 2 with SW measurement
conductor 2 with SW simulation

c)
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
conductor 3 without SW measurement
conductor 3 without SW simulation
conductor 3 with SW measurement
conductor 3 with SW simulation

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


41
d)
-0.5
1.5
3.5
5.5
7.5
9.5
11.5
13.5
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time [µs]
C
u
r
r
e
n
t


[
A
]
conductor SW measurement
conductor SW simulation

Fig. 3.8. – Comparison between the experimental results and simulations relevant to line configuration of Fig.
3.7: induced current on the line conductor; a) conductor 1; b) conductor 2; c) conductor 3; d) shielding wire.

An additional validation of the proposed model was performed by testing it with the
theoretical results of Yokoyama [1984]. The line configuration considered there is presented
in Fig. 3.9. Note that there is only one grounding of the shielding wire, at the line center, in
front of the stroke location. Our results of computation considering different values for the
grounding resistance are presented in Fig. 3.10. The results practically coincide with those
presented by Yokoyama, reported in Fig. 3.10b for convenience.



2 km
1
0

m

0.8 cm
Rg Matched
1000 m
1
0
0

m

Grounded
Conductor
Matched
0.5 m 0.5 m

Fig. 3.9. – Line geometry adopted to evaluate the effect of a shielding wire to a multi-conductor line
[Yokoyama, 1984]. Lightning current: peak value 100 kA; maximum time-derivative 50 kA/µs. Ideal ground.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


42
a)
0
100
200
300
0 2 4 6 8 10 12 14 16 18 20
Time [µs]
I
n
d
u
c
e
d

O
v
e
r
v
o
l
t
a
g
e

[
k
V
]
Rg=0 Ohm
Rg=30 Ohm
Rg=100 Ohm
Rg=200 Ohm
Rg=500 Ohm
Rg=1.E9 Ohm

b)
Fig. 3.10. – Induced voltage for a variable grounding resistance (R
g
). Line configuration of Fig. 3.9: a)
simulation results, b)Yokoyama [1984]-Fig. 4. Adapted from Paolone et al. [2000].

The results of Fig. 3.10 show that the value of the grounding resistance largely affects the
amplitude of the induced voltages. Such results can be ascribed, as we shall see better in
what follows, to the fact that the stroke location is situated just in front of the grounding
resistance.

Effect of the shielding wires groundings. To better assess the effect of the shielding wire,
of the number of the periodical groundings and of the value of the grounding resistance, we
have considered the line geometry shown in Fig. 3.11 in which the stroke location does not
‘face’ any of the grounding points.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


43
L
Shielding
Wire
Phase
conductor
Variable
9

m

1
0

m

1 cm
Rg Rg Rg Zc Zc Rg Rg
750 m
5
0

m

Stroke
Location

Fig. 3.11. – Line geometry adopted to evaluate the effect of the presence of a shielding wire. Lightning
current: peak value 30 kA maximum time derivative 40 kA/µs. Adapted from Paolone et al. [2000].

The adopted lightning current has the following characteristics: peak value 30 kA
maximum time derivative 40 kA/ms. The following results concern the effect on lightning
induced voltage peak value of the grounding step namely: 100 m; 200 m; 500 m and 1000
m, and of the grounding resistance namely: 0 Ω; 30 Ω, 100 Ω and 1 kΩ. In Figs 3.12, we
present the peak value of the induced overvoltages along the 2 km long line for a perfectly
conducting ground; the same results have been computed taking into account the finite
ground conductivity (σ
g
=0.001 S/m) and they are shown in Fig. 3.13.
In addition, the following figures which relate to the ideal ground case, the mitigation
effect as predicted by the Rusck formula is also shown:

g sw
i sw
i
sw
i
i
R Z
Z
h
h
U
U
2
1
'
+
⋅ − = =
−
η (3.48)

where:
! U’
i
is the lightning induced voltage in conductor i in presence of the shielding
wire;
! U
i
is the lightning induced voltage in the conductor in absence of the shielding
wire;
! h
sw
is the height of the shielding wire;
! h
i
is the height of conductor i;
! Z
sw-i
is the mutual surge impedance between the shielding wire and conductor i;
! Z
sw
is the surge impedance of the shielding wire;
! R
g
is the grounding resistance.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


44
It is worth reminding that the Rusck expression does not cover the case of multiple
groundings of the shielding wire (it assumes that the shielding wire is grounded at single
point along the line) and that is applicable in the assumption of perfectly conductive ground.

a)
0
50
100
150
200
250
0 500 1000 1500 2000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs
21 grs
Rusck

b)
0
50
100
150
200
250
0 500 1000 1500 2000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs
21 grs
Rusck

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


45
c)
0
50
100
150
200
250
0 500 1000 1500 2000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs
21 grs
Rusck

d)
0
50
100
150
200
250
0 500 1000 1500 2000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs
21 grs
Rusck

Fig. 3.12. – Induced voltage peak value along the line for a variable number of grounding step and variable
grounding resistance namely: a) 0 Ω, b) 30 Ω, c) 100 Ω, d) 1 kΩ. Line configuration of Fig. 3.11 with L=2 km.
Ideal ground. Adapted from Paolone et al. [2000].

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


46
a)
0
50
100
150
200
250
300
0 200 400 600 800 1000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs

b)
0
50
100
150
200
250
300
0 200 400 600 800 1000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SW
3 grs
5 grs
11 grs

Fig. 3.13. – Induced voltage peak value along the line for a variable number of grounding step and variable
grounding resistance namely: a) 10 Ω, b) 300 Ω. Line configuration of Fig. 3.11 with L=1 km. Stroke location
at 375 m from the left line termination. Lossy ground σ
g
=0.001 S/m. Adapted from Paolone et al. [2000].

The results reported in Fig. 3.10a show that, if the stroke location is located in front of a
grounding point, the attenuation of the induced overvoltage on the phase conductors is very
dependent from the value of the grounding resistance. Additionally, as confirmed by
Yokoyama [1984], the prediction of the maximum value of the induced overvoltages
obtained by the application of Rusck formula (3.48), is the same as the results obtained by
considering the shielding wire an illuminated conductor.
If, instead, the stroke location does not face any grounding point, which is the more
frequent case, the results show that for both cases of ideal and lossy ground, the mitigation
effect of the shielding wire depends, in general, more on the number of groundings than on
the value of the grounding resistance. The presence of a large number of groundings tends,
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


47
in fact, to shift the shielding wire potential to the ground one enhancing the mitigation effect
as predicted, for instance, by the Rusck approach.
For the case of lossy ground the number of groundings plays a decisive role too. This
differs from the case of direct stroke for which the effectiveness of the shielding wire
depends strongly on the grounding resistance.
The value of the grounding resistance which appear to influence the attenuation of the
induced overvoltage, for both considered ideal and lossy ground conductivity, is 100÷200
Ohms.
The comparison with the Rusck formula is possible only for the case of ideal ground: in
most cases, it tends to overestimate the effect of the shielding wire. This is due to the Rusck
formula assumption that considers the shielding wire as a conductor at ground potential.
Indeed only with a large number of grounding points (e.g. 11-21 groundings), the results
predicted by Rusck formula and the proposed model tend to provide the same values.

Effect of the presence of surge arresters. The presence of surge arresters along the line
can be treated in a similar way as for the discontinuities caused by the periodical groundings
of shielding wire. In this case, we have modeled the surge arresters representing them with a
non-linear V-I characteristic (the integro-differential operator Γ) placed in series with the
incident voltage.
To compare the mitigation effect achieved by employing surge arresters with that of the
shielding wire periodical grounding technique, we have considered the same single-
conductor line of the previous simulation (Fig. 3.14).


L
Phase
conductor
Variable
9

m

1 cm
Surge
Arrester
Surge
Arrester
Surge
Arrester
Surge
Arrester
Zc
Surge
Arrester
Zc
750 m
5
0

m

Stroke
Location

Fig. 3.14. – Line geometry adopted to evaluate the effect of surge arrester. Lightning current: peak value 30
kA maximum time derivative 40 kA/µs. Adapted from Paolone et al. [2000].



Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


48
The V-I characteristic of the surge arrester is shown in the following table.

Voltage [V] Current [A]
20820 0.09E-02
25980 0.12E-02
29100 0.06E-01
31140 0.06
33300 0.6
47460 3000
51000 6000
56640 12000

Tab. 3.1. – Surge arrester V-I characteristic.

We have considered a variable number of surge arresters placed along the line: 2 (at the
line terminal only) 3 (each 1000 m), 5 (each 500 m) and 11 (each 200 m). The induced
overvoltage amplitudes are shown in Figs. 3.15 and 3.16.

-250
-150
-50
50
150
250
0 500 1000 1500 2000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SA
2 SA
3 SA
5 SA
11 SA

Fig. 3.15. – Induced voltage peak value along the line for a variable number of surge arresters along the line.
Line configuration of Fig. 3.14 with L=2 km. Ideal ground. Adapted from Paolone et al. [2000].
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


49
-100
0
100
200
300
0 200 400 600 800 1000
Distance along the line [m]
P
e
a
k


V
a
l
u
e


o
f


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
without SA
2 SA
3 SA
6 SA

Fig. 3.16. – Induced voltage peak value along the line for a variable number of surge arresters along the line.
Line configuration of Fig. 3.14 with L=1 km. Stroke location at 375 m from the left line termination. Lossy
ground σ
g
=0.001 S/m. Adapted from Paolone et al. [2000].

The computed results show that an important reduction of the induced overvoltage can be
achieved only with a large number of surge arresters namely 1 surge arrester each 0.2 km. It
can also be seen that for some configurations, the presence of surge arresters could result in
important negative peaks which are due to the reflection coefficients, associated with the
surge arresters, when the induced voltage exceed the threshold voltage of this non-liner
component. Indeed, depending on the line configuration, stroke location and distance
between two consecutive surge arresters, the negative voltage wave generated by the non-
linear components can produce the maximum amplitude of the induced overvoltage not in
the point closest to the stroke location. In addition this overvoltage can be more severe than
the maximum voltage amplitude induced in the absence of surge arresters (see Fig. 3.15).
By increasing the number of surge arresters the maximum amplitude of the induced
overvoltage tends to stay within the range generated by the positive and negative values of
the threshold voltage of the surge arrester V-I non-linear characteristic (see Tab. 3.1).

3.2. – Distribution systems
This paragraph reports the developed illuminated distribution system model. It is
obtained by means of an interface between the developed complex overhead line model and
two popular programs used for the power system transient analysis, namely: the EMTP96
and the Power System Blockset within Matlab environment.
The developed illuminated distribution system model is aimed at the correct estimation
of the induced overvoltages for realistic distribution systems. This estimation is necessary in
order to optimize the number and location of protective devices (shielding wire groundings
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


50
and surge arresters) and to minimize the number of outages. In particular, the presence of
distribution transformers and of the relevant protection devices at the line terminations, as
well as the presence of surge arresters along the line, should be considered in the simulation.
The use of the developed illuminated distribution system model permits to perform the
analysis of the effect of the aperiodic disturbance caused by LEMP on the various power
system components (e.g. the distribution transformers). Indeed the lightning-induced
overvoltages on distribution transformers, and their propagation from the medium voltage to
the low voltage of the distribution system, is not yet considered in the literature on the
argument.
The aim of the proposed model is to gives an engineering tool enabling to analyze the
above mentioned aspects.
3.2.1. – Proposed approach
For evaluating the lightning performance of distribution networks the availability of a
tool for the calculation of lightning-induced voltages is crucial. Such a tool can be a) a
simple analytical formula (as for instance the Rusck [1958] one, adopted in the IEEE WG on
lightning performance of distribution lines [1997]), b) a computer code.
An analytical formula presents the advantage of short computational times; on the other
hand its application is limited to cases which tend to be unrealistic (as reported in the
comparison of Fig. 2.7 between the Rusck simplified formula and the Agrawal model). The
distribution system model that we are aimed at developing instead must potentially permit
the treatment of realistic cases reported schematically in Fig. 3.17.
In the LIOV code [Nucci et al., 1993; Rachidi et al. 1996,1997, Nucci, 2000] the
Agrawal et al. field-to-transmission line coupling model has been implemented for dealing
with the case of multi-conductor lines closed on resistive terminations. In principle, the
LIOV code could be suitably modified, case by case, in order to take into account the
presence of the specific type of termination, line-discontinuities (e.g. surge arresters across
the line insulators along the line) and of complex system topologies. This procedure requires
that the boundary conditions for the transmission-line coupling equations be properly re-
written case by case, as discussed in [Nucci et al., 1994].
However, as proposed by other authors too [Nucci et al., 1994; Orzan et al.,1996; Orzan,
1998; Høidalen, 1997,1999], we found more convenient to link the LIOV code with the
EMTP. With these LIOV-EMTP codes it is possible to analyze the response of realistic
distribution systems. In [Nucci et al., 1994], the distribution system model is considered, as
in the proposed approach (see next paragraph), as consisting of a number of illuminated
lines connected to each other through a shunt admittance. The LIOV code has the task of
calculating the response of the various lines connecting the two-ports (see Fig. 3.17); the
EMTP has the task of solving the boundary condition and presents the advantage of making
available a large library of power components.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


51
In this new version of the interface between the LIOV code and the EMTP96 (henceforth
called LIOV-EMTP96), each LIOV line is described by the Agrawal coupling model but the
partial differential equation are now solved using the previously described line model based
on the FDTD 2
nd
order scheme in order to improve the numerical stability when non-linear
phenomena are considered.
In the literature on the argument there are two other proposed models for the evaluation
of the LEMP response of distribution networks, proposed by Orzan [Orzan et al.,1996;
Orzan, 1998] and Høidalen [1997,1999] respectively.
In [Orzan et al.,1996; Orzan, 1998] for each illuminated line the coupling between the
external incident field and the phase conductors is reproduced, in the distribution network
model, by means of equivalent current generators whose value is pre-calculated by solving
the Agrawal transmission line coupling model. The current generators are then inserted in
an EMTP simulation where traditional lines models replace illuminated lines. Such an
approach cannot take into account non-linear local phenomena, like a variation in the line-
capacitance with space, as necessary for instance when taking into account the presence of
corona phenomenon [Nucci et al., 2000].
In the model proposed by Høidalen [1997,1999], the analysis of the response of the
illuminated distribution system uses the same concepts adopted in the model developed by
Orzan namely each illuminated line response is reproduced by means of equivalent voltage
generators. The main difference with Nucci et al. [1994] and [Orzan et al.,1996; Orzan,
1998] is in the evaluation of the coupling between the external incident field and each
illuminated line: Høidalen solves it using an analytical approach valid only for the ideal
ground case.
Finally, compared to [Nucci et al., 1994], the advantages of this new LIOV-EMTP96
program are that: (i) the illuminated line model is able to take into account more complex
line configuration, (ii) it does not require any modification of the EMTP source code, (iii) it
allows for the treatment of corona effect which was implemented.

3.2.2. – Interface between the developed line model and transient programs
Interface between LIOV and EMTP96. As previously mentioned, in order to perform in a
straightforward way the analysis of the LEMP response of real distribution systems
characterized by a certain topological complexity, the developed program based on the 2
nd

order FDTD scheme has been interfaced with the Electromagnetic Transient Program
(EMTP96). Note that in principle, the developed program could have been suitably enlarged
and extended case by case to take into account the specific system configuration, as
discussed in [Nucci et al., 1994]. As a matter of fact, paragraph 3.1 relevant to the transverse
discontinuities, report an example of such a concept.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


52
The concept at the basis of the new interface, a beta-version of which has been presented
in [Borghetti et al., 2000a; Paolone et al., 2001b] , is schematically described in Fig. 3.17.



Γ
0
LIOV lines
n-port

Fig. 3.17. – Electrical distribution system illuminated by LEMP.

The distribution system network is considered as consisting of a number of illuminated
lines connected to each other through a shunt admittance. This admittance represent the
presence of surge arresters, of groundings of shielding wires, of distribution transformers or
of other power components. Each section of the distribution system between two
consecutive shunt admittances is modeled as a single line henceforth called ‘LIOV-line’,
while the program which result from the link between the LIOV and the EMTP96 will be
called LIOV-EMTP96.
The difference from [Nucci et al., 1994] is that the new interface does not require any
modification to the source code of the EMTP: the modified LIOV code is indeed contained
in a dynamic link library (DLL), called within the TACS environment. The data exchange
between the LIOV code and EMTP96 is realize in the following way: the induced currents
at the terminal nodes, computed by the modified LIOV code are input to the EMTP via
current controlled generators, and the voltages calculated by the EMTP are input to the
modified LIOV code via voltage sources.
The scheme of the data exchange in the developed interface is shown in Fig. 3.18.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


53
EMTP
(1) Line terminals node
currents and voltages at
time step N
TACS
LIOV
LINE
(2) Line terminals node
currents and voltages at
time step N
(3) Line terminals voltage
at time step N+1
(4) Line terminals
voltage at time
step N+1
N=N+1
(6) Line terminals voltage at
time step N+1 inserted in
the EMTP by current
controlled generators
(5) EMTP time step
increment for node voltages
and branch currents
TACS – LIOV LINE
(2) (3)
(6) (6)
EMTP
(4) (5)
V (1)
V
V (1)
I (1) I (1)
V
Line
Termination
Line
Termination


Fig. 3.18. – Interface between LIOV code and EMTP96. Adapted from Paolone et al. [2001a].

The link between the LIOV-line and the EMTP96 is realized by means of a loss less
Bergeron line (see Fig. 3.19). The insertion of this line in necessary to write, in an
appropriate way, the link equation between the terminal nodes of the FDTD spatial
discretization of the illuminated line, and the line generic EMTP96 termination. In what
follows we shall refer to the left termination of a single-conductor line; the right line
termination is treated in the same way.


Node ‘0’
+
-
v
0
(t), i
0
(t)
G
2

+
-
+
-
Γ
0
(t)
v
1
(t), i
1
(t)
Z
c
v
0’
(t)
i
0’
(t)
v
e
(t)
Node ‘1’
LIOV line
Bergeron Line
EMTP termination
+
-
-v
e
(t)
G
1
Z
c

v
2
(t), i
2
(t)
Node ‘2’
∆x


Fig. 3.19. – Insertion of the Bergeron loss less line for the data exchange between the LIOV code and the
EMTP96 environment at the left line termination.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


54
where:
! G
1
and G
2
are the Bergeron equivalent generators;
! v
e
(t) is the incident voltage defined in equation (2.13);
! Z
c
is the line characteristic impedance;
! v
0
(t), v
1
(t), i
0
(t), i
1
(t) are the voltages and currents at nodes of the FDTD spatial
discretization;
! v
0’
(t), i
0’
(t) are the voltage and current at the EMTP node;
! Γ
0
(t) is the EMTP termination type.

The generators due to the incident voltages v
e
(t), can be introduced in the Bergeron G
1

and G
2
generators respectively. From the Bergeron equations in a fixed time step n we can
write the following equations:

h E i Z v G
n
z
n
c
n
0 ' 0 ' 0 1
+ − =
(3.49)
h E i Z v G
n
z
n
c
n 1
0
1
0
1
0 2
+ + +
+ − =
(3.50)

where:
! Z
c
is the characteristic impedance of the line;
! h is the conductor height;
! E
z0
is the vertical component of the electric field at the left line termination.

We can write, for node ‘0’ of the FDTD spatial discretization of the LIOV line, the
following equations. In particular if we consider the node ‘0’ as connected with the
Bergeron line we can write:

1
0
1
0 1
1
0
+ + +
− + =
n
c
n
z
n
i Z h E G v
(3.51)

Now if we consider node ‘0’ as connected to the LIOV line we can obtain the scattered
voltage v
0
n+1
from the discretization of the second equation of the Agrawal coupling model
for a single-conductor line (2.12):

0
0
1
0
1
0
1
1
=
∆
−
+
∆
−
+ + +
t
v v
C
x
i i
n n n n

(3.52)
( )
n n n n
v i i
x C
t
v
0
1
0
1
1
1
0
+ −
∆
∆
=
+ + +

(3.53)

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


55
From (3.52) and (3.53) we can write the following equation, from which we can obtain
variable i
0
n+1
:
|
.
|

\
|
− −
∆
∆
+
−
∆
∆
=
+ + + +
1
1
0
1
1
1
0
1
0
1
G h E i
x C
t
v
Z
x C
t
i
n
z
n n
c
n

(3.54)

In summary the algorithm for the data exchange between the LIOV line and the EMTP
consists of the following steps:

! calculation of the input voltage for the generators G
1
located at line left and right
terminations with the (3.49) (see Fig. 3.19);
! calculation, at time step n+1, of the scattered voltages and currents at all FDTD
spatial nodes;
! calculation of the currents i
0
n+1
and i
kmax
n+1

1
from (3.54);
! calculation of the scattered voltages v
0
n+1
and v
kmax
n+1
from (3.53);
! calculation of the values of the output variable for generators G
2
, located at line left
and right terminations, from (3.50).

Interface between LIOV and Power System Blockset within Matlab
environment. The procedure above described for the interface between the LIOV code with
EMTP96, has been also implemented in a Matlab core code (Mat-LIOV). In the Mat-LIOV
code, as in the LIOV one, the model adopted to describe the lightning return stroke is the
MTLE model, the calculation of the incident electromagnetic field is performed in the
presence of a finite conducting ground by using the Cooray-Rubinstein approach
[Rubinstein 1991,1996; Cooray 1992,1994], and the coupling between the external incident
field and a single multi-conductor line is modeled by means of the Agrawal et al. [1980]
coupling model.
The basic concept of the developed interface is the same developed for the EMTP96 one:
leave the task of solving the transmission line coupling equations relevant to each line to the
Mat-LIOV code, and that of solving the boundary conditions to the Power System Blockset.
The link between the Mat-LIOV and the Power System Blockset is realized also in this case
by using a Bergeron line, which accomplish the data exchange between the two mentioned
tasks (see Fig. 3.19. 3.20).
The Mat-LIOV task is contained in an s-function developed to exchange line terminals
voltages and currents with the Power System Blockset environment as shown in Fig. 3.20.
In particular, the s-function calculates the necessary line parameters (inductance,

1
kmax is the last FDTD spatial node at line right termination.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


56
capacitance and surge impedance matrix) and solves, for each line, the discretized
transmission line coupling equations by means of the 2
nd
order FDTD scheme.
The computational procedure is the following: at a given time step the Power System
Blockset sends the nodal voltages at each line termination to the s-function block which
solves the discretized transmission line coupling equations. Then, one time step later, the s-
function block sends the voltage at the line terminations back to the Power System Blockset;
this information exchange is accomplished step by step by means of a Bergeron line as
described in the previous paragraph for the interface with the EMTP96 (see Fig. 3.19).

S-Fuction
(Mat-LIOV)
V
V
V
I
I
V
Line
Termination
Line
Termination

Fig. 3.20. – Interface scheme between Mat-LIOV code and Power System Blockset in Matlab environment.
Adapted from Gutierrez et al. [2001].

3.2.3. – Comparison with experimental results and simulations
An experimental validation of the the developed LIOV-EMTP96 program and relevant
algorithm, is realized by means of the experimental results obtained by Piantini and
Janischewskyj [1992]. The measurements have been performed on reduced scale models set
up at the University of Sao Paulo in Brazil, which reproduce a typical overhead distribution
system (main feeder plus branches) including surge arresters, neutral grounding, T-junctions
(between line branches) and shunt capacitors aimed at modeling distribution transformers.
The surge arresters are simulated by means of a combination of diodes and resistances, and
their V-I non-linear characteristic is shown in Fig. 3.22. For cases 1.1 and 1.2 the lines of
such system have 4 conductors and the line geometry is represented in Fig. 3.21a in the real
scale. For cases 2.1, 2.2 and 2.3 the lines of the system have a simplified configuration with
2 conductors (see Fig 3.21b).
The system that simulates the lightning current generate a current wave shape that can be
approximated with a triangular profile with a time to peak value equal to 2 µs and a half
time equal to 85 µs. For all cases the lightning return stroke velocity is, in the real scale,
equal to 0.33⋅10
8
m/s, the lightning channel height is 600 m and the lightning return stroke
model can be represented with the TL model. For each case different value for the lightning
peak current are considered.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


57
a)
8

m

1
0

m

2 cm
0.75 m
b)
8

m

1
0

m

2 cm


Fig. 3.21. – Line geometries used for the reduced scale models in [Piantini and Janischewskyj, 1992]. a) line
geometry adopted for the cases 1.1 and 1.2, b) line geometry adopted for the cases 2.1, 2.2 and 2.3.

Fig. 3.22. – Surge arresters V-I non-linear characteristic. Adapted from Piantini and Janischewskyj [1992].

Case 1.1. The network configuration is shown in Fig. 3.23, scaled to real dimension. In
Figs. 3.24 are reported the connection types of the line terminations. The stroke location is
represented in Fig. 3.23 at point ‘r.s.m.’, two different value of lightning current are
considered for this case, their characteristics are given in Tab. 3.2.

Lightning current (kA)
Time (µs)
Case 1.1.a Case 1.1.b
0. 0. 0.
2. 34. 70.
85. 17. 35.

Tab. 3.2. – Lightning current for the case 1.1.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


58
90 m 210 m
150 m 148 m
210 m 148 m
346 m
42 m
132 m
284 m
210 m
152 m
210 m
150 m
170 m
1
5
0

m
1
5
0

m
r.s.m.
70 m
20 m
M1

Fig. 3.23. – Network configuration for the case 1.1.

1 2
3
N
R = 50 Ohm
L = 22 uH
SA SA
SA
Ct Ct Ct
Ct = 0.5 nF

1 2 3
N
R = 50 Ohm
L = 22 uH

M1
1 2 3
N
Cp Ct Ct
Cp = 0.65 nF
Ct = 0.5 nF
L = 22 uH

SA
L = 1 uH
Rpr
C = 13.5 uF
Ipr [A] Vpr [V]
0. 0.
36. 27000.
54. 27900.
360. 29700.
1620. 32220.


Fig. 3.24. – Termination types for the network configuration of Fig. 3.23.

Fig. 3.25 reports a comparison between the voltage measured at node M1 (see Fig. 3.23),
and the simulation performed with the LIOV-EMTP96 for lightning current peak values
equal respectively to 34 kA and 70 kA. The measurement and simulation regard the phase
conductor close to the stroke location.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


59
a)
-400
-300
-200
-100
0
100
200
300
400
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

b)
-800
-600
-400
-200
0
200
400
600
800
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Fig. 3.25. – Comparison between measurement and simulation for the case 1.1- a) Lightning current peak
value equal to 34 kA. b) Lightning current peak value equal to 70 kA.

Case 1.2. The network configuration is reported in Fig. 3.26. In Figs. 3.27 are shown the
connection types of the line terminations. The stroke location is represented in Fig. 3.26 at
point ‘r.s.m.’, three different value of lightning current are considered for this case, their
characteristics are given in Tab. 3.3.

Lightning current (kA)
Time (µs)
Case 1.2.a Case 1.2.b Case 1.2.c Case 1.2.d
0. 0. 0. 0. 0.
2. 34. 50. 60. 70.
85. 17. 25. 30. 35.

Tab. 3.3. – Lightning current for the case 3.2.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


60

90 m 210 m
150 m 148 m
210 m 148 m
346 m
42 m
132 m
284 m
210 m
152 m
210 m
150 m
170 m
1
5
0

m
1
5
0

m
r.s.m.
70 m
20 m
M1

Fig. 3.26. – Network configuration for the case 1.2.

1 2
3
N
R = 50 Ohm
L = 22 uH
SA SA
SA
Ct Ct Ct
Ct = 0.5 nF


SA
L = 1 uH
Rpr
C = 13.5 uF
Ipr [A] Vpr [V]
0. 0.
36. 27000.
54. 27900.
360. 29700.
1620. 32220.

M1
1 2 3
N
Cp Ct Ct
Cp = 0.65 nF
Ct = 0.5 nF
L = 22 uH

1 2 3
N
R = 50 Ohm
L = 22 uH
SA SA SA


Fig. 3.27. – Termination types for the network configuration of Fig. 3.26.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


61
Fig. 3.28 reports a comparison between the voltage measured in the node M1 (see Fig.
3.26), and the simulation performed with the LIOV-EMTP96 for various lightning current
peaks namely 34 kA (Fig. 3.28a), 50 kA (Fig. 3.28b), 60 kA (Fig. 3.28c) and 70 kA (Fig.
3.28d).
a)
-120
-100
-80
-60
-40
-20
0
20
40
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

b)
-200
-150
-100
-50
0
50
100
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


62
c)
-200
-150
-100
-50
0
50
100
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

d)
-250
-200
-150
-100
-50
0
50
100
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Fig. 3.28. – Comparison between measurement and simulation for the case 1.2- a) Lightning current peak value
equal to 34 kA. b) Lightning current peak value equal to 50 kA. c) Lightning current peak value equal to 60 kA.
d) Lightning current peak value equal to 70 kA.

Case 2.1. The network configuration is reported in Fig. 3.29. In Figs. 3.30 are shown the
connection types of the line terminations. The stroke location is represented in Fig. 3.29 at
point ‘r.s.m.’, the lightning current characteristics considered for this case are given in Tab.
3.4.

Lightning current (kA)
Time (µs)
Case 2.1 Case 2.2 Case 2.3
0. 0. 0. 0.
2. 70. 34. 50.
85. 35. 17. 25.

Tab. 3.4. – Lightning current for the cases 2.1, 2.2, 2.3.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


63
90 m 210 m
150 m 148 m
210 m 148 m
346 m
42 m
132 m
284 m
210 m
152 m
210 m
150 m
170 m
1
5
0

m
1
5
0

m
r.s.m.
M2
70 m
75 m

Fig. 3.29. – Network configuration for the case 2.1.


1
N
Ct
Ct = 0.5 nF
R = 50 Ohm
L = 22 uH

M2
1
N
Cp
Cp = 0.65 nF
L = 22 uH

SA
L = 1 uH
Rpr
C = 13.5 uF
Ipr [A] Vpr [V]
0. 0.
36. 27000.
54. 27900.
360. 29700.
1620. 32220.


1
N
R = 50 Ohm
L = 22 uH

1
N
R = 50 Ohm
L = 22 uH
SA
Ct
Ct = 0.5 nF

Fig. 3.30. – Termination types for the network configuration of Fig. 3.29.

Fig. 3.31 reports a comparison between the voltage measured in the node M2 (see Fig.
3.29), and the simulation performed with the LIOV-EMTP96 for lightning current peak
equal to 70 kA.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


64
-100
-50
0
50
100
150
200
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Fig. 3.31. – Comparison between measurement and simulation for the case 2.1. Lightning current peak value
equal to 70 kA.

Case 2.2. The network configuration is reported in Fig. 3.32. In Figs. 3.33 are shown the
connection types of the line terminations. The stroke location is represented in Fig. 3.32 at
point ‘r.s.m.’, the lightning current characteristics considered for this case are the same for
the case 2.1 and are given in the previous Tab. 3.4.

90 m 210 m
150 m 148 m
210 m 148 m
346 m
42 m
132 m
284 m
210 m
152 m
210 m
150 m
170 m
1
5
0

m
1
5
0

m
r.s.m.
M2
70 m
75 m


Fig. 3.32. – Network configuration for the case 2.2.


Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


65
1
N
Ct
Ct = 0.5 nF
R = 50 Ohm
L = 22 uH

SA
L = 1 uH
Rpr
C = 13.5 uF
Ipr [A] Vpr [V]
0. 0.
36. 27000.
54. 27900.
360. 29700.
1620. 32220.



1
N
R = 50 Ohm
L = 22 uH

M2
N
Cp
Cp = 0.65 nF
L = 22 uH
SA


Fig. 3.33. – Termination types for the network configuration of Fig. 3.32.

Fig. 3.34 reports a comparison between the voltage measured in the node M2 (see Fig.
3.32), and the simulation performed with the LIOV-EMTP96 for lightning current peak
namely 70 kA.
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Fig. 3.34. – Comparison between measurement and simulation for the case 2.2. Lightning current peak value
equal to 34 kA.

Case 2.3. The network configuration is reported in Fig. 3.35. In Figs. 3.36 are shown the
connection types of the line terminations. The stroke location is represented in Fig. 3.35 at
point ‘r.s.m.’, the lightning current characteristics considered for this case are the same for the
case 2.1-2.2 and are given in the previous Tab. 3.4.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


66
90 m 210 m
150 m 148 m
210 m 148 m
346 m
42 m
132 m
284 m
210 m
152 m
210 m
150 m
170 m
1
5
0

m
1
5
0

m
r.s.m.
M2
70 m
75 m

Fig. 3.35. – Network configuration for the case 2.3.


1
N
Ct
Ct = 0.5 nF
R = 50 Ohm
L = 22 uH

M2
1
N
Cp
Cp = 0.65 nF
L = 22 uH

SA
L = 1 uH
Rpr
C = 13.5 uF
Ipr [A] Vpr [V]
0. 0.
36. 27000.
54. 27900.
360. 29700.
1620. 32220.



1
N
R = 50 Ohm
L = 22 uH

1
N
R = 50 Ohm
L = 22 uH
SA
Ct
Ct = 0.5 nF


Fig. 3.36. – Termination types for the network configuration of Fig. 3.35.

Fig. 3.37 reports a comparison between the voltage measured in the node M2 (see Fig.
3.35), and the simulation performed with the LIOV-EMTP96 for lightning current peak
equal to 70 kA.
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


67
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
'LIOV-EMTP96'
Measurement

Fig. 3.37. – Comparison between measurement and simulation for the case 2.3. Lightning current peak value
equal to 50 kA.

The results above reported show a good agreement between the simulations and the
measurements. For case 1.1 the agreement between the simulations and the measurements is
better than for case 1.2. This difference is probably due to the presence, for case 1.2, of a
larger number of surge arresters than for case 1.1. This is supported by the results shown in
Fig. 3.34 for case 2.2 where the observation point is placed on a surge arrester. Indeed, at this
point the LEMP-response of the distribution system is due, mainly, to the surge arrester non-
linear characteristic, which means that the difference between the simulation and the results
are mainly affected by the surge arrester representation.
Which refer to the simplest line geometry and the reduced number of surge arresters, cases
2.1 and 2.3 show the better agreement between simulations and measurements.
Additional reasons for disagreement between measurements and simulations, for all cases,
can be addressed also to: a) measuring errors: the overall uncertainty of the measuring system
used in the tests was less than ±5%; b) high frequencies oscillations associated with the
switching device of the generating system; c) variation of the current propagation velocity, its
distortion and attenuation as it progresses upwards along the ‘stroke’ channel.
However, if we consider the complexity of the considered distribution systems and the
relative low disagreement between measurements and model simulations, the experimental
validation of the proposed model can be considered satisfactory.

Case of corona. The developed LIOV-EMTP96 interface allows also for the computation
of lightning-induced overvoltages in presence of corona. The same model described in
[Guerrieri, 1997; Correia de Barros et al., 1999; Nucci et al., 2000] has been indeed
implemented in the illuminated line model described in paragraph 3.1. According to this
model, from a macroscopic point of view, corona can be described by a charge-voltage
Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


68
diagram [Wagner et al., 1954; Gary et al., 1978]. After an initial linear increase of the
charge with the voltage, a threshold voltage ( ) t , x v
th
is reached and a sudden change of the
derivative of charge with respect to the voltage takes place. This derivative defines a
voltage-dependent dynamic capacitance (C
dyn
). We here consider a simplified corona model
given by [Correia de Barros and Borges da Silva, 1984; Correia de Barros, 1985] for a
single-conductor line:

( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )
|
.
|

\
|
> >
− + =
< =
0
2 1
t
t , x v
and t , x v t , x v for
t , x v / t , x v t , x v k k ' C t , x C
t , x v t , x v for ' C t , x C
th
th th dyn
th dyn
∂
∂

( )
( )
|
.
|

\
|
< = 0
t
t , x v
for ' C t , x C
dyn
∂
∂

(3.55)

where
! k
1
(≥1) is related to the sudden change of the capacitance when the voltage exceeds
the corona threshold v
th
(typical values are in the range 1.5-3);
! k
2
(≥0) is related to the gradual increase of the capacitance when the voltage is rising
above the threshold.

A value of k
2
=0 corresponds to the simplest approach to model corona, that is
considering that the dynamic capacitance switches between only two values.
The dynamic capacitance C
dyn
is then inserted in the Agrawal equations in lieu of C’.
Fig. 3.39 reports some results obtained using the LIOV-EMTP96 program for the line
geometry shown in Fig. 3.38 relevant to a lightning stroke with current amplitude of 35 kA,
maximum time-derivative of 42 kA/µs, ground conductivity of 0.01 S/m.


1 km
7
.
5

m

1 cm
Z
c

500 m
Stroke
Location
Z
c

5
0

m


Fig. 3.38. – Line geometry adopted to evaluate the effect of corona. Lightning current: peak value 35 kA;
maximum time-derivative 42 kA/µs. ground conductivity 0.01 S/m.

Chapter 3. – LEMP-to transmission-lines coupling models for distribution systems and their
implementation in a transient program pag.


69
-50
0
50
100
150
200
250
300
0 1 2 3 4 5
Time [µs]
I
n
d
u
c
e
d

O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
0 m
250 m
500 m

Fig. 3.39. – Induced voltages along the line in presence of corona. Line configuration of Fig. 3.38.
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


70
Chapter 4. – Influence of lightning-induced overvoltages
on power quality

The overvoltages induced by lightning on distribution and transmission networks, can
cause phase-to-ground and phase-to-phase flashovers that need, for their removal, the use of
recloser breakers. In the case of distribution networks the use of these breakers determines a
temporal decrease of consumers voltage feeding, known as ‘voltage sags’. The duration of
these voltage sags depends on the cycle type of the recloser breakers, which, in turn,
depends on the fault removal time.
As known, voltage sags cause serious malfunction of a large number of apparatus as, in
particular, of the electrical d.c. and a.c. motors. As a matter of fact, some types of speed
controls for electrical motors interrupt their normal operation for a 15% decrease of the
voltage supply that have a duration of one half cycle [Bollen, 1997].
There are different causes for voltage sags; for example, the start of electric motors with
large rated power, or inrush currents due to transformer insertion. There is, however, clear
evidence that in electrical systems located in regions with high value of isoceraunic level,
thunderstorm days with lightning activity are associated to more than 80% of the voltage
sags determining apparatus malfunction [Boonseng and Kinnares, 2001] (see Fig. 4.1 and
4.2). Nevertheless, lightning activity is not the only one responsible of the voltage sag
related to the thunderstorm days because, associated with the same days, there are other
natural causes [Dugan et al., 1996] such as is the wind effect, that may cause short circuits
(e.g. branch of trees-ground; conductor-conductor).
Therefore it results of fundamental importance the estimation of the lightning
performance of distribution lines in order to infer whether lightning is natural cause of
outages or voltage sags during thunderstorm days or not.
2
4
6
8
10
12
14
16
0 2 4 6 8
Lightning flash density (fl / km^2 / year)
S
a
g
s

/

m
o
n
t
h

/

n
o
d
e

Fig. 4.1. – Influence of lightning flash density on voltage sags. Adapted from Gunther et al. [1995].
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


71

0
2
4
6
8
10
12
14
16
18
Gen Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
%


Fig. 4.2. – Percentage of voltage sags, during one year, in 130 US and Canadian monitoring sites. Summer
months are characterized with the greater number of thunderstorm days. Adapted from Dugan et al. [1996]

The basic question for a power engineer is how many lightning faults per year a certain
distribution line may experience, as a function of its insulation and of its design. The issue
has been the object of several studies
1
.
In [Rusk, 1977; Pettersson, 1991] 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 taking into account its probability distribution,
while the front time of the lightning current and the return stroke velocity are considered
fixed.
In [Chowdhuri, 1989b; Jankov and Grzybowski, 1997] a statistical method is employed.
Both the probability distribution of amplitude and that of front time of the lightning current
are considered. In [Chowdhuri, 1989b] the coefficient correlation between the two above-
mentioned parameters is taken into account. The striking distance of the indirect stroke from
the line (lightning strokes occurring within a certain distance from the line will directly
strike the line) is evaluated as a function of the return stroke peak current (while in [Rusk,
1977] it is considered as independent of the current). The return stroke velocity is fixed and
the ground is considered as a perfectly conductive plane.
All the above studies deal with single conductor lines, of infinite length, above a
perfectly conducting ground.
In [Hermosillo and Cooray, 1995] the Monte Carlo method has been employed to solve
the problem. The induced voltages are calculated at the termination of a 2 km overhead line

1
[Rusk, 1977; Chowdhuri, 1989b; Pettersson, 1991; Hermosillo and Cooray, 1995; Jankov and
Grzybowski, 1997].
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


72
above a ground of finite conductivity. The Monte Carlo simulation involves 10
4
events
taking place over a surface covering the line and 1000 m away from it. The same striking
distance equation from the line as in [Chowdhuri, 1989b; Jankov and Grzybowski, 1997] is
adopted. The correlation between peak value and front time of lightning current
distributions is disregarded.
In this thesis, to estimate the frequency of lightning induced overvoltages exceeding a
given insulation level on an overhead distribution line, we propose a procedure based on the
Monte Carlo method, whose basis are found in [Borghetti and Nucci, 1998]. The proposed
procedure allows to evaluate the lightning performance of distribution lines taking into
account not only more realistic configurations than those usually considered in the literature,
namely lines provided with shielding wires or neutral conductors with periodical groundings
and surge arresters above a lossy ground, but also the line steady-state voltage and the type
of flashover induced by lightning, namely phase-to-ground or phase-to-phase. Besides,
according to the types of distribution transformer connections, the developed procedure
allows also to take into account the change of the coupling factor between the phase
conductors and the ground when a flashover occurs, and the relevant fault occurrence.
The proposed procedure will be first compared with the one proposed for the same
purpose by IEEE [IEEE WG on the lightning performance of distribution lines, 1997], and
eventually applied for a sensitivity analysis and for the lightning performance assessment of
a typical Italian distribution line.

4.1. – Statistical evaluation of the lightning performance of
distribution lines

4.1.1. – Procedure based on the Monte Carlo method and on the developed
LEMP-to-transmission-line coupling models for distribution lines
In the developed procedure lightning-induced voltages are calculated using the proposed
models for the LEMP illuminated distribution lines. For our purposes, and in order to reduce
the computation time, the wave shape of the return stroke current at the channel base is
approximated with a ramp until the peak value I
p
is reached at time t
f
, then the current value
is kept constant. The Monte Carlo method is applied to generate a significant number of
events (larger than 10÷20⋅10
3
), each characterized by four random variables: the peak value
of the lightning current I
p
, its front time t
f
, and the two coordinates of the stroke location.
We assume a correlation coefficient (ρ) between current amplitude and front duration
equal to 0.47 [Chowdhuri, 1989b] and the statistical parameters of log-normal distribution
of the peak and the front time published in [Anderson and Eriksson, 1980]. The stroke
locations are supposed uniformly distributed within a certain surface around the line. The
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


73
‘striking area’ relevant to the surface that we consider in our study, has to be chosen wide
enough to include all the lightning events that can induce a voltage along the line with
maximum amplitude greater than the considered insulation level. The maximum dimension
of this area (x
max
, y
max
of Fig. 4.3), is preliminarily determined by evaluating the induced
overvoltages due to the maximum lightning current of the statistical distribution at various
distances and position from the line.


Fig. 4.3. – Indirect stroke area to overhead line (top view).

The ‘striking area’ does not include the points whose distance from the line is less than
the values d
l
(lateral attractive distance, see Fig. 4.3).
In this study, the expression adopted by the IEEE Working Group on Lightning
performance of transmission lines [IEEE WG on the lightning performance of distribution
lines, 1997] is used for the evaluation of the lateral attractive distance d
l
. Such expression, is
based on the ‘Electro-Geometric’ model of the last step of the lightning flash, which gives
the following relation between the critical distances (the striking distances to the wire (r
s
)
and to ground (r
g
)) and the lightning current I [IEEE WG on estimation lightning
performance of transmission lines, 1985]):

65 . 0
10 I r
s
⋅ = ,
s g
r r ⋅ = 9 . 0

(4.1)

where r
s
and r
g
are expressed in m and the lightning current I in kA. As shown in Fig. 4.4,
the value d
l
is then determined from

( )
2
g
2
s l
h r r d − − =
(4.2)

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


74
where h is the line height (in m).
If the distance of the stroke location from the line is beyond the lateral distance, the event
is considered an indirect flash and the maximum amplitude of the induced overvoltages is
computed.
In order to infer the classical flashover vs BIL curves, for each event we calculate the
maximum of the voltage amplitudes induced along the line. Note that we do not limit the
calculation to the point closest to the stroke location as done in all procedures which assume
an infinitely long line and the ground as a perfect conductor, since for that case the
maximum amplitude occurs indeed at that point [Hermosillo and Cooray, 1995]. This
because, when the ground is not a perfect conductor, stroke locations close to one of the line
terminations can induce the larger overvoltages at the opposite one [de la Rosa et al., 1988]

r
s

r
g

indirect stroke
direct stroke
h
d
l


Fig. 4.4. – Striking distances to a conductor (r
s
) and to ground (r
g
) and lateral attractive distance (d
l
) of a line.
Adapted from Chowdhuri [1989b].

Then we compute the annual number of events that, for 100 km of line and for a given
ground flash density, induce an overvoltage with amplitude exceeding the various insulation
levels of the line. For all calculations the ground flash density is N
g
=1 flash/km
2
/year and,
otherwise specified, the line length is 2 km and its height 10 m.

4.1.2. – Comparison with the IEEE Std 1410-1997
A brief description of the IEEE Std 1410 IEEE WG on the lightning performance of
distribution lines, 1997] will be given. Concerning the model used for the calculation of the
lightning induced overvoltages, the IEEE Std 1410 adopts the simplified formula presented
by Rusck [1958] (hereafter called the Rusck formula). As earlier mentioned that formula has
been inferred by the same Rusck from the more general model he proposed in [Rusck,
1958]; it applies to the simple case of a step current and of an infinitely-long single-
conductor line above a perfectly conducting ground. It is worth remanding that the Rusck
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


75
formula (equation (2.24) of the paragraph 2.3) gives the maximum value V
max
(in kV) of the
induced overvoltages at the point of the line nearest to the stroke location.
Concerning the statistical procedure used to infer the lightning performance of a
distribution line, the IEEE Std 1410 follows the procedure presented in [Chowdhuri, 1989b],
which we summarize here for convenience. The amplitude of the stroke current is varied
from 1 to 200 kA in intervals of 1 kA. The number of annual insulation flashovers per km of
distribution line F
p
is obtained as the summation of the contributions from all intervals
considered as expressed by

( )
i g
i
i i
p
P N y y F ⋅ ⋅ − ⋅ =
∑
=
200
1
min max
2 (4.3)

where N
g
is the ground flash density and P
i
is the probability of current peak to be within
interval i; it is determined as the difference between the probability for current to be equal
or larger than the lower limit and the probability for current to reach or exceed the higher
limit of the interval. For the probabilistic distribution of the lightning current peak, the
following expression is adopted [Anderson, 1982]:

( )
kA 200
31 / 1
1
) (
*
6 . 2
*
*
≤
+
= ≥
p
p
p p
I
I
I I P
(4.4)

For each current value, y
min
and y
max
of equation (4.3) are the minimum distance for
which lightning will not divert to the line, and the maximum distance at which the stroke
may produce an insulation flashover, respectively. The value of y
min
is obtained by
expression (4.2). The value of y
max
is obtained by solving equation (2.24) for d, by taking I
p

as the lower limit of the interval and by taking V
max
= 1.5·CFO. The results for an
ungrounded overhead 10 m high line are shown in Fig. 4.5. The value of v in (2.29) is
chosen equal to 1.2⋅10
8
m/s. The results relevant to an ungrounded and grounded neutral or
shielding wire are obtained in [IEEE WG on the lightning performance of distribution lines,
1997] from the preceding ones by applying a scale factor of 0.75 to the induced voltages.

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


76

Fig. 4.5. – Number of annual induced flashovers vs distribution-line insulation level. Taken from IEEE WG on
the lightning performance of distribution lines [1997].

We have first verified that the proposed statistical procedure gives the same results of the
IEEE Std 1410 when the induced voltages are evaluated by using the Rusk formula instead
of using LIOV, and when the same probability distribution (4.4) of the lightning current, the
same values of return-stroke velocity (v=1.2⋅10
8
m/s), line height (10 m) and lateral distance
expression (4.2) are assumed. Then, we have found that the adoption of the peak distribution
of [Anderson and Eriksson, 1980] instead of (4.4) does not have a significant influence on
the results.
We have then compared the IEEE Std 1410 guide results of Fig. 4.5 with those obtained
by using our procedure (LIOV plus Monte Carlo), assuming the statistical distributions of
peak current and rise time of Fig. 2.3 (paragraph 2.1), a 2 km long line (beyond such a line
length the line illumination by the LEMP field for a perfectly conducting ground does not
affect the results [Borghetti and Nucci, 1999]). The comparison is shown in Fig. 4.6.

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


77
0.001
0.010
0.100
1.000
10.000
100.000
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
IEEE Guide
Proposed method

Fig. 4.6. – Comparison between the lightning performance evaluated by using the IEEE Std 1410 procedure
and the proposed one. t
f
is lognormally distributed with a median value of 3.83 µs. Adapted from Borghetti et
al. [2001a].

The difference between our results and those of the IEEE Std 1410 can be explained by
observing that the simplified Rusck formula applies to the case of a step wave shape for the
lightning current [Rusck, 1958]. We therefore repeated our computation by keeping constant
the value of t
f
throughout Monte Carlo simulation. Fig. 4.7 shows the results of four
different computations relevant to four different values of t
f
: 0.5, 1, 3 and 5 µs respectively.
From Fig. 4.7 we can observe that when t
f
decrease , the two procedures predict basically
the same results.

0.001
0.010
0.100
1.000
10.000
100.000
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
IEEE
Proposed method with tf=0.5us
Proposed method with tf=1us
Proposed method with tf=3us
Proposed method with tf=5us

Fig. 4.7. – Comparison between the lightning performances evaluated by using the IEEE Std 1410 procedure
and the proposed one, with different fixed value of t
f
.
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


78
4.1.3. – Sensitivity analysis
This paragraph describe the results of a sensitivity analysis carried out using the
proposed procedure. In particular we shall deal first with the case of a single conductor line
where we analyze the effect of the ground conductivity and the presence of surge arresters
along the line. Then for the multi-conductor case we analyze the effect of the grounding
resistance of the shielding wire, of the grounding intervals and of the different shielding
wire height.

Single conductor line. Since the maximum values of the induced voltage amplitudes
could be higher than for the case of a line above an ideal ground [Guerrieri et al.,
1996,1997] for these calculations a larger indirect stroke area is considered. In particular the
results refer to a 1 km long overhead line and to a 20 km
2
indirect stroke area, with x
max

equal to 5 km, y
max
equal to 2 km. The relative permittivity of the ground is assumed equal
to 10. Since, for line lengths shorter than about 2 km and for ground conductivity not lower
than 0.001 S/m, the effect of the lossy ground on surge attenuation along the line is not
significant [Rachidi et al., 1996], the presence of a lossy ground in our calculations is taken
into account only in the propagation of the incident electromagnetic field.
Fig. 4.8 shows the results obtained with two different ground conductivities, namely
σ
g
=0.01 S/m and σ
g
=0.001 S/m. These results are also compared with those obtained with
an ideal ground. It can be observed that lower values of ground conductivity result in higher
flashover rates, a result which is explained by the fact that, as already mentioned, the ground
resistivity enhances the induced voltages depending on the stroke location and on the
observation point [Guerrieri et al., 1996,1997].

0.010
0.100
1.000
10.000
100.000
1000.000
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
ideal ground
lossy ground 0.01 S/m
lossy ground 0.001 S/m

Fig. 4.8. – Effect of the ground conductivity, t
f
is lognormally distributed with a median value of 3.83 µs.
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


79
To illustrate the influence of the presence of surge arresters along the line, we have
considered the line configuration presented in Fig. 4.9 (we consider, for this case, only
common-mode voltages). The surge arresters are represented by the non-linear V-I
characteristic given in Tab. 3.1.

Phase conducor
h=9m d=1cm
Surge
Arrester
Surge
Arrester
Zc
Surge
Arrester
Zc
2 km

Fig. 4.9. – Line configuration with surge arresters.

As discussed in paragraph 3.1.3 the presence of surge arrester along the line, can be
produce the maximum amplitude of the induced overvoltage not in the point closest to the
stroke location. In order to reduce this effect, we have connected, at both line terminations
in parallel with the surge arresters, two resistive loads equal to the line characteristic
impedance.
The computed results are shown in Fig. 4.10. were we can observe that the presence of
surge arresters affects considerably the lightning performance of distribution lines.

0.010
0.100
1.000
10.000
100.000
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
phase conductor with 3 SA

Fig. 4.10. – Influence of the presence of surge arresters.


Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


80
Multi conductor line. In this section we first evaluate the lightning performance of a
typical three-phase Italian distribution line (see Fig. 4.11). For this purpose we consider a
1.8 km long line matched at both terminations, within an indirect stroke area of 7.6 km
2
.

10,8 m
a
b
c
1,3 m
10 m

Fig. 4.11. – Geometrical configuration of an Italian MV line.

The calculations have been carried out assuming two different values of ground
conductivity σ
g
(infinite and 0.001 S/m). They are plot in Fig. 4.12, which shows the
number of events resulting in overvoltages with amplitude exceeding the value indicated in
abscissa. We observe the larger number of overvoltages at phase b than at phases a and c,
due to the higher position of the phase b conductor. Note, again, the increase of the number
of induced voltages due to the poor ground conductivity.
0.01
0.10
1.00
10.00
100.00
1000.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase a or c with ideal ground
phase b with ideal ground
phase a or c with lossy ground
phase b with lossy ground

Fig. 4.12. – Statistical evaluation of lightning induced voltage in the Italian MV line of Fig. 4.11. Cases for
ideal ground and lossy ground (σ
g
=0.001 S/m). Adapted from Borghetti et al. [2001a].
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


81
We now consider a 2 km long, two-conductor line (phase conductor and shielding wire)
over a perfectly conducting plane, a case similar to the one discussed in [IEEE WG on the
lightning performance of distribution lines, 1997]. As shown in Fig. 4.13, the shielding
wire is grounded at the line terminations and at intervals along the line. The spacing
between two adjacent grounding is 2000 m (2 grs), 1000 m (3 grs), or 400 m (6 grs). The
simulations are carried out with the aim of evaluating the influence of the presence of the
shielding wire, of its height, of the spacing between two adjacent groundings as well as the
value of the grounding resistance.

Shielding Wire
Phase conductor
variable: 2km 1km 0.4km
9
m

1
0
m

1 cm
Rt Rt Rt Rt Rt Zc Zc


Fig. 4.13. – Line configuration with shielding wire.

Fig. 4.14 shows the results relevant to the phase conductor obtained with three different
shielding wire heights, namely 9.7 m, 10 m, 10.4 m. The results are affected by the presence
of the shielding wire, but the influence of the shielding wire height for the considered cases
is limited.

0.01
0.10
1.00
10.00
100.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
6 grs, Hsw=9.7 Rg=10ohm
6 grs, Hsw=10m Rg=10ohm
6 grs, Hsw=10.4 Rg=10ohm

Fig. 4.14. – Influence of shielding wire height. The spacing between to adjacent grounding is 400 m (i.e. 6
groundings overall) and the grounding resistance is equal 10 Ω.
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


82
a)
0.01
0.10
1.00
10.00
100.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
2 grs, Rg=0ohm
2 grs, Rg=10ohm
2 grs, Rg=100ohm


b)
0.01
0.10
1.00
10.00
100.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
3 grs, Rg=0ohm
3 grs, Rg=10ohm
3 grs, Rg=100ohm

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


83
c)
0.01
0.10
1.00
10.00
100.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
6 grs, Rg=0ohm
6 grs, Rg=10ohm
6 grs, Rg=100ohm

Fig. 4.15. – Influence of grounding resistance. a) The shielding wire is grounded only at its terminations
(overall 2 grs). b) The spacing between two adjacent grounding is 1 km (overall 3 grs). c) The spacing between
two adjacent grounding is 400 m (overall 6 grs).

Fig. 4.15 shows the influence of the grounding resistance (the shielding wire in this case
is at 10 m above ground). Three values of grounding resistance are considered, namely 0, 10
and 100 Ohm. The results do not differ considerably, at least when the number of grounding
is low (2 and 3 grs). On the other hand, the spacing between two adjacent grounding appears
to have a larger influence, as also illustrated in Fig. 4.16 for a value of grounding resistance
of 10 Ohm and a shielding wire height of 10 m.

0.01
0.10
1.00
10.00
100.00
0 50 100 150 200 250 300
CFO [kV]
N
o
.

o
f

i
n
d
u
c
e
d

o
v
e
r
v
o
l
t
a
g
e
s

w
i
t
h

m
a
g
n
i
t
u
d
e

e
x
c
e
e
d
i
n
g

t
h
e

B
I
L
/
(
1
0
0

k
m

y
e
a
r
)
phase conductor only
2 grs, Rg=10ohm
3 grs, Rg=10ohm
6 grs, Rg=10ohm

Fig. 4.16. – Influence of spacing between adjacent grounding.

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


84
It has been shown that the shielding wire appears to have a large influence on the
performance of the distribution line; additionally, as analyzed for single stroke location in
paragraph 3.1.3, the spacing between two adjacent grounding of the shielding wire seems to
have the major impact in decreasing the number of induced voltages exceeding a given
value, in accordance with the results presented in paragraph 3.1.3.
The results obtained by using the three values of grounding resistance do not differ
considerably, at least when the number of grounding is low (2 and 3 grs). On the other hand,
the spacing between two adjacent grounding appears to have a larger influence, as also
illustrated in Fig. 4.16 for a value of grounding resistance of 10 Ohm and a shielding wire
height of 10 m, again, in accordance with the results of paragraph 3.1.3.

4.2. – Case of a typical Italian distribution line
To improve the power quality assessment of a distribution line which experience induced
overvoltages, some additional features of the proposed models have been developed and are
described in this section. In particular, they are aimed at investigating the effects of two
parameters that are not considered in the previous model namely: the steady-state voltage
and the change of coupling factor among the line conductors and the ground at the location
where flashovers occur.
In the following paragraph the proposed model extensions are described, then an
application of procedure is made.

4.2.1. – Model extension
The steady-state voltage at industrial-frequency, taken into account both in the generation
procedure of the events and in the overvoltages calculations, is assumed constant, due to the
high frequency content and the short duration of the induced voltages.
In the generation procedure of the events, uniformly distributed random values of the
phase voltage are generated for one of the three phases; the voltages of the two remaining
phases are assumed to form, together with the first one, a positive system for each event.
In the overvoltages calculation, the steady-state values of each phase voltage is taken into
account by simply adding it to the incident voltage of equation (2.27). The reason for this is
that the current circulating in the line before any lightning event does not effect the
amplitude of the induced voltages, and therefore the coupling equations (2.25) and (2.26)
remain unvaried.
Another additional feature of the proposed procedure, compared to [Borghetti and Nucci,
1998], is that it takes into account also the change in the coupling factor among the line
conductors and the ground at the location where a flashover occurs. We consider each
phase-conductor connected to ground through a resistance R
gp
at each pole: in normal
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


85
conditions, the value of R
gp
is equal to infinite; when a phase-to-ground flashover occurs at
a certain pole, the relevant value of R
gp
is set at a value corresponding to the grounding
resistance of the specific pole and the fault impedance. With such a model, we are able to
investigate up to which extent a flashover at of one phase of the line can affect the
overvoltages on the other two conductors . The scattered voltage at the points x
p
of the phase
conductors where the induced voltage exceeds the CFO are calculated by means of

[ ] [ ] [ ]








+ ⋅ =
∫
i
h
e
z gp gp p
s
i
(x,z,t)dz E t i R t x v
0
) ( ) , ( (4.6)

where
! [i
gp
] is the matrix of the induced currents diverted to ground in correspondence of the
pole (see Fig. 3.1);
! [R
gp
] is the diagonal matrix of the phase-to-ground resistances of the poles.

We consider that an overvoltage causes a flashover when it exceeds the value of
1.5·CFO. The 1.5 factor is an approximation that accounts for the turn up in the insulation
volt-time curve, which is the same criterion proposed by the IEEE Std 1410 [IEEE WG on
the lightning performance of distribution lines, 1997].
Fig. 4.17 shows the voltages induced by a lightning event with I
p
=40 kA, t
f
=3 µs, on a
typical Italian distribution line (see Fig. 4.11). We consider a 1.8 km long line matched at
both terminations and stroke location equidistant to the line termination, 50 m from the line.
The line span between two consecutive poles is 150 m. The electromagnetic field
radiated by lightning is calculated assuming the ground conductivity equal to 0.01 S/m.
Also, we assume a phase to ground voltage equal to 16 kV on phase b (where the flahover
occur) and a resistance R
gp
of the poles during a phase to ground flashover equal to 10 Ω.
Figs. 4.17 a÷c shows the behavior of the induced voltage amplitude on phase conductors
a, b and c respectively at the various poles. The flashover of phase b at pole 6 at about 2 µs
causes a voltage reduction at the same pole, which propagates in both directions along the
line. This flashover causes an instant voltage reduction both on phases a and phase c where
the flashover does not occur, producing in turn a shielding effect similar to that of a
shielding wire.





Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.

86
a)
Poles
150 kV
112.5 kV
75 kV
37.5 kV
1
2
3
4
5
6
7
8
9
10
11
Time [µs]
0
1
2
3
4
5


Poles
150 kV
112.5 kV
75 kV
37.5 kV
1 2 3 4 5 6 7 8 9 10 11
T
i
m
e

[
µ
s
]

0
1
2
3
4
5
0 kV

b)

Poles
200 kV
150 kV
100 kV
50 kV
1
2
3
4
5
6
7
8
9
10
11
Time [µs]
0
1
2
3
4
5


Poles
200 kV
150 kV
100 kV
50 kV
1 2 3 4 5 6 7 8 9 10 11
T
i
m
e

[
µ
s
]

0
1
2
3
4
5
0 kV

c)
Poles
160 kV
120 kV
80 kV
40 kV
1
2
3
4
5
6
7
8
9
10
11
Time [µs]
0
1
2
3
4
5


Poles
160 kV
120 kV
80 kV
40 kV
1 2 3 4 5 6 7 8 9 10 11
T
i
m
e

[
µ
s
]

0
1
2
3
4
5
0 kV


Fig. 4.17. – Voltage amplitude induced on phase b by a lightning event with I
p
= 40 kA, t
f
= 3 µs, stroke
location equidistant to the line termination, 50 m from the line of Fig. 4.9. The LEMP is calculated assuming a
ground conductivity σ
g
=0.01 S/m. The grounding resistance of the poles R
gp
=10 Ω. a) phase a; b) phase b; c)
phase c. Adapted from Borghetti et al. [2001b].

Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


87
4.2.2. – Results
In this paragraph we evaluate the lightning performance of the typical Italian distribution
line previously described (see Fig. 4.9). We consider the same line length and indirect stroke
area assumed in the paragraph 4.2.1.: 1.8 km long line matched at both terminations, within
an indirect stroke area of 7.6 km
2
, the number of considered events is 2⋅10
5
. The poles are
made out of concrete or steel and are assumed to have a grounding resistance R
gp
in the
range 10-100 Ω. The line span between two consecutive poles is 150 m. The amplitude of
the r.m.s. value of the steady-state phase-to-phase voltage is 20 kV and the CFO of the line
is 125 kV.
We consider also the pole grounding resistance at those poles of the line where the CFO
is exceeded by the induced overvoltages and consequently a flashover occurs. The LEMP is
now calculated assuming two different values for the ground conductivity, namely 0.1 S/m
and 0.01 S/m.
In Figs. 4.18-4.21 we show the results of the statistical analysis (80000 events overall)
carried out on the Italian MV line configuration of Fig. 4.10, for two different values of
ground conductivity, and of pole grounding resistance. In particular, they report the
expected number of flashovers, distinguishing among one phase to ground, and two or three
phases to ground, caused by a single event.
To better assess the importance of taking into account the steady-state voltage in the
calculations, in Figs 4.18-4.21 we report also the results for the case in which the steady-
state voltage is disregarded.
To better assess the range of influence of the grounding resistance, Fig. 4.22 report the
number of flashovers for a fixed value of ground conductivity and three different values of
grounding resistances namely 100, 200 and 500 Ω.

0
0.5
1
1.5
phase a phase b phase c bi and/or three
phases
N
u
m
b
e
r


o
f


f
l
a
s
h
o
v
e
r
s

/

1
0
0

k
m

/

y
e
a
r
with steady-state voltage
without steady-state voltage

Fig. 4.18. – Number of flashovers along the line, R
gp
=10 Ω and σ
g
=0.1 S/m. Adapted from Borghetti et al.
[2001b].
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


88
0
0.5
1
1.5
2
2.5
3
3.5
phase a phase b phase c bi and/or three
phases
N
u
m
b
e
r


o
f


f
l
a
s
h
o
v
e
r
s

/

1
0
0

k
m

/

y
e
a
r
with steady-state voltage
without steady-state voltage

Fig. 4.19. – Number of flashovers along the line, R
gp
=10 Ω and σ
g
=0.01 S/m. Adapted from Borghetti et al.
[2001b].

0
0.5
1
1.5
phase a phase b phase c bi and/or three
phases
N
u
m
b
e
r


o
f


f
l
a
s
h
o
v
e
r
s

/

1
0
0

k
m

/

y
e
a
r
with steady-state voltage
without steady-state voltage

Fig. 4.20. – Number of flashovers along the line, R
gp
=100 Ω and σ
g
=0.1 S/m. Adapted from Borghetti et al.
[2001b].

0
0.5
1
1.5
2
2.5
3
phase a phase b phase c bi and/or three
phases
N
u
m
b
e
r


o
f


f
l
a
s
h
o
v
e
r
s

/

1
0
0

k
m

/

y
e
a
r
with steady-state voltage
without steady-state voltage

Fig. 4.21. – Number of flashovers along the line, R
gp
=100 Ω and σ
g
=0.01 S/m. Adapted from Borghetti et al.
[2001b]..
Chapter 4. – Influence of lightning-induced overvoltages on power quality pag.


89
0
0.5
1
1.5
2
phase a phase b phase c bi and/or three
phases
N
u
m
b
e
r


o
f


f
l
a
s
h
o
v
e
r
s

/

1
0
0

k
m

/

y
e
a
r
100 Ohm
200 Ohm
500 Ohm

Fig. 4.22. – Number of flashovers along the line, R
gp
=100-200-500 Ω and σ
g
=0.01 S/m.

As regard the presence of the steady-state voltage, a parameter that is not usually
considered in the literature, we can observe that the taking into account of it results in a
larger number of flashovers: 1.5 instead of 1.3 in Figs. 4.18 and 4.20, 3.7 instead of 3.2 in
Figs. 4.19 and 4.21. For all cases the average increase of the number of flashovers/100
km/yr obtained taking into account the steady-state voltage is about 15%.
Concerning the line geometry influence, we can observe the larger number of ground
flashovers at phase b than at the two others phases, independently of the steady-state
voltage. This is a consequence of the higher position of phase b (see also Fig. 4.11).
Concerning the increase of the pole grounding resistance we observe that produces, in
general, an increase of the number of simultaneous faults to ground (two or three phases).
The shielding effect produced by the phase where a flashover occurs tends in fact to
decrease with the increase of the grounding resistance [Paolone et al., 2000]. The value of
the poles grounding resistance starting from which this effect becomes important is of about
100÷200 Ω.

Chapter 5. – Conclusions pag.


90
Chapter 5. – Conclusions

Aims of this thesis are 1. – the development of models and their implementation in a
computer programs for an accurate estimation of the lightning-induced overvoltages on
distribution power networks, and their test against experimental results; 2. – the improved
assessment of the lightning performance of distribution systems in view of the solution of
power quality problems.
In what follows are summarized the contributions of this thesis thought to be original.

Development of a complex overhead line model illuminated by external electromagnetic
field. We have developed an overhead line model, based on a modification of the Agrawal
coupling model, originally introduced for the calculation of lightning-induced voltages on
uniform lines, to take into account the presence of overhead line transverse discontinuities,
such as those caused by periodically grounded shielding wires and surge arresters.
For solving the proposed model equations, we have proposed a 2
nd
order FDTD
integration scheme. The developed integration scheme applies to multi-conductor lines
above a frequency-dependent lossy ground. It has been compared with a similar one, also
developed within the framework of this thesis, based on the 1
st
order FDTD integration
scheme, and it has been shown to be numerically more stable when considering frequency-
dependent parameters and/or non-linearities like surge arresters.
Compared to similar models developed in the literature, which consider the shielding
wire at ground potential at any point along it and at any time, the proposed model permits a
more accurate evaluation of the lightning-induced voltages. The developed model allows to
evaluate the induced overvoltages on overhead lines above a lossy ground taking into
account the resistance of the groundings and the spacing between the groundings of
shielding wires. This feature is determinant if one has to found the ‘optimal’ distance
between two adjacent groundings of the shielding wire.
Its experimental validation has allowed to show its adequacy. The proposed model has in
fact been tested and validated versus experimental results obtained using a reduced scale
multi-conductor line model illuminated by an EMP simulator. The model was given
different line geometries of the conductors and of the groundings of the shielding wire.

Analysis of the effect of shielding wires and surge arresters on lightning-induced
overvoltages. The performed sensitivity analysis has shown that in the mitigation effect of
the shielding wire the number of groundings plays a decisive role: the mitigation effect has
been found to be more dependent on the number of groundings than on the value of the
grounding resistance, both in the case of lossy ground and of ideal ground. This differs from
Chapter 5. – Conclusions pag.


91
the case of direct strokes, for which the effectiveness of the shielding wire depends strongly
on the grounding resistance.
For the cases of an ideal ground, the mitigation effect as predicted by the Rusck formula
has been analyzed too, and the results have shown that the Rusck simplified formula tends
to overestimate the effectiveness of the shielding wires. It is worth reminding that the Rusck
expression does not cover the case of multiple grounded shielding wire (it assumes that the
shielding wire is grounded at single point along the line) and that is applicable only in the
assumption of perfectly conductive ground.
Concerning the effectiveness of surge arresters, it depends strongly on their number and
position along the line. An ‘optimum’ number appears, for the considered case, one surge
arrester each 200 m.

Development of a distribution power system model. In order to simplify the achievement
of the LEMP response of distribution systems, the developed line model has been interfaced
with the Electromagnetic Transient Program (EMTP96). Such an interface is based on the
use of a Bergeron loss less line which makes it possible the data exchange between the two
programs (LIOV and EMTP96). In addition, as opposed to [Nucci et al. 1994], the
developed interface does not require any modification of the source code of the EMTP96
because it use a Dynamic Link Library (DLL) called from the EMTP96-TACS environment.
Compared to other ones proposed in the literature, allows also for the treatment of the
corona phenomenon as opposed to [Orzan et al.,1996; Orzan, 1998], and line losses not
dealt with in the interface of Høidalen [1997,1999].
The advantages of merging the developed line model with the EMTP96 consists
essentially in the fact that (i) distribution systems of any complex configurations can be
modeled (the only limit being the memory of the computer and of EMTP96) and (ii) all
power component models of the EMTP library become available for lightning-induced
voltage simulations.
Calculations performed with the developed LIOV-EMTP96 program have been
compared with experimental results obtained by using a reduced-scale distribution line
model described in [Piantini and Janiszewski, 1992, Nucci et al., 1998]. The comparison
regards an overhead distribution system including surge arresters, shielding wire groundings
and shunt capacitors. Overall, the agreement between the numerical results and
measurements is quite satisfactory. We can conclude that LIOV-EMTP96 program represent
a promising tool for the analysis of the response of a complex distribution system to indirect
lightning discharge.

Estimation of the lightning performance of complex overhead distribution lines. We
have developed a procedure based on the LEMP-illuminated distribution system model, and
Chapter 5. – Conclusions pag.


92
on the Monte Carlo method that allows an improved evaluation of the lightning performance
of distribution networks.
Such a procedure has been compared with the one proposed in the IEEE Std 1410-1997
for the same purpose: it has been shown that for those cases in which a comparison is
possible (overhead single-wire line above a perfectly conducting ground, step function for
the channel base current wave shape) the two methods predict basically the same results.
Additionally, opposed to the IEEE procedure, the proposed one allows to take into account
the ground resistivity, which plays a fundamental role in the calculation of the lightning
electromagnetic field that excites the line, as well as any line configuration which may
include the presence of shielding wire and surge arresters.
The results obtained show that the shielding wire appears to have a large influence on the
performance of the distribution line; additionally, as analyzed for single stroke location, the
spacing between two adjacent grounding of the shielding wire, seems to have the major
impact in decreasing the number of induced voltages exceeding a given value.
Additional features of the proposed procedure have been developed in order to take into
account the steady-state voltage and the change of coupling factor among the line
conductors and the ground at the location where flashovers occur, which allows to estimate
type of flashovers and the effective number of failures.
The procedure has been eventually applied for assessing the lightning performance of a
typical Italian distribution line. The results obtained show that, in general, taking into
account the steady-state voltage results in a larger number of flashovers, and that the
increase of the pole grounding resistance produces, in general, an increase of the number of
simultaneous faults to ground (two or three phases): the phase where a flashover occurs
produces in fact the same shielding effect as a shielding wire; such effect tends to decrease
with the increase of the pole grounding resistance.

Future possible development of the research.
As mentioned at the beginning of chapter 4, although there is some evidence that
electrical distribution systems located in regions at high isoceraunic level are somewhat
more affected by voltage sags, there is however no clear evidence that lightning is the main
cause for these voltage sags.
An attempt to correlate indirect lightning activity with circuit breaker operation which, in
turn, depend on voltage sags, is described in [Bernardi et al., 1998] who carried out a study
to infer the possible correlations between lightning indirect events and breaker
interventions. They used on the one hand the lightning current amplitude and stroke location
coordinates provided by the Italian lightning location system (SIRF – Sistema Italiano di
Rilevamento Fulmini), and on the other hand the simplified Rusck equation to calculate the
corresponding lightning-induced overvoltage. The correlation they found between lightning
events and circuit breakers operations was poor.
Chapter 5. – Conclusions pag.


93
An explanation of this result, which is in part in contradiction with experimental
observations [Boonseng and Kinnares, 2001], may be the fact that the Rusck simplified
coupling model was adopted without considering the distribution system complexity and the
important parameters that modify, in a decisive way, the LEMP response of the distribution
system (e.g. the ground conductivity and power system configuration).
We expect that, as compared with the above mentioned study, the use of LIOV-EMTP
can improve the LEMP calculation, the evaluation of the coupling by means of realistic line
models and eventually the evaluation of entire distribution system response.
We think that the models developed in this thesis represent an improved tool for
performing the indirect lightning correlation with distribution system a-periodic
disturbances, as well as for the distribution networks protection and insulation coordination,
and power quality improvement.


References pag.


94
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Appendix pag.


104
Appendix A.1. – Numerical treatment of line transverse
discontinuities in 1
st
order FDTD scheme

The 1
st
order spatial Point Centered Finite Difference integration scheme for a single-
conductor line illuminated by an external electro magnetic field, around grounding point is
shown in Fig. A.1


∫
i
h
z
dz t z x E
0
) , , (
Γ
Voltage FD node
Current FD node
i
k
n

i
k+1
n

v
k+2
n

v
k+1
n

v
k
n
i
k+2
n

dx dx/2
k spatial discretization
n time discretization
dx spatial discretization
i
k-1
n

i
k’
n
i
k”
n

i
g
n

Known variables:
• All internal node scattered
voltages and currents at
time step n
• Ground resistance R
t

• Vertical Electric Field E
z


Unknown variables:
• v
k+1
n

• i
k
n

• i
k+1
n


Constant:
• Line Height h
i

• Operator Γ

Fig. A.1. – Insertion of discontinuity point, at a generic point along a single-conductor line, in the 1
st
order
Point Centered Finite Difference integration scheme.

The treatment of a grounding point, located anywhere along the line, involves a
modification of the numerical solution of the Agrawal model. It is handled as a discontinuity
point in the spatial grid of the Point Centered Finite Difference integration scheme.
In the numerical discretization of the Agrawal single-conductor coupling equations
[Nucci et al., 1994] the equation that allows to extract a generic current node at time step n,
known all variables at time step n-1, is the following:

(
¸
(

¸

+
∆
−
−
+
=
− +
−
1
4
1
1
3
2
n
k
n
k
n
k
n
k
n
k n
k
i A
x
v v Eh Eh
A i (A.1)

where the sense of the symbol that appear in (A.1) is the same of the paragraph 3.1.2. A3
and A4 are constants depending on the line inductance L’:

|
.
|

\
|
∆
=
|
.
|

\
|
∆
=
−
t
L
A
t
L
A
'
;
'
4
1
3
(A.2)
Appendix pag.


105
If we write equation (A.1) for nodes k and k+1 of Fig. A.1 we obtain the first two
equations of the solution system:


(
¸
(

¸

+
∆
−
−
+
=
− +
−
1
4
1
1
3
2
n
k
n
k
n
k
n
k
n
k n
k
i A
x
v v Ex Ex
A i (A.3)
(
¸
(

¸

+
∆
−
−
+
=
−
+
+ +
−
+ +
+
1
1 4
1 2
1
1 1
3 1
2
n
k
n
k
n
k
n
k
n
k n
k
i A
x
v v Ex Ex
A i (A.4)

To obtain the third equation of the solution system we apply the Kirchoff current
equation, at the grounding node, for currents i
gn
, i
k’
n
, i
k”

n
:

n
k
n
k
n
g
i i i
' ' '
− = (A.5)

Currents i
k’
n
, i
k”
n
can be expressed as a function of the adjacent current node assuming
the following linear spatial interpolation:

2
3
1
'
n
k
n
k n
k
i i
i
−
−
= (A.6)
2
3
2 1
' '
n
k
n
k n
k
i i
i
+ +
−
= (A.7)

Besides, the node voltage v
k+1
n
can be expressed as in (3.1):

∫
+ =
+
i
h
e
z
n n
k
(x,z,t)dz E ) Γ ( ig v
0
1
(A.8)

By introducing (A.5), (A.6) and (A.7) in (A.8) we obtain the third equation of the
solution system:

∫
+
|
|
.
|


\
| + − −
=
+ + −
+
i
h
e
z
n
k
n
k
n
k
n
k n
k
(x,z,t)dz E
i i i i
v
0
2 1 1
1
2
3 3
Γ (A.9)

FDTD 1
st
order multi-conductor line. The model for a multi-conductor line discontinuity
treatment is similar to that of the single-conductor line. We shall consider a discontinuity at
one of the conductors of the bundle as a discontinuity of all conductors, in that the
conductors which are not connected to ground are considered grounded through an infinite
resistance. We can rewrite (A.3), (A.4) and (A.9) for a multi-conductor line as follows (j is
the grounded conductor):
Appendix pag.


106
[ ] [ ]
[ ] [ ] [ ] [ ]
[ ][ ]
|
|
.
|


\
|
+
∆
−
−
+
=
− +
−
1
4
1
1
3

2


n
k
n
k
n
k
n
k
n
k n
k
i A
x
v v Ex Ex
A i (A.10)
[ ] [ ]
[ ] [ ] [ ] [ ]
[ ][ ]
|
|
.
|


\
|
+
∆
−
−
+
=
−
+
+ +
−
+ +
+
1
1 4
1 2
1
1 1
3 1

2


n
k
n
k
n
k
n
k
n
k n
k
i A
x
v v Ex Ex
A i (A.11)
[ ] [ ]
[ ] [ ] [ ] [ ]
(
(
¸
(


¸

+
|
|
.
|


\
|
+ − −
=
∫
+ + −
+
i
h
e
z
n
k
n
k
n
k
n
k n
k
(x,z,t)dz E
i i i i
ΓK v
0
2 1 1
1
2
3 3
(A.12)

where

[ ] [ ] [ ]
|
|
|
|
|
|
.
|






\
|
=
|
|
|
|
|
|
.
|






\
|
=
|
|
|
|
|
|
.
|






\
|
=
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
ExNC
Ex
Ex
Ex
Ex
vNC
v
v
v
v
iNC
i
i
i
i
:
3
2
1
;
:
3
2
1
;
:
3
2
1
(A.13)
|
|
|
|
|
|
|
|
|
|
|
|
.
|












\
|
=
(
(
¸
(


¸

∫
∫
∫
∫
∫
NC
i
h
e
z
h
e
z
h
e
z
h
e
z
h
e
z
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
(x,z,t)dz E
0
0
0
0
0
:
3
2
1
(A.14)
[ ]
[ ]
[ ]
[ ]
t
L
A
t
L
A
ij ij
∆
=
∆
=
−
4
1
3
;
(A.15)

and matrix [ ΓK ]:

[ ]
|
|
|
|
|
|
.
|






\
|
   
=
NCNC
K Γ |
|
| K Γ
| K Γ
| K Γ
K Γ
0 0 0
0 0 0
0 0 0
0 0 0
33
22
11
(A.16)


Appendix pag.


107
Appendix A.2. – Comparison between 1
st
and 2
nd
order
FDTD integration schemes

A first comparison of the newly proposed FDTD 2
nd
order integration scheme with the 1
st

order one has been performed making reference to a 2 km long, 10 m high single-conductor
line above a lossy ground shown in Fig. A.2. The ground conductivity is 0.001 S/m and its
relative permittivity is 10. The stroke location is at 50 m from the left-end line terminal. The
lightning channel base current peak value is 60 kA and its maximum time derivative is 120
kA/ms. The return stroke speed is 1.2⋅10
8
m/s. The LEMP is computed adopting the MTL
return stroke model and the value of the spatial and temporal steps adopted for the
simulations are 10 m and 10
-8
s respectively.


2 km
1
0

m

1 cm
Z
c

50 m
Stroke
Location
Z
c


Fig. A.2. – Line geometry for the comparison between FDTD 1
st
and 2
nd
order in presence of lossy ground.

In Fig. A.3 we show the results calculated both with the 1st order and 2
nd
order FDTD
algorithms. For both cases, we show the results considering the effect of the ground
resistivity both in the electromagnetic field and in the calculation of line parameters (‘lossy
line’) and the results obtained taking into account ground losses only in the electromagnetic
field calculation (‘ideal line’).
A comparison between the two methods has been performed also for the case of a line
with surge arresters. To perform these simulations, we have used the LIOV- EMTP96
program (see paragraph 3.2).
Appendix pag.


108
a)
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
0 m FDTD 1st order lossy line
0 m FDTD 2nd order lossy line
0 m FDTD 1st order ideal line
0 m FDTD 2nd order ideal line

b)
-550
-450
-350
-250
-150
-50
0 1 2 3 4 5 6
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
500 m FDTD 1st order lossy line
500 m FDTD 2nd order lossy line
500 m FDTD 1st order ideal line
500 m FDTD 2nd order ideal line

c)
-950
-850
-750
-650
-550
-450
-350
-250
-150
-50
4 5 6 7 8 9 10
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
2000 m FDTD 1st order lossy line
2000 m FDTD 2nd order lossy line
2000 m FDTD 1st order ideal line
2000 m FDTD 2nd order ideal line

Fig. A.3. – Lightning induced overvoltages calculated at three different observation points of the line of Fig.
A.2, a) x=0 m, b) x=500 m, c) x=2 km, using 1
st
and 2
nd
order FDTD scheme. Field calculation: lossy ground
(0.001 S/m), line impedance: ideal line and lossy line.
Appendix pag.


109
It can be seen that the wave-shapes computed using the 2
nd
order FDTD algorithm are
less affected by numerical oscillations, especially for observation points approaching the
line far-end.
Fig. A.4 show the geometry of the line used for the simulations and Fig. A.5 shows the
numerical results. Again, it can be seen that the proposed 2
nd
order scheme leads to an
improvement of the computed results, in terms of numerical stability.


2 km
1
0

m

1 cm
500 m
Stroke
Location
Surge
Arrester
Surge
Arrester
Surge
Arrester
Surge
Arrester
Surge
Arrester
500 m
5
0

m


Fig. A.4. – Line geometry for the comparison between FDTD 1
st
order ad 2
nd
order in presence of surge
arresters, using the developed interface between LIOV and EMTP96.

a)
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
0 m FDTD 1st order
0 m FDTD 2nd order

Appendix pag.


110
b)
-40
-30
-20
-10
0
10
20
30
40
50
0 5 10 15 20
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
500 m FDTD 1st order
500 m FDTD 2nd order

c)
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20
Time [µs]
I
n
d
u
c
e
d


O
v
e
r
v
o
l
t
a
g
e


[
k
V
]
2000 m FDTD 1st order
2000 m FDTD 2nd order

Fig. A.5. – Lightning induced overvoltage at two observation points a) x=0 m, b) x=500 m, c) x=2000 m of
Fig. A.4. Comparison between FDTD 1
st
order and 2
nd
order in presence of surge arresters.

With the presence of non-linear components the above reported comparison show, again,
that the proposed 2
nd
order scheme improve the computed results in terms of numerical
stability. Indeed the increase numerically stability is obtained without significant increase in
the computation time
1
.

1
The calculation of the exciting lightning electromagnetic field representing, for the problem of interest,
the bulk of the computation time.