Transient response of grounding systems under impulse lightning current

1. 978-1-5090-1564-1/16/$31.00 ©2016 IEEE
Abstract—Grounding systems are considered as an essential
part of power systems since they are designed to protect them
against fault currents according to some safety and power
quality standards. When lightning strikes a substation or a
transmission tower near the substation, high current generated
by the stroke would flow into the grounding system of the
substation and dissipate into the soil. The transient response of
the grounding system is a very important indicator to the safe
and reliable operation of the power system. A model with
lumped time-varying parameters is developed and is imple-
mented into software tools (Matlab/Simulink) to calculate the
transient behavior of the grounding systems during lightning
strokes. The effects of different parameters especially soil
resistivity and ionization of the soil around the electrode on the
transient behavior of the grounding system subjected to light-
ning impulse current are analyzed.
Index Terms—Power Quality, Grounding Grid, Atmos-
pheric Discharge, Soil Resistivity, Soil Ionization.
I. INTRODUCTION
Grounding systems, such as buried vertical, horizontal
electrodes and large grounding grids, are often part of the
lightning protection of the power systems. The complexity
of the soil structure and shape of grounding systems
introduced the development of different sets of numerical
methods for calculating the characteristics.
This characterization is a significant factor in determining
the overall grounding impedance behavior. Solutions based
on either circuit theory [2]–[5] or electromagnetic theory
[6]–[11] can be used to model the ground impedance
characteristic under transient conditions. In the circuit-based
model, the ground impedance is represented either as a
lumped or distributed parameter [10]. The elements of the
circuits are basically R, L and C; their values are computed
by using relevant formulas proposed by Sunde [12].
In recent years, numerical methodologies have been
developed and optimized for grounding systems transient
analysis. The FEM (Finite Element Method), the moment
method, the FDTD (Finite difference Time-Domain) method,
and the transmission line modeling method (TLM) [16] are
some of the state of the art techniques.
Injection of a current in the grounding system leads to the
appearance of time-varying voltage and current along the
grounding system. The impedance is determined as the ratio
of voltage at the node in which the current is injected [4]. Fig. 1
shows the profiles of the voltage, current and impedance.
I. Djamel and F. H. Slaoui are with the department of Electrical Engi-
neering at Université du Québec en Abitibi-Témiscamingue, Quebec,
Canada, J9X 5E4
(e-mail: Djamel.Idir@uqat.ca; fouad.slaoui-hasnaoui@uqat.ca)
S. Georges is with the department of Electrical, Computer and Commu-
nication Engineering at Notre Dame University – Louiaze, Zouk Mosbih,
Lebanon (e-mail: sgeorges@ndu.edu.lb).
Fig. 1. Time-varying impedance, voltage, and current along the grounding.
𝑧𝑧(𝑡𝑡) =
𝑣𝑣(𝑡𝑡)
𝑖𝑖(𝑡𝑡)
(1)
where
v(t) – voltage at the site of injection at the time t,
i(t) – injected current into the grounding system,
z(t) – impact impedance grounding at the time t.
In practice, the term conventional impedance earthing Zm
is often used:
𝑍𝑍𝑚𝑚 =
𝑉𝑉𝑚𝑚
𝐼𝐼𝑚𝑚
(2)
Based on the transmission line model, the increase in the
conductors diameter implies the need for a change in the
line parameters per unit length with consequent alteration of
the circuit elements present in the model [7], [8].
Despite these drawbacks, the circuit models are simple,
fast, and can be realized by standard simulation software
tools, such as the EMTP, PSCAD, and ATP.
Quantification of the grounding system characteristics
was made in the function of the parameters of atmospheric
discharges and soil characteristics [3].
The model developed was used to study the performance
of vertical rods, horizontal and will be extended to ground-
ing grids of different sizes. The results show that the esti-
mated values for the voltage are precise and they were
validated by the analytical method represented in [1].
II. MATHEMATICAL MODEL
Some analytical calculations show the voltage profile at the
injection node for different current pulses [1]. For a double
exponential impulse current, 𝐼𝐼(𝑡𝑡) = 𝐼𝐼𝑚𝑚�𝑒𝑒−𝛼𝛼𝛼𝛼
− 𝑒𝑒−𝛽𝛽𝛽𝛽
�, the
voltage at the injection point is given by (3) as:
Transient Response of Grounding Systems
Under Impulse Lightning Current
Idir Djamel, Fouad H. Slaoui, and Semaan Georges

2. 𝑒𝑒(𝑡𝑡) = 𝐼𝐼0
⎣
⎢
⎢
⎡�𝐿𝐿𝐿𝐿
𝐺𝐺
𝑒𝑒−𝛼𝛼𝛼𝛼
tan√𝐺𝐺𝐺𝐺𝐺𝐺
−
�𝐿𝐿𝐿𝐿
𝐺𝐺
𝑒𝑒−𝛽𝛽𝛽𝛽
tan�𝐺𝐺𝐺𝐺𝐺𝐺
(3)
+
𝛼𝛼 − 𝛽𝛽
𝐺𝐺2 𝐿𝐿
�
2𝑛𝑛2
𝜋𝜋2
𝑒𝑒−
𝑛𝑛2 𝜋𝜋2 𝑡𝑡
𝐺𝐺𝐺𝐺
�𝛼𝛼 −
𝑛𝑛2 𝜋𝜋2
𝐺𝐺𝐺𝐺
� �𝛽𝛽 −
𝑛𝑛2 𝜋𝜋2
𝐺𝐺𝐺𝐺
�
∞
𝑛𝑛=1
�
where
L – total rod inductance in Henry,
G – total ground conductance in Siemens,
Im – peak value of the injected current,
α, β – constants for the current wave shapes.
III. MODELING OF THE GROUNDING CONDUCTORS
After having estimated the parameters of the grounding
conductors per unit length, the transient behavior of ground-
ing systems can be simulated based on the conventional
transmission line approach. The exact effect of the distrib-
uted parameters is considered as the transmission line is
divided into many electrically small sections [14], [15] as
shown on Fig. 2.
Fig. 2. Transmission line [14].
Each linear element can be modeled with the appropriate
equivalent circuit (Γ, Π, T – equivalent circuit). Fig. 3 shows
a grounding grid with a single mesh with a marked linear
element i; Ri is the series resistance, Ls is the self induct-
ance, Gi is the transverse conductance, and Ci is the trans-
verse capacitance.
A lumped circuit model [14] has been proposed to simu-
late the grounding electrode under transient conditions. This
model includes all elements and can be easily simulated in
transient programs.
The lumped model has only one section and it does not
consider the wave propagation delay, the frequency- and
time-dependent phenomena.
The elements of the lumped model are determined by
Sunde [12] and Dwight [17]. The series resistance of con-
ductors with arbitrary shape and cross-section is determined
as follows:
𝑅𝑅𝑖𝑖 =
𝑙𝑙
𝜎𝜎𝑖𝑖 𝑆𝑆
(4)
while the resistance of a horizontal and vertical rodes in
uniform soil are determined by the following equations,
respectively
𝑅𝑅 =
1
𝐺𝐺
=
𝜌𝜌
2𝜋𝜋𝜋𝜋
�ln
2𝑙𝑙
√2𝑟𝑟ℎ
− 1� (5)
𝑅𝑅 =
1
𝐺𝐺
=
𝜌𝜌
2𝜋𝜋𝜋𝜋
ln
𝑙𝑙
𝑟𝑟
(6)
where
ρ – soil resistivity (in ohms-meters),
l – length of the electrode (in meters),
r – radius of the electrode (in meters),
h – burial depth (in meters).
Fig. 3. a) Grounding grid with one mesh for element i; b) T replacement
scheme; c) Γ replacement scheme; d) Π replacement scheme.
The inductance for horizontal and vertical electrodes in
uniform soil as given in (7) and (8), respectively (in Henry)
is calculated as
𝐿𝐿 =
𝜇𝜇0 𝑙𝑙
2𝜋𝜋
�ln
2𝑙𝑙
√2𝑟𝑟ℎ
− 1� (7)
𝐿𝐿 =
𝜇𝜇0 𝑙𝑙
2𝜋𝜋
ln
4𝑙𝑙
𝑟𝑟
(8)
where µ0 is the soil permeability 4π⋅10–7
H/m.
The grounding capacitance for horizontal and vertical
electrodes in uniform soil as given in (9) and (10), respec-
tively (in Farad) is calculated as
𝐶𝐶 =
2𝜋𝜋𝜋𝜋𝜋𝜋
ln
2𝑙𝑙
√2𝑟𝑟ℎ
− 1
(9)
𝐶𝐶 =
2𝜋𝜋𝜋𝜋𝜋𝜋
ln
2𝑙𝑙
𝑟𝑟
(10)
where ε is the permittivity of soil (in farads/meter). The
relative permittivity εr = ε/ε0 varies as a function of water
content and frequency.
IV. MODELING THE IONIZATION EFFECT AROUND
GROUNDING ELECTRODES
In some conditions, electrical current discharge in the soil
around the grounding electrodes can cause soil ionization
which is a nonlinear effect that arises when high magnitude
current are injected in grounding systems located in a poorly
conductive soil.
Due to the nature of the studied phenomenon, a time-
domain analysis is more suitable to analyze the grounding
system considering the soil ionization.
Several authors have modeled the soil ionization effect by
using the transmission-line approach [9]–[13], but the mutual
couplings among conductors were not taken into account.
Fig. 4 shows the effect of ionization around an electrode, the
largest diameter is where the high current is injected.

3. Fig. 4. Ionized soil around the electrode [18].
When the electric field strength exceeds a critical value, it
leads to the formation of highly conductive channel around
grounding electrode. The influence of the ionized zone can
be modeled using a fictitious increase in the radius of the
grounding element. According to the results of theoretical
and experimental research, soil ionization around grounding
electrode leads to a reduction in shock impedance [15].
Under this procedure, when determining the radius of the
equivalent linear element, it is assumed that the current is
driven to the ground along an element where the current
density is almost constant.
Surface density of the current is determined by the
following equation:
𝐽𝐽𝑖𝑖 =
𝐼𝐼𝑚𝑚𝑚𝑚
2𝜋𝜋𝑎𝑎𝑖𝑖 𝑙𝑙𝑖𝑖
(11)
where
Ji – current surface density,
Imi – maximum value of the current drainage,
ai – equivalent radius of the element,
li – length of the grounding element.
The intensity of the electric field is determined by:
𝐸𝐸𝑖𝑖 =
𝜌𝜌𝜌𝜌𝑚𝑚𝑚𝑚
2𝜋𝜋𝑎𝑎𝑖𝑖 𝑙𝑙𝑖𝑖
≥ 𝐸𝐸𝑘𝑘 (12)
where Ek is the intensity of the critical electric field.
If the dissipated current is large enough for a segment, the
electric field intensity on the surface of this segment will
exceed the critical electric field intensity value for soil
ionization Ek , then the soil ionization is initiated on that
segment. The radius of the ionization region is increased to
some certain distance where the electric field intensity
finally falls to the critical value, Ek . This radius is calculated
by [15], [7]:
𝑎𝑎𝑒𝑒 =
𝜌𝜌𝜌𝜌𝑚𝑚
2𝜋𝜋𝜋𝜋𝑘𝑘 𝑙𝑙𝑢𝑢
(13)
where
ae – equivalent radius of all elements of grounding,
lu – total length of the grounding element,
Im – the maximum value of the lightning current injected.
The radius of each segment aei is determined by (14).
Imi is the dissipation current from the conductor segment to
the soil and li is length of each segment.
𝑎𝑎𝑒𝑒𝑒𝑒 =
𝜌𝜌𝜌𝜌𝑚𝑚𝑚𝑚
2𝜋𝜋𝜋𝜋𝑘𝑘 𝑙𝑙𝑖𝑖
(14)
V. CURRENT WAVEFORM
The current waveform that is injected in the groundings
can be represented as a double exponential function.
The bi-exponential waveform is defined mathematically
by the difference between two decaying exponentials:
𝐼𝐼(𝑡𝑡) = 𝐼𝐼𝑚𝑚�𝑒𝑒−𝛼𝛼𝛼𝛼
− 𝑒𝑒−𝛽𝛽𝛽𝛽
� (15)
α and β are two constant numbers in s–1
and Im is a constant
number in kiloamperes. Note that for β > α, Imax = Im.
Rise time Tr defined as the time difference that exists
when the signal rises up from 10% to 90% of its maximum
amplitude:
𝑇𝑇𝑟𝑟 =
2.746
𝛽𝛽
(16)
Td defined as the time difference for which the rising
waveform and the decaying waveforms are equal to half of
the maximum:
𝑇𝑇𝑑𝑑 =
0.396
𝛼𝛼
(17)
The transient behavior of horizontal grounding, vertical
and three grounding grids are chosen for computations.
VI. VERTICAL AND HORIZONTAL GROUNDING
Assume that the two earthing electrodes (Figs. 5 and 6)
receive a lightning current shape 2/12.5 μs/μs and amplitude
of Im = 12.5 kA. The current wave profile is shown in Fig. 7
while Figs. 8 and 9 show the transient voltage and imped-
ance respectively for a vertical and horizontal buried elec-
trode. Data for the resistivity of the soil and horizontal and
vertical electrodes are presented on Table I.
The horizontal electrode is buried at 0.6 m depth in the
homogeneous soil.
Fig. 5. Grounding electrode horizontal buried in the soil.
Fig. 6. Grounding electrode vertical buried in the soil.
TABLE I. DATA FOR HORIZONTAL AND VERTICAL ROD
Horizontal and Vertical Electrode Soil
l = 10 m ρ = 100 Ωm
r = 7 mm ɛr = 15
ρCu = 0.0178 Ωmm2
/m µr = 1

4. Fig. 7. Current waveform at the site of injection.
Fig. 8. Transient voltage at the site of injection A.
Fig. 9. Transient impedance at the site of injection A.
VII. GROUNDING GRID
Fig. 10 shows a grounding grid 1 × 1 and with dimensions
12 by 12 m. The diameter of the conductors is 14 mm and
the grid is buried at 0.6 m depth in the homogeneous soil,
εr = 15 and 𝐼𝐼(𝑡𝑡) = 𝐼𝐼𝑚𝑚�𝑒𝑒−𝛼𝛼𝛼𝛼
− 𝑒𝑒−𝛽𝛽𝛽𝛽
� the current impulse is
injected at point A and has Tr /Td = 2/12.5 µs wave shape,
Im = 12.5 kA and resistivity ρCu = 0.0178 Ωmm2
/m.
The analysis was performed for different values of soil
resistivity, ρ (Ωm) = {100, 300, 600, 1000}.
Figs. 10 and 11 represent a grounding grid with one mesh
in the ground and the circuit equivalent of a square mesh of
this grid in SIMULINK.
Fig. 10. Grounding grid 1×1 buried in the soil.
Fig. 11. Equivalent circuit of a square mesh of the grid in Matlab
SIMULINK.
The simulation results are illustrated on Figs. 12 and 13.
Fig. 12. Transient voltage at the site of injection A, with different resis-
tivity.
Fig. 13. Transient impedance at point A, with different resistivity.
VIII. IONIZATION EFFECT
Based on the procedure described, a systematic analysis
was conducted of the ionization influence on the grounding
grid.
Assume that the depth of the buried grounding grid is
0.6 m, while the conductors diameter is 14 mm. The relative
dielectric constant of soil is εr = 15. The calculation results
are shown in Table II.
In both trials presented here, for the same resistivity and
a current of 50 kA and 25 kA in soil without ionization,
transient impedance of the two cases given the same shape
unlike taking into account the phenomenon of soil ioniza-
tion. Increasing the radius of the electrode caused by the
influence of this phenomenon leads to the decrease of the
voltage and consequently the impedance.
Table II shows the results obtained form each side of the
grounding grid after injecting.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10
-5
0
2000
4000
6000
8000
10000
12000
t(s)
I(A)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
x 10
-5
0
2
4
6
8
10
12
14
x 10
4
t(s)
V(Volt)
Hotizontal
Vertical
0 1 2 3 4 5 6
x 10
-6
0
5
10
15
20
25
t(s)
Z(ohm)
Horizontal
Vertical
0 0.2 0.4 0.6 0.8 1 1.2 1.4
x 10
-5
0
0.5
1
1.5
2
2.5
3
3.5
x 10
5
t(s)
V(Volt)
100 ohm.m
300 ohm.m
600 ohm.m
1000 ohm.m

5. TABLE II. RESULTS FROM INJECTING A CURRENT IN THE
GROUNDING GRID
ρ
(Ωm)
Im
(kA)
Tr/Td
(µs/µs)
Im1
(kA)
Im2
–
Im3
–
Im4
–
ae
(mm)
Ek
(kV/m)
50 25 2/12.5 12.95 4.990 4.514 4.990 6.217 800
50 50 2/12.5 25.90 9.980 9.028 9.980 12.43 800
300 25 2/12.5 6.510 5.236 5.215 5.236 24.86 1200
300 50 2/12.5 13.02 10.47 10.43 10.47 49.73 1200
1000 25 2/12.5 5.254 5.240 5.242 5.240 76.516 1300
1000 50 2/12.5 10.51 10.48 10.48 10.48 153.33 1300
IX. CONCLUSION
This paper presents a method to simulate the transient
response of grounding system following a lightning stroke
on the power system. The obtained results show that soil
resistivity and soil ionization around the grounding elec-
trode may affect the calculated value of the grounding
impedance during the transient state of the system.
Transient impedance is highly affected by the value of the
resistivity at the point of the current injection. It increases
with the resistivity.
The study of the effects of various factors on the response
of grounding systems show that the impulse performance of
grounding systems depends on three factors: the geometry
of the grounding electrode, the electrical properties of the
soil, and lightning current (waveform) or the current inten-
sity.
This contribution through simulation tools may be
applied to the engineering practices in order to assess
temporal changes in the behavior of the grounding systems
characteristics
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Idir Djamel received his Master’s Degree in electrical engineering from
the University of Quebec in Abitibi-Temiscamingue in 2016. He developed
a remarkable industrial experience through working on different engineer-
ing project.
Slaoui Hasnaoui Fouad received his B.Sc.A. and M.Ing. in Electrical
Engineering in 1986 and 1995 respectively from École Polytechnique de
Montréal. He received also his Ph.D. degree from Ecole de Technologie
Superieure, Montreal Canada in 2003. Dr. Slaoui worked as a power
industrial engineer for few years in Morocco where he was responsible of
the electrical maintenance department in one of the largest fish industries in
the country. Then he worked in Montreal, Canada on various research
projects in power systems, grounding systems and ground fault distribution
in substations, towers and ground wires. He is currently a Professor in
power systems at the University of Quebec in Abitibi-Temiscamingue. His
research interests are numerical analysis in power system stability and
grounding.
Semaan W Georges was born in Lebanon, in 1963. He received the
Bachelor Degree in electrical engineering from the Higher Institute of
Electrical and Mechanical Engineering, Sofia, in 1989, the Master’s Degree
in electrical engineering from the Ecole Polytechnique, Montreal Canada in
1995 and the Ph.D. degree from the Ecole de Technologie Superieure,
Montreal, Canada in 2001. From 1990 to 1996, he worked on various
research projects in power engineering including grounding systems for
high voltage power networks, electric power quality and power system
stability. He joined the teaching staff of the department of electrical
engineering at the Ecole de Technologie Superieure from 1996 to 2001.
Since then, he joined Notre Dame University – Lebanon where he is
presently a Professor in the department of Electrical, Computer and
Communication Engineering. His research interests include active power
filters, power systems, digital signal processing and grounding systems.