Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
Modelling of Electrical Cable Parameters and Fault Detection
using Fourier Series
*Hachimenum N. Amadi, Kingsley O. Uwho, Dennis M. Kornom
Department of Electrical Electronic Engineering, Rivers State University,
Port-Harcourt, Nigeria
*Corresponding Author
E-Mail Id: hachimenum.amadi@ust.edu.ng
ABSTRACT
The choice of a reliable and efficient cable has been of utmost concern to power Engineers
and researchers for it enables an adequate energy supply. So many models have been
considered to design either voltage or current and/or fault locations in cables. In this work,
the focus is to model three-phase voltage and current as parameters of a high-voltage cable
and to detect possible faults along its length with the aid of MATLAB /Simulink. The
parameters of the cable-three-phase voltage and current are measured with their respective
Fourier transformations. Equations parameters are determined using the Fourier transform.
The system is then checked for the presence of fault and if yes, the type of fault is displayed. It
is found that the presence of λ in the modelled voltage equation is significant because it
represents how far the current travels along the length of the dendrite. The greater the rm, the
membrane resistance, the more the current will remain inside the cytoplasm (internal fluids)
of the dendrite to travel longitudinally. The more rL, or intracellular resistance, the harder it
is for current to travel through the cytoplasm; hence the shorter the current can travel.
Keywords: Modelling, Cable parameters, Fault detection, Fourier transformation,
Matlab/Simulink
1.0 INTRODUCTION
High-quality electrical power supply is
necessary for infrastructural development,
job opportunities, wealth creation, and
human capital development. Electrical
cables constitute a significant part of the
electric power system [1].
However, the reliability of an electrical
cable rests majorly on its insulation which
is affected mostly by partial discharges.
These discharges occur due to mistakes
(errors) in the insulation of the cable during
the manufacturing; and also installation
stage. One of the notable defects in
electrical cables is the cavity. When there
is electrical stress, partial discharges are
noticed and can give rise to cable failure.
Modelling and of course simulation of
partial discharges as a cavity defect in
cable insulation is crucial in studying the
cable's parameters. Ref [2] used the plasma
model in simulating the partial discharges
in cavity defect by showing the parameters'
influence e.g. type of gas in the cavity,
voltage magnitude and frequency imposed
on the insulation mechanism. Ref [3] used
the
field-to-transmission-line
(FTL)
frequency domain approach to model
multi-conductor transmission lines in 3D
arrangement and other types of electrical
cable
bundles
subjected
to
an
electromagnetic field (EM).
When it is applied to the numerical
modelling process, the incident EM fields
at the stage of the wires, scattered by the
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Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
whole 3D structure can be calculated and
collected in the 3D model on the path of
the multi-conductor transmission lines. It is
found that the field of the multi-conductor
transmission line network is huge. The aim
of the field to the transmission line is that
the incident field source terms can be
estimated as identical for all the wires of
the medium transmission line (MTL)
model.
variable speed drive systems. Ref [7]
considered a mathematical model of
XLPE(Cross-Link Polyethylene) insulated
cable power line with underground
installation. The equivalent circuit of the
heat process is made using homogeneous
bodies method and it takes into cognizance
dielectric losses, the ambient temperature
and the temperature dependence of the
cable core.
Ref [4] studied the modelling of ageing
distribution
cable
for
replacement
planning. The work proposes a reliability
model which combines the following
models: IEC cable heat, Arrhenius ageing
and Weibull probability distribution. The
proposed technique is deployed on two
medium voltage distribution systems and is
found that ageing models have a great
impact on the system's reliability.
The evaluation of the mathematical model
adequacy is carried out by comparing the
obtained results with the calculation of
thermal and electrical processes using the
finite element method via the ANSYS
software workbench. The model can be
used to control the capacity of cable lines
with XLPE insulation and reduce their
lifespan due to temperature, and ageing of
the insulation.
A revised way for calculating high
voltage(HV) cable sheath currents and for
fault location of cross-bounded high
voltage cable systems has been looked at
[5]. From the study, the equivalent circuit
and the cable impedance per unit length
under short circuit fault conditions are
revised from those under non-fault
conditions. The study found that there is a
connection between the fault position and
the sheath fault currents, it is therefore easy
to spot a fault by looking at or considering
the characteristics of the sheath currents
when a fault occurs.
To tackle the
frequency dependency of unshielded power
cables per unit length parameters for EMC
(electromagnetic cable).
Parametric cable designs can be adjudged
on measurements or calculated as a
function of the cable features and the
physical characteristics of the cables by
finite-element analysis or by using specific
mathematical equations.
French Institute of Technology SaintExupery developed an approach where
fast, prediction models are compared to
different shapes of numerical models [6].
The approach is applied to unshielded two
and three-wire cables. Common mode(CM)
emissions modelling is therefore proposed
to forecast the CM noise currents which are
found to be the most turbulent in any
The last method results in certain errors
due to approximations whereas parametric
models adjusted using finite-element
analysis need large preparation and
simulation durations, so measurementbased parametric models outperform the
other options. However, assuming the
parameters of cable from measurements is
not a simple task, but the models can be
accurate for a wide frequency range even
in the face of resonance phenomena.
The researchers applied iterative genetic
algorithm (IGA) optimization in their
analysis [8]. Cable models that have
distributed parameters can be more exact
(accurate) than the ones with lumped
parameter models though, it is a function
of cable length and signal frequency.
However, the later ones give a better
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Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
physical interpretation of the problem
while simplifying the mathematical
complexity so that they can be easily suited
in any electromagnetic transients program.
In the past, cables were modelled by
discretising the telegrapher's equations into
cascaded lumped circuit networks that is
based on their per-unit-length (p.u.)
parameters and can be gotten by using
direct measurements or analytical formulas
or with the aid of electromagnetic
simulations. Straight measurements of
frequency-dependent per unit (p.u.)
parameters of a multiconductor cable are
rather cumbersome and inaccurate at high
frequencies. Analytical formulas are used
only for a given set of cross-sectional
geometries and model the frequency
dependence of the unit length parameters
only
approximately.
Electromagnetic
solvers require the cable cross-sectional
geometry and electric parameters which are
not readily available [9].
Ref [10] presents a new tool for the
computation of per-unit-length (p.u)
parameters for transmission line and cable
models
used
for
simulating
electromagnetic transients (EMT). The
intended approach is based on the MoMSO theory and state-of-the-art formulations
for the computation of the series
impedance
and
shunt
admittance
parameters.
The new tool has major benefits in
comparison to ancient approaches available
in EMT-type software. These advantages
include accurate skin and proximity effect
modelling, above-ground cable modelling,
modelling of stranded wires in cables,
representation of multilayer soil, coupled
overhead lines and underground cables,
and so on. The researchers present the new
tool together with demonstrations of
transient
simulations
for
practical
examples.
Middle voltage cables convey signals
poorly with great frequency spectrum e.g.
partial discharges. The radiating signals are
strongly attenuated and distorted based on
the transmitted distance. To solve this
problem, Ref [11] provides a model to
simulate the transmission of such energy
(signals) on middle voltage cables.
When compared to earlier approaches, this
model neglects not the wave character of
the high-frequency signals.
Consequently, to explain the transmission
signal, a holistic solution of the
telegrapher's equations is given. In this
solution, the propagation constant of the
middle voltage cable used must also be
stated; and this can be determined based on
the individual cable layers, taking into
cognizance all ohmic and dielectric losses.
Contrary to past methods, the constants of
the primary lines are all modelled to
depend on frequency.
The model can forecast the spectrum of a
signal transmitted at any distance from its
origin. From re-evaluation, it is obvious
that prediction and measurement agree
with reasonable accuracy. The model
developed is checked by investigating the
transmission of partial discharges on
middle voltage cables. Computer-based
approach is proposed for electromagnetic
transient simulations in power cables
dominated by an arbitrary cross-section
geometry. The parameters of the cables that
are hinged on frequency are calculated via
the finite element method, and the threephase cable modelling is done using modal
decoupling and fitting techniques.
The representation of cables with multiple
layers is easier from the calculation of a
constant; and a real modal transformation
matrix, which gives rise to four
independent propagation modes (three
phases and cable shield), which are
modelled from the addition of frequency
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Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
effect in the classic Bergeron technique.
The currents and voltages are stated as a
system of differential equations, which are
given as state equations and solved using
numerical integration methods.
The intended modelling technique helps to
add time-variable and non-linear elements
during
electromagnetic
transient
simulations in the time domain, which is
not possible from frequency domain
models that are solved with the aid of
inverse transforms. The intended model is
validated from results simulated using
numerical Laplace transform and exact
modal
transformation
matrix
for
calculation of phase currents and voltages
[12].
The major material for cable insulation is
XLPE (Cross-linked polyethylene). The
breakdown of cable insulation is of safety
concern to nuclear power plants. Ref [13]
proposed deterministic and probabilistic
models to quantitatively forecast the
decline in electrical resistance of the
insulation as a function of time in thermal
degradation.
The activation energy of the degradation
reaction is determined. The researchers
modelled one specimen as the embodiment
of two parts-degraded and non-degraded
parts having disparate resistivity based on
the Dichotomy approach. The volume ratio
of the two parts gives the sum resistance.
The cumulative density function (CDF) of
an exponential distribution is employed to
determine the change in ratio as a function
of time.
The objective of this research is to model
the parameters of a high voltage cable
three-phase voltage and current using
Fourier transformation and to detect faults
along its length.
2.0
MATERIALS AND METHODS
2.1
Materials
The materials used include the following:i. 50m of 150kV crosslinked polyethylene
(XLPE) cable
ii. Digital
multimeter
PCE-DC
41
(voltages up to 600 V, currents up to
600 A and electrical resistance up to
1000 Ω)
iii. Frequency meter of 10kHz to 150MHz
iv. HP4194A impedance analyzer
v. HP41941A impedance probe set
vi. Bandpass filter
vii. overvoltage protection (TVS-WE-TVS
diodes)
viii. Digital oscilloscope
2.2
Methods
The cable is modelled mathematically with
the aid of MATLAB/Simulink. The
parameters of the cable-three-phase voltage
and current are measured with their
respective
Fourier
transformations.
Equations parameters are determined using
the Fourier transform. The system is then
checked for the presence of fault and if yes,
the type of fault is displayed.
2.1
Design of a Flow Chart to Define
Fault Detection/Location Indicators
Phasor measuring techniques are used for
the identification and localization of faults
in every three-phase system. Based on the
equations developed in this research, a
general framework for a comparative
Fourier
transform
detection/location
technique is shown in Figure 1.
Below is the flowchart diagram of this
work which shows how the project is
achieved.
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Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
Fig 1: Based on Fourier Transforms, Modal, and Indicator Definition, a Functional
Schematic of Transmission Line Detection and Location Design is Shown
2.2
The Cable Equation's Derivation
The cable equation must be derived from fundamental ideas. The goal of this research is to
distinguish between membrane potential, voltage, and V(x,t) along the dendrite. Branch-like
structures called dendrites transport electrochemicals from axons to the neuron.
The cable is assumed to be passive in this model, which means that resistance is independent
of voltage.
r
( L  0 ).
(1)
V
Although either axons or dendrites can be
and it has the shape of a cylinder similar to
modeled by this equation, dendrites are a
the heat-conducting rod that was discussed
better fit for the passivity assumption than
in class. It is presumed that the crossaxons are. The potential is a function of the
sectional area will remain constant along
spatial dimension x, and of time t, as
the length of the neuron. The components
shown by the notation. No matter the
of the LRC model may be familiar to
string's radial coordinate, the string's
physicists and engineers. An illustration of
thinness enables each point x of the
the cable model and the equation's
cylinder to have the same tension. The
parameters are shown below.
dendrite has a cross-sectional area of πa2,
Fig 2.: Dendrite Model
(Source: [14])
Where;
rL = intracellular resistance found in the neuron’s cytoplasm (internal fluid)
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DOI: https://doi.org/10.5281/zenodo.11096455
RL= resistance to the longitudinal flow of current (IL).
cm = capacitance of a membrane of a unit length (Δx) of the cable
Cm = total capacitance
IL = longitudinal current
ie=
ion flow per unit of membrane area
Ιe = total ion flow
As was already mentioned, two components make up series resistance: rL, the resistivity
between cells, or RL, or the longitudinal resistance of a cable segment. The following
equation relates the two:
r x
RL  L 2
a
(2)
The cable's area serves as the denominator. This ratio demonstrates that longitudinal
resistance decreases with surface area. The voltage change is defined as follows:
V  V ( x  x)  V ( x)
(3)
Then, using Ohm's law, which states that the voltage change is equal to the current plus the
resistance (V=IR), we introduce the longitudinal current IL, and plug it into the equation.
V ( x  x)  V ( x)  I L RL
Replacing (2) by (4), we get:
I rx
V(x+Δx) – V(x) = L 2
a
The result of moving Δx over the left side is:
I L rL x
V ( x  x)  V ( x)

x
 2
Now the derivative on the left is Δx 0,, which gives us:
V I L rx

x
a 2
The longitudinal
current results in:
2
IL 
(4)
(5)
(6)
(7)
Va
xrL
(8)
As a result of ion flux in and out of the cell, we denote 𝑖𝑒 as the current per unit area. The
equation below can be used to represent the total ion flux per unit Δx across the membrane:
Ie(x,t) = (2πaΔx)ie
(9)
Since the total ion flux is simply the ion flux on the membrane times the area, this can be
understood. Let cm be the capacitance of the membrane and Cm = (2πaΔx)cm be the
capacitance overall. Icap stands for the amount of current necessary to alter the membrane
potential at a rate of ∂V/∂t.
V
(10)
t
Kirchoff's law, which states that the amount of current that crosses the membrane equals the
change in axial current within the cell, results in:
Icap (x,t) + Ie (x,t) = -IL(x+ Δx,t) + IL(x,t)
(11)
Adding (10) and (11) results in:
Icap(x,t) = (2πaΔx)cm
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DOI: https://doi.org/10.5281/zenodo.11096455
(2πaΔx)cm V +
t
V
( x  x , t )
a 2 V

( x., t )
(2πaΔx)𝑖𝑒 = a 2 x
rL
rL x
(12)
Dividing both sides of the equation by 2πaΔx, and taking Δx0 gives us:
V
a  2V
cm

 ie
t 2r L x 2
The ie equals:
𝑖𝑒 = V/rm
Where rm denotes the membrane's resistivity. This results in:
V
a  2V V
cm


t 2r L x 2 rm
For simplicity’s sake, we substitute
arm
τm = cmrm and λ =
to get the Cable Equation:
2rL
2
V
2  V
m

V
t
x 2
2.
3
Fourier Solution to the Bounded
Cable Equation
We now solve the infinite cable equation
after first solving the finite cable equation
using PDE reasoning. The Fourier method
with variable separation is used initially.
The cable has a length L. The boundary
and initial conditions available to us are the
following:
(13)
(14)
(15)
(16)
i.
ii.
No current flows where the dendrite
ends or branches.: Vx(0,t) = Vx(L,t) = 0;
and
The current at startup is solely
determined by the cable's spatial
coordinate: V(x,0) = f(x).
The problem mimics an insulated rod with
thermally conductive ends according to
assumption (I). The equation of the form is
taken using the variable separation
:
V(x,t) = X(x)T(t).
Connecting this to the PDE in (17), we get:
(17)
τmXT' = λ2X"T – XT
Rearranging terms and setting them equal to the negative characteristic (-k2) gives us:
(18)
= 𝑋 − −1 = −𝑘 2
𝑇
This yields equations (20) and (21):
(19)
τmT' = k2T
(20)
λ2X"= (1−k2)X
Equation (21) yields the solution:
(21)
𝜏𝑚 𝑇 ′
𝜆2 𝑋 ′′
T (t )  ce ( k t /  m )
2
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(22)
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Volume 7 Issue 2
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DOI: https://doi.org/10.5281/zenodo.11096455
As for the solution to (23), we assume thatk2>1, so that the roots of the equation are complex.
If k2<1, the roots would be positive and the solution would be in the form of:
1 k
2

x
1k
x
X(x) = a e
+ be 
(23)
That is not desirable because in that case the boundary conditions would not be helpful since
they would not yield unique values of a and b, resulting in only trivial solutions. So assuming
that k2>1, we get the following result:
X(x) = acos( 12k x) + bsin( 12k x)
(24)
2


The boundary conditions V'(0) = 0 and V'(L) = 0 mean that
X'(0)T(t) = X'(L)T(t) = 0
(25)
In order to avoid trivial solutions again, we need:
X'(0) = X'(L) = 0
(26)
Plugging in X'(0) = 0 gives
X'(0) = -
1 k
2
sin(
1 k
2
(0)) + bcos(
1 k
2
(0))= 0
(27)
Everything associated with the sine term drops out, and leaving us with b = 0, giving us:
X(x) = acos( 12k x)
(28)

Using the second boundary condition X'(L) = 0
X'(L) = - 12k sin( 12k (L)) = 0

(29)

In general, since sin(α) =0 means that α =πn, we have:
 n 
πn/L and k2= 1  
(30)


 L 
Now we plug these two equations back into (22) and (18), respectively, resulting in the
following:
2
1 k
=
2
T(t) = c e(1(n / L) )t / m
and
nx
X(x) = acos(
)
L
Plug (31) and (32) into (16) and we have:
2
nx
Vn(x,t) = acos(
) c∙ e(1(n / L) )t / m
L
Now we substitute dn = ac
nx (1(n / L)2 )t / m
Vn(x,t) = dncos(
)e
L
2
Consequently, we impose this in anticipation of the Fourier cosine expansion:

nx (1(n / L)2 )t / m
V(x,t) =  dncos(
)e
L
n 1
After adding the initial condition V(x,0) = f(x), the exponential term drops out.

1
nx
V(x,0) = d 0   dncos(
)= f(x)
2
L
n 1
This is the Fourier cosine expansion. So we have:
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(31)
(32)
(33)
(34)
(35)
(36)
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2
n
)d
dn =  f ( ) cos(
L0
L
Plugging this back into (33) gives us:
L
L
2
n
nx (1(n / L)2 )t / m
2 
f ( )d   (  f ( ) cos(
)d ) cos(
V(x,t) =
) e

L
L0
L
L n1 0
L
This is the elegant solution to the bounded
Cable Equation. Thus Fourier methods
with the proper boundary and initial
conditions enable us to obtain an analytical
solution for the conductance of membrane
potential along the dendrite. Moving on,
we now consider a cable of infinite length.
2.4
Fourier Transform of Infinite
Cable Equation
(37)
(38)
Neuroscientists often examine the idealized
“infinite cable.” With such a notion, the
spatial coordinate x extends over [-∞, ∞].
The Fourier transform is used to find the
answer. Even though O'Neill's telegraph
equation has a different form and produces
a very different outcome, we partially rely
on its solution in this section. We represent
the PDE with this slightly different
notation:
τm vt = λ2vxx –v
(39)
To use a Fourier transform we need an initial condition. The initial voltage for the infinite
cable depends only on the spatial coordinate x, or in other words, where along the cable:
v( x,0) = f(x)
(40)
To the PDE we apply the Fourier transform.
τm {vt } =λ2 {vxx}  {v}
(41)
Then we represent the transformed voltage variable with capital letters.
{vt } = V'(ω,t)
(42)
Applying the Fourier Transform to the PDE gives us
τmV’(ω,t) = -λ2ω2V(ω,t) – V(ω,t)
(43)
which is an ODE with the prime in the equation representing derivation of the transform with
respect to t. The derivatives of the variable operating with respect to x are represented by the
terms on the right side of the equation. The Fourier transform for the v(x,t) by definition is:
V(ω,t) =

 v(, t )e

 i x
d
(44)
We derive the following two equations using the Operational Rule for Fourier Transforms:
∞
∞
𝜕
𝜕
=𝑖𝜔𝑥
𝑉(𝜔,
𝑡)
=
𝑣(𝜔,
𝑡)𝑒
𝑑𝜔
=
𝑖𝜔
𝑣(𝜔, 𝑡)𝑒 𝑖𝜔𝑥 𝑑𝑤 = −𝑖𝜔𝑉(𝜔, 𝑡)
(45)
∫
∫
∞
∞
𝜕𝑥
𝜕𝑥
𝜕2
𝜕𝑥 2
∞
𝜕
∞
𝑉(𝜔, 𝑡) = 𝜕𝑥 𝑖𝜔 ∫∞ 𝑣(𝜔, 𝑡)𝑒 =𝑖𝜔𝑥 𝑑𝜔 = 𝑖 2 𝜔2 ∫∞ 𝑣(𝜔, 𝑡)𝑒 𝑖𝜔𝑥 𝑑𝑤 = −𝜔2 𝑉(𝜔, 𝑡)
With this in mind, we rearrange (43) into canonical form.
(1  2 2 )
V'(ω,t) −
V (, t ) = 0
m
(46)
(47)
This ODE yields the following solution:
 (1 2 2 )
t
V(ω,t)=k e  m
(48)
Recall our initial condition in (40), v( x,0) = f(x). We assume that f(x) has a Fourier Transform
G(ω).
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G(ω) =  {f(x)}
Applying the initial condition (40), we have:
 (1 2 2 )
m
V(ω,0) = k e
( 0)
(49)
(50)
= G(ω), or k = G(ω)
Solving for the Fourier term V(ω,t) gives:
 (1 2 2 )
t
V(ω,t) = G(ω) e  m
Applying the inverse Fourier transform gives us:

1
v(x,t) =
G( ) e
2 
 (1 2 2 )
m
t
(51)
e ix dω
(52)
So we found a v(x,t) for which its Fourier transform is G(ω) e
(12 2 )
m
function s(x,t) whose Fourier transform is S(ω,t) = e
would be:
s(x,t) = 1

S ( , t )e
2 

it

d   e
 (1 2 2 )
m

t

 2 2
e ix d  e1/ m  e  m
t ix
t
 (1 2 2 )
t
m
. We need to find the
. The Fourier transform of s(x,t)
(53)
d

We can evaluate the integral using completing the square, with the following rules.


 az 2
2
2
2
2
ax – b = a(x - b/(2a)) – b /(4a), e dz 

a


e b /( 4a )  e a ( x-b/(2a)) dx  e b /( 4a )  e az dz  e b /( 4a )
2
2
2

2
2


a
(54)
(55)
We plug the parameters into the equation.
 (2 2 )
a=
, b=ix
(56)
m
Solving gives us:
x 2 m
t / m  2
1
𝑠(𝑥, 𝑡) −
e 4 t
2𝜋
 m
2
x 2 m
t / 
m
4 t
e
2 2
4  t
m
=
(57)
2
Using the function u = s * f and that Fourier transforms turn convolutions into multiplication,
we get:
( x  ) 
v(x,t) = (s(.,)t * f(.))x) =  s( x   ) f ( )d   m  f ( )et /  4 t d
(58)
2

4 t 
m
m
2
2


And to state it straight up, the solution to the unbounded Cable Equation is:
v(x,t) =
t / 
m 
f ( )e
2 
4 t 
m
( x  ) 2  m
4 2t
d
(59)
3.0 Algorithm for the Cable and Fault
Model (Open-Circuit)
function [Zin] = TIL (f, cable_len) % f frequency, cable length % Cable characteristics
(Distributed parameters)
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Journal of Recent Trends in Electrical Power System
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e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
l_m=240e-9;
c_m=138e-12;
r_m=1.6/10000. *sqrt (f);
g_m=0.1. *c_m. *f;
fault_loc=17;
% velocity of propagation typical
vp = 0.535.*3e8;
% Characteristic impedance vector
Z0=42;
% Attenuation coefficient vector
alfa = 0.5*1e-6. *f.^0.61;
% Propagation coefficient vector
beta = 2*pi*f./vp;
% Propagation constant
gamma = alfa + j*beta;
% Fault circuit characteristics (variable parameters)
Zc=(1/(2*pi*f.*c_m));
% Capacitive reactance
Z1=2*pi*f.*l_m;
% Inductive reactance
Zf=(Zc*1.5+Z1*0.2);
% Fault resistance (zc+Z1*0.4); %
% The remaining cable's impedance at the location of the cable fault
Zcable = Z0. *coth (gamma.*(cable_len-fault_loc); % 33
% The load impedance at the location of failure
% Zload = (Zcable.*Zf)./(Zcable + Zf);
% Zload = Zcable.*100e9;
% Matrix for the first segment of the cable before the fault
TL1 (1,1) = cosh (gamma.*(fault_loc));
TL1 (1,2) = Z0.*sinh (gamma.*(fault_loc));
TL1 (2,1) = (1/Z0).*sinh (gamma.*(fault_loc));
TL1 (2,2) = TL1 (1,1);
% Matrix for the second part of the cable after fault
TL2(1,1) =cosh (gamma.*(cable_len-fault_loc));
TL2(1,1) =Z0.*sinh (gamma.*(cable_len-fault_loc));
TL2(1,1) =((1/Z0).*sinh (gamma.*(cable_len-fault_loc));
TL2(1,1) = TL2(1,1);
% Fault circuit matrix (shunt impedance at the fault location)
TL3 (1,1) =1;
TL3 (1,2) = 0;
TL3 (2,1) = (1/Zf);
TL3 (2,1) = TL3 (1,1);
% Cascading together various parts of the transmission line
% TL=TL1*TL2; % For healthy cable
TL = TL1*TL3*TL2; % For faulty cable
% Input impedance
Zin= ((TL(1,1). *Zload) + (TL(1,2))) / ((TL(2,1).*Zload) + (TL (2,2))); end
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Journal of Recent Trends in Electrical Power System
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DOI: https://doi.org/10.5281/zenodo.11096455
4.0 CONCLUSION
Conclusively, Fourier methods with the
proper boundary and initial conditions
enable obtaining an analytical solution for
the conductance of membrane potential
along the dendrite. It is found that the
presence λ in the modelled voltage
equation is significant because it represents
how far the current travels along the length
of the dendrite.
The greater the rm, the membrane
resistance, the more the current will remain
inside the cytoplasm (internal fluids) of the
dendrite to travel longitudinally. The more
rL, or intracellular resistance, the harder it
is for current to travel through the
cytoplasm; hence the shorter the current is
able to travel. Thus a Fourier
transformation enables us to solve for the
voltage conduction along an infinitely-long
cable fiber.
REFERENCES
1. Amadi. H.N., Uwho, K.O. & Tebepah,
E.A. (2024). Improvement
of
Power Quality: A Case Study of
Eneka Town Hall, Eliozu and Akani
Distribution
Substations.
International Journal of Engineering
Research And Development. 20(1).
16-24.
2. Elagoun, A. & Seghier, T. (2023).
Parameters that Influence the Partial
Discharge Occurrence in Cavity
Defect
with
Electrical
Cable
Insulation. Diagnostyka, 24(2), 1-11.
https://doi.org/10.29354/diag/166378
3. Parmantier, J., Guiffaut, C., Roisse,
D., Girard, C., et al. (2020). Modelling
EM-Coupling on Electrical
CableBundles with a Frequency-Domain
Field-to- Transmission line Model
Based on Total
Electric
Fields.
Advanced
Electromagnetics, 9
(3), 15-31.
4. Buhari, M., Levi, V. & Awadallah, S.
K. E. (2015). Modelling of Ageing
Distribution Cable for replacement
Planning.
IEEE Transactions on
Power
Systems,
1-8.
DOI:10.1109/TPWRS.2015.2499269.
5. Li, M. Zhou, C. & Zhou, W. (2019). A
Revised Model for Calculating HV
Cable Sheath
Current under ShortCircuit Fault Condition and its
Application for Fault Location-Part 1:
The
Revised
Model.
IEEE
Transactions on Power Delivery,
34(4),16741683.https://doi.org/10.1109/TPWRD.
2019.2918159
6. Santos, V.D., Roux, N., Revol, B.,
Sareni, B., Cougo, B. & Carayon, J.
(2017). Unshielded Cable Modeling
for Conducted Emissions Issues in
Electrical Power Drive Systems.
2017 International Symposium on
Electromagnetic Compatibility (EMC
EUROPE), September 2017, France,
1-6. hal- 01656080.
7. Kropotin,
O.,
Tkachenko,
V.,
Shepelev, A., Petrova, E., et al. (2019).
Mathematical
Model of XLPE
Insulated Cable Power Line
with
Underground Installation. OMSK
State Technical University, Russian
Federation.
doi.10.15199/48.2019.06.14
8. Bogarra, S., Riba, J.R., Sala-Caselles,
V. & Garcia, A. (2017). Optimal
Fitting of High Frequency Cable
Model Parameters by Applying
Evolutionary Algorithms. International
Journal of Electrical Power & Energy
Systems.
doi:10.1016/j.ijepes.2016.11.006
9. Wunsch, B., Stevanovic, I. & Skibin,
S.
(2017).
Length-Scalable
Multiconductor Cable Modeling for
EMI Simulations in Power Electronics.
IEEE
Transactions on Power
Electronics, 32 (3), 1908-1916.
10. Morales, J., Xue, H., Mahseredjian, J.
& Kocar, I. (2023). A New Tool for
Calculation of
Lines and Cable
Parameters. International Conference
on Power Systems
Transients
HBRP Publication Page 44-56 2024. All Rights Reserved
Page 55
Journal of Recent Trends in Electrical Power System
Volume 7 Issue 2
e-ISSN: 2584-2404
DOI: https://doi.org/10.5281/zenodo.11096455
(IPS72023) in Thessaloniki, Greece,
June 12-15,
2023.
11. Fritsch, M. & Wolter, M. (2022).
Transmission Model of Partial
Discharges on Medium Voltage
Cables. IEEE Transactions on Power
Delivery, 37 (1), 395-404.
12. Hafner, A. A., Caballero, P. T.,
Monteiro, J. H. A., Costa, E. C. M., et
al. (2017). Modelling of Power Cables
with Arbitrary Cross Section: from
Parameter
Calculation
to
Electromagnetic
Transients
Simulation. Journal of Control,
Automation and Electrical Systems
(2017), 28, 405-417.
13. Chang, Y. & Mosleh, A. (2018).
Predictive Model on the Degradation
of the Electrical Resistance of Cable
Insulation.
Probabilistic
Safety
Assessment and Management PSAM
14, September 2018, Los Angeles,
California.
14. White, J. A., Manis, P. B. & Young, E.
D.
(1992).
The
Parameter
Identification Problem for the Somatic
Shunt Model, Biol. Cybern. 66, 307318.
https://doi:10.1109/TPWRS.2015.2499
269.
Cite as :
Hachimenum N. Amadi, Kingsley O.
Uwho, & Dennis M. Kornom. (2024).
Modelling of Electrical Cable Parameters
and Fault Detection using Fourier Series.
Journal of Recent Trends in Electrical
Power
System,
7(2),
44–56.
https://doi.org/10.5281/zenodo.11096455
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