power line tap modeling at power-line carrier frequencies with
TRANSCRIPT
Power line tap modeling at power-line carrier frequencies with radial-basis function network
N.Suljanović, A. Mujčić, M. Zajc, J.F. Tasič University of Ljubljana, Faculty of Electrical Engineering, Laboratory for Digital Signal Processing
Tržaška 25, 1000 Ljubljana, Republic of Slovenia [email protected], [email protected], [email protected],
[email protected] Abstract
This paper deals with modeling of tapped high voltage power line using radial basis networks. Model of power-
line tap appropriate for power-line communication was investigated. It is common to approximate power line
conductor sag with the stair step line where every lump is replaced with one multiport. The approach presented
in this paper introduces a modification of the method of multiports where radial-basis function (RBF) network is
used to estimate A parameters of one line tap lump. Simulations are conducted in MATLAB for 400 kV power
line with horizontal dispositions of conductors. Results of simulations presented at the end show that RBF
network can be used in power line tap modeling with acceptable accuracy.
Key words: tapped power line model, power line carrier (PLC), neural networks, radial-basis networks.
1.Introduction
High voltage power line modeling at power-line carrier frequencies (up to 500 kHz) significantly differs from
the model used at the power frequency (50/60 Hz). Power line model that is used in power-line carrier
applications refers frequency dependant propagation function γ. In general, propagation function is a complex
variable that depends on frequency and can be represented as γ = α + jβ. α and β are amplitude and phase
frequency characteristics, respectively. Amplitude characteristic defines signal attenuation when propagating
through the channel while phase characteristics corresponds to the phase shift of spectra components in the
signal . Group delay is often used instead of phase characteristic since it directly provides traveling time of
spectra components [1].
Overhead power lines are multiconductor lines and their modeling is usually based on equations of telegraphy in
matrix form and modal analysis [2,3]. Modal analysis was developed for homogeneous power lines, however,
while power lines in practice appear nonhomogeneous. Therefore, modal analysis must be extended in the sense
to cover these inhomogeneities.
Power line taps represent distributed inhomogeneity where conductor’s height varies due to sag. Tapped power
lines are usually described with equations of telegraphy with variable coefficients [4]. Solution of these
equations in general requires complex mathematical tools. Solution for two-conductor case is available in [3,4].
An alternative approach presents the method of multiports that is used for solution of discrete inhomogeneities
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[5,6,7]. According to this method, line tap is divided into n segments (lumps) at different heights that can be
considered as a homogeneous and each lump can be represented as a multiport. Power line tap amplitude and
phase characteristics are then obtained from a cascade of n multiports.
This paper proposes a modification of the method of multiports that introduces a radial-basis function (RBF)
network.
Every multiport is characterized with its parameters. Since these multiports are in the cascade, A parameters are
the most appropriate parameters. Relation between A parameters and primitive power line parameters is
nonlinear [6,7] and can be efficiently modeled with neural networks.
Neural network is used to model one lump of the line tap in the way to estimate A parameter matrix of the lump
multiport. Such tap lump model based on the neural network is later used in calculation of power line tap
characteristics. Alternative to this approach is to use neural network for modeling power line tap amplitude and
phase characteristics for different power-line tap lengths.
If neural network is used to estimate A parameters of one line lump, training data scope is obtained through
computation of lump A parameters for different conductor heights and frequencies. Since responses of the
model are known for the scope of input vectors, we can use nonlinear input-output mapping property of the
neural networks [8]. Usually we use two types of neural networks for this purpose: multilayer perceptron (MLP)
and radial basis-function (RBF) network [8].
RBF network requires less training time but it has a large number of neurons in hidden layer due to local
approximation of nonlinear input-output mapping. MLP is characterized with distributed representation and less
neurons where neurons in the hidden layer contribute to the output for a given input.
We deal with larges scope of data and low-dimensional input space therefore we gave advantage to the RBF
network.
Theoretical considerations are implemented in MATLAB and simulation results are presented for the example
of 400 kV overhead power line. The analysis conducted in the paper is limited to the high voltage overhead
power lines while the simulations are completed in the frequency range from 50 to 550 kHz. Influence of
power-line poles, representing a discrete inhomogeneity, wasn’t taken into account.
The paper is organized in the following manner. Second section contains basic principles of homogeneous
power line analysis while third section gives a method of multiport usually used in modelling of power lines
with discrete inhomogeneities. RBF network application to power line tap modelling and modification of
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method of multiport in this sense are provided in this the fourth section. Simulation results are located at the
end.
2. Homogeneous high voltage power line model
2.1 Formulation of equation of telegraphy
The high voltage power line modeling at power line carrier frequencies is usually carried out through the modal
analysis [2,3]. The wave propagation phenomena along the power line are described by equations of telegraphy.
Since the power line is considered as a multiconductor line, the equations of telegraphy take the matrix form:
)x(dx
)x(ddxd ZJEZYUU
+−=−2
2 (1.1)
)x(dx
)x(ddxd YEJYZII
+−=−2
2 (1.2)
Elements of matrices Z and Y form the set of primary parameters of the power line parallel to the earth plane.
Those parameters are determined by geometrical design and electrical parameters of the conductors and earth
[6,7]. Influence of the earth with the finite conductivity is taken in the manner described in [9]. Any primary
parameter is frequency dependent. Matrices E(x) and I(x) represent voltage and current sources distributed
along the line. In the case of the passive network their values are equal to zero.
2.2 Solution of equations of telegraphy: Modal analysis
Equations (1.1) and (1.2) are written in original (phase) coordinates. The solution of these equations is found
through the modal transformation:
, I (2) (s)SUU = (s)QI=
which introduces modal coordinates where voltage and current are described with vectors U(s) and I(s). Matrix S
is chosen to be a matrix of eigenvectors of product Γ2=ZY while Q . Γ denotes propagation function
matrix in original coordinates. Applying transformation defined with expression (2), equations (1.1) and (1.2)
are becoming a set of ordinary nonhomogeneous differential equations. Solution of such differential equation
set defines the wave (natural) modes that are independent to each other and having their own propagation
functions and characteristic impedances. It means that those modes propagate independently to each other.
Thus, through this transformation all mutual couplings between original coordinates are discarded. Number of
conductors N defines the number of natural modes.
1)( −= TS
3
Every mode has it’s own propagation function. The eigenvalues of the matrix ZY define square of modal
propagation function . Propagation function 2γ γ is a diagonal matrix NxN whose elements are modal
propagation functions.
ab
c
ab
c
ab
c
Ground mode Interphasemode 1
Interphasemode 2
Figure 1. Natural modes of a three-phase power line
If we consider, for instance, three-phase power line without shield wires, number of natural modes is three. Each
mode has a physical meaning that is represented in Figure 1.
The highest attenuation characterizes the ground mode. Energy propagates through three phase conductors
(Figure 1) and returns via the ground. This mode is equivalent to the signal loss due to emission into the air.
Interphase modes 1 and 2 have a significantly lower attenuation since their energy propagates and returns
through phase conductors. Interphase mode 1 corresponds to the case when signal propagates and returns via
outer phases. Interphase mode 2 is mode when signal propagates through outer phases and returns via inner
phase.
Since high voltage power line is a multiconductor line, there are many solutions how to connect transmitter and
receiver to power line. Coupling scheme defines how transmitter and receiver are connected to the power line.
Different coupling schemes excite different modes. The most common coupling scheme are single conductor to
earth and two-outer phase coupling.
The previous analysis goes for homogenous power line with conductors parallel to each other and to the ground
plane.
Propagation function together with characteristic impedance of the line form secondary power line parameters.
The characteristic impedance matrix in modal coordinates is calculated as:
(3) ZQSγZ 11c(s)
−−=
Homogeneous line is often presented as a symmetric multiport. The A parameters of such multiport are defined
with:
(4)
=
2
2
2221
1211
1
1IU
AAAA
IU
4
where:
, (5.1) 111
−= SγSA lch 1c(s)12
−= QZγSA lsh
, (5.2) 11c(s)21
−−= SγQZA lsh 122
−= SγQA lch
Variable l denotes the line length. Index 1 denotes input variables while 2 output variables. Matrices in A
parameters (A11, A12, A21, A22) are NxN matrices where N denotes number of conductors.
2.3 Major geometrical and electrical parameters of overhead power line
Mikutski [7] defined major geometrical and electrical parameters influencing overhead power line attenuation:
• Distance between conductors. Increase of this distance results in higher attenuation of interphase modes
due to increase of ground losses.
• Conductor height. Attenuation of all modes decrease when conductor height increases.
• Conductor diameter. Interphase mode attenuation decreases when conductor diameter increase.
• Number of conductors in a bundled phase. The increased number in a bundled conductor decreases
attenuation due to decrease of conductor’s internal impedance while, in the other hand, increases
attenuation due to decrease of characteristic impedance. Thus, in channels, where attenuation is determined
with losses in conductors, increase of number of conductors results in decrease in attenuation. If channel
attenuation is determined with losses in the ground, increase of number of conductors in the bundled phase
brings an increase in attenuation.
• Ground resistance per unit length. Attenuation of ground channel linearly depends on the
ground resistance. Attenuation of interphase modes depends on ratio frequency to ground
resistance. Poor ground conductivity can increase line loss. It is especially significant if single-
phase-to-ground coupling scheme was used.
3. Tapped high voltage power line model
When any flexible conductor is supported at the ends, the conductor usually takes the shape of catenary, what
power line make nonhomogeneous (Figure 2). This section describes method of multiports usually used for
modeling of tapped power lines [3,6,7].
The secondary power line parameters are determined under the assumption that all line conductors are parallel
to each other and to the earth plane, ideally insulated from the earth. Power lines in the practice, however, differ
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from the idealistic model in that height of phase and shield wires above the earth changes between two poles.
Additionally, influence of chain insulators supporting wires and capacity between pole and wires should be
calculated.
tl
2/tltf
minh
z
x
2/tl
Figure 2. The catenary curve
The sag is described by the inverse hyperbolic cosine function, that is:
),12/(min −−
+=alxchahz t a ,)8/(2
tt fl= tlx ≤≤0 (6)
where:
z - height of the line above the earth;
tl - length of the tap;
tf - sag, that is the maximal distance between the line and line passing support points;
minh - minimal height.
Sag may be replaced with the stair step line (Figure 3) [6,7]. Each horizontal element is replaced with a
multiport while normal components of the line are neglected.
The sag is divided into 2p lumps, where symmetry between right and left side exists. Arbitrary i-th horizontal
lump may be presented as a multiport and described with A parameter equations. A parameters are then
calculated as for a long horizontal homogenous line with conductor height equal to the average conductor height
of that lump. Equivalent matrix Aeq of the line tap, considering symmetry, can be defined as:
(7)
==
= 2
22221
212
211
122
121
112
111
212221
1211
AAAA
AAAAAA
AAAA
Aeq
The every single element of power line tap (Figure 3) has length l and it’s own height. Matrix describes
equivalent nonsymmetrical multiport representing left half of the line tap and which consist of p homogeneous
elements. Since these elements are in cascade, matrix is found as a product of matrices of p elements:
t 1A
1A A
(8) 113
12
111 .... pAAAAA =
6
tl
1 2
p p+1
2p
Figure 3. Power line tap
The matrix describes the right half of the line tap, which is the mirror image of the left side. Due to
symmetry .
2A
2 = T1AA
The propagation function of one half of the line tap is found from the expression characterizing image
parameters of the nonsymetric multiport:
(9) T
lch 122
1111
2 )( AAΓ =
The diagonalization of the resulting matrix via eigenvalue analysis leads to the equation:
ch (10) 1111
2 )( −⋅⋅= SDSΓ l
where D is a diagonal matrix of eigenvalues and matrix of corresponding eigenvectors of the matrix
. Diagonal elements of matrix d define modal propagation functions of the first half of line tap, that
is:
1S
)l(ch 12 Γ
(11) 0)Re( ),( 12/1
1 >= lArchl γDγ
Surge impedance in modal coordinates is found to be:
Z (12) TTCM l )()tanh( 1
1112
111
111
−−= SAASγ
Matrix of modal attenuation coefficients per unit length of line tap is:
α (13) teq ll /)Re(2 1γ=
Imaginary part of propagation function defines the matrix of modal phase coefficients per unit length as follows:
β (14) teq ll /)Im(2 1γ=
If power line contains k taps, which are characterized by matrices , ieqA ki ,...2,1= , it is possible to define A
parameter matrix describing overall line, that is correlating voltages and currents at both sides a power line. For
long power lines, application of (9) to matrix A may lead to the lack of accuracy. Thus, it is recommended to
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calculate Y parameters for every line tap and Y parameters of their cascade. The matrix propagation function of
complete power line is then calculated from Y matrix.
4. Power line tap model based on the RBF network
Radial-basis function network is used to model one lump of the line tap in the way to estimate A parameter
matrix of the lump multiport. Such model based on the neural network is later used in calculation of power line
tap characteristics in the manner described in Section 3.
We settle two different models:
• Model of the general power line tap lump (for arbitrary power line);
• Model of the known line tap lump.
4.1 Radial-basis function network
A radial-basis function (RBF) network is a feedforward network with three layers (Figure 4):
• Input layer of source nodes;
• Hidden layer of nonlinear processing units (neurons);
• Output layer of linear weights.
RBF network in Figure 4 is presented as a network with one output in the sake of simplicity. In the general case,
number of outputs is arbitrary. Output y(t) for given input vector x(t), that is input-output mapping, is defined
as:
(15) ∑ +==
N
kkk wwy
10);( txϕ
where );( ktxϕ is a radial-basis function that compute the “distance” between input vector x and its own center
tk. Output of kth neuron is a nonlinear function of that distance. The weight wk is a complex number that
connects the kth hidden neuron with the network output. Term w0 is a constant that can be complex and is called
bias. N is the number of neurons in the hidden layer.
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.
.
.
.
.
.
1x
Mx
2xx
),( 1txϕ
),( 2txϕ
),( Ntxϕ
1w
2w
Nw
0w
y
Σ
Figure 4. RBF network structure
Behavior of neurons in the hidden layer is defined by radial-basis function. Gaussian radial-basis function is
defined as:
)1exp();( 22 kk
k txtx −−=σ
ϕ , k N,...2,1= (16)
where is the width and 2kσ ktx − is Euclidean distance between x and tk. Neurons in output layer have a linear
activation function.
The RBF network completes input-output mapping through two transformations. First, nonlinear
transformation from input space into the hidden-layer space and second, linear transformation from hidden-layer
space into the output space. The radial-basis functions define the character of nonlinear transformation while
linear transformation is defined by the weights w.
The learning procedure of RBF network comprises algorithms for selection of centers of radial-basis functions
and supervised algorithms for the weight computations [8]. The simplest procedure is to choose centers
randomly from the training set and to use method of least squares in computation of weights in the output layer
[10].
4.2 Proposed RBF network structure
Radial-basis function network is used to model one lump of the line tap in the way to estimate A parameter
matrix of the lump multiport. Such model based on the neural network is later used for calculation of power
line tap characteristics in the manner described in Section 3.
We define two different models:
• Model of a general power line tap lump (for arbitrary power line);
• Model with known line tap lump.
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All information that neural network receives about the process that models is incorporated in input vector. Thus,
input attributes to neural network must contain all information necessary for process modeling but with minimal
redundancy. When neural network is trained in the supervised manner, training scope contains input-output
vector pairs.
Input vector of the first model contains all power line parameters, listed in Section 2, needed for calculation of
matrices Z and Y. These matrices are essential for A parameter computations. The important issue is definition
of the parameter value ranges used in the training scope. The advantage of this approach is that neural network
models lump element of an arbitrary power-line tap. The disadvantage is a large training scope and time,
dimension of the network and selection of input parameter range.
If we model a tap lump of a known power line, input vector to the neural network reduces and contains only a
conductor’s height, characterizing sag, and frequency.
RBF NetworkReA
RBF NetworkImA
Σ
⊗j
Aparameters
ih
jf
Training andvalidation
...
...
...
.
.
.
Inputvector
Outputmatrix
Vector intomatrix
...
Figure 5. Two RBF networks in parallel used to estimate A parameters of line lump
In both cases, elements of the output vector correspond to the A parameters. Since A parameters contain
complex numbers, two neural networks are used in parallel. First, to estimate a real part and second, an
imaginary part of A matrix.
Model illustrated in Figure 5 estimates A parameters of line lump at height hi and frequency fj. Outputs of RBF
networks are vectors and it is necessary to rearrange them into a matrix to obtain A parameters. Neural network
parameters are adjusted through process of training and validation.
For the considered power line, training data are gathered via calculation of A parameters of the lump at
different heights and frequencies. The conductor’s height varies in the range defined by the support height and
maximal sag (Section 3). The height and frequency spans are chosen to reduce number of calculations but to
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provide good neural network generalization. Neural network lump model is then used in the calculation of
power line tap characteristic.
Since we deal with larges scope of data and low-dimensional input space, we gave advantage to the RBF
network. RBF network requires less training time but it has a large number of neurons in hidden layer due to
local approximation of nonlinear input-output mapping.
1f2fnf ...Desired frequency range
ph
kh
1h......
Range of heightsin catenary
......
......
A parametersestimator withRBF networks
kA
A parameters of lump at kh
peq AAAA ...211 =
Teqeqeq 11AAA =
)( ),( ff βαChoose next frequencyfrom the span
Figure 6. Algorithm for computation of power line tap frequency characteristics
5. Simulation results
MATLAB was chosen as a modeling and simulation environment, mainly due to the following:
• Matrix analysis is in the core of this problem, and MATLAB owns an excellent routine for eigenvalue
analysis which is crucial for modal analysis;
• Neural net toolbox, for design and training of RBF neural network;
• Visualization of obtained results.
We applied this approach to 400kV transmission line with three phase conductors and two shield wires, with
horizontal disposition of conductors (Figure 7). The line height is 20 m with maximal sag 5 m while tap is 400
m long.
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igure 7. 400 kV overhead power line pole with horizontal conductor disposition
Data about 400 kV power line is listed in Table 1 and Table 2.*
Table 1. Table 2.
Disposition of conductors Horizontal
1 2 3
45
15dx 25dx 24dx 34dx
12dx 23dx
321 hhh == 54 hh =
400 mm
F
Height of phase conductors [m] = 20 h1 = h2 = h3(average) h4 = h5 = 2(average)
[m]
dx(1-2) = 1dx(1-3) = 20 dx(1-4) = 16 dx(1-5) = 4 dx(2-3) = 10 dx(2-4) = 6 dx(2-5) = 6 dx(3-4) = 4 dx(3-5) = 16 dx(3-6) = 12 [1 or 2 or 3] – 3.7
Height of shield wires [m] 3.7
Horizontal distance between conductors dx 0
Vertical distance between conductors [m] [4 or 5]
*) From the project documentation with permission of power utility ELES (designed by IBE)
Since shield wires are grounded there are three natural modes. We incorporated their influence in calculation of
primitive matrices Z and Y. In the case that the power line is infinitely long, attenuation in dB per kilometer for
all three modes is calculated in MATLAB and presented in Figures 8, 9 and 10, while group delay of the
interphase mode 2 is presented in Figure 11.
Rated voltage [kV] 380 Length [km] 50 Number of phase conductors 3 Number of conductors in the bundle 2 Distance between conductors in bundle 0 [mm]
40
Phase conductor material Al Fe 490/65 mm2 Phase conductor diameter [mm] 30.6 Number of shield wires 2 Shield wire diameter [mm] 18 Shield wire material Al Mg FE 170/70
2mm 100 (approxim
Ground resistance per unit length [Ωm] ately)
12
50 100 150 200 250 300 350 400 450 500 5500.2
0.4
0.6
0.8
1
1.2
1.4
1.6Attenuation [dB/km]
Frequency [kHz] 50 100 150 200 250 300 350 400 450 500 550
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22Attenuation [dB/km]
Frequency [kHz]
Figure 8. Ground mode attenuation of Figure 9. Interphase mode 1 attenuation of infinitely long 400kV power line infinitely long 400kV power line
50 100 150 200 250 300 350 400 450 500 5500.005
0.01
0.015
0.02
0.025Attenuation [dB/km]
Frequency [kHz] 50 100 150 200 250 300 350 400 450 500 550
-5.855
-5.85
-5.845
-5.84
-5.835
-5.83x 10-5 Group delay [s/km]
Frequency [kHz]
Figure 10. Interphase mode 2 attenuation of Figure 11. Interphase mode 1 group delay infinitely long 400kV power line The attenuation of long homogeneous line parallel to the earth plane changes with the height of conductors. This
change is different for different modes. Influence of power line height above the earth plane to attenuation per
unit length of 400kV power line as a function of height and frequency is showed on the figures 12, 13 and 14.
Figure 15 represents the phase shift per unit length of interphase mode 2 versus height and frequency.
0
200
400
600
14
16
18
200
0.5
1
1.5
2
2.5
Frequency [kHz]
Attenuation [dB/km]
Height [m] 0
200
400
600
14
16
18
200
0.1
0.2
0.3
0.4
Frequency [kHz]
Attenuation [dB/km]
Height [m]
Figure 12. Attenuation of ground mode Figure 13. Attenuation of interphase mode 1
13
0
200
400
600
14
16
18
200
0.01
0.02
0.03
0.04
Frequency [kHz]
Attenuation [dB/km]
Height [m] 0
200
400
600
14
16
18
200
2
4
6
8
10
12
Frequency [kHz]
Phase shift [rad/km]
Height [m]
Figure 14. Attenuation of interphase mode 2 Figure 15. Phase shift of interphase mode 2
In the manner explained in the previous sections, we generated training data for the second type of RBF network
based model. The step in the height change is 5 m and frequency span is 10 kHz, while accepted tap lump
length is 5 m since sag is divided into an even number of lumps. One tap lump is considered as a homogeneous
line, with different heights from the selected range. Calculation of A parameters was completed according to
the Section 2.
Since RBF network is trained with supervised learning algorithms, training scope consists of input-output vector
pairs. Input vector contains height and frequency, while output vector correspond to A parameters of the lump.
Two RBF networks with identical input attributes are designed and trained with MATLAB Neural Net Toolbox.
First, for estimation of the real part of A parameters of the line tap lump and second, for imaginary part.
Trained neural network is then used in calculation of 400 m long line tap attenuation. Neural network estimated
A parameters of tap lumps with the heights not used in the training process.
Attenuation of one line tap for three natural modes obtained in simulations with the neural net and in the classic
manner, is presented on the figures 16, 17 and 18 (dotted line is characteristic obtained with NN model).
50 100 150 200 250 300 350 400 450 500 5500.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Attenuation [dB]
Frequency [kHz]50 100 150 200 250 300 350 400 450 500 550
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14Attenuation [dB]
Frequency [kHz]
Figure 16. Line tap attenuation – ground mode Figure 17. Line tap attenuation - interphase mode 1
14
50 100 150 200 250 300 350 400 450 500 5500
0.002
0.004
0.006
0.008
0.01
0.012
0.014Attenuation [dB]
Frequency [kHz]
Figure 18. Line tap attenuation - interphase mode 2
We can conclude that curve obtained with neural network model doesn’t significantly deviate from the curve
calculated in the classical manner. This deviation is more evident at higher frequencies since dependence of
attenuation of the height is more expressed at higher frequencies, what can be seen from figures 12, 13 and 14.
We conclude from the simulation results that attenuation of the line tap is not characterized with monotone
change. It shows rapid changes near the frequency corresponding to half of the wavelength that is a multiple of
line tap length. The frequency at which the monotone change is violated is called resonant frequency. The value
of the resonant frequency can be approximated with:
t
res lvkf
2)12( += (17)
where is a wave propagation velocity. v
Conclusion
From presented study of RBF network application to tapped power line modeling and simulation results we can
conclude the following:.
1. The tapped power line appears as a practice line with a systematic inhomogeneity. The known frequency
characteristics of these power lines (attenuation, phase shift and group delay) are required for design of
digital power line communication systems.
2. This paper described basics of tap power line modeling and introduced RBF network model of the lump,
which is later, used for secondary parameter calculation of the line tap. If the RBF network model of tap
lump is used, it is not necessary to use modal analysis for every line tap lump in calculation of its A
parameters. Neural network is characterized by generalization property and is able to estimate A parameters
of the lump for the lump heights that were not used in the training process.
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3. Simulation results show that lump model based on the RBF network can be used in the calculation of line
tap frequency characteristics with satisfying accuracy.
4. Attenuation dependence on conductor’s height is more articulated at higher frequencies. This is the reason
why deviation between attenuation characteristics obtained in two manners described above increases at
higher frequencies. Phase shift (and group delay as a consequence) dependence on conductor’s height can
be neglected.
Acknowledgements
This work was supported by the Ministry of the Economy, Republic Slovenia and Iskra Sistemi d.d. under the
project “Digital Power Line Carrier Communications”. Authors are also grateful to power utility ELES for
access to electric power facilities and all technical support.
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