Three-dimensional simulation and

localization of power cable fault point by

combining wavelet transform and neural

network

Yuning-Tao*

• College of Electrical Engineering and New Energy, China Three Gorges

University, Yichang, Hubei, 443000, China

•taoyuning12@163.com

Reception: 20 April 2024 | Acceptance: 6 May 2024 | Publication: 14 June 2024

Suggested citation:

Tao Yuning (2024). Three-dimensional simulation and localization of

power cable fault point by combining wavelet transform and neural

network. 3C Tecnología. Glosas de innovación aplicada a la pyme. 13 (1), 35-55.

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143

Ed.45 | Iss.13 | N.1 April - June 2024

35

ABSTRACT

This study centers on the 3D simulation localization technology of power cable fault

points, using the combination of wavelet transform and neural network, aiming to

improve the accuracy and efficiency of cable fault localization. This paper adopts the

method of combining wavelet transform and genetic algorithm optimization of back

propagation (GA-BP) neural network. First, by constructing a three-dimensional

simulation model of power cables, the wavelet transform is applied to extract fault

features and the GA-BP neural network is utilized for fault point localization. The

experimental results show that the average localization errors of this method in single-

phase ground fault and two-phase grounded short-circuit fault are 0.112km and

0.126km, respectively, which are significantly better than the traditional method. Under

different fault initial phase angle conditions, the proposed method shows strong

adaptability and the error is controlled within 1%. Meanwhile, the present algorithm

exhibits strong noise suppression ability under white and colored noise backgrounds,

especially in low signal-to-noise ratio environments. In summary, this study

demonstrates the effectiveness of the 3D simulation localization technique for power

cable fault points combining wavelet transform and GA-BP neural network in

improving the localization accuracy and noise resistance.

KEYWORDS

Power cable, three-dimensional simulation, wavelet transform, GA-BP neural network

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143

Ed.45 | Iss.13 | N.1 April - June 2024

36

ABSTRACT

This study centers on the 3D simulation localization technology of power cable fault

points, using the combination of wavelet transform and neural network, aiming to

improve the accuracy and efficiency of cable fault localization. This paper adopts the

method of combining wavelet transform and genetic algorithm optimization of back

propagation (GA-BP) neural network. First, by constructing a three-dimensional

simulation model of power cables, the wavelet transform is applied to extract fault

features and the GA-BP neural network is utilized for fault point localization. The

experimental results show that the average localization errors of this method in single-

phase ground fault and two-phase grounded short-circuit fault are 0.112km and

0.126km, respectively, which are significantly better than the traditional method. Under

different fault initial phase angle conditions, the proposed method shows strong

adaptability and the error is controlled within 1%. Meanwhile, the present algorithm

exhibits strong noise suppression ability under white and colored noise backgrounds,

especially in low signal-to-noise ratio environments. In summary, this study

demonstrates the effectiveness of the 3D simulation localization technique for power

cable fault points combining wavelet transform and GA-BP neural network in

improving the localization accuracy and noise resistance.

KEYWORDS

Power cable, three-dimensional simulation, wavelet transform, GA-BP neural network

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

INDEX

ABSTRACT .....................................................................................................................2

KEYWORDS ...................................................................................................................2

1. INTRODUCTION .......................................................................................................4

2. THREE-DIMENSIONAL SIMULATION OF POWER CABLE FAULT POINT

LOCALIZATION METHODS .....................................................................................5

2.1. Constructing three-dimensional simulation models of power cables .................5

2.2. Extraction of three-dimensional features of power cables .................................7

2.3. Cable fault point localization based on wavelet transform and GA-BP neural

network ...................................................................................................................8

2.3.1. Wavelet Transform and Mode Maxima Search Module .............................10

2.3.2. GA Optimization Module ............................................................................12

2.3.3. BP Neural Network Training and Prediction Module .................................13

3. EXPERIMENTAL ANALYSIS OF FAULT POINT LOCALIZATION ALGORITHMS

FOR POWER CABLES ..........................................................................................14

3.1. Simulation analysis under different fault initial phase angle conditions ...........14

3.2. Comparative error analysis between different algorithms ................................16

3.3. Error analysis in a noisy background ...............................................................18

4. CONCLUSION ........................................................................................................19

REFERENCES ..............................................................................................................20

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1. INTRODUCTION

China's urban power distribution network distribution form can be mainly divided

into cable lines and overhead lines, the former has higher requirements for the

installation environment and installation technology, but has the advantages of small

power loss and strong stability, so power cables are widely used in undersea tunnels,

transformer stations and other complex environments [1-3]. However, the installation

environment of power cables determines the difficulty index of its fault detection

increases linearly, compared with overhead lines, it is very difficult to diagnose and

repair power cable faults in complex hidden environments such as undersea,

underground, etc. [4-6].

With the rapid development of electric power science and technology, power cables

have gradually become the "main artery" of power transmission in major cities. The

overloaded operation of the power grid or long time use, will bring great burden to the

power cable, and then appear low resistance failure, high resistance failure and other

conditions. After the cable failure, rapid and accurate fault location, rapid restoration of

reliable power supply, not only effectively reduce the fault outage time, but also reduce

economic losses [7-8]. Due to the construction cost, cable channel conditions, power

supply, accuracy and reliability and other factors, cable online monitoring and fault

location technology has not yet been fully implemented, is still mainly offline testing for

cable fault location [9-10].

Power cable fault diagnosis and localization techniques have been improved to

reduce the outage time and maintenance cycle, as well as to enable fast power supply

to the faulty local grid, to detect the operation of transmission lines and to reduce the

economic losses.Li, G. et al. explored how to diagnose and locate the aging or

deterioration of power cables by using impedance spectroscopy, and pointed out that

the optimal frequency range of impedance spectrum analysis and a set of criteria for

assessing the condition of cables can be determined by the reference frequency and

the characteristic frequency, respectively. The reference frequency and characteristic

frequency can be used to determine the optimal frequency range for impedance

spectral analysis and a set of criteria for assessing the condition of cables, which is

easier to diagnose and locate the aging of cables than previous methods, and the

measurement process is simpler [11]. Li, M. et al. proposed a fault location method

based on traveling waves, and improved it with an autonomous learning algorithm.

Simulation experiments of the improved localization method on typical power cable

circuits were carried out using PSCAD, and the results confirmed that the improved

algorithm is fully capable of identifying short-circuit faults in power cable systems

[12].Sommervogel, L. pointed out several types of faults that may be encountered

during the detection of a cable's entire lifecycle and used a new methodology for

modeling to conduct the study by combining measurement and simulation to improve

the fault identification process [11]. measurement and simulation to improve the

accuracy of fault localization, and the conclusions of the study provide an important

reference for the practical application of cables [13].Tian, Y et al. conceptualized a

method to conductive frequency protection of underground cables in closed-loop

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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143

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38

1. INTRODUCTION

China's urban power distribution network distribution form can be mainly divided

into cable lines and overhead lines, the former has higher requirements for the

installation environment and installation technology, but has the advantages of small

power loss and strong stability, so power cables are widely used in undersea tunnels,

transformer stations and other complex environments [1-3]. However, the installation

environment of power cables determines the difficulty index of its fault detection

increases linearly, compared with overhead lines, it is very difficult to diagnose and

repair power cable faults in complex hidden environments such as undersea,

underground, etc. [4-6].

With the rapid development of electric power science and technology, power cables

have gradually become the "main artery" of power transmission in major cities. The

overloaded operation of the power grid or long time use, will bring great burden to the

power cable, and then appear low resistance failure, high resistance failure and other

conditions. After the cable failure, rapid and accurate fault location, rapid restoration of

reliable power supply, not only effectively reduce the fault outage time, but also reduce

economic losses [7-8]. Due to the construction cost, cable channel conditions, power

supply, accuracy and reliability and other factors, cable online monitoring and fault

location technology has not yet been fully implemented, is still mainly offline testing for

cable fault location [9-10].

Power cable fault diagnosis and localization techniques have been improved to

reduce the outage time and maintenance cycle, as well as to enable fast power supply

to the faulty local grid, to detect the operation of transmission lines and to reduce the

economic losses.Li, G. et al. explored how to diagnose and locate the aging or

deterioration of power cables by using impedance spectroscopy, and pointed out that

the optimal frequency range of impedance spectrum analysis and a set of criteria for

assessing the condition of cables can be determined by the reference frequency and

the characteristic frequency, respectively. The reference frequency and characteristic

frequency can be used to determine the optimal frequency range for impedance

spectral analysis and a set of criteria for assessing the condition of cables, which is

easier to diagnose and locate the aging of cables than previous methods, and the

measurement process is simpler [11]. Li, M. et al. proposed a fault location method

based on traveling waves, and improved it with an autonomous learning algorithm.

Simulation experiments of the improved localization method on typical power cable

circuits were carried out using PSCAD, and the results confirmed that the improved

algorithm is fully capable of identifying short-circuit faults in power cable systems

[12].Sommervogel, L. pointed out several types of faults that may be encountered

during the detection of a cable's entire lifecycle and used a new methodology for

modeling to conduct the study by combining measurement and simulation to improve

the fault identification process [11]. measurement and simulation to improve the

accuracy of fault localization, and the conclusions of the study provide an important

reference for the practical application of cables [13].Tian, Y et al. conceptualized a

method to conductive frequency protection of underground cables in closed-loop

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

distribution networks by using the current phase comparison method, which is

executed through the remote unit (rtu) in the DAS, and the PSCAD simulation model

is used, which verifies that the method can be used to locate and isolate the faults in

time, and the application scenarios. locate and isolate faults with a wide range of

application scenarios [14].Li, Z. et al. envisioned a deep-learning based cable infrared

image state assessment method to measure the temperature of power cables, and

based on the results of the statistical analysis, it was noted that this method can

provide an effective and accurate state identification of the infrared maps of the cables

[15].Siodla, K. et al. elaborated on the management of the medium-voltage cable

network and the fault rate analysis of the current state of the art to manage MV cable

networks by evaluating and diagnosing the operational status of individual cable lines

obtained, these studies and conclusions are important references for cable line

diagnostic studies [16].Samet, H. et al. conceptualized a method for the detection of

early faults and differentiated such faults from other kinds of similar faults in the power

system. The feasibility of the method was confirmed by simulation detection of four

different arc models and analysis of actual fault data [17].Li, C. et al. investigated for

the first time the effect of arc size on fire and flame propagation of 110 kV cross-linked

polyethylene cables under the setting of constant power in arc discharge. After analog

simulation testing, it was found that the larger the arc size, the more rapid the ignition

rate of the cable fire, and the flame propagation in both vertical and horizontal

azimuths increased significantly, which is of great significance for the prevention of

and education about cable fires [18].A new distance-protection algorithm was

proposed by Vinícius A. et al. to improve the protection of multi-terminal high-voltage

direct current (MTDC) systems, and the feasibility of this algorithm was demonstrated

by four-segment MTDC system detection and real-time simulation, which can provide

more program choices for multi-terminal DC protection [19].

In this paper, the cable discharge data are collected and processed to build a

model to simulate the actual fault scenario. Then, the wavelet transform technique is

used to extract fault features, and the weights and thresholds of the backpropagation

(BP) neural network are optimized by genetic algorithm (GA) to improve the training

efficiency and accuracy of the network. Ultimately, the optimized GA-BP neural

network is used to accurately locate the fault point. This process not only focuses on

improving the accuracy of localization, but also considers the robustness of the

algorithm in different noise environments.

2. THREE-DIMENSIONAL SIMULATION OF POWER

CABLE FAULT POINT LOCALIZATION METHODS

2.1. CONSTRUCTING THREE-DIMENSIONAL SIMULATION

MODELS OF POWER CABLES

The design of the three-dimensional simulation model of power cable needs to be

based on data, that is, the collection of three-dimensional data, the establishment of

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visualization information collection program, and the summary of the collected data to

build a database. Power cable discharge data acquisition process, inevitably contains

part of the noise information, resulting in poor authenticity of the three-dimensional

model, to which this paper adopts the use of the Internet of Things and ZigBee

networking real-time adjustment of the discharge signal data, to generate a three-

dimensional visualization model of the power cable. After the partition block

processing of the power cable, the fuzzy pixel value of the three-dimensional visual

information acquisition is expressed as:

（1）

Where denotes the pixel value, g

denotes the visual information sampling

result, h

denotes the set of association rules for 3D information sampling of power

cables, * denotes the convolution, and denotes the interference information.

Based on the above calculation results, fusing the collected data of each block, the

template matching method is applied to establish a statistical analysis model that can

be used for fault point localization. The vector of three-dimensional features of the

power cable model is represented as:

（2）

Where t denotes the 3D feature vector, n

denotes the dimension of the 3D

simulation model of power cables, s denotes the partition block, and

denotes the

partition block vector fusion value. Figure 1 shows the three-dimensional simulation

design architecture, according to which the purpose of human-computer interaction is

achieved in the process of cable fault point localization. Using the results of the above

vector value calculation, the three-dimensional simulation model of power cables is

constructed in the embedded control platform. In order to strengthen the authenticity

of the three-dimensional simulation model of power cables, it is necessary to enhance

the efficiency of the screen rendering, and the comprehensive consideration of a

variety of modeling software such as 3DStudio, Lightware3D, Multigen Creator and

other modeling software is analyzed to analyze the rendering efficiency of the

software modeling, and then the Multigen Creator software is selected to meet the

three-dimensional simulation design requirements. After analyzing the rendering

efficiency of the software modeling. The application of Multigen Creator software

combined with the three-dimensional feature vectors of the power cable model, the

output vector modeling results, to meet the virtual real-time and efficient requirements

of virtual three-dimensional model building.

g(x,y)=h(x,y)∗η(x,y)

(x,y)

η

t=As+ns

As

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visualization information collection program, and the summary of the collected data to

build a database. Power cable discharge data acquisition process, inevitably contains

part of the noise information, resulting in poor authenticity of the three-dimensional

model, to which this paper adopts the use of the Internet of Things and ZigBee

networking real-time adjustment of the discharge signal data, to generate a three-

dimensional visualization model of the power cable. After the partition block

processing of the power cable, the fuzzy pixel value of the three-dimensional visual

information acquisition is expressed as:

（1）

Where denotes the pixel value, g denotes the visual information sampling

result, h denotes the set of association rules for 3D information sampling of power

cables, * denotes the convolution, and denotes the interference information.

Based on the above calculation results, fusing the collected data of each block, the

template matching method is applied to establish a statistical analysis model that can

be used for fault point localization. The vector of three-dimensional features of the

power cable model is represented as:

（2）

Where t denotes the 3D feature vector, n denotes the dimension of the 3D

simulation model of power cables, s denotes the partition block, and denotes the

partition block vector fusion value. Figure 1 shows the three-dimensional simulation

design architecture, according to which the purpose of human-computer interaction is

achieved in the process of cable fault point localization. Using the results of the above

vector value calculation, the three-dimensional simulation model of power cables is

constructed in the embedded control platform. In order to strengthen the authenticity

of the three-dimensional simulation model of power cables, it is necessary to enhance

the efficiency of the screen rendering, and the comprehensive consideration of a

variety of modeling software such as 3DStudio, Lightware3D, Multigen Creator and

other modeling software is analyzed to analyze the rendering efficiency of the

software modeling, and then the Multigen Creator software is selected to meet the

three-dimensional simulation design requirements. After analyzing the rendering

efficiency of the software modeling. The application of Multigen Creator software

combined with the three-dimensional feature vectors of the power cable model, the

output vector modeling results, to meet the virtual real-time and efficient requirements

of virtual three-dimensional model building.

g(x,y)=h(x,y)∗η(x,y)

(x,y)

η

t=As+ns

As

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Figure 1 3D simulation design architecture

2.2. EXTRACTION OF THREE-DIMENSIONAL FEATURES OF

POWER CABLES

Aiming at the three-dimensional simulation model of power cables, the analysis

yields a three-dimensional feature distribution model for cable fault location:

（3）

Where K denotes the three-dimensional feature distribution information of power

cable, denotes the three-dimensional feature distribution model, denotes the

exponential function, e denotes the natural constant, E denotes the three-dimensional

information reconstruction error, and , ,denote the statistical

eigenvalues. The specific constraints are:

（4）

Where denotes the maximum feature screening amount. Combined with the

principal component analysis algorithm, the database centered on 3D visual

reconstruction data is constructed to ensure that the reconstruction results have a

small error. The high-precision reconstruction of the 3D simulation model is completed

by the above calculation results, and the feature information within the reconstructed

model is collected using the information interaction mode, and the feature component

output results are:

（5）

Where denotes the feature component of the information interaction output,

denotes the spatial region, denotes the modal value of the pixel point distribution

μ

(n) =

β1

[

1−exp(−α1en2)

]

, E[en2]≥K

β2[1−exp(−α2en2)], Other

μ

α1

α2,β1β2

{α

1

≥0, α

2

≥0

0≤β1≤1

λmax

, 0 ≤β2≤1

λmax

λmax

f=

n

∑

i=1

∂iK+

b

f

i

∂i

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within the spatial b

region, and denotes the dimensionality of the 3D visual

reconstruction. The blurred center of mass of the view image is determined using 3D

imaging technology, and the probability density function calculation formula for feature

extraction is derived through 3D visual simulation:

（6）

Where denotes the probability density function, denotes the skew angle of 3D

feature extraction, and denotes the feature point edge roundness function. Using

the continuous difference reconstruction algorithm, the power cable fault region is

made to exhibit smoothness, and the three-dimensional smooth region is denoted as:

（7）

Where P denotes the 3D smoothing region, V denotes the continuous difference

function, and (p,q) denotes the coordinates of the fault localization region after 3D

reconstruction of the power cable.

In order to ensure the accuracy of the three-dimensional feature extraction results,

the image is divided into multiple sub-pixel blocks, and the feature extraction

optimization formula is derived by superposition calculation:

（8）

Formula， denotes the modulus of the 3D eigenvolume at sub-pixel points.

2.3. CABLE FAULT POINT LOCALIZATION BASED ON

WAVELET TRANSFORM AND GA-BP NEURAL NETWORK

The proposed wavelet transforms and GA-BP based power cable fault location

model is shown in Fig. 2, which mainly contains wavelet transform and mode maxima

search module, GA optimization module, BP neural training and prediction module.

σ

(x,y∣μK)=∏θK

1

2exp −

(

x−μK

)2

2

σ

μ

θ

P

=

∑(

fp

)

+

∑

Vp,q

(

fp,fq

)

f=

n

∑

i=1

(∂i−¯

∂i)K+

b

¯

∂i

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within the spatial b region, and denotes the dimensionality of the 3D visual

reconstruction. The blurred center of mass of the view image is determined using 3D

imaging technology, and the probability density function calculation formula for feature

extraction is derived through 3D visual simulation:

（6）

Where denotes the probability density function, denotes the skew angle of 3D

feature extraction, and denotes the feature point edge roundness function. Using

the continuous difference reconstruction algorithm, the power cable fault region is

made to exhibit smoothness, and the three-dimensional smooth region is denoted as:

（7）

Where P denotes the 3D smoothing region, V denotes the continuous difference

function, and (p,q) denotes the coordinates of the fault localization region after 3D

reconstruction of the power cable.

In order to ensure the accuracy of the three-dimensional feature extraction results,

the image is divided into multiple sub-pixel blocks, and the feature extraction

optimization formula is derived by superposition calculation:

（8）

Formula， denotes the modulus of the 3D eigenvolume at sub-pixel points.

2.3. CABLE FAULT POINT LOCALIZATION BASED ON

WAVELET TRANSFORM AND GA-BP NEURAL NETWORK

The proposed wavelet transforms and GA-BP based power cable fault location

model is shown in Fig. 2, which mainly contains wavelet transform and mode maxima

search module, GA optimization module, BP neural training and prediction module.

σ(x,y∣μK)=∏θK

1

2exp −(x−μK)2

2

σ

μ

θ

P=∑(fp)+∑Vp,q(fp,fq)

f=

n

∑

i=1

(∂i−¯

∂i)K+b

¯

∂i

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

Figure 2 Fault location model of power cable based on wavelet transform and GA-BP

The flowchart of the wavelet transforms and GA-BP based power cable fault point

localization algorithm is shown in Fig. 3, and the detailed steps are as follows.

The fault traveling wave mode maximum value data

collected on both sides of the

cable is used as the feature value, and the set fault distance

is

used as the label value, which is normalized, and then the processed data is divided

into the training data and the test data

, which are inputted into the GA-BP

model.

Determine the topology of the BP neural network, the number of input layers

inputnum, the number of hidden layers hiddennum and the number of output layers

outputnum and initialize the BP neural network weights W1,W2 , threshold length B1,

B2 .

GA optimizes the BP neural network module, determines the chromosome length

and calculates the fitness by encoding the initial weight threshold of the BP neural

network in real numbers, and constantly updates the fitness by selecting the function,

crossover function and variation function.

M={(M11,M12),(M21,M22),⋯(Mn1,Mn2)}

X={X1,X2,⋯X3}

P1

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T1

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Determine whether the end condition is satisfied, the number of termination

iterations or a set threshold is reached. If it is not satisfied revert to the GA

optimization step, if it is satisfied, the optimal weights and thresholds obtained are

given to the BP neural network to complete the optimization purpose.

Retrain the BP neural network, calculate the error, and determine whether the end

conditions are satisfied, if not, update the weights and thresholds. If the set error

range is satisfied, simulation prediction is performed.

Figure 3 Power cable fault location flow chart

2.3.1. WAVELET TRANSFORM AND MODE MAXIMA SEARCH

MODULE

When a power cable fault occurs, the transient traveling wave containing fault

information will propagate in the cable, but because of its non-smooth and high

frequency characteristics lead to the traveling wave head is difficult to identify, so the

wavelet transform is used to analyze the good time-frequency domain characteristics

and noise canceling ability to analyze the transient traveling wave, and the mode

maxima theory is used to analyze the time of arrival of the initial head of faulty

traveling wave.

（1）wavelet transform

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Determine whether the end condition is satisfied, the number of termination

iterations or a set threshold is reached. If it is not satisfied revert to the GA

optimization step, if it is satisfied, the optimal weights and thresholds obtained are

given to the BP neural network to complete the optimization purpose.

Retrain the BP neural network, calculate the error, and determine whether the end

conditions are satisfied, if not, update the weights and thresholds. If the set error

range is satisfied, simulation prediction is performed.

Figure 3 Power cable fault location flow chart

2.3.1. WAVELET TRANSFORM AND MODE MAXIMA SEARCH

MODULE

When a power cable fault occurs, the transient traveling wave containing fault

information will propagate in the cable, but because of its non-smooth and high

frequency characteristics lead to the traveling wave head is difficult to identify, so the

wavelet transform is used to analyze the good time-frequency domain characteristics

and noise canceling ability to analyze the transient traveling wave, and the mode

maxima theory is used to analyze the time of arrival of the initial head of faulty

traveling wave.

（1）wavelet transform

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Assuming that the Fourier transform of satisfies the admissibility

condition, can be called a mother wavelet:

（9）

And for a continuous signal g(t) , its corresponding continuous wavelet transform

(CWT) is:

（10）

Where denotes the complex conjugate function of denotes the scale

factor of the frequency-dependent wavelet function. b denotes the time-dependent

displacement factor. However, in the process of practical application, it is necessary to

discretize the continuous wavelet, assuming that , , j, and

can be obtained as the discrete wavelet transform (DWT):

（11）

（2）Db6 Wavelet basis functions with mode maxima

Suppose there exists a simple step signal with the following functional expression:

（12）

This is an obvious signal with a mutation point, and according to the singularity

principle, a singularity point is generated at the point.

The waveforms are more concentrated at the d1 scale and the moment of the step

signal change can be clearly distinguished at 300s, so the modal maxima at the d1

scale will be collected as the characteristic vectors of the dataset. The method of

determining the mode maxima is shown below.

Under a certain decomposition scale a0, if there exists a point (a0, b0) such that

, then (a0, b0

) is called the local mode maxima. If

exists in any field of B0, then (a0, b0) is the

mode maxima point of the wavelet transform coefficients. Therefore, in this paper, we

use GA-BP neural network to train the mapping relationship between mode maxima

ψ(ω)

ψ(t)

ψ(t)

C

ψ=

∫R

|ψ(ω)|2

ω

dω<

∞

C

W T(a,b) =

1

a

∫R

g(t)ψ

(t−b

a

)

d

t

ψ(t)

ψ(t)⋅a

a=aj

0

b=kaj

0xb0

k∈Z

D

W T(j,k) = a−j/2

0

ψ

(

a−j

0

t−kb0

)

S

(t) =

{0 1≤t≤300

10 301 ≤t≤600

∂

(

Wψf

)(

a0,b0

)

∂b

=

0

(

Wψf

)

(a0,b)≤

(

Wψf

)

(a0,b0)

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and fault location by taking the mode maxima data under the acquisition scale as the

feature quantity and the distance to the fault is set as the label.

2.3.2. GA OPTIMIZATION MODULE

The use of traditional BP neural network for fault location in cables may cause the

algorithm to fall into the local optimum or converge slowly because of the random

initialization parameters of the network, in order to improve the convergence speed of

the BP algorithm and the ability of global optimization searching, therefore, the

introduction of GA to optimize the weights and thresholds of the BP neural network,

the specific steps are as follows.

(1) Real number coding. The initial weights and thresholds of the BP neural

network are encoded in real numbers with faster operation speed, utilizing the

following linear transformation:

（13）

The initial change interval for the interval of the j optimization variable

x(j) corresponds to the interval on the real number in the genetic

algorithm is expressed as genes, all the variables corresponding to the genes

sequentially linked together to form a coded form of the solution to the problem

is then called the individual or chromosome.

(2) Calculate the individual fitness. The initial weights and thresholds of the BP

neural network are obtained according to the individual, and the output of the system

is predicted after training the BP neural network with the training data, and the

absolute value of the error between the predicted output and the desired output of the

individual is taken as the value of the individual fitness F:

（14）

n is the number of output nodes of the network. is the desired output of the i th

node of the BP neural network and is the actual output of the i th node. K is the

coefficient.

(3) Selection operation. Genetic algorithm selection operation betting roulette

selection method, tournament selection method and geometric planning sorting

selection and other methods, when selecting the roulette method, that is, based on

the fitness ratio selection strategy, the selection probability Pi of each individual i is:

（15）

x(j)=p(j)+a(j)(q(j)−p(j)( j= 1, 2, ⋯,n)

[p(j), q(j)]

[0, 1]

a(j), a(j)

(a1,a2,⋯ak)

F

=K

(n

∑

i=1

abs(yi−oi)

)

Oi

yi

fi=

K

Fi

,pi=

f

i

∑N

j=1

fj

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46

and fault location by taking the mode maxima data under the acquisition scale as the

feature quantity and the distance to the fault is set as the label.

2.3.2. GA OPTIMIZATION MODULE

The use of traditional BP neural network for fault location in cables may cause the

algorithm to fall into the local optimum or converge slowly because of the random

initialization parameters of the network, in order to improve the convergence speed of

the BP algorithm and the ability of global optimization searching, therefore, the

introduction of GA to optimize the weights and thresholds of the BP neural network,

the specific steps are as follows.

(1) Real number coding. The initial weights and thresholds of the BP neural

network are encoded in real numbers with faster operation speed, utilizing the

following linear transformation:

（13）

The initial change interval for the interval of the j optimization variable

x(j) corresponds to the interval on the real number in the genetic

algorithm is expressed as genes, all the variables corresponding to the genes

sequentially linked together to form a coded form of the solution to the problem

is then called the individual or chromosome.

(2) Calculate the individual fitness. The initial weights and thresholds of the BP

neural network are obtained according to the individual, and the output of the system

is predicted after training the BP neural network with the training data, and the

absolute value of the error between the predicted output and the desired output of the

individual is taken as the value of the individual fitness F:

（14）

n is the number of output nodes of the network. is the desired output of the i th

node of the BP neural network and is the actual output of the i th node. K is the

coefficient.

(3) Selection operation. Genetic algorithm selection operation betting roulette

selection method, tournament selection method and geometric planning sorting

selection and other methods, when selecting the roulette method, that is, based on

the fitness ratio selection strategy, the selection probability Pi of each individual i is:

（15）

x(j)=p(j)+a(j)(q(j)−p(j)( j= 1, 2, ⋯,n)

[p(j), q(j)]

[0, 1]

a(j), a(j)

(a1,a2,⋯ak)

F=K(n

∑

i=1

abs(yi−oi))

Oi

yi

fi=K

Fi

,pi=fi

∑N

j=1 fj

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Fi is the fitness value of individual i. Since smaller fitness values are better, the

inverse of fitness is taken before individual selection. K is the coefficient. N is the

number of individuals in the population.

(4) Crossover operation. If the real number crossover method is used, the

crossover operation of the k th chromosome ak and the l th chromosome a1 in the j

dimension is as follows:

（16）

Where b is a random number between .

(5) Mutation operation. The j th gene aij of the i th individual was selected for

mutation:

（17）

（18）

amax is the upper bound of the gene aij. Amin is the lower bound of the gene. r2 is a

random number. g is the current iteration number. Gmax is the maximum number of

evolutions. r is a random number between .

(6) Calculate the individual fitness and determine whether the set end condition of

minimum error is satisfied, if not, the operation of steps 3 to 5 is repeated. If the end

condition is satisfied, the optimized weights and thresholds of the BP neural network

are given.

2.3.3. BP NEURAL NETWORK TRAINING AND PREDICTION

MODULE

The optimal weights and thresholds obtained through the GA optimization module

are given to the BP neural network, and then the BP neural network is re-trained and

predicted in the steps shown below.

(1) Select the transfer function and training function. Set the implicit layer transfer

function, the output layer transfer function as and select the training function.

(2) Set other parameters of the BP neural network. The number of iterations

epochs, learning rate learningrate, training target minimum error goal.

a

kj

=a

kj

(1 −b)+a

lj

b

alj =

a

lj(1 −

b

) +

a

kj

b

a

ij =

aij +

(

aij −amax

)

∗f(g) r>

0.5

aij +(amin −aij)∗f(g) r≤

0.5

f(g)=r2

(

1−

g

Gmax )2

[0, 1]

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47

(3) Start training the BP neural network, and determine whether to meet the end

conditions of setting the minimum error, if not, the weights and thresholds need to be

re-updated. If satisfied, simulation prediction is carried out.

3. EXPERIMENTAL ANALYSIS OF FAULT POINT

LOCALIZATION ALGORITHMS FOR POWER CABLES

3.1. SIMULATION ANALYSIS UNDER DIFFERENT FAULT

INITIAL PHASE ANGLE CONDITIONS

This paper builds a power cable line with a total length of 70km, and the simulation

model diagram of the power cable line is shown in Figure 4. Its voltage level is 110kV,

50Hz dual-source power supply system, single-phase ground fault occurs in phase A

at 0.016s, the transition resistance is 100Ω, and the total simulation time is 0.05s. The

total length of the hybrid line is 70km, of which the length of the overhead line is 50km

and the length of the cable line is 20km.

Figure 4 Schematic diagram of power cable line model

Faults may occur at any time in the power system, so the size of the initial phase

angle of the fault has randomness. In order to verify that under different fault initial

phase angle conditions, the power cable fault point localization algorithm in this paper

has strong adaptability. In this subsection, different initial phase angles of faults at the

same location are simulated and analyzed, and the values of initial phase angles of

faults are set to be 0°, 30°, 45°, and the fault occurs at the 6km overhead line section

as an example. Figure 5 shows the zero-mode component waveform curves of

different faults, where (a)~(c) are the zero-mode component waveforms of the

traveling wave of the faults on the M side of the bus, the cable junction point P and the

N side of the bus, respectively. Although the zero-mode components at the same point

may be slightly different for different initial phase angles of the fault, the general trend

is the same, and the overall error does not exceed 5%. Therefore, for the case of

short circuits with different initial phase angles of faults, the three-dimensional

simulation and localization method based on wavelet transform and GA-BP power

cable fault point proposed in this paper is still applicable to fault point localization.

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48

(3) Start training the BP neural network, and determine whether to meet the end

conditions of setting the minimum error, if not, the weights and thresholds need to be

re-updated. If satisfied, simulation prediction is carried out.

3. EXPERIMENTAL ANALYSIS OF FAULT POINT

LOCALIZATION ALGORITHMS FOR POWER CABLES

3.1. SIMULATION ANALYSIS UNDER DIFFERENT FAULT

INITIAL PHASE ANGLE CONDITIONS

This paper builds a power cable line with a total length of 70km, and the simulation

model diagram of the power cable line is shown in Figure 4. Its voltage level is 110kV,

50Hz dual-source power supply system, single-phase ground fault occurs in phase A

at 0.016s, the transition resistance is 100Ω, and the total simulation time is 0.05s. The

total length of the hybrid line is 70km, of which the length of the overhead line is 50km

and the length of the cable line is 20km.

Figure 4 Schematic diagram of power cable line model

Faults may occur at any time in the power system, so the size of the initial phase

angle of the fault has randomness. In order to verify that under different fault initial

phase angle conditions, the power cable fault point localization algorithm in this paper

has strong adaptability. In this subsection, different initial phase angles of faults at the

same location are simulated and analyzed, and the values of initial phase angles of

faults are set to be 0°, 30°, 45°, and the fault occurs at the 6km overhead line section

as an example. Figure 5 shows the zero-mode component waveform curves of

different faults, where (a)~(c) are the zero-mode component waveforms of the

traveling wave of the faults on the M side of the bus, the cable junction point P and the

N side of the bus, respectively. Although the zero-mode components at the same point

may be slightly different for different initial phase angles of the fault, the general trend

is the same, and the overall error does not exceed 5%. Therefore, for the case of

short circuits with different initial phase angles of faults, the three-dimensional

simulation and localization method based on wavelet transform and GA-BP power

cable fault point proposed in this paper is still applicable to fault point localization.

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

Figure 5. Waveform diagram of zero mode component of different faults

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49

Different fault location fault initial phase angle of 0 °, 30 °, 45 °, power cable fault

point localization simulation results shown in Table 1, in different fault initial phase

angle, the algorithm proposed in this paper can accurately determine the fault section,

and determine the results of the fault section with the occurrence of faults are

consistent with the positioning results of the relative error in the majority of the 1% or

less, to meet the demand for three-dimensional simulation of fault point localization

accuracy of power cables.

Table 1 Short-circuit fault location results for different initial phase angles

3.2. COMPARATIVE ERROR ANALYSIS BETWEEN DIFFERENT

ALGORITHMS

Comparing this paper's algorithm with EMD, Wavelet decomposition, SKT

algorithm, Table 2 shows the error analysis of four algorithms for fault localization, and

the data show that the error of the fault localization method proposed in this paper is

lower than that of the other two fault localization errors. In single-phase ground fault,

the average error of this paper's algorithm is 0.112km, and the average errors of EMD

decomposition, traditional wavelet, and synchronous squeezed SWT decomposition

are 0.32km, 0.282km, and 0.241km, respectively. in two-phase grounded short-circuit

fault, the average error of this paper's algorithm is 0.126km, the average error of EMD

Fault distance /km

Initial fault phase

Angle/°

Fault location/km Ranging error/%

6

0 5.961 0.65

30 6.047 0.78

45 5.957 0.72

11

0 11.102 0.93

30 11.105 0.95

45 11.096 0.87

34

0 34.021 0.06

30 34.024 0.07

45 33.948 0.15

54

0 54.014 0.03

30 54.021 0.04

45 53.984 0.03

63

0 63.038 0.06

30 62.896 0.17

45 63.026 0.04

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Different fault location fault initial phase angle of 0 °, 30 °, 45 °, power cable fault

point localization simulation results shown in Table 1, in different fault initial phase

angle, the algorithm proposed in this paper can accurately determine the fault section,

and determine the results of the fault section with the occurrence of faults are

consistent with the positioning results of the relative error in the majority of the 1% or

less, to meet the demand for three-dimensional simulation of fault point localization

accuracy of power cables.

Table 1 Short-circuit fault location results for different initial phase angles

3.2. COMPARATIVE ERROR ANALYSIS BETWEEN DIFFERENT

ALGORITHMS

Comparing this paper's algorithm with EMD, Wavelet decomposition, SKT

algorithm, Table 2 shows the error analysis of four algorithms for fault localization, and

the data show that the error of the fault localization method proposed in this paper is

lower than that of the other two fault localization errors. In single-phase ground fault,

the average error of this paper's algorithm is 0.112km, and the average errors of EMD

decomposition, traditional wavelet, and synchronous squeezed SWT decomposition

are 0.32km, 0.282km, and 0.241km, respectively. in two-phase grounded short-circuit

fault, the average error of this paper's algorithm is 0.126km, the average error of EMD

Fault distance /km

Initial fault phase

Angle/°

Fault location/km

Ranging error/%

6

0

5.961

0.65

30

6.047

0.78

45

5.957

0.72

11

0

11.102

0.93

30

11.105

0.95

45

11.096

0.87

34

0

34.021

0.06

30

34.024

0.07

45

33.948

0.15

54

0

54.014

0.03

30

54.021

0.04

45

53.984

0.03

63

0

63.038

0.06

30

62.896

0.17

45

63.026

0.04

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

decomposition, traditional wavelet, and synchronous squeezed SWT decomposition

are 0.304km, respectively. 0.304km, 0.272km, 0.227km respectively. the proposed

method based on wavelet transform and GA-BP power cable fault point 3D simulation

localization is better than the traditional wavelet decomposition method, EMD

decomposition method, and synchronous squeezing SWT algorithm.

Table 2 Fault location error analysis of the four algorithms

In order to further clarify the advantages and disadvantages of the method

proposed in this paper among different algorithms under different fault types, two-

phase short circuit and three-phase short circuit fault simulation are set up

respectively, and the fault point localization errors are analyzed as shown in Table 3,

and the fault localization errors of the method proposed in this paper, which is based

on the wavelet transform and the three-dimensional simulation localization of the fault

point of GA-BP power cables, are 0.109km and 0.12km, and are smaller than the fault

localization errors of EMD decomposition method, traditional wavelet decomposition,

and synchronous squeezed SWT algorithm. 0.109km and 0.12km, which are smaller

than the fault localization errors of EMD decomposition method, traditional wavelet

decomposition, and synchronous squeezing SWT algorithm, so the method proposed

in this paper is suitable for fault point localization of power cables with different fault

types.

A

Ground

connecti

on is

faulty

AB Two-

phase

ground

connecti

on fault

Fault

distance

/km

EMD

Wavelet

decomp

osition

SKT Textual

algorithm EMD

Wavelet

decompositi

on SKT Textual

algorithm

10 486 469 395 250 427 402 337 250

20 415 375 320 199 389 351 297 190

30 424 380 328 192 410 367 314 222

40 351 309 273 109 357 317 271 161

50 318 290 249 126 310 286 222 140

60 290 254 222 96 286 252 216 113

70 240 198 183 44 253 221 195 67

80 294 229 159 78 241 230 150 83

90 205 173 153 17 193 156 140 22

100 173 142 129 4 170 139 128 10

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Table 3 Fault location error analysis of four algorithms

3.3. ERROR ANALYSIS IN A NOISY BACKGROUND

Under two background noise environments, white noise and colored noise, the

signal-to-noise ratios are taken as 0dB, -10dB, -15dB, -20dB, and 300 Monte Carlo

experiments are carried out for the generalized correlation, quadratic correlation, and

this paper's algorithms, respectively, and the statistical results of the simulation of time

delay estimation are obtained as shown in Table 4. The results use the time delay

estimation mean and root mean square error ( is the estimated value and a

is the

actual time delay value, which is 300) as the performance evaluation criteria. The "-"

horizontal line indicates that the value is no longer of reference value because the

estimation error is too large. Analyzing the data in the table, it can be seen that at not

too low signal-to-noise ratios, all three types of algorithms can give accurate

estimation results regardless of whether the background noise is white or colored

noise. However, in the case of low signal-to-noise ratio, when the white noise signal-

to-noise ratio is -20dB, only the error of this algorithm is still within the controllable

range, at this time, the mean square error is 0.057, while the white noise signal-to-

noise ratio is -15dB, the generalized correlation and the quadratic correlation error is

no longer a reference value, at this time, the mean square error of this algorithm is

0.076. The algorithm of this paper has a certain inhibition effect on the noise, and its

effect is more desirable than that of the other two methods. The effect is more ideal

than the other two methods. Under the same signal-to-noise ratio, for the three delay

AB

Two-

phase

short-

circuit

fault

ABC

Three-

phase

short

circuit

fault

Fault

distanc

e /km

EMD

Wavelet

decomp

osition

SKT

Textual

algorith

m

EMD

Wavelet

decomp

osition

SKT

Textual

algorith

m

10 455 444 370 238 408 381 320 239

20 396 350 305 190 373 330 285 214

30 405 359 308 180 394 352 298 213

40 336 297 288 124 342 302 260 155

50 293 280 238 122 294 279 213 134

60 284 252 222 102 284 254 217 113

70 222 194 179 46 241 214 183 68

80 223 218 168 56 227 220 148 30

90 194 168 146 21 182 152 138 20

100 168 136 122 8 162 134 123 11

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Table 3 Fault location error analysis of four algorithms

3.3. ERROR ANALYSIS IN A NOISY BACKGROUND

Under two background noise environments, white noise and colored noise, the

signal-to-noise ratios are taken as 0dB, -10dB, -15dB, -20dB, and 300 Monte Carlo

experiments are carried out for the generalized correlation, quadratic correlation, and

this paper's algorithms, respectively, and the statistical results of the simulation of time

delay estimation are obtained as shown in Table 4. The results use the time delay

estimation mean and root mean square error ( is the estimated value and a is the

actual time delay value, which is 300) as the performance evaluation criteria. The "-"

horizontal line indicates that the value is no longer of reference value because the

estimation error is too large. Analyzing the data in the table, it can be seen that at not

too low signal-to-noise ratios, all three types of algorithms can give accurate

estimation results regardless of whether the background noise is white or colored

noise. However, in the case of low signal-to-noise ratio, when the white noise signal-

to-noise ratio is -20dB, only the error of this algorithm is still within the controllable

range, at this time, the mean square error is 0.057, while the white noise signal-to-

noise ratio is -15dB, the generalized correlation and the quadratic correlation error is

no longer a reference value, at this time, the mean square error of this algorithm is

0.076. The algorithm of this paper has a certain inhibition effect on the noise, and its

effect is more desirable than that of the other two methods. The effect is more ideal

than the other two methods. Under the same signal-to-noise ratio, for the three delay

AB

Two-

phase

short-

circuit

fault

ABC

Three-

phase

short

circuit

fault

Fault

distanc

e /km

EMD

Wavelet

decomp

osition

SKT

Textual

algorith

m

EMD

Wavelet

decomp

osition

SKT

Textual

algorith

m

10

455

444

370

238

408

381

320

239

20

396

350

305

190

373

330

285

214

30

405

359

308

180

394

352

298

213

40

336

297

288

124

342

302

260

155

50

293

280

238

122

294

279

213

134

60

284

252

222

102

284

254

217

113

70

222

194

179

46

241

214

183

68

80

223

218

168

56

227

220

148

30

90

194

168

146

21

182

152

138

20

100

168

136

122

8

162

134

123

11

https://doi.org/10.17993/3ctecno.2024.v13n1e45.35-55

estimation algorithms, the estimation performance in the case of white Gaussian

background noise is significantly better than that in the case of colored Gaussian

background noise.

Table 4 Simulation statistical results of delay estimation

4. CONCLUSION

In this study, a new 3D simulation localization method for power cable fault points is

developed by combining wavelet transform and neural network techniques.

Experimental results show that the method shows high accuracy and stability in

different types of fault localization. Specifically, in the case of single-phase ground

fault and two-phase grounded short-circuit fault, the average localization errors of the

proposed method are 0.112 km and 0.126 km, respectively, which are much lower

than those of the traditional EMD decomposition, traditional wavelet, and synchronous

squeezed SWT decomposition methods. In addition, the proposed method shows

good adaptability under different fault initial phase angles, and the error is controlled

within 1%, which proves its effectiveness under different fault scenarios.

The method also shows excellent performance in noisy environments. Especially in

the case of low signal-to-noise ratio, the method proposed in this paper shows better

noise immunity and maintains lower localization error compared with traditional

methods. This feature makes the method more valuable in practical power system

operation and maintenance, especially in complex noise environments.

The proposed three-dimensional simulation localization technique for power cable

faults based on wavelet transform and GA-BP neural network not only achieves

significant results in improving the localization accuracy, but also significantly

SNR

White noise

Colored noise

Generalized

correlation

Secondary

correlation

Textual

algorithm

Generalized

correlation

Secondary

correlation

Textual

algorithm

0dB

Mean 300.04 299.96 300.02 299.95 300.05 300.02

Mean

square

error

13 11 12 34 49 25

-10dB

Mean 299.89 300.07 299.96 299.87 300.116 300.05

Mean

square

error

42 42 22 214 198 53

-15dB

Mean 301.394 298.83 300.28 — — 299.18

Mean

square

error

85 44 34 — — 76

-20dB

Mean — — 298.18 — — 298.14

Mean

square

error

— — 57 — — 122

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53

improves the ability of anti-noise interference. The development of this technology will

bring important practical application value to the fault detection and maintenance of

the power system, and help to improve the reliability and stability of the power system.

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(12) Li, M., Bu, J., Song, Y., Pu, Z., & Xie, C. (2021). A novel fault location method for power

cables based on an unsupervised learning algorithm. Energies, 14(4), 1164.

(13) Sommervogel, L. (2020). Various models for faults in transmission lines and their

detection using time domain reflectometry. Progress in Electromagnetics Research C,

103, 123-135.

(14) Tian, Y., Zhao, Q., Zhang, Z., Li, L., & Crossley, P. (2018). Current-phase-comparison-

based pilot protection for normally closed-loop distribution network with underground

cable. International Transactions on Electrical Energy Systems, 28(9), e2733.1-

e2733.19.

(15) Li, Z., Yang, H., Yang, F., Tan, T., Lu, X., & Tian, J. (2022). An infrared image based state

evaluation method for cable incipient faults. Electric Power Systems Research.

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improves the ability of anti-noise interference. The development of this technology will

bring important practical application value to the fault detection and maintenance of

the power system, and help to improve the reliability and stability of the power system.

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