INTELLIGENT CABLE FAULT DETECTION

TECHNOLOGY CONSIDERING CURRENT

HARMONIC CHARACTERISTICS

Yuning-Tao*

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

University, Yichang, Hubei, 443000, China

•taoyuning12@163.com

Reception: 8 April 2024 | Acceptance: 6 May 2024 | Publication: 8 June 2024

Suggested citation:

Yuning-Tao. (2024). Intelligent cable fault detection technology

considering current harmonic characteristics. 3C Tecnología. Glosas de

Innovación aplicadas a la pyme. 13(1), 99-128. https://doi.org/

10.17993/3ctecno.2024.v13n1e45.99-128

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

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

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

99

ABSTRACT

With the rapid development of the power system, cable faults have become an

important factor affecting the stable operation of the power system. In this paper, for

the problem of cable faults, we improve the overshoot, undershoot phenomenon and

sieve speed of the envelope fitting in the Hilbert-Huang transform algorithm, and

extract the harmonic characteristics of the current of cable faults by using the

improved HHT model. Then, we utilize the information entropy and wavelet singular

entropy algorithm to integrate semi-parametric support vector machine algorithm, S-

SVM, and construct the wavelet singular entropy and S-SVM model. The information

entropy and wavelet singular entropy algorithms are fused with semiparametric

support vector machine algorithm, S-SVM, and constructed into wavelet singular

entropy and S-SVM models, which are applied to the cable fault identification

experiments for detecting different faults in cables. The experimental results show

that, when the cables are short-circuited, the currents of different short-circuited

cables are all lower than the normal currents, and the wavelet singular entropy and S-

SVM models reach more than 92% of the accuracy of the identification of the

degradation of the cable line and short-circuited faults. The accuracy of the wavelet

singular entropy and S-SVM model for the identification of cable line deterioration and

short circuit faults reaches more than 92%, and the overall cable fault detection

reaches 98.04%.The maximum error value of the wavelet singular entropy and S-

SVM model for detection is 0.5329, and there are only two groups of data more than

0.5.The algorithms in this paper are able to detect the various localization of the cable

faults quickly and accurately, and they have a high practical value.

KEYWORDS

HHT algorithm; Wavelet singular entropy; S-SVM model; Current harmonic

characteristics; Fault detection

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

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

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

100

ABSTRACT

With the rapid development of the power system, cable faults have become an

important factor affecting the stable operation of the power system. In this paper, for

the problem of cable faults, we improve the overshoot, undershoot phenomenon and

sieve speed of the envelope fitting in the Hilbert-Huang transform algorithm, and

extract the harmonic characteristics of the current of cable faults by using the

improved HHT model. Then, we utilize the information entropy and wavelet singular

entropy algorithm to integrate semi-parametric support vector machine algorithm, S-

SVM, and construct the wavelet singular entropy and S-SVM model. The information

entropy and wavelet singular entropy algorithms are fused with semiparametric

support vector machine algorithm, S-SVM, and constructed into wavelet singular

entropy and S-SVM models, which are applied to the cable fault identification

experiments for detecting different faults in cables. The experimental results show

that, when the cables are short-circuited, the currents of different short-circuited

cables are all lower than the normal currents, and the wavelet singular entropy and S-

SVM models reach more than 92% of the accuracy of the identification of the

degradation of the cable line and short-circuited faults. The accuracy of the wavelet

singular entropy and S-SVM model for the identification of cable line deterioration and

short circuit faults reaches more than 92%, and the overall cable fault detection

reaches 98.04%.The maximum error value of the wavelet singular entropy and S-

SVM model for detection is 0.5329, and there are only two groups of data more than

0.5.The algorithms in this paper are able to detect the various localization of the cable

faults quickly and accurately, and they have a high practical value.

KEYWORDS

HHT algorithm; Wavelet singular entropy; S-SVM model; Current harmonic

characteristics; Fault detection

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

INDEX

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

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

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

2. CONSTRUCTION OF INTELLIGENT CABLE FAULT DETECTION MODEL .........7

2.1. Construction of Improved Hilbert-Huang Transform Models ..............................7

2.1.1. Hilbert-Huang transform algorithm ..............................................................7

2.1.2. Improvement of Envelope Fitting Algorithm ...............................................10

2.1.3. Improved HHT algorithm flow ....................................................................12

2.2. Construction of wavelet-based singular entropy sum and S-SVM network models

14

2.2.1. Semiparametric Support Vector Machines ................................................14

2.2.2. Semi-parametric support vector machines based on least squares ..........15

2.2.3. Sparse greedy matrix approximation .........................................................15

2.2.4. Iterative reweighted least squares .............................................................17

2.2.5. Wavelet Transform ....................................................................................18

2.2.6. Shannon information entropy and wavelet singular entropy .....................19

2.2.7. Cable fault detection strategy based on wavelet singular entropy and S-SVM

models ................................................................................................................20

3. ANALYSIS OF INTELLIGENT CABLE FAULT DETECTION RESULTS

CONSIDERING CURRENT HARMONIC FEATURES ...........................................21

3.1. Analysis of intelligent detection results of cable line deterioration faults ..........21

3.1.1. Harmonic feature extraction for cable line degradation .............................21

3.1.2. Analysis of cable line deterioration identification test results ....................22

3.2. Analysis of intelligent detection results of cable short-circuit faults ..................24

3.2.1. Cable short circuit fault analysis ................................................................24

3.2.2. Harmonic feature extraction for cable short-circuit faults ..........................26

3.2.3. Identification and detection results of cable short-circuit faults .................27

3.3. Analysis of intelligent detection results of cable faults .....................................28

4. CONCLUSION ........................................................................................................28

REFERENCES ..............................................................................................................29

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1. INTRODUCTION

Power resources have become one of the most important energy sources in China,

the rapid increase in the consumption of electric power resources in small, medium

and large cities, in order to meet the needs of the masses of electricity, widely used

power cables as a transmission tool and connecting lines [1-2]. the current

development of China's electric power industry projects continue to increase, in order

not to take up too much land resources, cables are usually generally buried in the

ground, which increases the difficulty of troubleshooting power cable faults to a certain

extent. If the maintenance work is not timely, then it is easy to increase the probability

of power outages, bringing difficulties to the people's lives, directly affecting people's

production and life [3-4]. combined with the current social development trend in China,

the most critical type of power system failure is cable failure, to ensure the stability

and safe operation of China's power system, it is necessary to carry out power cable

failure at the first time, the power cable failure is the most critical type of power cable

failure. To ensure the stable and safe operation of China's power system, it is

necessary to carry out power cable fault inspection and testing at the first time,

accurately put forward the cable inspection method, and effectively put forward the

measures to solve the fault, repair the power cable faults, so as to promote the

stability and safety of the power project [5-6].

In order to reduce the use of land resources, to bring higher economic benefits, so

the cable is usually buried in the ground, but this to a certain extent also brings certain

problems, due to the cable buried deep underground, if a fault occurs, it is difficult to

investigate, which increases the difficulty of investigation [7-8]. In the detection of

cable faults, the principle that must be adhered to is to be green, while maximizing the

economic benefits, not only to require the practicality of strong, but also to ensure that

the scientific nature of the use of advanced technology to fully reduce losses, and

constantly improve the efficiency of the power grid in the operation of the power

industry to a certain extent will promote the progress and development of China's

electric power industry, and play a huge significance of the promotion of the [9-10].

With the continuous development of the market economy, people's living standards

continue to improve, the structure of the urban power grid is more complex, the

number of cables in use is increasing day by day. the safety of the cable operation,

directly on the power system to bring a direct impact on the cable management and

maintenance of the power sector has become the focus of the attention of the electric

power sector. Baranowski, J. and other researchers proposed a new method to detect

different signals based on the depth distribution of the Bayesian function to diagnose

cable faults, the results of the study confirm that the method has great potential for

diagnosis in unknown situations [11]. Liu, X. and other researchers designed a short

cable line fault location method based on the theory of transmission line and the

circuit theorem, and the test was carried out on 50 coaxial cables, and the test verified

that the accuracy of this method is not affected by the impedance of faults and the

terminating impedance [12]. Xuebin, Q. and other researchers proposed an on-line

cable fault diagnosis method to address the need for on-line diagnosis of cable faults,

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

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

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1. INTRODUCTION

Power resources have become one of the most important energy sources in China,

the rapid increase in the consumption of electric power resources in small, medium

and large cities, in order to meet the needs of the masses of electricity, widely used

power cables as a transmission tool and connecting lines [1-2]. the current

development of China's electric power industry projects continue to increase, in order

not to take up too much land resources, cables are usually generally buried in the

ground, which increases the difficulty of troubleshooting power cable faults to a certain

extent. If the maintenance work is not timely, then it is easy to increase the probability

of power outages, bringing difficulties to the people's lives, directly affecting people's

production and life [3-4]. combined with the current social development trend in China,

the most critical type of power system failure is cable failure, to ensure the stability

and safe operation of China's power system, it is necessary to carry out power cable

failure at the first time, the power cable failure is the most critical type of power cable

failure. To ensure the stable and safe operation of China's power system, it is

necessary to carry out power cable fault inspection and testing at the first time,

accurately put forward the cable inspection method, and effectively put forward the

measures to solve the fault, repair the power cable faults, so as to promote the

stability and safety of the power project [5-6].

In order to reduce the use of land resources, to bring higher economic benefits, so

the cable is usually buried in the ground, but this to a certain extent also brings certain

problems, due to the cable buried deep underground, if a fault occurs, it is difficult to

investigate, which increases the difficulty of investigation [7-8]. In the detection of

cable faults, the principle that must be adhered to is to be green, while maximizing the

economic benefits, not only to require the practicality of strong, but also to ensure that

the scientific nature of the use of advanced technology to fully reduce losses, and

constantly improve the efficiency of the power grid in the operation of the power

industry to a certain extent will promote the progress and development of China's

electric power industry, and play a huge significance of the promotion of the [9-10].

With the continuous development of the market economy, people's living standards

continue to improve, the structure of the urban power grid is more complex, the

number of cables in use is increasing day by day. the safety of the cable operation,

directly on the power system to bring a direct impact on the cable management and

maintenance of the power sector has become the focus of the attention of the electric

power sector. Baranowski, J. and other researchers proposed a new method to detect

different signals based on the depth distribution of the Bayesian function to diagnose

cable faults, the results of the study confirm that the method has great potential for

diagnosis in unknown situations [11]. Liu, X. and other researchers designed a short

cable line fault location method based on the theory of transmission line and the

circuit theorem, and the test was carried out on 50 coaxial cables, and the test verified

that the accuracy of this method is not affected by the impedance of faults and the

terminating impedance [12]. Xuebin, Q. and other researchers proposed an on-line

cable fault diagnosis method to address the need for on-line diagnosis of cable faults,

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

which is more advantageous than the traditional shallow neural network-based cable

fault identification method [13]. Cataldo, A. C. G. described the time-domain

reflectance (TDR) based localization method along the permittivity variations (DPVs)

of cable systems, and pointed out that due to the increase in the number of DPVs, the

fault location is not as accurate as the fault impedance and termination impedance of

the cable system. It was pointed out that increasing the pulse width to study the

localization of cables over longer distances leads to focusing on different TDR

reflectance maps each time, which is very time-consuming and does not guarantee

optimal performance [14].Wang, F. and other researchers envisioned a fault

diagnostic method based on the BP-Adaboost algorithm to solve the problem of faults

on aeronautical cables, and the results of the algorithm were studied by using the

Matlab software for analysis and example feedback to validate the results. Matlab

software is used to analyze the algorithm results and feedback examples to verify the

feasibility of the proposed fault diagnosis method [15]. Sian, H. W. and other

researchers designed a hybrid diagnostic algorithm based on the Discrete Wavelet

Transform (DWT) and Symmetric Dot Plot (SDP) analysis of Convolutional

Probabilistic Neural Networks (CPNN) to solve the insulation faults in XLPE cables.

Simulation tests show that the accuracy of this method is more than 96%, and the

accuracy of this method is higher than 96%. Simulation tests have shown that the

method can diagnose power cable faults with an accuracy of more than 96% and a

short detection time, which makes it fully capable of detecting insulation faults in

cables [16]. Marriott, N. discusses the role and advantages of Megger's new SMART

THUMP ST25-30 Portable Cable Tester, which provides an automated test sequence

with the ability to identify, prelocate, and pinpoint cable faults, and to automatically

supplement the interpretation of the test results. Non-specialized users can obtain

reliable results in a safe and easy way [17]. Lowczowski, K. and other researchers

analyzed the application of cable shielding currents in the identification and location of

ground faults, and in this way gave phase ground fault currents in different power

system configurations, in order to test the ability to detect faults based on the above

principles, and proved the reliability of this method for fault detection. Finally, a

solution to improve the localization capability is proposed, and the feasibility of the

improved solution is confirmed by simulation tests with PSCAD software [18]. Hu, C.

and other researchers investigate the ship cable fault information acquisition model

based on the automatic identification technology, and simulation tests are conducted

to confirm the feasibility of the Hilbert-Huang Transform (HHT) to automatically identify

cable faults, and to construct an automatic cable fault information acquisition model

[19]. Lai, Q. and other researchers studied the 110kv transmission line cable terminal

tail pipe breakdown fault, and give the cable installation improvement measures to

reduce the probability of cable terminal breakdown faults, and after simulation and

disassembly test, the feasibility analysis, for similar cable faults, is carried out. The

feasibility analysis is carried out to provide important reference opinions for the

investigation and solution of similar cable faults [20]. Liu, N. and other researchers

conceived a deep neural network-based cable fault signal classification and

identification method to accurately identify the early cable fault problems, and the

reliability of the method is demonstrated through experiments [21]. Wang, Y. and other

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researchers designed a method based on the constrained Boltzmann machine

( Wang, Y. and other researchers designed a cable early fault identification method

based on Restricted Boltzmann Machine (RBM) and Stacked Auto-Encoder (SAE),

which demonstrated higher accuracy after simulation tests comparing with traditional

methods such as convolutional neural networks [22]. A new algorithm for accurate

location of early faults in underground cables using double-ended synchronized zero

sequence waveforms was proposed by Qu, K. et al. A large number of simulation

experiments were conducted at PSCAD/EMTDC to support the accuracy and

reliability of the algorithm [23]. Kwon, G. Y. et al. discussed an improved fault

localization technique, instantaneous frequency-domain reflectance and tangent-

distance pattern recognition, for fault diagnosis and protection of cables, and

simulation data verified that this method can improve the reliability of high-voltage DC

power systems. For fault diagnosis and protection of cables, and simulated test data

verified that this method can improve the reliability of HVDC power systems [24].

In this paper, the Hilbert transform is applied to the IMF components to obtain the

Hilbert-Huang transform model, and then the local extreme points are densified by

using the cut-contact mean points, and then the interpolated curve segments are

spliced by the segmented power function to improve the insufficiency of the envelope

fitting in the HHT transform model, and then the current harmonic characteristics of

the cable faults are extracted and analyzed, and the wavelet singularity is obtained by

combining the wavelet transform with the information entropy. The wavelet singular

entropy is obtained by combining the information entropy and wavelet transform,

which is integrated into the S-SVM algorithm, and the sparse greedy matrix

approximation is used to select the basic element set, and the iterative reweighted

least squares algorithm is introduced to iterate and optimize the S-SVM model, which

finally constitutes the wavelet singular entropy and the S-SVM model. the signal is

decomposed by the wavelet singular entropy and the signal components are

extracted, and the extracted current harmonic features are screened and

reconstructed, and the current harmonic features are extracted and analyzed. After

that, the extracted current harmonic feature vectors are filtered and reconstructed,

and then the cable fault samples are input to the S-SVM model for training, and finally

the wavelet singular entropy and S-SVM models are realized to accurately identify

different faults of cables.

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

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researchers designed a method based on the constrained Boltzmann machine

( Wang, Y. and other researchers designed a cable early fault identification method

based on Restricted Boltzmann Machine (RBM) and Stacked Auto-Encoder (SAE),

which demonstrated higher accuracy after simulation tests comparing with traditional

methods such as convolutional neural networks [22]. A new algorithm for accurate

location of early faults in underground cables using double-ended synchronized zero

sequence waveforms was proposed by Qu, K. et al. A large number of simulation

experiments were conducted at PSCAD/EMTDC to support the accuracy and

reliability of the algorithm [23]. Kwon, G. Y. et al. discussed an improved fault

localization technique, instantaneous frequency-domain reflectance and tangent-

distance pattern recognition, for fault diagnosis and protection of cables, and

simulation data verified that this method can improve the reliability of high-voltage DC

power systems. For fault diagnosis and protection of cables, and simulated test data

verified that this method can improve the reliability of HVDC power systems [24].

In this paper, the Hilbert transform is applied to the IMF components to obtain the

Hilbert-Huang transform model, and then the local extreme points are densified by

using the cut-contact mean points, and then the interpolated curve segments are

spliced by the segmented power function to improve the insufficiency of the envelope

fitting in the HHT transform model, and then the current harmonic characteristics of

the cable faults are extracted and analyzed, and the wavelet singularity is obtained by

combining the wavelet transform with the information entropy. The wavelet singular

entropy is obtained by combining the information entropy and wavelet transform,

which is integrated into the S-SVM algorithm, and the sparse greedy matrix

approximation is used to select the basic element set, and the iterative reweighted

least squares algorithm is introduced to iterate and optimize the S-SVM model, which

finally constitutes the wavelet singular entropy and the S-SVM model. the signal is

decomposed by the wavelet singular entropy and the signal components are

extracted, and the extracted current harmonic features are screened and

reconstructed, and the current harmonic features are extracted and analyzed. After

that, the extracted current harmonic feature vectors are filtered and reconstructed,

and then the cable fault samples are input to the S-SVM model for training, and finally

the wavelet singular entropy and S-SVM models are realized to accurately identify

different faults of cables.

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

2. CONSTRUCTION OF INTELLIGENT CABLE FAULT

DETECTION MODEL

2.1. CONSTRUCTION OF IMPROVED HILBERT-HUANG

TRANSFORM MODELS

2.1.1. HILBERT-HUANG TRANSFORM ALGORITHM

The HHT algorithm is to sieve the various frequency components or trends

contained in the signal layer by layer in order to obtain a series of IMF components

containing different feature information, and then these IMF components are

subjected to the Hilbert transform, which can obtain the Hilbert spectra and the

marginal spectra, and then obtain the amplitude distribution pattern of the signal in the

spatial or temporal scale.

1. Instantaneous frequency

In the process of analyzing nonlinear signals, the instantaneous characteristics

have a very important role, in the traditional Fourier transform, less than a wavelength

signal will not be able to give the definition of the frequency, that is, it cannot be used

to accurately describe the instantaneous parameters of the non-smooth signal, the

instantaneous frequency mentioned in the Hilbert transform has the actual physical

meaning, and the basic definition of the frequency is consistent.

When analyzing a smooth signal, the frequency of the signal refers to the in the

Fourier transform.

(1)

In the formula , becomes the Fourier frequency, and the time is not related.

When analyzing non-stationary signals, the instantaneous frequency changes with

time, and it is not possible to accurately describe the changing frequency, the Fourier

frequency loses its significance, so a new definition of the instantaneous frequency is

needed to describe this change.

Let be a signal of any time series, is the Hilbert transform of the signal,

then can be expressed by as follows.

(2)

Meanwhile, can be expressed by as follows:

f

X(f) =

∫∞

−∞

x(t)e−jπftdt

f

X(t)

Y(t)

Y(t)

X(t)

Y

(t) =

1

π∫∞

−∞

X(t)

t−τ

dτ

X(t)

Y(t)

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(3)

From the above two equations, we can know that and are complex

conjugate pairs, and and are correlated with the time series, so we can get

the following analyzed signals.

(4)

(5)

(6)

Where is the instantaneous amplitude and

is the phase. another

instantaneous parameter is known from the phase and frequency.

(7)

As can be seen from the above equation, the Hilbert transform obtains a unique

function, is the convolution of the Hilbert transform with and , which

emphasizes the limitation of the characteristics of

The three instantaneous

parameters of the signal can be found out through the Hilbert transform, i.e., the

instantaneous amplitude, the instantaneous phase, and the instantaneous frequency.

Each of the parameters is analytic, thus, the Hilbert transform has a real-life

instantaneous characteristic.

2. Intrinsic Modal Functions

In general, a data often contains more than one oscillation mode, a simple Hilbert

transform can not be decomposed into all the frequencies of a signal, so the data

must first be decomposed into the intrinsic modal function. the definition of the

instantaneous frequency of a signal needs to have the following necessary conditions,

firstly, in the entire data segment, the function is symmetric, and secondly, the number

of zeros and the number of extrema are the same, and finally, the local mean value of

the signal is zero. Finally, the signal is locally zero-mean.

N.E. Huang defined that the intrinsic modal function must satisfy the following two

conditions.

One is that the number of zero crossing points and the number of extreme points in

the signal data are the same or differ by at most one.

X(t) =

1

π∫∞

−∞

Y(t)

τ−t

d

τ

X(t)

Y(t)

X(t)

Y(t)

Z(t)=X(t)+jY(t) = A(t)ejθ(t)

A(t) = X2(t)+Y2(t)

θ

(t) = arctan

(Y(t)

X(t))

A(t)

θ(t)

f(t) =

1

2π

dθ(t)

dt

Y(t)

X(t)

1

t

X(t)

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(3)

From the above two equations, we can know that and are complex

conjugate pairs, and and are correlated with the time series, so we can get

the following analyzed signals.

(4)

(5)

(6)

Where is the instantaneous amplitude and is the phase. another

instantaneous parameter is known from the phase and frequency.

(7)

As can be seen from the above equation, the Hilbert transform obtains a unique

function, is the convolution of the Hilbert transform with and , which

emphasizes the limitation of the characteristics of The three instantaneous

parameters of the signal can be found out through the Hilbert transform, i.e., the

instantaneous amplitude, the instantaneous phase, and the instantaneous frequency.

Each of the parameters is analytic, thus, the Hilbert transform has a real-life

instantaneous characteristic.

2. Intrinsic Modal Functions

In general, a data often contains more than one oscillation mode, a simple Hilbert

transform can not be decomposed into all the frequencies of a signal, so the data

must first be decomposed into the intrinsic modal function. the definition of the

instantaneous frequency of a signal needs to have the following necessary conditions,

firstly, in the entire data segment, the function is symmetric, and secondly, the number

of zeros and the number of extrema are the same, and finally, the local mean value of

the signal is zero. Finally, the signal is locally zero-mean.

N.E. Huang defined that the intrinsic modal function must satisfy the following two

conditions.

One is that the number of zero crossing points and the number of extreme points in

the signal data are the same or differ by at most one.

X(t) = 1

π∫∞

−∞

Y(t)

τ−tdτ

X(t)

Y(t)

X(t)

Y(t)

Z(t)=X(t)+jY(t) = A(t)ejθ(t)

A(t) = X2(t)+Y2(t)

θ(t) = arctan(Y(t)

X(t))

A(t)

θ(t)

f(t) = 1

2π

dθ(t)

dt

Y(t)

X(t)

1

t

X(t)

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One is that the average value of the upper and lower envelopes formed by the local

extreme value points and the local extreme value points of the IMF at any moment is

zero, i.e., the local signal is symmetric about the time axis.

3. Empirical modal decomposition method

Empirical modal decomposition (EMD) method is the essence of the Hilbert-Huang

transform, this method is a non-stationary complex signal from the separation of

several IMF process, the process is known as the screening process, screened out of

the various components superimposed on the actual data series. at any time, most of

the signals are unable to meet the conditions of the IMF, so it is necessary to screen

the signal first and then use the Hilbert transform on the signal. At any moment, most

of the signals cannot meet the conditions of IMF, so the signals need to be filtered first

and then processed by Hilbert transform for each IMF component, each IMF can be

linear or nonlinear.

The decomposition process is based on three assumptions: one is that the time-

domain characteristics are determined by the interval between the maxima and

minima, two is that the original signal should have at least one maxima and one

minima, three is that if the original signal contains only inflection points, the maxima

and minima can be calculated by taking the first derivative or multiple derivatives of

the signal, and then the signal can be reduced by integrating the signal. The specific

steps of the empirical mode decomposition are as follows.

The first step is to write the original signal as

, determine all the maximum and

minimum value points of

, and fit the maximum value points to the upper

envelope with the three times spline sampling function, and the minimum value points

to the lower envelope with the three times spline sampling function, and the upper and

lower envelopes should contain all the data.

In the second step, the average value of the upper and lower envelopes is

calculated and denoted as , and the original signal X(t), is subtracted from

to

obtain a new sequence .

(8)

In this decomposition, the low-frequency quantities of the signal are separated out,

and is the high-frequency quantity of the signal. Ideally, if

satisfies the two

necessary conditions of IMF, then

is the first IMF component of the original signal

, and is denoted as .

In the third step, if the high-frequency quantity

does not satisfy the two

necessary conditions of IMF, then

is continued as the original signal, repeat the

above two steps, first calculate the average value of , denoted as

, and then

calculate , and judge whether

satisfies the two necessary

conditions of IMF, if it does, then is denoted as

, if it doesn't, then continue to

X(t)

X(t)

m1

m1

h1

h1=X(t)−m1

h1

h1

h1

X(t)

c1=h1

h1

h1

h1

m11

h11 =h1−m11

h11

h11

c1

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calculate, and repeat the cycle of the first and the second steps, until the obtained

satisfies the two necessary conditions and is denoted as , and is the first

IMF component of the original signal .

In the fourth step, c1, is separated from the original signal to obtain a signal

that removes the high-frequency quantityv , denoted as .

(9)

The signal as the original signal, repeat the above steps, until we get a

component that meets the two necessary conditions of IMF, which is recorded as ,

and have been so looped times, we can get the IMF components of the original

signal , as shown in Eq. (10).

(10)

In the fifth step, the loop to the end, there is a residual , the end of the loop

termination conditions for the residual is a monotonic function, that is, can no longer

be extracted from it to meet the two necessary conditions of the IMF components, the

end of the decomposition. the final decomposition of the form as shown in Equation

(11).

(11)

Where is the IMF of the original signal and is a residual that converges to a

constant.

2.1.2. IMPROVEMENT OF ENVELOPE FITTING ALGORITHM

When the upper and lower envelopes of the signal are approximately symmetric, it

is inappropriate to use the upper and lower envelopes to find the average envelope,

and for the problems of overshoot, undershoot and sieving speed in the process of

EMD, this paper proposes to firstly use the tangent mean point to densify the local

extreme point, and then use the segment power function to interpolate and fit the

tangent mean point and local extreme point, and finally splice the interpolated curve

segments together, which is the final average envelope. Then the segmented power

function is used to fit the interpolation of the tangential mean and local extreme points,

and finally the interpolated curve segments are spliced together, which is the final

mean envelope.

1. Tangential mean points

h1k

c1=h1k

c1

X(t)

X(t)

c1

r1

r1=X(t)−c1

r1

c2

n

n

X(t)

r

1

−c

2

=r

2

⋮

rn

−1

−cn=r

n

rn

rn

X(t) =

n

∑

i=1

ci(t)+r

n

c1

rn

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calculate, and repeat the cycle of the first and the second steps, until the obtained

satisfies the two necessary conditions and is denoted as , and is the first

IMF component of the original signal .

In the fourth step, c1, is separated from the original signal to obtain a signal

that removes the high-frequency quantityv , denoted as .

(9)

The signal as the original signal, repeat the above steps, until we get a

component that meets the two necessary conditions of IMF, which is recorded as ,

and have been so looped times, we can get the IMF components of the original

signal , as shown in Eq. (10).

(10)

In the fifth step, the loop to the end, there is a residual , the end of the loop

termination conditions for the residual is a monotonic function, that is, can no longer

be extracted from it to meet the two necessary conditions of the IMF components, the

end of the decomposition. the final decomposition of the form as shown in Equation

(11).

(11)

Where is the IMF of the original signal and is a residual that converges to a

constant.

2.1.2. IMPROVEMENT OF ENVELOPE FITTING ALGORITHM

When the upper and lower envelopes of the signal are approximately symmetric, it

is inappropriate to use the upper and lower envelopes to find the average envelope,

and for the problems of overshoot, undershoot and sieving speed in the process of

EMD, this paper proposes to firstly use the tangent mean point to densify the local

extreme point, and then use the segment power function to interpolate and fit the

tangent mean point and local extreme point, and finally splice the interpolated curve

segments together, which is the final average envelope. Then the segmented power

function is used to fit the interpolation of the tangential mean and local extreme points,

and finally the interpolated curve segments are spliced together, which is the final

mean envelope.

1. Tangential mean points

h1k

c1=h1k

c1

X(t)

X(t)

c1

r1

r1=X(t)−c1

r1

c2

n

n

X(t)

r1−c2=r2

⋮

rn−1−cn=rn

rn

rn

X(t) =

n

∑

i=1

ci(t)+rn

c1

rn

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

Since the density of the distribution of the points to be interpolated directly

determines the interpolation accuracy and fitting effect, the densification of the

extreme points by the tangent mean points can optimize the fitting effect, and then

inhibit or eliminate the overshooting and undershooting phenomena in the envelope

fitting.

All the local extreme points are arranged in a time series, notated as

, where is the moment when the extreme point

appears, and is the amplitude of the extreme point. Let

be the three neighboring extreme points, and

the tangent touching the mean point is shown in Eq. (12).

(12)

Where, is the extreme point of the signal, denotes the tangential

mean point.

2. Segmental power function interpolation

Set the point to be interpolated as as

the interpolation function, and use the segmented power function to interpolate any

three neighboring points .

(13)

The interpolation function satisfies equation (14).

(14)

The value curve is shown in equation (15).

(15)

Finally, the interpolated line segments are spliced together to obtain the mean

envelope, which is constructed by the segmented cubic function after many

experiments and comparisons.

p(t1,y1),p(t2,y2), …, p(tn,yn)

ti

yi

p(ti−1,yi−1),p(ti,yi),p(ti+1,yi+1)

P

(ti,yi)=1

2

[

p(ti,yi)+

t

i

−t

i−1

ti

+1

−ti

−1

p(ti+1,yi+1)+

t

i+1

−t

i

ti

+1

−ti

−1

p(ti−1,yi−1)

]

p(ti,yi)

P(ti,yi)

P(t1,y1),P(t2,y2), …, P(tn,yn),y=f(x)

P(ti−1,yi−1),P(ti,yi),P(ti+1,yi+1)

Q

i=

(t

i+1

−t

i−1

)(y

i−1

−y

i

)−(t

i−1

−t

i

)(y

i+1

−y

i−1

)

ti+1 −ti−1

.

fi(t)

fi(t) =

(

t−ti

ti−1−ti

)β

Qi+yi+1 −yi−1

ti+1 −ti−1(t−ti)+yi,t≤ti,

fi(t)=(t−ti

ti+1 −ti)β

Qi+yi+1 −yi−1

ti+1 −ti−1(t−ti)+yi,t≥ti.

fi,i+1(t) =

t

i+1

−t

ti+1 −ti

fi(t) +

t−t

i

ti+1 −ti

fi+1(t) .

f(t)IMF

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2.1.3. IMPROVED HHT ALGORITHM FLOW

The theory of the HHT algorithm and the operation of the specific improvement

measures have been introduced, the improved HHT algorithm flowchart is shown in

Fig. 1. firstly, after inputting the signal

, find out all the local maxima and minima

contained in the signal

, and then use the cubic spline interpolation method to

construct the envelope of the local maxima and minima, and then find out the average

value of the envelope. If the

meets the condition of IMF component, then find out

the energy of the remaining signal and proceed to the next step, if not, then re-screen

repeatedly until the condition of IMF can be determined, and repeat the above steps

until there is only one extreme point, then stop iterating.

x(t)

x(t)

f(t)

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110

2.1.3. IMPROVED HHT ALGORITHM FLOW

The theory of the HHT algorithm and the operation of the specific improvement

measures have been introduced, the improved HHT algorithm flowchart is shown in

Fig. 1. firstly, after inputting the signal , find out all the local maxima and minima

contained in the signal , and then use the cubic spline interpolation method to

construct the envelope of the local maxima and minima, and then find out the average

value of the envelope. If the meets the condition of IMF component, then find out

the energy of the remaining signal and proceed to the next step, if not, then re-screen

repeatedly until the condition of IMF can be determined, and repeat the above steps

until there is only one extreme point, then stop iterating.

x(t)

x(t)

f(t)

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Figure 1 HHT algorithm improves the process

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111

2.2. CONSTRUCTION OF WAVELET-BASED SINGULAR

ENTROPY SUM AND S-SVM NETWORK MODELS

2.2.1. SEMIPARAMETRIC SUPPORT VECTOR MACHINES

Semi-parametric support vector machine, S-SVM, perfectly combines the

advantages of parametric and non-parametric support vector machines, and improves

the computational efficiency. Compared with linear support vector machine, S-SVM

can handle non-linear sample data, and has better classification performance than

non-linear support vector machine using soft intervals. S-SVM can avoid the problem

of slow classifier computation caused by a large number of support vectors by using a

predefined model. S-SVM avoids the slow computation problem caused by a large

number of support vectors by using a predefined model.

The predefined model selects a representative set of basic elements

from the sample data to reduce the size of the sample data set,

and estimates the normal vector , and then calculates the discriminant function to

construct the classifier. The expression of the normal vector estimated according to

Equation (16) is as follows.

(16)

Where is the mapping function, is similar to the Lagrange multiplier, and

the objective function is similar to Eq. (17), denoted as.

(17)

Where the matrix denotes the kernel matrix of the basic elements, i.e.,

. The discriminant function is

constructed as described in the previous section.

(18)

It can be seen that by limiting the size of the basic element set , the complexity

and operation speed of the classifier can be effectively limited. However, the element

set plays an important role for this kind of classifier, which has a great influence on

C={c1,c2,…,cm}

ω

ω

≃

m

∑

i=1

βiφ(ci

)

φ(⋅)

β

min

β

1

2βTKCβ+C

m

∑

i=1

ξi

s.t.

yi(βTKij +b)≥1−ξi∀i= 1, 2, …,

m

ξi≥0, i= 1, 2, …, m

KC

n×n

(

KC

)i,j

=k

(

ci,cj

)

,∀i,j= 1, 2, …, n,Kij =k

(

xi,cj

)

f(xi)=

m

∑

j=1

βjk

(

xi,cj

)

+

b

m

C

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112

2.2. CONSTRUCTION OF WAVELET-BASED SINGULAR

ENTROPY SUM AND S-SVM NETWORK MODELS

2.2.1. SEMIPARAMETRIC SUPPORT VECTOR MACHINES

Semi-parametric support vector machine, S-SVM, perfectly combines the

advantages of parametric and non-parametric support vector machines, and improves

the computational efficiency. Compared with linear support vector machine, S-SVM

can handle non-linear sample data, and has better classification performance than

non-linear support vector machine using soft intervals. S-SVM can avoid the problem

of slow classifier computation caused by a large number of support vectors by using a

predefined model. S-SVM avoids the slow computation problem caused by a large

number of support vectors by using a predefined model.

The predefined model selects a representative set of basic elements

from the sample data to reduce the size of the sample data set,

and estimates the normal vector , and then calculates the discriminant function to

construct the classifier. The expression of the normal vector estimated according to

Equation (16) is as follows.

(16)

Where is the mapping function, is similar to the Lagrange multiplier, and

the objective function is similar to Eq. (17), denoted as.

(17)

Where the matrix denotes the kernel matrix of the basic elements, i.e.,

. The discriminant function is

constructed as described in the previous section.

(18)

It can be seen that by limiting the size of the basic element set , the complexity

and operation speed of the classifier can be effectively limited. However, the element

set plays an important role for this kind of classifier, which has a great influence on

C={c1,c2,…,cm}

ω

ω≃

m

∑

i=1

βiφ(ci)

φ(⋅)

β

min

β

1

2βTKCβ+C

m

∑

i=1

ξi

s.t. yi(βTKij +b)≥1−ξi∀i= 1, 2, …, m

ξi≥0, i= 1, 2, …, m

KC

n×n

(KC)i,j=k(ci,cj),∀i,j= 1, 2, …, n,Kij =k(xi,cj)

f(xi)=

m

∑

j=1

βjk(xi,cj)+b

m

C

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the classification performance and accuracy, and a processing tool that can accurately

select the element set is needed.

2.2.2. SEMI-PARAMETRIC SUPPORT VECTOR MACHINES

BASED ON LEAST SQUARES

For the selection of the basic element set, this subsection adopts the sparse

greedy matrix approximation, SGMA algorithm, and introduces the iterative

reweighted least squares algorithm, IRWLS, to calculate the weights of the kernel

function.

The kernel function used in this paper is Gaussian kernel function, as shown in Eq.

(19).

(19)

The kernel matrix of the sample data set is denoted.

(20)

In this paper, we use the hinge loss function to construct a soft interval classifier.

(21)

Where y is the predicted classification output.

2.2.3. SPARSE GREEDY MATRIX APPROXIMATION

Given that the semiparametric support vector machine needs to select a suitable

basic element set , this section introduces the sparse greedy matrix approximation

algorithm to find the element set . The basic element set selected by using the

SGMA algorithm can represent the whole sample dataset with the most representative

features, which is very useful for solving the support vectors, constructing classifiers,

and improving the classification performance.

C

k

(xi,xj)=exp −∥xi−xj∥

2

2σ2

K

T={x1,x2, …, xn}

K

=

k(x1,x1)…k

(

x1,xj

)

…k(x1,xn)

⋮ ⋱ ⋮ ⋱ ⋮

k(xi,x1)…k(xi,xj)…k(xi,xn)

⋮ ⋱ ⋮ ⋱ ⋮

k(xn,x1)…k(xn,xj)…k(xn,xn)

ℓhinge (y) = max(0, 1 −y)

C

C

C

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For the sample dataset , a subset is

randomly selected; for optimal performance, the range of needs to be larger than .

The optimal set of basic elements is

selected by the elements in the subset .

Assuming that the kernel function of the column vectors in the subset N is

estimated to be , set a value to be a linear combination of and the

kernel function , of the elements in .

(22)

Where is noted as the weight value.

The approximation error of the weights can be determined by the kernel function

of the kernel matrix trace of the sample dataset and the column vectors in the

subset as .

(23)

After the new element is added to the basic element set by the SGMA

algorithm, the new weight error is shown in Eqs. (24) and (25).

(24)

(25)

is the kernel matrix of the basic element , and is the

kernel matrix of the subset and the basic element set .

From the above analysis, the SGMA algorithm can calculate the element with the

maximum value in the data set by the formula (25), and add this metamethod to

the basic data set , so as to find out a group of subsets that can more accurately

represent the distribution of the whole feature space.

T={x1,x2, …, xn}

N={x1,x2, …, xk}

k

log0.05/log0.95 = 59

C={c1,c2,…,cm}

N

k(xi,.)

λ,k(xi,.)

λ

k(ci,.)

C

k

(xi,.)=

κ

∑

j=1

λijk(ci, .

)

λij

λij

T

N

k(xi,.)

Err

(λ) = trK−

n

∑

i=1

m

∑

j=1

λijk

(

xi,cj

)

cm+1

C

Err(λn,m+1)= Err(λn,m)−η−1∥Kn,mz−k

Sn

∥2

(kNC)i=k(xi,cm+1)

z=K−1

c⋅kmc

η= 1 −zTkmc

(

kmC

)i

=k

(

ci,cm

+1)

ED = Err(λn,m)−Err(λn,m+1)

KC

m×m

K

Kn,m

n×m

N

C

ED

C

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114

For the sample dataset , a subset is

randomly selected; for optimal performance, the range of needs to be larger than .

The optimal set of basic elements is

selected by the elements in the subset .

Assuming that the kernel function of the column vectors in the subset N is

estimated to be , set a value to be a linear combination of and the

kernel function , of the elements in .

(22)

Where is noted as the weight value.

The approximation error of the weights can be determined by the kernel function

of the kernel matrix trace of the sample dataset and the column vectors in the

subset as .

(23)

After the new element is added to the basic element set by the SGMA

algorithm, the new weight error is shown in Eqs. (24) and (25).

(24)

(25)

is the kernel matrix of the basic element , and is the

kernel matrix of the subset and the basic element set .

From the above analysis, the SGMA algorithm can calculate the element with the

maximum value in the data set by the formula (25), and add this metamethod to

the basic data set , so as to find out a group of subsets that can more accurately

represent the distribution of the whole feature space.

T={x1,x2, …, xn}

N={x1,x2, …, xk}

k

log0.05/log0.95 = 59

C={c1,c2,…,cm}

N

k(xi,.)

λ,k(xi,.)

λ

k(ci,.)

C

k(xi,.)=

κ

∑

j=1

λijk(ci,.)

λij

λij

T

N

k(xi,.)

Err(λ) = trK−

n

∑

i=1

m

∑

j=1

λijk(xi,cj)

cm+1

C

Err(λn,m+1)= Err(λn,m)−η−1∥Kn,mz−kSn ∥2

(kNC)i=k(xi,cm+1)

z=K−1

c⋅kmc

η= 1 −zTkmc

(kmC)i=k(ci,cm+1)

ED = Err(λn,m)−Err(λn,m+1)

KC

m×m

K

Kn,m

n×m

N

C

ED

C

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2.2.4. ITERATIVE REWEIGHTED LEAST SQUARES

The iterative reweighted least squares, IRWLS, process is a solution method that

has been widely used in the field of support vector machines. Compared with the

weighted least squares algorithm, IRWLS can gradually correct for the effects of

anomalous sample data and set the weights always within a relatively optimal range.

For the Lagrangian function with the penalty factor added.

(26)

By taking the partial derivation of Eq. (26) and eliminating the term about and

calculating , Eq. (27) is converted as shown below.

(28)

In which:

(29)

(30)

is denoted as the error of the sample data points, and is denoted as the weight

coefficient associated with it. Substituting into Eq. (30) to obtain Eq.

[112] with respect to , the objective function is obtained.

(31)

(32)

Cp

L(ω,b,a,ξ,μ) =

1

2∥ω∥2+Cp

n

∑

i=1

ξi

+

n

∑

i=1

ai(1−ξi−yi(ωTφ(xi)+b))−

n

∑

i=1

μiξ

i

ξi

Cp=ai+μi

L(ω,b,a,ξ,μ) =

1

2∥ω∥2+

n

∑

i=1

ai

(

1−yi

(

ωTφ(xi)+b

))

=1

2∥ω∥2+1

2

n

∑

i=1

2ai

1−yi(ωTφ(xi)+b)(yi(ωTφ(xi)+b))

2

=1

2

∥ω∥2+1

2

n

∑

i=1

aie2

i

ei=yi(ωTφ(xi)+b)

α

i=

2a

i

1−

y

i(

ωTφ

(

x

i)+

b

)

ei

αi

ω

≃

n

∑

i=1

βiφ(xi

)

αi

s.t.e

ei=yi−

m

∑

j=0

βiK(xi,ci)+b

α

i=

0, y

i

e

i

≤0

Cp

eiyi, yiei>

0

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Determine the weight through equation (33).

(33)

Where , vectors ,

The iterative process allows each round of training and learning to gradually correct

the weight values, so that the weights and eventually converge to a fixed value,

and the iterative calculation steps are as follows.

Step 1: Pre-set a weight , solve the least squares problem to obtain .

Step 2: Based on the correlation between weight BB and , re-calculate the

value of by calculating weight .

Step 3: Repeat the first two steps until the weights and converge to a fixed

convergence value.

At this point, the discriminant function of the classifier is:

(34)

2.2.5. WAVELET TRANSFORM

The continuous wavelet transform of the function is shown in equation (35).

(35)

Where, is the scale factor, is the translation factor and is the mother

wavelet.

Continuous wavelet transform can accurately extract the characteristics of the

signal, but in each possible scale discrete points to calculate the wavelet coefficients,

will be a huge project. if only a small part of these scales, and part of the time point,

will greatly reduce the workload, and without loss of accuracy, the use of such an

approximation will be obtained by the discrete wavelet transform, the definition of

which is shown in equation (36).

β

[K

C

+KT

sC

D

a

K

sc

KT

sc

D

α

1

1TD

a

K−1

C

1TD

a

1

][

β

b

]

=

[

K

T

scDαy

1Day

]

(

KC

)ij

=k

(

ci,cj

)

,KSC =k

(

xi,cj

)

,

(

Dα

)i

=α

i

1 = [1, …, 1]T

y=[

y

1,

y

2, …,

y

n]T

αi

β

αi

β

αi

αi

β

β

αi

f(xi)=

m

∑

j=1

βjk

(

xi,cj

)

+

b

f(t)

W

(f,a,b) =

1

a∫+∞

−∞

f(t)ψ∗

(t−b

a

)

d

t

a

b

ψ(t)

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Determine the weight through equation (33).

(33)

Where , vectors ,

The iterative process allows each round of training and learning to gradually correct

the weight values, so that the weights and eventually converge to a fixed value,

and the iterative calculation steps are as follows.

Step 1: Pre-set a weight , solve the least squares problem to obtain .

Step 2: Based on the correlation between weight BB and , re-calculate the

value of by calculating weight .

Step 3: Repeat the first two steps until the weights and converge to a fixed

convergence value.

At this point, the discriminant function of the classifier is:

(34)

2.2.5. WAVELET TRANSFORM

The continuous wavelet transform of the function is shown in equation (35).

(35)

Where, is the scale factor, is the translation factor and is the mother

wavelet.

Continuous wavelet transform can accurately extract the characteristics of the

signal, but in each possible scale discrete points to calculate the wavelet coefficients,

will be a huge project. if only a small part of these scales, and part of the time point,

will greatly reduce the workload, and without loss of accuracy, the use of such an

approximation will be obtained by the discrete wavelet transform, the definition of

which is shown in equation (36).

β

[KC+KT

sC DaKsc KT

scDα1

1TDaK−1

C1TDa1][β

b]=[KT

scDαy

1Day]

(KC)ij =k(ci,cj),KSC =k(xi,cj),(Dα)i=αi

1 = [1, …, 1]T

y=[y1,y2,…,yn]T

αi

β

αi

β

αi

αi

β

β

αi

f(xi)=

m

∑

j=1

βjk(xi,cj)+b

f(t)

W(f,a,b) = 1

a∫+∞

−∞

f(t)ψ∗(t−b

a)dt

a

b

ψ(t)

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(36)

In Eq. (36), the parameters and of continuous wavelet transform are replaced

by and , and MM are integers, and A02 is taken in general. The wavelet

decomposition tree of the discrete wavelet transform is shown in Fig. 2. The original

signal sequence of SN is shown in Fig. 2. denotes the low-frequency coefficients

of the decomposition of the layer, and denotes the high-frequency coefficients

of the decomposition. The signal components obtained from the decomposition of

the decomposition coefficients of each layer by the single-branch reconstruction are

denoted as and , and then the original signal is the sum of the signal

components obtained by the reconstruction, and the formula is as follows.

(37)

Figure 2 The wavelet decomposition tree

2.2.6. SHANNON INFORMATION ENTROPY AND WAVELET

SINGULAR ENTROPY

Shannon information entropy and its state characteristics can be expressed by a

variable that takes a finite number of values, the probability that the state

characteristics of the source take the value of is and

, then represents the information obtained from the result

of , and the information entropy is defined as shown in Equation (38).

DWT

(f,m,n) = 1

am

0

∑

k

f(k)y∗

(n−ka m

0

am

0)

a

b

am

0

kam

0

k

CAk

k

CDk

k

ak

Dk

S(n)

x

(n)=D1+A1=D1+D2+A2=

n

∑

j=1

Dj+A

n

X

xj

pj=P

{

X=xj

}

j= 1, ⋯,

L

L

∑

j=1

pj=

1

I

j= log

(

1/pj

)

j

X

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(38)

Wavelet singular entropy is a new data processing method obtained by combining

wavelet transform with Shannon's information entropy theory, which is defined as

follows.

(39)

In Eq. (40), is the incremental wavelet singular entropy of order .

(40)

The is calculated as follows.

After the signal wavelet decomposition and reconstruction of the components

at the th scale is , then the components of the signal can

form a matrix , by the theory of signal singular value decomposition, for

the above matrix , there exists a -dimensional matrix and a

-dimensional diagonal matrix and a -dimensional matrix , so that the matrix

Dmxn decomposition is as shown in Eq. (41).

(41)

Where, is the element on the main diagonal of the diagonal matrix

, i.e., the singular value of the matrix formed after the signal is

decomposed by the wavelet decomposition of the layer and reconstructed.

2.2.7. CABLE FAULT DETECTION STRATEGY BASED ON

WAVELET SINGULAR ENTROPY AND S-SVM MODELS

In this section, wavelet singular entropy and semiparametric support vector

machine are combined and applied to cable fault detection, and the cable fault

detection strategy based on wavelet singular entropy and S-SVM model is shown in

Fig. 2. firstly, the cable current signal is extracted, then wavelet singular entropy is

used to decompose the signal, extract the meaningful sub-signal components, and

then the correlation analysis is used to filter the components to reconstruct the fault

signal, and then wavelet singular entropy is used to reduce the influence of non-

Gaussian noise on the fault signal, extract the wavelet energy entropy of harmonic

components as feature vectors, and then the fault sample data is inputted into the S-

SVM model to be trained. Then the wavelet singular entropy is used to reduce the

H

(X)=−

L

∑

j=1

pjlog

(

pj

)

W

k=

k

∑

i=1

Δp

i

Δpi

j

Δ

pi=−λi/

l

∑

j=1

λjlog λi/

l

∑

j=1

λj

λi(i= 1, ⋯,l)

S(n)

j(j= 1, ⋯m)

Dj(n)

m

S(n)

m×n

Dm×n

Dm×n

m×l

U

l×l

Λ

l×n

V

Dm×n=Um×1ΛI×1Vl×n

λi(i= 1, ⋯,l)

Λ

Dm×n

S(n)

m

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(38)

Wavelet singular entropy is a new data processing method obtained by combining

wavelet transform with Shannon's information entropy theory, which is defined as

follows.

(39)

In Eq. (40), is the incremental wavelet singular entropy of order .

(40)

The is calculated as follows.

After the signal wavelet decomposition and reconstruction of the components

at the th scale is , then the components of the signal can

form a matrix , by the theory of signal singular value decomposition, for

the above matrix , there exists a -dimensional matrix and a

-dimensional diagonal matrix and a -dimensional matrix , so that the matrix

Dmxn decomposition is as shown in Eq. (41).

(41)

Where, is the element on the main diagonal of the diagonal matrix

, i.e., the singular value of the matrix formed after the signal is

decomposed by the wavelet decomposition of the layer and reconstructed.

2.2.7. CABLE FAULT DETECTION STRATEGY BASED ON

WAVELET SINGULAR ENTROPY AND S-SVM MODELS

In this section, wavelet singular entropy and semiparametric support vector

machine are combined and applied to cable fault detection, and the cable fault

detection strategy based on wavelet singular entropy and S-SVM model is shown in

Fig. 2. firstly, the cable current signal is extracted, then wavelet singular entropy is

used to decompose the signal, extract the meaningful sub-signal components, and

then the correlation analysis is used to filter the components to reconstruct the fault

signal, and then wavelet singular entropy is used to reduce the influence of non-

Gaussian noise on the fault signal, extract the wavelet energy entropy of harmonic

components as feature vectors, and then the fault sample data is inputted into the S-

SVM model to be trained. Then the wavelet singular entropy is used to reduce the

H(X)=−

L

∑

j=1

pjlog(pj)

Wk=

k

∑

i=1

Δpi

Δpi

j

Δpi=−λi/

l

∑

j=1

λjlog λi/

l

∑

j=1

λj

λi(i= 1, ⋯,l)

S(n)

j(j= 1, ⋯m)

Dj(n)

m

S(n)

m×n

Dm×n

Dm×n

m×l

U

l×l

Λ

l×n

V

Dm×n=Um×1ΛI×1Vl×n

λi(i= 1, ⋯,l)

Λ

Dm×n

S(n)

m

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effect of non-Gaussian noise on the fault signal, and the wavelet energy entropy of the

harmonic components is extracted and used as the feature vectors. After that, the

fault sample data are inputted into the S-SVM model for training, and the S-SVM

model is finally used to recognize and diagnose the test samples.

Figure 3 Cable fault detection strategy based on wavelet singular entropy and S-SVM model

3. ANALYSIS OF INTELLIGENT CABLE FAULT

DETECTION RESULTS CONSIDERING CURRENT

HARMONIC FEATURES

3.1. ANALYSIS OF INTELLIGENT DETECTION RESULTS OF

CABLE LINE DETERIORATION FAULTS

3.1.1. HARMONIC FEATURE EXTRACTION FOR CABLE LINE

DEGRADATION

In this section, according to the structural characteristics of the cable, 40,000 sets

of harmonic diagnostic data of the same power cable are selected, and the harmonic

content of the main part of the cable from the 2nd to the 10th harmonic is multiplied

with its corresponding contribution rate, and 9 harmonic vectors are obtained as the

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input data, and the degree of deterioration of the insulation, shielding, protective layer

and cable joints are derived through the improved HHT transformation model. The

energy spectrum of the harmonic vectors of different parts of the cable is shown in

Fig. 4, and the relative energy of each harmonic is 1. The relative energies of the

harmonic vectors are obviously different in diagnosing the operation status of different

parts of the cable, the operation status of the cable insulator mainly depends on the

change of the 2nd harmonic vector, with the relative energy value of 0.32, and the

operation status of the shielding layer mainly depends on the change of the 2nd, 3rd,

and 5th harmonic vectors, with the relative energy value of 0.24, 0.25, 0.2, 0.5, and

0.6, respectively. 0.25 and 0.2 respectively, and the operating state of the protective

layer and the roving state of the cable joints mainly depends on the changes of the

2nd, 4th, 7th, 8th and 9th harmonic vectors, with the relative energy values of 0.3,

0.25 and 0.26, 0.26, 0.3 respectively. The harmonic vectors obtained based on the

improved HHT transformation model completely characterize the operating state of

the different parts of the cables.

Figure 4 The shoe wave vector energy spectrum of the different parts of the cable

3.1.2. ANALYSIS OF CABLE LINE DETERIORATION

IDENTIFICATION TEST RESULTS

In this section, based on the wavelet singular entropy and S-SVM model to identify

and detect the degradation of cable lines, in order to verify the accuracy of the wavelet

singular entropy and S-SVM model, the harmonic characteristics database of different

fault loss currents of cables is formed by taking the harmonic total aberration rate, the

fundamental content and the harmonic contents of each harmonic as the

characteristics.Firstly, we extracted the sample data of the loss currents in the

database to form the sample data set of the wavelet singular entropy and S-SVM

model. Then some data in the sample data set are randomly extracted as the training

sample set, and the rest of the data in the sample data set are used as the test

sample set. finally, different ratios of the training set to the test set are set, and

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input data, and the degree of deterioration of the insulation, shielding, protective layer

and cable joints are derived through the improved HHT transformation model. The

energy spectrum of the harmonic vectors of different parts of the cable is shown in

Fig. 4, and the relative energy of each harmonic is 1. The relative energies of the

harmonic vectors are obviously different in diagnosing the operation status of different

parts of the cable, the operation status of the cable insulator mainly depends on the

change of the 2nd harmonic vector, with the relative energy value of 0.32, and the

operation status of the shielding layer mainly depends on the change of the 2nd, 3rd,

and 5th harmonic vectors, with the relative energy value of 0.24, 0.25, 0.2, 0.5, and

0.6, respectively. 0.25 and 0.2 respectively, and the operating state of the protective

layer and the roving state of the cable joints mainly depends on the changes of the

2nd, 4th, 7th, 8th and 9th harmonic vectors, with the relative energy values of 0.3,

0.25 and 0.26, 0.26, 0.3 respectively. The harmonic vectors obtained based on the

improved HHT transformation model completely characterize the operating state of

the different parts of the cables.

Figure 4 The shoe wave vector energy spectrum of the different parts of the cable

3.1.2. ANALYSIS OF CABLE LINE DETERIORATION

IDENTIFICATION TEST RESULTS

In this section, based on the wavelet singular entropy and S-SVM model to identify

and detect the degradation of cable lines, in order to verify the accuracy of the wavelet

singular entropy and S-SVM model, the harmonic characteristics database of different

fault loss currents of cables is formed by taking the harmonic total aberration rate, the

fundamental content and the harmonic contents of each harmonic as the

characteristics.Firstly, we extracted the sample data of the loss currents in the

database to form the sample data set of the wavelet singular entropy and S-SVM

model. Then some data in the sample data set are randomly extracted as the training

sample set, and the rest of the data in the sample data set are used as the test

sample set. finally, different ratios of the training set to the test set are set, and

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different numbers of test samples are taken to identify the deterioration of cables with

the wavelet singular entropy and the S-SVM model trained in the above steps. red

part indicates the classification of the predicted test set, and blue part indicates the

classification of the actual test set. red part indicates the classification of the predicted

test set, and blue part indicates the classification of the actual test set. red part

indicates the classification of the actual test set. The red part indicates the

classification of the predicted test set, and the blue part indicates the classification of

the actual test set, and the coincidence of the red part and the blue part indicates that

the test set data matches with the prediction result, and the prediction result is

accurate, and vice versa.The accuracy results of the wavelet singular entropy and S-

SVM models with different ratios of the training set to the test set are shown in Fig. 5,

and the results of the accuracy results of the models with the ratio of the training set to

the test set 6:4, 7:3, 8:2, and 9:1 are shown in Figs. 5(a) to (d) respectively. Vertical

coordinates 1, 2, 3 and 4 indicate the degradation of insulator, shield, protective layer

and cable connector, respectively.When the ratio of training set to test set is 6:4, there

are 12 groups of prediction failures, 6 groups of samples predicted the protective layer

to be insulator, and 4 groups of samples predicted the shield to be insulator, with an

identification accuracy of 93.93%.When the ratio of training set to test set is 7:3, there

are 3 groups of samples predicted the protective layer sample to be insulator, with an

identification accuracy of 93.93%. When the ratio of training set to test set is 7:3, there

are 3 groups of protective layer samples predicted to be shielding layer and 2 groups

of shielding layer samples predicted to be insulators, the recognition accuracy is

94.38%. when the ratio of training set to test set is 8:2, there are 1 group of protective

layer and 1 group of shielding layer samples predicted to be insulators, the recognition

accuracy is 96.58%. when the ratio of training set to test set is 9:1, there is only 1

group of protective layer sample predicted to be insulators, the recognition accuracy is

97.36%. The more training samples, the higher the accuracy of wavelet singular

entropy and S-SVM model in recognizing different cable line degradation, and the

recognition rate is above 93%, which indicates that this model can well identify the

categories of cable line degradation.

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Figure 5 Results of different training sets and test sets

3.2. ANALYSIS OF INTELLIGENT DETECTION RESULTS OF

CABLE SHORT-CIRCUIT FAULTS

In order to provide data support for cable fault diagnosis, this paper adopts the

10KV cable line model to construct the fault data set, and utilizes the electrical

components in the tool library to build the simulation circuit.

3.2.1. CABLE SHORT CIRCUIT FAULT ANALYSIS

Cable short-circuit fault is the largest proportion of cable faults occurring, cable a

phase connected to another phase or one of the phases connected to the earth is

called a short-circuit fault, in the case of mixed faults do not take into account the

occurrence of 10 types of faults may occur: A_G, B_G, C_G were A-phase, B-phase

and C-phase ground faults, AB, AC, BC, respectively, on behalf of AB, AC two-phase

and BC two-phase short-circuit faults, AB_G, AC_G, BC_G were AB-phase, AC phase

and BC two-phase short-circuit ground faults, ABC is a three-phase short-circuit fault,

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Figure 5 Results of different training sets and test sets

3.2. ANALYSIS OF INTELLIGENT DETECTION RESULTS OF

CABLE SHORT-CIRCUIT FAULTS

In order to provide data support for cable fault diagnosis, this paper adopts the

10KV cable line model to construct the fault data set, and utilizes the electrical

components in the tool library to build the simulation circuit.

3.2.1. CABLE SHORT CIRCUIT FAULT ANALYSIS

Cable short-circuit fault is the largest proportion of cable faults occurring, cable a

phase connected to another phase or one of the phases connected to the earth is

called a short-circuit fault, in the case of mixed faults do not take into account the

occurrence of 10 types of faults may occur: A_G, B_G, C_G were A-phase, B-phase

and C-phase ground faults, AB, AC, BC, respectively, on behalf of AB, AC two-phase

and BC two-phase short-circuit faults, AB_G, AC_G, BC_G were AB-phase, AC phase

and BC two-phase short-circuit ground faults, ABC is a three-phase short-circuit fault,

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ABC is a three-phase short circuit. AB_G, AC_G, BC_G are AB phase, AC phase and

BC two-phase short-circuit ground faults, and ABC is three-phase short-circuit

faults.Figure 6 shows the current waveform curves of the cable in normal operation

and various short-circuit faults, and Fig. 6(a)~(d) shows the current waveform curves

of the cable in normal operation, the single-phase grounded short-circuit faults, and

two-phase indirect short-circuit current waveforms, and Fig. 6(a)~(d) shows the

current waveform curve of the cable in normal operation, single-phase ground faults,

and two-phase short-circuit faults, and Fig. 6(a)~(d) shows the current waveform

curves of the cable in normal operation. Indirect ground short-circuit current waveform

curve, three-phase ground short-circuit waveform curve. cable normal operation

current waveform is more symmetrical, the overall current between -10K-10K

amperes. cable short-circuit faults, short-circuit current is lower than the normal

current. single-phase ground short-circuit, fault A-phase current in the 0.07s current is

smaller than the non-fault B, C-phase current, and the waveform is more chaotic. two-

phase short circuit in the 0.075s, the AB phase current is lower than the non-fault C

phase current, AB two short-circuit faults occur. three-phase short-circuit fault, in

0.075s, the three-phase current is lower than the normal current, and the three-phase

current sum is equal to zero.

Figure 6 The current waveform of the cable's various circuited faults

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3.2.2. HARMONIC FEATURE EXTRACTION FOR CABLE SHORT-

CIRCUIT FAULTS

In this section of the experiment, the sampling frequency is set to 12.8kHz, the

sampling time is 0.2s, and the fault is imposed after 0.05s of the system's normal

operation state, each simulation samples 2780 points of voltage, each line has three

phases, and each line collects 8670 points, i.e., the length of each sample is 8670,

and then collects the operation state of 10 cables, 80% of them are used as training

samples, 20% as test samples, and 18 harmonic vectors are obtained by using the

improved HHT transform model. Figure 7 shows the energy spectrum of harmonic

vector during various short-circuit faults of cable. 20% are used as test samples, and

18 harmonic vectors are obtained by using the improved HHT transform model. Fig. 7

shows the energy spectrum of harmonic vectors in various short-circuit faults of

cables. In single-phase cable short-circuit, phase A mainly looks at the 1st and 9th

harmonics, with the relative energy values of 0.18 and 0.12, respectively, and phases

B and C mainly look at the 1st and 2nd harmonics, with the relative energy values of

0.19 and 0.12, 0.24 and 0.14, respectively. Two-phase cable short-circuit mainly looks

at the 1st and 2nd harmonics, with the relative energy values of 0.19 and 0.12, 0.24

and 0.14, respectively. When two-phase cable is short-circuited, phase A mainly looks

at the 18th harmonic with a relative energy value of 0.38, phase B mainly looks at the

13th and 16th harmonics with relative energy values of 0.17 and 0.14, and phase C

mainly looks at the 15th harmonic with a relative energy value of 0.21. When three-

phase cable is short-circuited, phase A mainly looks at the 2nd harmonic with a

relative energy value of 0.05, phase B mainly looks at the 1st and 2nd harmonics with

relative energy values of 0.19 and 0.12, and phase B mainly looks at the 1st and 2nd

harmonics with relative energy values of 0.19 and 0.14, and phase C mainly looks at

the 1st and 2nd harmonics with relative energy values of 0.19 and 0.14 respectively.

The relative energy values are 0.19 and 0.13 for phase B, and 0.13 and 0.1 for phase

C, mainly for the 2nd and 3rd harmonics.

Figure 7 The harmonic energy spectrum of all kinds of short-circuit failures of the cable

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3.2.2. HARMONIC FEATURE EXTRACTION FOR CABLE SHORT-

CIRCUIT FAULTS

In this section of the experiment, the sampling frequency is set to 12.8kHz, the

sampling time is 0.2s, and the fault is imposed after 0.05s of the system's normal

operation state, each simulation samples 2780 points of voltage, each line has three

phases, and each line collects 8670 points, i.e., the length of each sample is 8670,

and then collects the operation state of 10 cables, 80% of them are used as training

samples, 20% as test samples, and 18 harmonic vectors are obtained by using the

improved HHT transform model. Figure 7 shows the energy spectrum of harmonic

vector during various short-circuit faults of cable. 20% are used as test samples, and

18 harmonic vectors are obtained by using the improved HHT transform model. Fig. 7

shows the energy spectrum of harmonic vectors in various short-circuit faults of

cables. In single-phase cable short-circuit, phase A mainly looks at the 1st and 9th

harmonics, with the relative energy values of 0.18 and 0.12, respectively, and phases

B and C mainly look at the 1st and 2nd harmonics, with the relative energy values of

0.19 and 0.12, 0.24 and 0.14, respectively. Two-phase cable short-circuit mainly looks

at the 1st and 2nd harmonics, with the relative energy values of 0.19 and 0.12, 0.24

and 0.14, respectively. When two-phase cable is short-circuited, phase A mainly looks

at the 18th harmonic with a relative energy value of 0.38, phase B mainly looks at the

13th and 16th harmonics with relative energy values of 0.17 and 0.14, and phase C

mainly looks at the 15th harmonic with a relative energy value of 0.21. When three-

phase cable is short-circuited, phase A mainly looks at the 2nd harmonic with a

relative energy value of 0.05, phase B mainly looks at the 1st and 2nd harmonics with

relative energy values of 0.19 and 0.12, and phase B mainly looks at the 1st and 2nd

harmonics with relative energy values of 0.19 and 0.14, and phase C mainly looks at

the 1st and 2nd harmonics with relative energy values of 0.19 and 0.14 respectively.

The relative energy values are 0.19 and 0.13 for phase B, and 0.13 and 0.1 for phase

C, mainly for the 2nd and 3rd harmonics.

Figure 7 The harmonic energy spectrum of all kinds of short-circuit failures of the cable

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3.2.3. IDENTIFICATION AND DETECTION RESULTS OF CABLE

SHORT-CIRCUIT FAULTS

This section extracts local features of the input target based on wavelet singular

entropy and S-SVM model, achieving target detection and improving diagnostic

efficiency. Figure 7 shows the confusion matrix of the cable fault diagnosis model,

where the vertical axis represents the true category of the cable fault signal and the

horizontal axis represents the predicted category of the network for the cable fault

signal, The diagonal values on the matrix represent the probability of correctly

classifying cable fault signals. Wavelet singular entropy and S-SVM models can fully

identify 10 types of cable short circuit faults. For A_ G, B_ G, C_ G, AB, AC, BC, AB_

G, AC_ G, BC_ The correct classification probabilities for 10 types of short-circuit

faults, including G and ABC, are 0.94, 0.95, 0.96, 0.96, 0.98, 0.99, 0.94, 0.92, and 1,

respectively. Although the correct classification probability for BC phase short-circuit

grounding is relatively low, the overall probability is above 90%, indicating that the

diagnosis of cable faults based on wavelet singular entropy and S-SVM model is

completely feasible

Figure 8 Confusion matrix of cable fault diagnosis model

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3.3. ANALYSIS OF INTELLIGENT DETECTION RESULTS OF

CABLE FAULTS

In order to verify the diagnostic effect of wavelet singular entropy and S-SVM model

on all faults of cables, 180 samples are selected for test experiments in this section.

The maximum error of wavelet singular entropy and S-SVM model on the test

samples is 0.5329, only two samples have the absolute error value more than 0.5,

and the other samples have the absolute error value lower than 0.5, and the average

error is 0.1124.Among 180 sets of test samples, the correct rate of wavelet singular

entropy and S-SVM model to diagnose various cable faults reached 98.04%, and the

performance is good. In 180 sets of test samples, the correct rate of the wavelet

singular entropy and S-SVM model in diagnosing various cable faults reaches

98.04%, which is good, and the wavelet singular entropy and S-SVM model can be

used in the actual cable fault diagnosis.

Figure 9 Wavelet singular entropy and S-SVM model diagnose cable fault curve

4. CONCLUSION

In this paper, by improving the envelope fitting algorithm of HHT algorithm, the HHT

transform model is proposed to extract the current harmonic features of the cable.

wavelet singularity algorithm and S-SVM algorithm are utilized to construct into the

wavelet singular entropy and S-SVM model for detecting the faults of the cables. the

experimental results are as follows.

In the detection of cable line deterioration, the current harmonic vectors

characterize the operation status of different parts of the cable, and the operation

status of cable insulator, shield, protective layer and cable joints are shown in the

changes of 2nd, 2nd, 3rd, 5th, 2nd, 4th and 7th, 8th, 9th harmonic vectors,

respectively, and the accuracy of the Wavelet Singularity Algorithm and S-SVM

algorithm on the detection of the cable line deterioration is 93.93%, 94% and 94%,

respectively. are 93.93%, 94.38%, 96.58% and 97.36%.

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3.3. ANALYSIS OF INTELLIGENT DETECTION RESULTS OF

CABLE FAULTS

In order to verify the diagnostic effect of wavelet singular entropy and S-SVM model

on all faults of cables, 180 samples are selected for test experiments in this section.

The maximum error of wavelet singular entropy and S-SVM model on the test

samples is 0.5329, only two samples have the absolute error value more than 0.5,

and the other samples have the absolute error value lower than 0.5, and the average

error is 0.1124.Among 180 sets of test samples, the correct rate of wavelet singular

entropy and S-SVM model to diagnose various cable faults reached 98.04%, and the

performance is good. In 180 sets of test samples, the correct rate of the wavelet

singular entropy and S-SVM model in diagnosing various cable faults reaches

98.04%, which is good, and the wavelet singular entropy and S-SVM model can be

used in the actual cable fault diagnosis.

Figure 9 Wavelet singular entropy and S-SVM model diagnose cable fault curve

4. CONCLUSION

In this paper, by improving the envelope fitting algorithm of HHT algorithm, the HHT

transform model is proposed to extract the current harmonic features of the cable.

wavelet singularity algorithm and S-SVM algorithm are utilized to construct into the

wavelet singular entropy and S-SVM model for detecting the faults of the cables. the

experimental results are as follows.

In the detection of cable line deterioration, the current harmonic vectors

characterize the operation status of different parts of the cable, and the operation

status of cable insulator, shield, protective layer and cable joints are shown in the

changes of 2nd, 2nd, 3rd, 5th, 2nd, 4th and 7th, 8th, 9th harmonic vectors,

respectively, and the accuracy of the Wavelet Singularity Algorithm and S-SVM

algorithm on the detection of the cable line deterioration is 93.93%, 94% and 94%,

respectively. are 93.93%, 94.38%, 96.58% and 97.36%.

https://doi.org/10.17993/3ctecno.2024.v13n1e45.99-128

In the detection of cable short circuit faults, the current of different short circuit

faults in the cable is lower than the normal current, and in the case of three-phase

short circuit faults, the three-phase currents add up to 0. For the extraction of the

different short circuit current harmonics, in the case of a single-phase short circuit in

the cable, the operating state of phases A, B, and C is shown in the first and ninth,

first and second, and first and second harmonics of the current respectively, in the

case of a two-phase short circuit in the cable, the operating state of phases A, B, and

C is shown in the 18th, 13th, and 16th, and 15th harmonics of the current respectively.

When three-phase cable is short-circuited, the operating states of phases A, B and C

are characterized by 2, 1 and 2, 2 and 3 harmonic vectors, respectively. The wavelet

singularity algorithm and S-SVM algorithm have an accuracy of more than 92% in

identifying 10 different cable short-circuit faults, such as A_G, B_G, C_G, AB, AC, BC,

AB_G, AC_G, BC_G, ABC, and so on.

Overall for cable fault detection, the wavelet singularity algorithm and S-SVM

algorithm have reached 98.04% correct rate for detecting all kinds of pegged accounts

of cables, and the error value is only more than 0.5 for two groups of data out of 180

groups of sample data. The improved HHT transform model and the wavelet singular

entropy and S-SVM models proposed in this paper have high accuracy and

practicability, and provide a a new method.

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