[ieee 2007 international conference on wavelet analysis and pattern recognition - beijing, china...

Proceedings of the 2007 International Conference on Wavelet Analysis and Pattern Recognition, Beijing, China, 2-4 Nov. 2007

1-4244-1066-5/07/$25.00 ©2007 IEEE 1853

APPLICATION OF THE B-SPLINE WAVELET IN RAILWAY CATENARY INTELLIGENT FAULT DIAGNOSIS

YAN-LING GAO1 , DE-YING ZHANG2

1Department of Electronic Engineering, Beijing Institute of Petro-chemical Technology Beijing 102617, china 2School of Civil Engineering, Shijiazhuang Railway Institute, Shijiazhuang

E-MAIL: [email protected], [email protected]

Abstract

A new method based on cubic B-Spline dyadic mother wavelet of centro-symmetric about zero is presented, the principle of singularity detection by using wavelet transform modulus maximum is applied in railway catenary fault feature extraction by using the mother wavelet, at last, the project is concluded by using BP neural network to recognize pattern. The result shows that it can confirm position accurately and identify the fault types effectively

Keywords: cubic B-Spline; mother wavelet; modulus maximum; feature extraction; BP neural network

1. Introduction

With the electrical railway developing rapidly, we are facing with the sixth speed raising, while railway catenary accidents are happening frequently, it has a higher requirement to the running security of the railway catenary. In the situation of speed, quantity and density is increasing, once happen accident, influence surface is wide, according to the problems in rush repairing, it is very important that how to confirm position accurately and identify the fault types effectively.

There is plentiful information contained in the fault traveling waves of the railway Catenary. It is un-balanced, and varying. Because wavelet transform has good resolution capability in both time domain and frequency domain, it can extract any fault feature by using smaller and smaller sample step for different frequency, it has perfect result to high frequency and low frequency, so wavelet transform is adapted to analyze the fault signal, and can locate the fault position accurately [1].

In this paper, according to compare the property of different kinds of mother wavelets, present a new method based on cubic B-Spline dyadic mother wavelet of centro-symmetric about zero, and used for analyzing the fault traveling waves by multi-scale decomposing. According to the different maximum mode values in different scales of wavelet transform, we can locate the

position accurately and singular degree of fault signal, and realize feature extraction. It can recognize the fault types accurately. Results show that the approach is effective.

2. Theory of Wavelet Transform

2.1. Definition of wavelet transform

The function satisfying the permit condition as follows is called mother waveletϕ (x)∈L2

0)( =∫+∞

∞−

dxxϕ (1)

Wavelet transform is a kind of expression of wavelet function f(x), and it is defined as:

∫+∞

∞−

−== dxs

xxfsxfxfW ss )()(/1)(*)( τϕϕ (2)

Where the s denotes the scale factor, τ is correspondence to shift, )(xsϕ is obtained from the mother wavelet using extension or compression for the scale factors. It is called wavelet as follows:

)(1)(sx

sxs ϕϕ = (3)

When s adopts 2j as the scale factor, this kind of wavelet is called dyadic wavelet.

2.2. Construction of Cubic B-Spline Dyadic Mother Wavelet

The choice limit of wavelet function is decreased hugely when using the dyadic wavelet transform. Here the derived of smooth function is chosen as a wavelet function.

Suppose )( xθ is a smooth function, )(xϕ is wavelet function, and

dxdxs

θϕ =)( (4)

Because different wavelet has different property, constructing or selecting distinct mother wavelet will achieve distinct analysis result. Which determine wavelet property is orthogonality, the orders of fade moment, the


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length of compact support and symmetry[2]. According to the properties, we can compare different mother wavelets: Daubechies wavelet has property of compact support in time domain, orthogonality and high fade moment, but symmetry and gliding property is poor; Shannon wavelet and meyer wavelet’s property of compact support in time domain is poor; Haar wavelet only has one order fade moment, so it isn’t suitable for analyzing the railway catenary fault signal. To sum up the above arguments，and according to the dyadic wavelet property, the characteristic of fault traveling wave and the rapidity of algorithm synthetically, it constructs the derived of Cubic B-Spline centering as a mother wavelet function[3-6], it can assure liner filter, and simplify of decomposition and reconstruction and prevent handling phase from distorting.Fig.1 shows the mother wavelet.

2.2.1. Definition of B-Spline Function

Suppose a nodal sequence knit

+}{ , define function 1

1, )](,,)[()( −++++ −⋅⋅⋅⋅⋅⋅−= k

kiiiikiki xttttttxB (5) as the No. I nodal sequence {ti}’s k order B-spline function [7-8].

Concluding from the fifth formula, when k=1, No. i 1 order B-spline function is

other 0

1)( 1

1,⎩⎨⎧ ≤≤

= +iii

ttttB (6)

Because i only expresses spline function parallel transl- ation on time axis, suppose B-spline function is zero center-symmetry and nodal distance is 1, it is defined as:

other 0

1/21/2 1)(1

⎩⎨⎧ ≤≤−

=t

tN (7)

2.2.2. Construction of Dyadic Wavelet

Dyadic wavelet transform analyses mainly the sudden

changing signal’s edge and sudden changing position. The signal has two kinds of mutability, pulse mutability and jump mutability, so construct two different kinds of dyadic spline wavelets: (1) zero center-symmetry (2) zero center- anti-symmetry.

(1) Construct zero center-symmetry dyadic wavelet. Suppose )(tϕ is a dyadic wavelet, )(ωH is its

generating element, N is random positive constant,

),( 21 Naaa ⋅⋅⋅ which satisfies 11

<∑=

N

kka is random N

dimensional real number vector, so )(ˆ)()2(ˆ ωϕωωψ OO G= is dyadic wavelet defined in )(2 RL , where

2

10 )(1])cos(1[)( ωωω HkaG

N

kk −+= ∑

=

(8)

When 0,1 == kaN , 20 )(1)( ωω HG −= (9)

When select different N and ka , we can construct different dyadic wavelet with different filtering coefficients according to the eighth formula.

(2) In the same way, we can construct zero center- antisymmetry dyadic wavelet.

According to )()( 11 tNNtN mm ∗= − and convolution theorem, we can conclude

)(ˆ)()(ˆ)2

(cos

)sin()2(ˆ)2(ˆ

ωωωωω

ωωωϕ

mmm

mm

NHN

N

=

=== (10)

So mH )2

(cos)( ωω = (11)

According to the tenth formula and Euler's formula, cubic B-Spline dyadic wavelet has the result:

)6(161)( 22 ωωωωω iiii eeeeH ++++= −−

The corresponding generating element’s filtering coefficients are

125.0,375.0,125.0 2101 ====− hhhh where h is low-pass filter coefficient, g is high pass filter, summarize as table1:

Table1. cubic B-Spline dyadic wavelet filter coefficient

n zero center-symmetry gn zerocenter-antisymmetry gn

0.

8048.

8833.

9160.

9225 .

0 .

0 .

0 .0

1-.

1634-.

3331-

.4038-.

4179 .

5564 .

5236.

5181.

5067

2-.

0980-.

0476-

.0266-.

0224 -

.1128 .

0804.

1137.

1804

3-.

0464-.

0175-

.0055-.

0030 .

1101 .

0541.

0445.

0252

4-.

0223-.

0080-

.0020-.

0008 -

.0801 -.

0296-

.0209-.

0036

(a) (b) Figure 1. (a) wave mode of smooth function

(b) wave mode of wavelet function


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3. The Feature Extraction about Fault Signal

3.1. Theory of Modulus Maximum

There may be plenty of fault information included in the section of sudden variety of fault traveling waves, which is very important to detect the sudden variety signals[9]. The signals of traveling waves have abrupt changes as the traveling waves reaching the fault point. The wavelet transform will appear maximum mode value, so the analysis of traveling waves will be transacted into the analysis of the maximum mode value of wavelet transform.

Maximum mode value is defined as follow: Suppose )(2 xfW d

j is the series of order coefficients of wavelet, if there are:

)()()( 22122 −− ≥≥ kd

kd

kd xfWxfWxfW jjj (12)

)()()( 22122 ＋＋ kd

kd

kd xfWxfWxfW jjj ≥≥ (13)

The above formulas are not permitted of taking equal sign simultaneously. The wavelet coefficients possess the maximum mode value at the point xk, the )(

2 kxfW j is called the modulus maximum of wavelet transform.

The maximum mode values of traveling wave signals are increasing as the scale increased, while the maximum mode values of noise are decreasing as the scale increased. The traveling wave signals on this ground can be extracted from noise by using modulus maximum.

3.2. The Wavelet Decomposition Algorithm

In the process of multi-definition analyses, we draw into discrete smooth approach and discrete detail approach. In order to apply it in the practice engineering and engage signal disposal conveniently, a fast pyramid decompose algorithm is published by Mallat[10].

Suppose the input signal is f(n), the mother wavelet function is described as above, in a certain scale, the discrete smooth approach component of correspondence signal (expressed as )(2 nfS jd ) is:

∑∈

−−= −

Zk

jdk

d knfShnfS jj )2()( 122 1 (14)

When j=0, )()(02 nfnfSd = , it is just prototype signal. The dyadic wavelet function should be discrete form

( )(2 nfW jd ), as follows:

∑∈

−−= −

Zk

jdk

d knfSgnfW jj )2()( 122 1 (15)

equation (14)(15) are just the Mallat algorithms.

4. Detection Example and Result Analysis

Simulation fault being detected and recorded by using the special circuit of HuaNeng ShangAn electrical power

plant, at Shi-Tai railway ShangAn switch station. Adopting the test step-up transformer, electrical railway overhead touch system sphere gap charge. Testing voltage is 13.7KV. Shorting the arm support and the overhead earth wire, the point is at a distance 600m from the switch station. By using of MATLAB, the fault wave mode and the results of analysis are lumped shown in Fig.2

Fig.2 showed that the position of the singular point of fault signal can be located accurately by using wavelet transform. The singular degree of singular point is shown in

the Figures clearly. The wavelet decompose of low scale(1-3 order) can extract fault features efficiently. When the scale of wavelet transform is increased, the effect is not obvious. So 3 order decompose is the best. The analyzing results of test are summarized as follows:

(1) The maximum mode values coming from fault point of traveling wave have the same polar, compared with the maximum mode values of the prototype traveling wave. The fault point distance x can be calculated by the following formula:

vx = △t/2 (16) Where v denotes the wave velocity.

The △ t represents the distance between the two maximum mode values.

From figure 3,we can find discharge pulse and reflect pulse appearing on ①sampled point 110 and ②sampled point 320. The distance between two points is 210 points. I did the experiment with cable fault flashover ranger, its sample frequency is 30MHz,the time interval between two points is 7us,the traveling wave speed of railway catenary is 170m/us, the fault distance by calculating is 600m, error is only 5m, the error results in low sample frequency, this is

s:test record wave mode; name ca1~ca5:5 orderapproach signals decomposed by wavelet; cd1~cd5: 5order detail signals; cm1~cm5:5ordermodulus maximum;

Figure 2. signal wavelet


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the method that traveling wave distance protection and traveling wave fault distance measurement to railway catenary current traveling wave is based on wavelet transform maximum mode values, which is significance.

(2) The transient traveling wave signals and noise appear different characteristics when using wavelet transform, the maximum mode values of traveling wave signals increased as the scale increased. The maximum mode values of noise decreased as the scale increased. The traveling wave signals on this ground can be extracted from noise by using wavelet transform.

5. Pattern Recognition

BP Neural Network is a kind of nonlinear mapping system with powerful pattern recognition ability, which can classify and recognize any complex status or process. Regarding all kinds of fault of railway catenary as a special running status, and using BP Neural Network as classifier, we can recognize fault property and giving an accurate triggering signal for re-closing equipment, then giving the effective act of re-closing equipment, avoiding the un- necessary drawback of secondary impact.

I adopt 3 layers BP Neural Network with an s type hidden layer and a liner-output. The input signal is the third order maximum mode values after extracting characteristics by wavelet transform. Important fault characteristics from many characteristics can be found, which both strengthened classified information and compressed characteristic data, thus reducing classified time.

Fig.3. showed that the Neural Network’s training output sum-squared error reduced gradually. The learning rate increased with training steps increasing. When training 94 steps, learning rate reaches its maximum and meets the demand for sum-squared error less than 0.02, now the error is 0.0192, learning rate is 0.0672. After training in the practice engineering, the results indicate that the BP Neural Network has perfect classified ability. By applying the BP Neural Network in railway catenary fault pattern recognition, we can distinguish fault types accurately with perfect results.

6. Summary

The cubic B-Spline Wavelet can be used for extracting the characteristics of transient signals accurately. It can also be used to check the details of fault signals after multi-scale decomposing. The position and the level of fault can be detected out efficiently by using the method of modulus maximum multi-scale wavelet transform, and improve study efficiency of Neural Network. The classifier of pattern- recognition of railway catenary by using BP Neural Network can recognize the fault types accurately. Results show that the approach is effective.

The system realizes intelligent fault diagnosis, which can locate the fault position and recognize the fault types accurately in time. The system plays an important role in assuring high-speed railway to run safely.

References

[1] YuHua Peng.Wavelet Transform and Engineering Application BeiJing：Science Press,2002.

[2] XiangChu Feng, XiaoBing Gan,GuoXiang Song. Numeric Functional Analysis and Wavelet Theory .XiAnXidian University Press,2003.57－60

[3] L. T. Liu, H. T. Hsu, B. X. Gao. A new family of orthonormal wavelet bases. Journal of Geodesy. 1998. 72:294-303.

[4] G.Plonka. Generalized spline wavelets. ConstructiveApproximation.1996.12:27-55

[5] Gang Zhao, Shu Hong Xu, Wei Shi Li. International Journal of Computational Engineering Science. 2004, 5(2): 403-415

[6] DaYi Yi, DaoQi Chen. numerical analysis Theory. Hang Zhou: Zhejiang University Press，2002.73-84

[7] Michael Unser, Akram Aldroubi, Murray Eden. A family of polynormial spline wavelet transforms. Signal Processing. 1993. 30:141-162

[8] Cheng Hong,S.Elangovan.A B-Spline wavelet based fault classification scheme for high speed protection relaying. Electric Machines and Power Systems. 2000 28:313-324.

[9] XinZhou Dong, YaoZhong Ge etc.Fault Characteristic Transient Current Travelling Waves and Its Analysis with Wavelet Transform. Transactions of China Electrotechnical Society.1999.14(1)

[10] Mallat S, zhong s.Wacelet transform maxima and multiscale edges in wavelet and their applications.Beykin G.leds Jonesand Baftlett Cambridge,1991

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