2006 International Conference on Power System Technology

Wavelet Entropy Measure Definition and ItsApplication for Transmission Line Fault

Detection and Identification(Part I: Definition and Methodology)

HE Zheng-you1 2, Chen Xiaoqing1, Luo Guoming11. School of Electrical Engineering Southwest Jiaotong University Chengdu 610031, Sichuan Province, China;

2.Key Laboratory of Power System Protection and Dynamic Security Monitoring and Control(North ChinaElectric Power University), Ministry of Education

Abstract-- Shannon entropy in time domain can measuresignal or system uncertainty. Spectrum entropy based onShannon entropy can be taken as a measure of signal or systemcomplexity. Use for reference, wavelet entropy measures built onwavelet analysis can signify the complexity of unsteady signal orsystem in both time domain and frequency domain. Beginningwith the thought of information mergence and post-analysis inthis paper, fundamental definitions of wavelet entropy measureare discussed, calculation methods including wavelet energyentropy, wavelet time entropy, wavelet singular entropy, wavelettime frequency entropy, wavelet average entropy and waveletdistance entropy are put forward, and their physical meaningsare analyzed. Considering wavelet entropy measure applied wellin field of EEG signal and mechanical fault diagnosis, thepotential and approach that it is applied in transmission line faultdetection and identification are analyzed.

Index Terms-- Shannon entropy, Wavelet Analysis, Waveletentropy, Fault detection, Electric Transmission line

I. INTRODUCTION

In Electric power system, when transmission line faultoccurring, plenty of transient components of differentfrequency will be generated. Lots of fault information is

included in the transient components. So it can be used topredict the fault or abnormity of equipments or power system,deal with the fault and analyze the reason of fault orabnormity, the reliability of the power system will beconsiderably improved. Today, to accurately obtain largeamounts of various fault transient information in time hasbecome the reality. But the key problem is how to use thosetransient signals to detect fault or to classify fault. Thereforethe new information mergence methods and the effectivetechnology used in detection and classification of electric

This work was supported by Chinese National Science FundNo.50407009, Sichuan Province Distinguished Scholars Fund No.006ZQ026-012 and the Science Fund of Key Laboratory of Power System Protectionand Dynamic Security Monitoring and Control, Ministry of Education, ChinaNo: KW02002

Email: hezyghome.swjtu.edu.cn

power system faults transient is need to studied.Information mergence usually includes a lot of

information mergence techniques, such as the estimation,statistics, information theory, artificial intelligence and so on.The information theory technique includes clustering analysis,relativity analysis, entropy theory and template methods, etc.[1]. Before analyzed by the methods above, feature picking-upis an important approach to information mergence, and thekey to detection and identification of transient signals.Recently there are many techniques used in picking-up thefeature of signals, e.g. time domain analysis, frequencydomain analysis, time-frequency domain analysis and bi-spectrum analysis [2]. A lot of paper has been published inthese fields, e.g. time-sequence analysis and wavelet analysisetc. These technologies have applied broadly in manyindustrial fields such as system parameter recognition,structure optimization, device operation-state detection andfault diagnosis. All above techniques in use are broughtforward and on effective a certain kind of engineeringproblems, but there are flaws in the signal synthetic featureanalysis or information mergence.

The purpose of this paper is to find an effective techniquewhich is not only suitable for the application of the transientsignal feature analysis (eg. information quantity, uncertaintyand complexity) but also suitable for the fault detection anddiagnosis of power system transmission lines. Based on thewavelet decomposition technology and in reference to thedefinition of information entropy, various wavelet entropymeasures and according calculation methods are put forward.Meanwhile, the unique capability of wavelet entropy measuresin the application of the signal synthetic feature analysis andthe potentials in transmission line fault detection andclassification are also shown. The application of waveletentropy measures in power system is developed in part II:fault detection and part III: fault identification.

II. WAVELET TRANSFORM AND ENTROPY

A. Wavelet Average EntropyTransient signals have some characteristics such as high

frequency and instant break, so wavelet transform is strong

1-4244-0111-9/06/$20.00c2006 IEEE.

I

2

tool for them in feature picking-up, and it satisfies the analysisneed of electric power transient signals. Usually wavelettransform of transient signal is expressed by multi-revolutiondecomposition fast algorithm which utilizes the orthogonalwavelet bases to decompose the signal to components underdifferent scales. It is equal to recursively filtering the signalwith a high-pass and low-pass filter pair. Filtering by high-pass filter produces details and filtering by low-pass producesapproximations. The band width of these two filters is equal.After each circle of decomposition, the sampling frequency isreduced by half. Then recursively decompose the low-passfilter outputs, both components of the next stage are produced.Given discrete signal x(n) being fast transformed, at instant kand scale j it has high-frequency component coefficientd1(k) and low-frequency component coefficientaj(k). The

frequency band of the information contains in signalcomponents Dj (k), Aj (k) obtained by reconstruction is [3, 4],

SDi.(k) [2 -(j+')F,2 -jF,]y1 m 1

UAj (k): [0,2-(j+') F, ]j=12 M 1

Where f is the sampling frequency. The original signalsequence x (n) can be represented by the sum of allcomponents, namely

x(n) D1(n) + A1 (n) = D1(n) + D2(n) + A2(n)X (2)Z Dj(n) + A,(n)j=I

For the purpose of unification, denote A. (n) by D.+1 (n) andwe get

J+1x(n) = DjV (n) (3)

j=I

Dj (n) represents the component of transient signal x(n) at

each scale (frequency band), it is also the multi-resolutionrepresentation of the signal which can act as feature subset ofclassification.

For continuous wavelet transform, series of discretewavelet coefficients Di under the different scales

j (j = 1..,m) are obtained, which can reflect time-frequencydistribution to some extent. Below is the partial definitionsand calculations based on wavelet transform result Dj(k)using multi-resolution analysis, which can be extended todiscrete result of the continue wavelet transform.

B. Information EntropyThe uncertainty of any event is associated with its states

and probabilities. The aggregation of all possible states iscalled sample space {x1, x2, ..., xn4. Each piece of informationhas a probability P(xi)=Pi, OPi<l, ZPi = 1. The self-information quantity of the event xi is,

I(Xi) =-log, P(Xi) =-log, P (4)I(xi) is a random variable changing with different information,so it is not suitable for measuring the whole informationsource. Therefore, we define the mathematical expectation ofthe self-information as the mean self-information of the

information source, which is entropy denoted by H(X).H(X) = E[I(x1)] = E[-log, F] =Z-P1 logI P1 (5)

The base a of the logarithm defines the unit of the entropy.When a is 2, e and 10 , the unit of the entropy is bit, nat andHartely respectively. Customarily, we choose a=e. Theinformation entropy above is used to measure the meaninformation quantity of the information source. When allevents have the same probabilities, the uncertainty of a certainevent reaches its maximum, so does the entropy. The entropyof any certain event is zero. Therefore, entropy is the measureof the uncertainty.

C. Spectrum EntropyBased on conception of information entropy and power

spectrum, the spectrum entropy is defined in the frequencydomain [5].Given X(co) as the DTF of signal x(n), the power

spectrum is S() 1 2X( Because of the conversion of

energy in time and frequency domain, namelyEx2 (t)At = y |X()2An, S={S, S2, , Sn is a partition of

original signal, so the proportion of i-th power spectrumoccupied in whole is p S The correspondingPX n

i=l

information entropy namely power spectrum entropy is thefollowing,

(6)n

H = -, Pi1og Pii=l

Spectrum entropy is a measure of the signal complexity.Narrower the peak of the signal power spectrum is, smallerthe spectrum entropy is. Which means the signal is moreregular and less complex. Flatter the power spectrum is, largerthe spectrum entropy is. For example, the white noise isirregular random signal, it has flat power spectrum and largespectrum entropy, which means the signal has highcomplexity.

III. VARIOUS WAVELET ENTROPY MEASURES AND THEIRCALCULATION METHODS

When it is used in fault picking-up, the amplitude andfrequency of the test signal will change significantly as thesystem change from the normal state to fault. The Shannonentropy will change accordingly. But it is not capable ofdealing with some abnormal signals, while wavelet can solvethis question. Wavelet combined entropy can make full use oflocalized feature at time-frequency domains which waveletanalysis deals with unsteady signal and embody the abilitywhich information entropy expresses information of signal. Sowavelet entropy not only can touch the purpose of informationmergence, but also can analyze fault signals more efficiently.Like the spectrum entropy, below various wavelet entropymeasures are defined. (In each definition, Ek=lDjk) 2 is the

wavelet energy spectrum at scale j and instant k, E ZEk isk

the wavelet spectrum at scale j, Fig.1-Fig.6 are respectivelythe fundamental drawing of their corresponding wavelet

3

entropy).

A. Wavelet Energy Entropy

in W(m; w, 8) . Thus we get the definition wavelet timeentropy (WTE) under scale j as,

Definition 1: Giv E = 2 m as wavelet spectraof signal x(t) on m scales. Then E is a partition of signalenergy at scale domain. According to the orthogonal wavelettransform, at certain time window (window width is w C N)

£total signal power E is the sum of power

E of each

=E. IE p. =1component. IfP E ,/E then i We thus defineaccordingly the wavelet energy entropy (WEE),

WEE =- pj log pj (7)J

When the window sliding, rule that the WEE moves withtime can be get. In the above expression, the scale space iscorresponding to the frequency space. The WEE defined inExpression (7) indicate the energy distribution of signals.Wavelet function does not have pulse selectivity at eitherfrequency or time domain, whereas it has a support region. Sothe partition of current or voltage energy at scale space alsoindicates the signals' energy distribution at time andfrequency domain.

t

22 W-E-- ---

EEm1 E,2EIw E,gE EI

Fig. 1 Fundamental of wavelet energy entropy

B Wavelet Time EntropyA sliding-window w E N is defined under the wavelet

transform result D1 (k), and the sliding factor is 8 E N. Then

W(m;w,6) ={IDj(k),k =1+md,...w+md} should bethe sliding window, m 1,2,N i,M Divided the sliding-

L

window into the following L sections W(m; w,d) = UZ/,/1=

therein {Z1 = [s/_11 s), / = 1,2,.--L} do not intersect.

Moreover, so <SI<s2<.2 <SL, and so =min[W(m;w,5)]

SL =max[W(m; w, 8)] .

Definition 2: Given ptm (ZI ) as the probability that

wavelet coefficient D1 (k) E W(m; w, 3) falls into section Z1according to the classic probability theory, it is the proportionof the amount of Di (k) within Z1 to the total coefficients

WTE() - pm(Z,)log(pm(z,)) m = 1,2,---M (8)iL t~~~~~~~~

:,

p'tZ1Illfz2)Ep'(4ZI) **- 1\ 1 Zk M

tFig.2 Fundamental of wavelet time entropy

WhereM = (N - w) / CE N, Accordingly WTEj (m) at

each scale can be calculated and the WTEJ figure

{w/2+md,WTEj(m)},m =1,2,**-M1 also can be drawn.

WTEJ is capable of detecting and locating the change of

signals or system parameters, even its calculation load is low.Lest the noise disturbance when j=1, usually we choose scale j=2 or greater.

C. Wavelet Singular Entropy

Given wavelet decompositions Dj (k) of the signal

constitute a m x n matrix D. According to the signal singularvalue decomposition theory, for any m x n matrix, there exista m x I matrix U, a I x n matrix V and a l xl matrix A,which make D n Ur xA1 1VTxn, therein the diagonal

elements A, (i = 1,2,..., 1) of the diagonal matrix A are no

minus and in descent order, namely 4 > 22 > ....* > 0°these diagonal elements are the singular values of the wavelettransform result Dmxn In reference to the signal singularvalue decomposition theory, when signal has no noise or highsignal-to-noise, only a few nonzero diagonal singular values.The singular value of the wavelet transform result Dmnxobserves the similar rule. Furthermore if there are fewerfrequency components, then there are fewer nonzero singularvalues of the wavelet decomposition.

Definition 3: To describe the signal frequency componentsand their distributions quantitatively, define the waveletsingular entropy (WSE) as follows,

WSE = E Ap,i=l

(9)

Fig .3 Fundamental of wavelet singular entropy

E )g i E07

I

Li I

L4u .1

.w

.

Fig .4 Fundamental of wavelet time-frequency entropy

Where Ap, A 0A l that is the j-th

rank increment wavelet singular entropy. The singular valuedecomposition of the wavelet transform result is equal to mapthe correlated wavelet space to linearly independent feature-space. Combining the redundant information, the waveletspace singular entropy indicates the distribution uncertainty ofthe energy in signal time-frequency space. Simpler theanalyzed signal is, fewer modes the energy congregates to andsmaller the wavelet singular entropy is. Vise versa. Thereforethe wavelet singular entropy that we defined previously is theindex to measure the signal complexity or uncertainty.

D. Wavelet Time-Frequency Entropy

Definition 4: The discrete wavelet presentation Dj (k)

mentioned before is in fact a two dimension matrix. Alongwith variable k and j two vector sequences can be get.Therefore we define wavelet time-frequency entropy (WTFE)as,

WTFE(k, j) = [WTFEt(t = kT),WTFEf(a = 2' )]Where

WTFEt(t = kT)

WTFEf (a = 2') =

PD(a=2j) 1nPD(a=2

E PD(t=kT) InPD(t=kT)k

Where PD(=2j) Dj (k) / Dj (k)|

(10)

(11)

(12)

PD(t=kT) =

Dj (k) i2 / D (k) 2. The result of WTFE measure consistsk

of two vectors or sequences. The first vector stretches on thewhole time space and the second vector stretches on the wholefrequency space. A large entropy value at instant kT indicatesthere are widely distributed wavelet coefficients extend allover the frequency space. On the other hand, a small entropyvalue indicates wavelet coefficients congregate at a fewfrequency points or segments. WTFE is able to measure thesignal information feature at any given instant and frequency.Therefore it can be used to classify different signals and haspotential in the fault detection and diagnosis field.

E. Wavelet average entropy

Different signal distributions at the wavelet time-frequency domain are presented by the differences of theenergy distribution on a small time-frequency plane within thetime-frequency (phase space) domain. The homogenization ofenergy distribution in each time-frequency area reflects thedifference of the physical state in the observed system. Basedon this idea, the definition of entropy based on the waveletanalysis is given on the entire time-frequency plane. We call itwavelet average entropy because it is has the property ofaverage.

Definition 5: Dividing the signal wavelet transform time-frequency plane (t = kT,a = 2i) by N scaled time-frequencywindow area [kT- aA,,, KT+aA,,]x[(0*/a-Ala,co*/a+A la], ( A, ,I As are the time

domain radius and frequency radius of the mother wavelet,co* is the mid-point of the mother wavelet, T is the discretestep and k is the discrete sequence 0,1,2... .,N). If the energy ofeach area is E (i = 1,2,.**, N) and the total energy

N

is E Ei, standardize each energy and we get Pi E, / E,

NSo p = 1 . Therefore the definition of the wavelet average

i=l

entropy (WAE) is,N

WAE = - ZP, lnPi=l

WAE indicates the average complexity of the whole signal,it does not change when time or frequency changes. In certainapplications, different signals have close energy distributions,the signal feature pick-up and classification become difficult.According to the property of the information entropy, themore equal the energy distributes, the larger the total WAE is.And vise versa. In fault detection, when system (ortransmission line) is normal, the fundamental wave is a maincomponent, and energy distributes on few frequency sectionunequally, so WAE is smaller. While when fault occurs, the

(13)

4

Imv m Ai -liiiiiiiiiiiiiiiiiiiiillillillillilli

T'%

5

high frequency transient distributes widely, so WAE increases..

-...

- - n

,,... l z .2 1 oin wfSin,wEi=- --;LZ_ _2;;L i-;,_ - _ _ _

IW .. . ...i N l:S

Fig .5 Fundamental of wavelet average entropy

F. Wavelet distance entropy

Definition 6: For the discrete wavelet presentation Dj (k)along with variable k we can get a vector sequence D(k). Inreference to the definition of distance and introduce theinformation calculation methods then we define waveletdistance entropy (WDE) as follows,

m m

WDE -E d Ind (14)k=l1=1k=/=Il

N NWhere d1 = dkl Z Z dkl

k=l 1=1

k,lI = 1,2,---N.

power system. Like the EEG signal and mechanical faultsignal, fault transient signals on transmission line are alsobreaking signals composed of different frequency components.Inspired by that, the potential of application using waveletentropy in transmission fault detection and classification areanalyzed below.

Taking the two fault signals for example, after multi-revolution analysis, the wavelet energy and wavelet time-frequency entropy (including WTFEt and WTFEf) arecalculated. Seeing Fig.7 and Fig.8, some conclusions are get,OBecause information of signal under the different frequencyband varies, so the wavelet energy spectrum also varies asfrequency change. (g)WTFEt along time has an singularityvariety at instant of frequency change, so the break can bedetected; (®) For different signals, such as signal 1 and signal 2,WTFEf along frequency stretches on different trends. So, if itis input to one classifier, various signals will be recognized.

20

<?o-20

0 2 4 6 8 10 12 14 16 18 20tVms

t5

:- 1 -.

pmC-M

O.5 -

dkl D(kT) - D(IT)o

1,l 4 6 ° '- A

Fig.7 signal 1 and its wavelet energy and WTFE

0.5

I2 Al iA._/) n/LI~~~~~~~~~~~~~~~~~~.. us "I

Fig .6 Fundamental of wavelet distance entropy

IV. POTENTIAL ANALYSIS OF WAVELET ENTROPY APPLIED INTRANSMISSION FAULT DETECTION AND IDENTIFICATION

The definitions above, some of which have been applied inthe fields of biological EEG Signal and mechanical faultdiagnosis [6-10]. Thus, wavelet entropy measures have someunique capabilities in feature picking-up of transient signals.Meanwhile, in reference [I1, 12], it is pointed out that therewill be a good foreground if wavelet entropy is applied in

-0.50 2

0.06

EDM40 H.02oUU

4 6 8 10tims

12 14 16 18 20

30ig.8 se

Fig.8 signal and its wavelet energy and WTFE

lb 1o16 is

< o -- -

6

From the simulations above which take the wavelet time-frequency entropy for example, we can know that, 0) Waveletentropy distributing along time can detect fault transient. Butin practice, threshold is must be set to judge whether faultoccurs. The flow is shown in Fig.9. (2) Wavelet entropydistributing along frequency can reflect the difference ofvarious signals entropy values, it is hopeful for recognizingtransient signals, such as the common short circuit andlightning strike, breaker operating and capacitance switching.Now, if neural network acts as a classifier, the flow of signalrecognition is shown in the Fig. 10, which indicates twoapproaches, the one touches a purpose of waveformidentification by picking-up wavelet entropy feature, the othertouches a purpose of recognition by computer after thefeatures are input to neural network.

Obtainig thecanalyzed signmis

Picking up the WEIhaxur of tiesigrl

Settlig ti Lhold value and compai ringit uwthcayputing vanue ofWE o

WDEae putforwadandthircgalculainmthderpresented.th treh

y

wFandt NomfaultFig 9 The flow of detection

VoltLS or Waveleta xtwMio.

and roidetsifiatio are aalyzed.- Thenetw tp il etdyo

.~~~~~O Signls

a prsontlinefldtctio n a Idificat

Fig.wa The flow of identification

V. CONCLUSION

Wavelet entropy measure based on wavelet analysis is ableto observe the unsteady signals and complexity of the systemat time-frequency plane. Based on this, beginning withinformation mergence and post-analysis in this paper, wavelettheory and information entropy are combined, six waveletentropies including WEE, WTE, WSE, WTFE, WAE andWDE are put forward and their calculation methods are

presented.From the aspect that wavelet entropy measure is applied

well in EEG signal and mechanical fault diagnosis, thepotential that it is applied in transmission line fault detectionand identification are analyzed. The next step will be study ofa practical transmission line fault detection and identificationmethod based on wavelet entropy.

VI. REFERENCES[1] R.C.Luo, Datafusion and sensor integration: State ofthe art 1990s.Data

fusion in robotics and machine intelligence. Academic Press.Inc1992,pp. 127-135

[2] Zeng Xiangjun, K.K.Li, W.L.Chan, Chen Deshu, "Discussion onapplication of information fusion techniques in electric power systemfault detection", Electric Power. 2003, 3(6), pp.48-12

[3] I.Daubechies. "The wavelet transform, time-frequency location andsignal analysis". IEEE Trans, IT. 1990, 36(5), pp. 961-1005.

[4] S. Mallat. "A theory for multiresolution signal decomposition: Thewavelet representation". IEEE Trans. PAMI. 1989, 11(7), pp. 674-693.

[5] Wang Taiyoong, LIU Xingrong, Qin Xuda, et al. "Spectrum entropy andits application in characteristics abstraction of magnetic flux leakagesignals". Journal ofTianjin University.2004, 37(3), pp.216-220.

[6] Daniel Lemire, Chantal Pharand, Jean-Claude Rajaonah, et.al. "WaveletTime Entropy, T Wave Morphology and Myocardial Ischemia". IEEETransactions on Biomedical engineering. 2000, 47(7):967-970.

[7] A. M. Petrock, Dr. S. Reisman, Dr. I. Dardd. "Total Wavelet EntropyAnalysis of Cyclic Exercise Protocol on Heart Rate Variability". IEEE.2004:91-92.

[8] Feng Zhouyan, Dynamic analysis of the rat EEG using wavelet entropy.ACTA Biophysical Sinica. 2002,18(3), pp.325- 330.

[9] Quiroga RQ,Rosso OA ,Basar E,et al. "Wavelet entropy in event-ralatedpotential:a new method shows ordering of EEG oscillations". BiologicalCybernetics 2001,84(4),pp.291-299.

[10] Zhang Wenju, Su Qingzu. "Exploration of shannon entropy for faultdiagnosis is of vehicle gearbox", Chinese Journal of the AgricultureMachine, 2002, 33(1), pp.80-83

[11] Zhimin Li, Weixing Li, Ruiye Liu. Applications ofentropy principles inpower systems: A Survey. IEEE/PES Transmission and Distribution:Asia and Pacific Dalian, China. 2005:1-4.

[12] He Zhengyou, Cai Yumei, Qian Qingquan. "A study of wavelet entropytheory and its application in electric power system fault detection".Proceedings ofthe CSEE,2005, 25(5), pp.23-43.

VII. BIOGRAPHIES

He Zhengyou, received the PH.D. degree in electricalengineering at Southwest Jiaotong University, in Mar.2001. B.S degree in computation engineering fromChongqing University, P.R.China, in 1992. Hereceived M.S. degree in the same department fromChongqing University in june 1995. His researchinterests are in the area of Signal Process andInformation Theory and its application in electrical

.......... power system.

Chen Xiaoqin, is a graduate student, who is studyingfor a M.S degree in college of electrical engineeringSouthwest Jiaotong University. Her research interestsare in the signal processing and information theory inelectrical power system.

Luo Guomin, is a graduate student who is workingfor a M.S degree in college of electrical engineeringin Southwest Jiaotong University. Her researchinterests are in application of information theory in

power system.