Research Article Application Research of the Sparse …downloads.hindawi.com/journals/ijap/2016/1343194.pdf · 2019-07-30 · Research Article Application Research of the Sparse Representation

Research ArticleApplication Research of the Sparse Representation ofEigenvector on the PD Positioning in the Transformer Oil

Qing Xie12 Dan Liu2 Ying Zhang2 Shuguo Gao3 Tong Li2

Xinjie Wang4 and Fangcheng Luuml2

1State Key Laboratory of Alternate Electrical Power System with Renewable Energy SourcesNorth China Electric Power University Baoding Hebei 071003 China2Hebei Provincial Key Laboratory of Power Transmission Equipment Security DefenseNorth China Electric Power University Baoding Hebei 071003 China3State Grid Hebei Electric Power Research Institute Shijiazhuang 050021 China4Department of English North China Electric Power University Baoding Hebei 071003 China

Correspondence should be addressed to Qing Xie xq ncepu126com

Received 8 April 2016 Revised 23 August 2016 Accepted 27 September 2016

Academic Editor Elias Aboutanios

Copyright copy 2016 Qing Xie et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The partial discharge (PD) detection of electrical equipment is important for the safe operation of power system The ultrasonicsignal generated by the PD in the oil is a broadband signal However most methods of the array signal processing are used forthe narrowband signal at present and the effect of some methods for processing wideband signals is not satisfactory Thereforeit is necessary to find new broadband signal processing methods to improve detection ability of the PD source In this paperthe direction of arrival (DOA) estimation method based on sparse representation of eigenvector is proposed and this methodcan further reduce the noise interference Moreover the simulation results show that this direction finding method is feasible forbroadband signal and thus improve the following positioning accuracy of the three-array localization method And experimentalresults verify that the direction finding method based on sparse representation of eigenvector is feasible for the ultrasonic arraywhich can achieve accurate estimation of direction of arrival and improve the following positioning accuracy This can provideimportant guidance information for the equipment maintenance in the practical application

1 Introduction

Electrical equipment working status is directly related to thereliable operation of power system And the practice hasproved that the PD is the main reason for the high voltageelectrical equipment insulation breakdown finally In orderto avoid accidents and timely find the potential danger itis necessary for the electrical equipment partial dischargetesting to ensure system reliability [1ndash6]

In the PD detection an array sensor is used to collectultrasonic signals generated by the PD Then the arraysignal processing technology is used to complete the sourcedirection of arrival (DOA) estimation and positioning Thismethod not only has strong anti-interference ability butalso has high positioning accuracy and it has been widelyused in many areas [7ndash9] However the main processing

object of the traditional array signal processing methods isa narrowband signal and the corresponding variety of spacespectrum estimation (direction of arrival DOA) methodsthat have high resolution and fast computing speed havebeen successfully appliedThe electrical equipment ultrasonicsignal generated by the PD in transformer oil is a typicalbroadband signal [10 11] so the study on theDOA estimationalgorithm that is suitable for wideband signal has extremelyimportant significance

The more classical wideband direction finding algorithmis mainly divided into two categories

The first kind ofmethod is incoherent subspace algorithm(ISM algorithm) [12 13] It is that a broadband signal isdivided into a number of narrowband signals and theaverage value is obtained after estimating the DOA of eachnarrowband signal This method is a simple average of the

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 1343194 13 pageshttpdxdoiorg10115520161343194

2 International Journal of Antennas and Propagation

narrowband signal processing results which has a largeamount of calculation and cannot overcome the shortcom-ings of subspace algorithms adopted by the narrowbandsignal such that it is easily affected by noise and samplingpoints and cannot solve the coherent sources The secondmethod is the coherent signal subspace algorithm (CSMalgorithm) [14ndash17] A focusing matrix is used to focus onall frequency components on a single reference frequencyNarrowband signal processing method is used to estimatethe DOA of the covariance matrix after focusing whichreduces the correlation coefficient between signals and canachieve the goal of coherent solution Moreover the existingCSM algorithm has to use the traditional narrowband signalprocessing method after focusing which is still unable toavoid the disadvantages of subspace algorithms

Mallat and Zhang in 1990s proposed the theory of signalsparse decomposition [18 19] It can be constructed by usingdifferent ways according to the specific signal form and differ-ent research purposes Although the signal is represented bya handful of basis functions the information in the signal alsofocuses on these few basis functions so it is more conduciveto extract and explain the essential characteristics of signalsAt present the signal sparse decomposition has been widelyused in signal noise reduction compression coding andimage processing and other fields [11] In this paper thesparse decomposition theory is applied to the PD signal DOAestimation According to the array signal direction vectors anovercomplete atom dictionary is established The matchingpursuit (MP) algorithm is used to choose the appropriateatoms and the angle information contained in the atoms isthe DOA of signal sources

Thiswork studies the PDpositioningmethod in the trans-former oil based on the sparse representation of eigenvectorsTaking a nine-element circular ultrasonic array sensor as anexample the mathematical model of ultrasonic array signalsis given Firstly the broadband PD signals are received byan ultrasonic array sensor and the covariance matrix of asingle frequency is obtained by using RSS focusing method[20] Then an eigenvector corresponding to the maximumeigenvalue is obtained through eigendecomposition of thecovariance matrix obtained the eigenvector is as the amountto be decomposed According to the reference frequency andthe steering vector form of an array signal a step and stepovercomplete dictionary is established and thus the DOAestimation of the PD signal can be obtained byMPMoreoverthismethod can further reduce the noise interference Finallyaccording to the results the PD source is located by using thethree-array cross positioning principle The simulation andexperimental results show that the direction finding methodbased on sparse representation of eigenvectors can get higheraccuracy of the DOA estimation results and improve thesubsequent positioning precision

2 Broadband PD Ultrasonic Array Signal

21 The Mathematical Model of Array Signal The researchresults show that the ultrasonic frequency produced by thePD in transformer oil is mainly concentrated in the range of

50 kHz to 400 kHz the center frequency is between 70 kHzand 200 kHz and so the PD ultrasonic signal source is atypical broadband signal

Assuming that a uniform array consists of M equallyspaced elements and there is a space with 119875 broadbandsignals the incident angle is respectively 1205931 1205932 120593119875 andthe signal received from the119870th element can be expressed as

119909119896 (119905) =119875sum119894=1

119904119894 [119905 minus 120591119896 (120593119894)] + 119899119896 (119905) (1)

where 119904119894(119905) (119894 = 1 2 119875) is incident broadband signal119899119896(119905) is additive noise 120591119896(120593119894) is time difference relative to thereference node when the 119894th signal source is received by the119896th element

The time shift theorem of Fourier transform is as followsa signal is carried on Fourier transform after the signal hasa time shift equal to that of the signal that has a phase delayafter Fourier transform If 119904(119891) is the Fourier transform formof 119904(119905) that is

FFT [119904 (119905)] = 119904 (119891) (2)

then the Fourier transform form of 119904(119905 + 120591) isFFT [119904 (119905 minus 120591)] = 119904 (119891) 119890minus1198952120587119891120591 (3)

For the signal received by the 119896th element both sides of(1) are analyzed based on Fourier transform

119909119896 (119891) =119875sum119894=1

119904119894 (119891) 119890minus1198952120587119891120591119896(120593119894) + 119899119896 (119891) (4)

The Fourier transform for 119872 elements can be written inmatrix form which is

[[[[[[[

1199091 (119891)1199092 (119891)

119909119872 (119891)

]]]]]]]

=[[[[[[[[

119890minus1198951205961205911(1205931) 119890minus1198951205961205911(1205932) sdot sdot sdot 119890minus1198951205961205911(120593119875)119890minus1198951205961205912(1205931) 119890minus1198951205961205912(1205932) sdot sdot sdot 119890minus1198951205961205912(120593119875)

d

119890minus119895120596120591119872(1205931) 119890minus119895120596120591119872(1205932) sdot sdot sdot 119890minus119895120596120591119872(120593119875)

]]]]]]]]

[[[[[[[

1199041 (119891)1199042 (119891)

119904119875 (119891)

]]]]]]]

+[[[[[[[

1198991 (119891)1198992 (119891)

119899119872 (119891)

]]]]]]]

(5)

And they can be written as

X (119891) = A (119891 120579) S (119891) + N (119891) (6)

International Journal of Antennas and Propagation 3

Among them the steering vector matrix is

119860 (119891 120593) =[[[[[[[[

119890minus1198951205961205911(1205931) 119890minus1198951205961205911(1205932) sdot sdot sdot 119890minus1198951205961205911(120593119875)119890minus1198951205961205912(1205931) 119890minus1198951205961205912(1205932) sdot sdot sdot 119890minus1198951205961205912(120593119875)

d

119890minus119895120596120591119872(1205931) 119890minus119895120596120591119872(1205932) sdot sdot sdot 119890minus119895120596120591119872(120593119875)

]]]]]]]] (7)

The signal direction matrix 119860(119891 120593) is different fromnarrowband direction matrix Here the frequency is thewhole band of the signal while the frequency is a single fixedvalue in a narrowband model

When the signal is analyzed based on the discrete Fouriertransform (DFT) with 119869 points the frequencies are 119869 discretepoints and then (6) can be discrete as

119883(119891119895) = 119860 (119891119895) 119878 (119891119895) + 119873(119891119895) 119895 = 1 2 119869 (8)

The steering vector matrix is

119860(119891119895 120593) = [119886 (1205931 119891119895) 119886 (1205932 119891119895) sdot sdot sdot 119886 (120593119875 119891119895)] (9)

where a(120593119894 119891119894) (119894 = 1 2 119896) is a steering vector

a (120593i fj) =[[[[[[[[

eminusj2120587fj1205911(120593i)

eminusj2120587fj1205912(120593i)

eminusj2120587fj120591M(120593i)

]]]]]]]] (10)

22The Structure of the Circular Ultrasonic Array Sensor Thecircular ultrasonic array sensor is composed of 119872 identicalelements evenly distributed on the circumference with aradius of 119877 in the 119909-119910 plane the elements are arranged asshown in Figure 1 (eg taking nine element) The coordinatesystem of the sphere is used to express the DOA of theincident plane wave and o is in the center of the array whichis the origin of the coordinate system Consequently it istaken as a reference point In addition when the incidentsignal direction is (120572 120579) azimuth 120572 is expressed as the anglebetween the 119909-axis and a projection in the 119909-119910 plane andthe projection is wired from the reference point to the sourceof the signal The pitch angle 120579 is the angle between the 119911-axis and the wired one that is from the reference point to thesource of signal Then the delay time 120591119898 in which the signalarrives at the119898th element relative to the reference element is

120591119898 = 119903119888 (cos(2120587 (119898 minus 1)

119872 minus 120572) sin 120579)119898 = 1 2 119872

(11)

1

2

34

5

6

7 89

z

y

x

120579

120572

r

p

Figure 1 The structure of a nine-element circular ultrasonic array

Then according to (10) and (11) the steering vector of an119898-element circular array can be expressed as

a (120572 120579 119891)

=

exp [minus1198952120587119891 cos120572 sin 120579119903119888 ]

exp [minus1198952120587119891 cos(2120587119872 minus 120572) sin 120579119903

119888 ]

exp [minus1198952120587119891 cos(2120587 (119898 minus 1)119872 minus 120572) sin 120579119903

119888 ]

(12)

where the frequency 119891 is the whole frequency band of thesignal

3 DOA Estimation Based onSparse Decomposition

31 The Mathematical Expression of Sparse RepresentationGiven an overcomplete dictionary D = Φ119894 119894 = 1 2 119868there are 119868 atoms which is a whole Hilbert space 119867 = 119877119889and 119868 gt 119889 Therefore for any signal expressed as 119910 119910 isin 119867the 119896 atoms can be selected adaptively in D to make sparseapproximation with the signal 119910 that is

119910 = sum119903isin119868119896

119888119903Φ119903 (13)

where 119868119896 is index set ofΦ119903 and the corresponding coefficientsare expressed as 119862 = 119888119903119903isin119868119896 The atomic number 119896 selectedis usually much smaller than the atomic number 119868 in theatomdictionary A few atoms can express the signal so-calledsparse representation

The matrix is used to express 119910 isin 119877119889 and D isin 119877119889times119868 the119894th column ofD is Φ119894 and then (13) can be written as

Y = Dc (14)

where c isin 119877119868 is a sparse vector

4 International Journal of Antennas and Propagation

The ways in which the atoms in the overcomplete dic-tionary are used to express the signal have infinite varietyof forms Therefore how to effectively solve the sparsecoefficient vector c is an important problem the sparse repre-sentation which is the basic problem of sparse representationand the specific expression is as follows

argmin 1003817100381710038171003817c01003817100381710038171003817 st 119910 = Dc (15)

where c0 is 1198970-norm of c which is the number of the nonzeroelements in the coefficient vector c

32 Application of Matching Pursuit Algorithm in DOA Esti-mation In the application process of sparse representationmethod different overcomplete dictionaries are constructedaccording to different research purposesWhen an ultrasonicarray is used to estimate theDOAof the broadbandPD signalthe overcomplete dictionary can be structured according toa steering vector matrix form of the received signal Thesteering vector matrix contains the wave direction of signalstherefore constructing a group of atomic vectors coveredspace at any angle inevitably includes the DOA of signalsBased on sparse representation theory these atoms thatinclude the DOA of signals can be selected by usingmatchingpursuit (MP) algorithm and they can be used to realize thedirection finding

The principles of the MP algorithm are similar to theadaptive projection decomposition algorithm Firstly theatoms that match with the signal mostly are selected fromthe overcomplete dictionary which is the idea that theseatoms have the maximum inner product with signal Herethe projection coefficient is that the signal on the atom is thelargest and the rest of energy on the atom after decompo-sition is minimum Next the same method is used to findout the best matching atoms with the remaining amount andthen make decomposition Repeat the above steps When theremaining energy of decomposition is small enough or thebest matching atom combinations can represent the originalsignal stop the decomposition The flow chart of the sparserepresentation by using MP algorithm is shown in Figure 2

When estimating the DOAs of the ultrasonic signalgenerated by 119901 PD sources the incidence angles of the signalcan be searched on 119873 angle vectors that have been set Ingeneral the number of the PD sources is much smaller thanthe number of angles to be searched that is 119875 ≪ 119873 A searchvector of the angle is constructed and the vector is coveringall space angle which is 120573 = [1205731 1205732 sdot sdot sdot 120573119873] There are 119875components equal to 1205931 1205932 120593119875 respectively Accordingto (9) the search matrix of angle is constructed as

119860 119904 (120573) = [119886V (1205731) 119886V (1205732) 119886V (120573119873)] (16)

The direction finding by using the sparse representationtheory is to decompose the received signals on the atomswith different directions The projection value is maximumwhen the incident signal has the same direction with theatoms According to the relevant knowledge of the vectorprojection theory in mathematics the projection of an array

Start

Input the signal that needs to bedecomposed

Set decomposed parameters

Formed overcomplete dictionary

Search for the optimal atom inthe overcomplete dictionary

The component of the optimal atom is reduced inthe signal or residual signal completed a stepof decomposition

Complete decompositionN

Y

Save the results of decomposition

End

Figure 2 The flow chart of sparse representation based on MPalgorithm

signal on the atom is maximum which means that theinner product module between the array signal and thecorresponding atom is maximum Firstly the parameter tobe decomposed is set to beX andXmakes the inner productwith each atom 120572V(120573119899) (119899 = 1 2 119873) then the optimalatom 119886V(1205731205740) 1205740 isin 1 2 119875 is selected by the absolutevalue of the inner product and the optimal atom meets thefollowing conditions

10038161003816100381610038161003816⟨X aV (1205731205740)⟩10038161003816100381610038161003816 = sup 1003816100381610038161003816⟨X aV (120573119899)⟩1003816100381610038161003816 (17)

The received signalX is decomposed into the componentof projection on aV(1205731205740) and the remains of the signal

119883 = 119875119886V(1205731205740)119883 + 119877119883 (18)

where 119875119886V(1205731205740)119883 is the projection of signal on the optimalatom And with the definition of the matrix projection the

International Journal of Antennas and Propagation 5

part of projection can be obtained by using the followingequation

P119886V(1205731205740)X

= aV (1205731205740) ⟨aV (1205731205740) aV (1205731205740)⟩minus1 aV119867 (1205731205740)X= ⟨X aV (1205731205740)⟩ aV (1205731205740)

(19)

Repeat the above steps with the residual signal and afterthe decomposition for P times the residual signal is smallenough tomeet the requirements of the allowable error so thedecomposition results of the array signal X can be obtained

X = 119875sum119899=1

⟨R119899X aV (120573119899)⟩ aV (120573119899) + 119877119896X (20)

When the decomposition of the received signal has beenfinished a group of orientationmatrixes120573 = [1205731 1205732 sdot sdot sdot 120573119875]can be obtained And 119875 elements are wave directions of 119875signals respectively

The number of PD sources is previously unknown andaccording to the signal sparse representation in the processof the change in energy the iterative termination conditionsfor DOA estimation of ultrasonic array signals based on MPalgorithm are obtained However if the difference of theenergy variation for the adjacent decomposition is particu-larly large and the value of the energy variation is small in theprocess of subsequent classification then the iteration can beterminated

33 The Principle of Direction Finding Based on SparseRepresentation of the Eigenvectors According to the intro-duction of Section 21 assuming that the signal and noise areindependent of each other the center frequency is119891 the arraycovariance matrix of received data is

R (119891) = 119864 X (119891)X119867 (119891)= A (119891)R119878 (119891)A119867 (119891) + 1205752I (21)

where I is identity matrix 1205902I = 119864[N(119891)N(119891)119867] and R119878(119891)is the covariance matrix of the source signal Moreoverthe signal subspace composed by the signal eigenvectorand the noise subspace composed by the noise eigenvectorcan be obtained respectively by the decomposition of thecovariance of the ultrasonic array signal

Theorem 1 Suppose that 119873 (119873 le 119872 minus 1) narrowband far-field signal is incident on the array that consists of119872 elementsthe order of the array manifold matrix is 119873 and the order ofthe signal covariance matrix is 119870 (119870 le 119873) Assuming that thenoise covariance matrix R119873 is a matrix with full rank so thefollowing linear relationship meets

R119873e119896 =119873sum119899=1

120572119896 (119899) a (120579119899) (22)

where 1 le 119896 le 119870 e119896 is an eigenvector of covariancematrix thatreceives the data 120572119896(119899) is a factor of linear combination and

a(120579119899) is a steering vector The proof process is in the literature[21]

Based on the theorem when the noise covariance matrixis the ideal white noise (22) can be simplified as

e119896 =119873sum119899=1

120572119896 (119899) a (120579119899) 1 le 119896 le 119870 (23)

Equation (23) shows that whether the source of signal iscoherent the eigenvectors corresponding to the maximumeigenvalue is a linear combination of the steering vectors foreach signal source And the biggest eigenvector of the datacovariance matrix contains the information of all signals

Consequently the eigenvector corresponding to themax-imum eigenvalue can be sparse representation thereby theDOA estimation for the signal is obtained Compared withthe DOA estimation of the received data based on sparserepresentation the eigenvalue decomposition canweaken theinterference caused by noise and the eigenvector correspond-ing to themaximumeigenvalue is selected to be as the amountto be decomposed and the estimation results will be moreaccurate

34 The Steps of Direction Finding Based on Sparse Represen-tation of Eigenvectors Firstly in order to obtain the narrowcovariance matrix of a single frequency the received dataneed to be focused on because the ultrasonic signal is abroadband signalThe rotate signal subspace (RSS) algorithmpresented in literature [20] is used to focus on the receiveddata of the array in this work

Therefore the detailed steps of theDOA estimation basedon sparse representation of eigenvectors can be expressed asfollows

(1) The data X received by the ultrasonic array sensor isanalyzed based on the DFT to obtain the X1015840 and thisis the preparation for the subsequent focus

(2) The reference frequency is selected as 1198910 and X1015840 isfocused by RSS and the covariance matrix of a singlefrequency P is obtained by using the focus algorithm

(3) The corresponding eigenvector emax of the maximumeigenvalue can be obtained through the eigenvaluedecomposition of the covariance matrix P

(4) According to Section 22 the overcomplete dictionaryis established in the form of the steering vectorand the frequency 119891 of the atom in the step andstep overcomplete dictionary is replaced with thereference frequency 1198910

(5) By using the MP algorithm the eigenvector emax isto make sparse representation and the optimal atomis selected Then the DOA estimation of the signal isobtained which is contained in the angle (120572 120579) of theoptimal atom

35 Three-Array Cross Localization Method After the DOAestimation of the signal the position of the PD source cannotbe sure because the distance between the PD source and the

6 International Journal of Antennas and Propagation

X

Y

Z

(x3 y3 z3)

d1

d2

d3

S (x y z)997888S1

997888S2

997888S3

(x2 y2 z2)

A2

A1

(x1 y1 z1)

A3

Figure 3 The map of three-array cross localization principle

array sensor is unknown [22 23] The space position of thePD source is obtained according to themethod of three-arraycross localization method and the results of the directionfinding The principle of three-array localization method isshown in Figure 3

The space positions of the three sets of array sensors arerespectively 1198601(1199091 1199101 1199111) 1198602(1199092 1199102 1199112) and 1198603(1199093 1199103 1199113)and using the direction angle and the positions of theultrasonic array sensor the equation of the direction line canbe obtained Suppose that spatial coordinates of the signalsource are 119878(119909 119910 119911) In the ideal situation the three differentdirection lines should intersect in the 119878(119909 119910 119911) But theselines are on different surfaces because there are many actualmeasurement errors Therefore the sum function that is asum of the vertical distance from a point in the space to thethree lines is

119889 = 3sum119905=1

119889119905 = 1198891 + 1198892 + 1198893 (24)

Through searching in the space by using ChaoticMonkeyalgorithm [24] when the sum of distance is minimum thepoint with the minimum value d can be regarded as the spaceposition of the PD source

4 The Simulation Study

41 The Simulation of the PD Signal The length (119909) width(119910) and height (119911) of the electrical equipment model arerespectively 150 cm 100 cm and 120 cm (they are matchedwith the size of the experimental equipment)The simulationparameters of signal are set as the wavelength 120582 = 10mmthe amplitude is 5mm the center frequency is 150 kHzthe equivalent velocity is 1500ms the acoustic attenuationcoefficient of the signal is 120572 = 50 times 10minus7 cmminus1 the number ofsampling snapshots is 1024 the sampling frequency is 2MHzthe noise-signal ratio is 10 dB Moreover in order to verify

the validity of the array signal direction finding based onsparse representation the simulation research is carried outon a nine-element circular ultrasonic array sensor and theinterval between array elements is 119889 = 1205822 = 5mm

The form of simulated signal [25 26] is

119891 (119905) = 119860119890 (1198961 (1199050 minus 119905)) cos (2120587119891119905) 0 le 119905 le 1199050119860119890 (1198962 (119905 minus 1199051)) cos (2120587119891119905) 1199050 le 119905 le 1199051 (25)

where 119891 is the central frequency of the signal 119860 is theamplitude of the signal and 1199050 is the time division pointFirstly because of the randomness the PD ultrasonic signal isin electrical equipment so 100 frequency points of the signalare generated according to the average probability in thebandwidth and they formed frequency distributionThen onthe basis of center frequency of ultrasonic signal the ampli-tude of the signal corresponding to each frequency point isformed by the normal distributionmethod Finally the initialphase of each frequency point is randomly generated andthe white Gaussian noise is added in the signal and the PDultrasonic signal in the oil can be simulated

The map of the time waveform of the simulated signal isshown in Figure 4(a) by using the Fourier transform themapof the frequency domain is shown in Figure 4(b) when thenoise is large the PD signal is submerged in the waveform ofthe time domain and it is shown in Figure 4(c)

The map of the frequency domain shows that the sim-ulated PD signal is a broadband signal and the centerfrequency is 150 kHz

The oscillogram of the simulated signal received by anine-element circular ultrasonic array sensor is shown inFigure 5

42 The Simulation of Location For the broadband signalsimulated the received data by ultrasonic array sensor issegmented according to the observation time and the arraycovariance matrix of each frequency point can be obtainedby the DFT in every period 1198910 is selected as focusingfrequency and the covariance matrix of a single frequencycan be acquired The step and step overcomplete dictionaryis established according to the focusing frequency and thesteering vector form of Section 22 After the focusing and theeigendecomposition of the covariancematrix the eigenvectorcorresponding to the maximum eigenvalue can be acquiredand the eigenvector is the parameter to be decomposed

Then taking a circular ultrasonic array sensor for exam-ple the position of the source is set at (35 50 60) cm andthe positions of the three-array sensors are set at position1 (40 0 10) cm position 2 (80 0 0) cm position 3 (030 50) cmTherefore the theoretical values of the DOAs arerespectively (57∘ 451∘) (1320∘ 483∘) and (297∘ 761∘)

In accordancewith the steps of the Section 34 the processof searching for the optimal atom is that the array signalmakes inner products with each atom respectively and thevalue of the inner product is maximum with the optimalatom In order to figuratively present this process the scattergram of absolute value of the inner product in the angle spacecan be made

International Journal of Antennas and Propagation 7

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10A

mpl

itude

(mV

)

2 4 6 8 10 120Time (ms)

(a) The waveform of the time domain

Ener

gy (d

B)

times105

0

50

100

150

200

250

15105 2 30 25Frequency (Hz)

(b) The waveform of the frequency domain

200 400 600 800 1000 12000minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

(c) The PD signal with the noise

Figure 4 The simulated PD signal

The DOA estimation result of the PD ultrasonic signalreceived by the array sensor at the position 1 is (959∘ 48∘)and the scatter gramof the absolute value of the inner productin the space is shown in Figure 6

The DOA estimation result of the PD ultrasonic signalreceived by the array sensor at position 2 is (1316∘ 85∘) andthe scatter gram of the absolute value of the inner product inthe space is shown in Figure 7

The DOA estimation result of the PD ultrasonic signalreceived by the array sensor at position 3 is (301∘ 64∘) andthe scatter gram of the absolute value of the inner product inthe space is shown in Figure 8

Then using the three DOA estimation results above theobjective function equation (24) is calculated by the three-array cross positioning principle And when the objectivefunction is minimum by using the search of the optimizationalgorithm the position of the PD source in the space can beacquired Consequently the result is (331 518 587) cm theerror is 29 cm and the location diagram is shown in Figure 9

Changing the positions of the PD source and the ultra-sonic array sensors the five groups of the PD source posi-tioning simulation are conducted The positioning results ofthe circular ultrasonic array are shown in Table 1

The table shows that after direction of the eigenvectorwith the sparse representation the average error for position-ing is 308 cm And it illustrates that the eigenvector with thesparse representation can obtain the better direction findingresults and reduce the errors in the positioning

5 The Experimental Study

51 The Experimental System The experimental system forresearch includes discharge device the array sensor the dataacquisition system and the data processing system

The simulated electrical equipment is a tank welded bysteel plates the body length is 150 cm the width is 100 cm theheight is 120 cm and the thickness of the steel plate is 5mm

8 International Journal of Antennas and Propagation

Am

plitu

de (m

V)

times10minus450

Time (s)

minus5

0

5

Am

plitu

de (m

V)

times10minus450

Time (s)

minus10

0

10

Am

plitu

de (m

V)

times10minus450

Time (s)

minus10

0

10

Am

plitu

de (m

V)

times10minus450

Time (s)

minus5

0

5

Am

plitu

de (m

V)

times10minus450

Time (s)

minus5

0

5

Am

plitu

de (m

V)

times10minus450

Time (s)

minus2

0

2

Am

plitu

de (m

V)

times10minus450

Time (s)

minus5

0

5

Am

plitu

de (m

V)

times10minus4

50Time (s)

minus5

0

5

Am

plitu

de (m

V)

times10minus4

50Time (s)

minus2

0

2

Figure 5 The oscillogram of the signal received by a full ultrasonic array

0

1

2

3

4

5

Abso

lute

val

ue

times104

Azimuth ( ∘)Pitch angle (∘ )

0

50

100150

200

050

100150

200

(a) The stereogram

20 40 60 80 100 120 140 160 1800Pitch angle (∘)

0

20

40

60

80

100

120

140

160

180

Azi

mut

h (∘ )

(b) The contour map

Figure 6 The scatter gram of the absolute value of the inner product in the space

Moreover a three-capacitor discharge tube is used to simulatethe PD source of the internal electrical equipment And thearray sensor is put in the preset position

A nine-element circular ultrasonic array sensor is usedto receive signal it is fixed on the outer wall of the tankand the shielding lines are used to transfer the data then

the data are processed by the computer combined withultrasonic detection software The principle diagram of thewhole experiment system is shown in Figure 10

The scalemodel and the physicalmap of the nine-elementcircular ultrasonic array sensor are respectively shown inFigures 11(a) and 11(b)

International Journal of Antennas and Propagation 9

times104

Azimuth ( ∘)Pitch angle (∘ )

0

50

100150

200

050

100150

200

0

2

4

6

8

Abso

lute

val

ue

(a) The stereogram

20 40 60 80 100 120 140 160 1800Pitch angle (∘)

0

20

40

60

80

100

120

140

160

180

Azi

mut

h (∘ )

(b) The contour map

Figure 7 The scatter gram of the absolute value of the inner product in the space at position 2

times104

Azimuth ( ∘)Pitch angle (∘ )

050

100150

200

050

100150

200

0

2

4

6

8

10

Abso

lute

val

ue

(a) The stereogram

20 40 60 80 100 120 140 160 1800Pitch angle (∘)

020406080

100120140160180

Azi

mut

h (∘ )

(b) The contour map

Figure 8 The scatter gram of the absolute value of the inner product in the space at position 3

0 50100

150

020406080100

XY

020406080

100120

Z

lowast

Figure 9Themap of simulated location of the circular array sensorldquolowastrdquo refers to the position of the PD source

In order to avoid the influences of discharge instabilityand other factors on the experimental results a three-capacitor discharge tube is used to simulate the PD source

it can generate the ultrasonic that is similar to the ultrasonicsignal of the real partial discharge and it has good stabilityand repeatability and the discharge voltage is low and easyto meet the insulation The EPSON discharge tube is usedto be a discharge device and the critical discharge voltage is230VGenerally the discharge frequency of ultrasonic signalsemitted by the discharge tube is in the range of 50 kHz to280 kHz the center frequency is 150 kHz and the equivalentvelocity is 1500ms while thewavelength is about 10mmThediagram of the three-capacitor discharge principle is shownin Figure 12

In Figure 12 1198620 is a coupling capacitor and1198621 is equiva-lent capacitance of the other parts of the insulating medium1198622 is equivalent capacitance of the insulating medium thatis in series with the PD source and the gas-discharge tubeis used to be a discharge device when the voltage of the tube

10 International Journal of Antennas and Propagation

Table 1 The location results of the circular ultrasonic array

GroupThe positionof the PDsourcecm

The positionof the arraysensorcm

Theoreticalangle(∘)

Directionangle(∘)

The result of thethree-array cross

positioningcm

The errorcm

1 (35 50 60)(40 0 10) (957 451) (959 448)

(331 518 587) 29(80 0 0) (1320 483) (1316 485)(0 30 50) (297 761) (301 764)

2 (30 80 95)(35 0 50) (936 607) (941 604)

(320 785 932) 31(40 0 0) (971 403) (975 395)(0 20 30) (634 459) (643 462)

3 (25 70 20)(50 0 10) (1097 823) (1103 816)

(232 684 223) 33(0 50 0) (387 580) (383 574)(0 50 60) (387 1413) (383 1398)

4 (60 30 50)(30 0 20) (450 547) (443 541)

(623 317 490) 30(65 0 35) (995 637) (987 640)(0 45 0) (1660 510) (1652 505)

5 (75 20 45)(0 0 80) (149 1143) (144 1148)

(732 217 431) 31(40 0 0) (297 419) (303 408)(55 0 0) (450 322) (441 327)

(4)

(2) (3)

(6)

(7)

(10) (11)(9)(8)

(5)

Z

X

Y

(1)middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 10 The structure of the PD positioning experimental system (1) AC power regulator (2) voltage regulator (3) test transformer (4)coupling capacitor (5) impedance measurement (6) inlet bushing (7) PD model (8) oil tank (9) Partial Discharge Ultrasonic array sensor(10) transmission lines (11) data acquisition unit

(a) The scale model (b) The physical map

Figure 11 A nine-element circular ultrasonic array sensor

International Journal of Antennas and Propagation 11

Table 2 The positioning results based on sparse representation of the eigenvector

GroupThe positionof the PDsourcecm

Theoreticalangle(∘)

Directionangle(∘)

The result of thethree-array crosspositioningcm

The errorcm

1(45 0 20) (369 320) (360 312)

(677 174 626) 45(80 0 20) (1350 279) (1342 289)(0 30 40) (1670 733) (1681 735)

2(55 0 35) (563 358) (552 366)

(625 122 571) 47(0 25 35) (1630 698) (1621 692)(45 0 20) (369 320) (375 329)

3(0 45 45) (1552 782) (1540 794)

(620 176 622) 45(70 0 45) (1083 465) (1076 454)(45 0 40) (563 514) (556 520)

4(55 0 45) (563 503) (555 512)

(674 125 627) 44(0 40 25) (1589 633) (1578 640)(45 0 20) (369 320) (375 311)

5(55 0 35) (563 358) (572 350)

(678 176 624) 45(70 0 35) (1083 323) (1074 312)(0 40 20) (1589 601) (1597 612)

Measuredimpedance

Dischargingtube

Powersource

C0

C1

C2

C3

R1

R2

R3

Figure 12 The diagram of the three-capacitor discharge principle

reaches a certain value the gas is breakdown conduction andthen the discharge effect is produced

52 The Signal Processing In the process of the experimentthe position of the PD source was preset And a pluralityof preset fixed locations were for the array sensors on theouter wall of the electrical equipment Then the ultrasonicarray sensor was used to collect the PD signal In order toensure that the experiment was performed under the sameconditions and to facilitate subsequent data processing thePD source location was fixed during the experiment onlychanging the position of ultrasonic array sensor so that therelative space position was changed The location of the PDsource in experiment was (65 15 60) cm

The oscillogram of the PD signal received by the nine-element ultrasonic array sensor is shown in Figure 13

Three different locations were selected to place ultrasonicarray sensors that were used to receive the PD signal and theultrasonic signal was used to be focusing and the eigenvector

corresponding to the maximum eigenvalue was acquired byusing the eigendecomposition of the covariance matrix witha single frequency This eigenvector was the amount to bedecomposed and the optimal atom was selected by usingMP algorithm then the DOA estimation can be realizedBesides the noise interference can be reduced based onsparse representation of the eigenvectors and improve theaccuracy of positioning

Using the methods of direction finding and positioningin the work the results are shown in Table 2

The table shows that in the experimental process theaverage error of positioning for the circular ultrasonic arraysensor based on sparse representation of the eigenvectors is452 cm which meets the requirement of practical engineer-ing

6 Conclusion

Theaccurate detection of the PD in oil of electrical equipmentis the key to the maintenance and repairs of equipmentThe positioning method for the PD based on the sparserepresentation of the eigenvectors is studied in this workFirst of all the mathematical model of a wideband PD signaland the steering vector of a circular ultrasonic array sensoris given Then the sparse representation theory is applied tothe DOA estimation The principle and process of the sparsepresentation of the eigenvectors is introduced in detail andthe three-array cross positioning method is also introducedLastly the simulation study and experimental research areconducted on this method and the results show that thepositioning method in the work can achieve the accuratepositioning of the PD source

12 International Journal of Antennas and PropagationA

mpl

itude

(mV

)A

mpl

itude

(mV

)A

mpl

itude

(mV

)

Am

plitu

de (m

V)

Am

plitu

de (m

V)

Am

plitu

de (m

V)

Am

plitu

de (m

V)

Am

plitu

de (m

V)

Am

plitu

de (m

V)

1 2 3 4 5 60Time (s)

minus01

minus005

0

005

01

1 2 3 4 5 60Time (s)

minus005

0

005

1 2 3 4 5 60Time (s)

minus005

0

005

01

015

1 2 3 4 5 60Time (s)

minus01

minus005

0

005

01

1 2 3 4 5 60Time (s)

minus01

minus0050

00501

015

1 2 3 4 5 60Time (s)

minus01

minus005

0

005

01

1 2 3 4 5 60Time (s)

minus004

minus002

0

002

004

1 2 3 4 5 60Time (s)

minus01

minus005

0

005

01

1 2 3 4 5 60Time (s)

minus004minus002

0002004006

times10minus4 times10minus4 times10minus4

times10minus4 times10minus4 times10minus4

times10minus4 times10minus4 times10minus4

Figure 13 Nine channel ultrasonic signal waveforms

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by Project Supported by NationalNatural Science Foundation of China (51307060) ProjectSupported by Natural Science Foundation of Hebei Province(E2015502081) State Key Laboratory of Alternate ElectricalPower System with Renewable Energy Source (LAPS16009)

References

[1] C R Qiu and N Q Wang Electrician Equipment Local Dis-charge and Its Test Technology Mechanical Industry PublishingHouse Beijing China 1994

[2] Z LiuTheUltra-High Voltage Grid China Electric Power PressBeijing China 2005

[3] J YWang R J Liao Y Y Zhang and F J Meng ldquoEconomic lifeassessment of power transformers using an improved modelrdquoCSEE Journal of Power and Energy Systems vol 1 no 3 pp 68ndash75 2015

[4] W J Chen and X Cui ldquoForeword for the special section on ACand DC ultrahigh voltage technologiesrdquo CSEE Journal of Powerand Energy Systems vol 1 no 3 pp 1ndash2 2015

[5] Q Xie S Cheng F Lu and Y Li ldquoA new sparse design methodon phased array-based acoustic emission sensor for partialdischarge detectionrdquo Measurement Science amp Technology vol25 no 3 Article ID 035102 11 pages 2014

[6] S Chen F Lv Q Xie et al ldquoThe transformer partial dischargepositioning method based on transient voltage to earth andultrasonic array signalsrdquo Transactions of China ElectrotechnicalSociety vol 27 no 4 pp 255ndash262 2012

[7] Y L Wang H Chen Y N Peng et al The Spatial SpectrumEstimation Theory and Algorithms Tsinghua University Press2004

[8] E SkudrzykThe Foundations of Acoustics Springer New YorkNY USA 1971

[9] Q Xie S Cheng F Lu and Y Li ldquoLocation of partial dischargein transformer oil using circular array of ultrasonic sensorsrdquoIEEE Transactions on Dielectrics and Electrical Insulation vol20 no 5 pp 1683ndash1690 2013

[10] Q Xie X Liu J Tao T Li S Cheng and F Lu ldquoExperimentalverification of the sparse design of a square partial dischargeacoustic emission array sensorrdquo Measurement Science amp Tech-nology vol 26 no 4 Article ID 045101 2015

[11] M R Rao and B P Singh ldquoDetection and localization ofinterturn fault in the HV winding of a power transformerusing waveletsrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 8 no 4 pp 652ndash657 2001

[12] M Wax T J Shan and T Kailath Spatio-Temporal SpectralAnalysis by Eigenstructure Methods Stanford Univ Ca Informa-tion Systems Lab 1984

[13] S Valaee B Champagne and P Kabal ldquoLocalization ofwideband signals using least-squares and total least-squaresapproachesrdquo IEEE Transactions on Signal Processing vol 47 no5 pp 1213ndash1222 1999

[14] S Valaee and P Kabal ldquoWideband array processing using a two-sided correlation transformationrdquo IEEE Transactions on SignalProcessing vol 43 no 1 pp 160ndash172 1995

International Journal of Antennas and Propagation 13

[15] S Valaee and P Kabal ldquoThe optimal focusing subspace forcoherent signal subspace processingrdquo IEEE Transactions onSignal Processing vol 44 no 3 pp 752ndash756 1996

[16] M Allam and A Moghaddamjoo ldquoTwo-dimensional DFTprojection for wideband direction-of-arrival estimationrdquo IEEETransactions on Signal Processing vol 43 no 7 pp 1728ndash17321995

[17] M A Doron and A J Weiss ldquoOn focusing matrices for wide-band array processingrdquo IEEE Transactions on Signal Processingvol 40 no 6 pp 1295ndash1302 1992

[18] S G Mallat and Z Zhang ldquoMatching pursuits with time-frequency dictionariesrdquo IEEE Transactions on Signal Processingvol 41 no 12 pp 3397ndash3415 1993

[19] S Mallat and Z Zhang ldquoAdaptive time-frequency decompo-sition with matching pursuitsrdquo in Proceedings of the IEEE-SP International Symposium Time-Frequency and Time-ScaleAnalysis pp 7ndash10 Victoria Canada October 1992

[20] H Hung and M Kaveh ldquoFocussing matrices for coherentsignal-subspace processingrdquo IEEE Transactions on AcousticsSpeech and Signal Processing vol 36 no 8 pp 1272ndash1281 1988

[21] J A Cadzow Y-S Kim andD-C Shiue ldquoGeneral direction-of-arrival estimation a signal subspace approachrdquo IEEE Transac-tions on Aerospace amp Electronic Systems vol 25 no 1 pp 31ndash471989

[22] Q Xie J Tao Y Wang J Geng S Cheng and F Lu ldquoUseof ultrasonic array method for positioning multiple partialdischarge sources in transformer oilrdquoReview of Scientific Instru-ments vol 85 no 8 Article ID 084705 2014

[23] Q Xie Y-Q Li F-C Lu C-R Li N Wang and Y-J DingldquoMethod for PD location in oil combining ultrasonic phasedarray with wideband array signal processingrdquo Proceedings of theChinese Society of Electrical Engineering vol 29 no 28 pp 13ndash17 2009

[24] Y Q Xu H W Zhang T Li Y Yang J Tao and Q XieldquoApplication of chaotic monkey algorithm in partial dischargelocation with ultrasonic array signalsrdquo Insulating Materials vol47 no 5 pp 92ndash95 2014

[25] T Wang G Cheng and S Wan ldquoFourierminuswavelet regular-ized deconvolution in medical ultrasound imagingrdquo TechnicalAcoustics vol 30 no 6 pp 501ndash504 2011

[26] Z Tan D Feng G Chen et al ldquoPre-processing in medicalultrasonic lesions imagesrdquo Foreign Electronic MeasurementTechnology vol 33 no 3 pp 89ndash91 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of