sensors Article
An Ultrahigh Frequency Partial Discharge Signal De-Noising Method Based on a Generalized S-Transform and Module Time-Frequency Matrix Yushun Liu 1 , Wenjun Zhou 1, *, Pengfei Li 1 , Shuai Yang 1 and Yan Tian 2 1 2
*
School of Electrical Engineering, Wuhan University, No. 299, Bayi Road, Wuhan 430072, China;
[email protected] (Y.L.);
[email protected] (P.L.);
[email protected] (S.Y.) Guangzhou Power Supply Bureau Co. Ltd., No. 38, Huangshidong Road, Guangzhou 510620, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-27-6877-2283 (ext. 217)
Academic Editors: Changzhi Li, Roberto Gómez-García and José-María Muñoz-Ferreras Received: 18 May 2016; Accepted: 20 June 2016; Published: 22 June 2016
Abstract: Due to electromagnetic interference in power substations, the partial discharge (PD) signals detected by ultrahigh frequency (UHF) antenna sensors often contain various background noises, which may hamper high voltage apparatus fault diagnosis and localization. This paper proposes a novel de-noising method based on the generalized S-transform and module time-frequency matrix to suppress noise in UHF PD signals. The sub-matrix maximum module value method is employed to calculate the frequencies and amplitudes of periodic narrowband noise, and suppress noise through the reverse phase cancellation technique. In addition, a singular value decomposition de-noising method is employed to suppress Gaussian white noise in UHF PD signals. Effective singular values are selected by employing the fuzzy c-means clustering method to recover the PD signals. De-noising results of simulated and field detected UHF PD signals prove the feasibility of the proposed method. Compared with four conventional de-noising methods, the results show that the proposed method can suppress background noise in the UHF PD signal effectively, with higher signal-to-noise ratio and less waveform distortion. Keywords: ultrahigh frequency antenna sensor; partial discharge; signal de-noising; generalized S-transform; module time-frequency matrix; periodic narrowband noise; Gaussian white noise
1. Introduction Partial discharge (PD) is a major cause and manifestation of insulation degradation. PD detection has been utilized for insulation condition assessment of high voltage (HV) apparatus [1]. When PD occurs in a HV apparatus, ultra-high frequency (UHF) electromagnetic waves are radiated from the PD source and propagate to the space of the substation through objects with no shielding effect [2]. External UHF antenna sensors can be utilized to detect UHF PD signals [3], however, in field PD tests, the UHF PD signals detected by antenna sensors are vulnerable to interference due to the numerous electromagnetic waves present in space [4]. The background noise interferes with the PD detection and causes PD pulse shapes to be distorted, which may negatively affect HV apparatus fault diagnosis and localization accuracy [5,6].Therefore, noise suppression is a vital issue of field UHF PD signal testing. The background noise mainly includes the periodic narrowband noise and white Gaussian noise [7]. The periodic narrowband noise generally originates from wireless communication systems and may overwhelm the original PD signals. Several techniques have been proposed to suppress the narrowband noise [8–13]. Methods based on adaptive filters have been proposed to effectively suppress the narrowband noise [9,10], however, when the frequencies of the interferences and the
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PD signal are mixed, the de-noised PD signal is distorted due to the calculation error of the Fourier transform spectral analysis and the characteristics of the impulse response filter [11]. The methods based on wavelet transform (WT) were reported to extract PD signals from excessive narrowband noise [12], but due to the diversity of detected UHF PD signals, it is difficult to select the suitable mother wavelet and the number of decomposition-reconstruction levels [13]. The Gaussian white noise is generated by the heating effect of HV apparatus and will distort the UHF PD signals. Many methods have been proposed to suppress the white noise, such as the WT method [14], empirical mode decomposition method [15], and mathematical morphology method [16]. Recently, a novel method based on singular value decomposition (SVD) has been utilized for Gaussian white noise suppression, and more accurate recovery of the original PD signal was obtained [17]. A hard thresholding method based on standard deviation was employed to select effective singular values in this method. In this paper, a novel de-noising method based on the module time-frequency matrix (MTFM) is proposed to suppress periodic narrowband noise and Gaussian white noise. The MTFM of detected UHF PD signals is constructed by employing the generalized S-transform time-frequency analysis technique. Frequencies and amplitudes of periodic narrowband noise can be calculated by utilizing the sub-matrix maximum module value method. The periodic narrowband noise is separated from the noisy UHF PD signals through the reverse phase superposition of the MTFM. Different from the conventional SVD de-noising method in [17,18], the singular values is calculated through decomposing the 2-dimensional MTFM. The fuzzy c-means (FCM) clustering method is employed to select effective singular values for recovering the UHF PD signals without Gaussian white noise. The de-noising results of simulated and field test UHF PD signals demonstrate the validity of the proposed method. 2. Algorithm of the Generalized S-Transform As a non-stationary signal, the localized feature information of an UHF PD signal cannot be expressed only in time domain or frequency domain [19,20]. To extract information related to the time-frequency variation of a PD signal, Stockwell proposed the S-transform [21] to map time domain signal into the time-frequency domain, the S-transform of signal x(t) is defined as: ż8 Spτ, f q “
xptqwpt ´ τ, f qe´j2π f t dt
(1)
´8 f 2 pt´τq2 |f| wpt ´ τ, f q “ ? e´ 2 2π
(2)
where t and τ are time, f is frequency, w(t ´ τ, f ) is the Gaussian window function. The inverse S-transform is expressed as: ż 8 „ż 8 xptq “ Spτ, f q ej2π f t d f (3) ´8
´8
The S-transform combines the separate strengths of the short-time Fourier transform and WT, which has provided an alternative approach to process non-stationary signals [22]. Because the height and the width of Gaussian window are varied by changing the frequency, the S-transform overcomes the defect of constant time-frequency resolution of the short time Fourier transform [23]. Because the form of the S-transform window function is invariant, the application is limited in some cases. Pinnegar proposed a generalized S-transform [24] by adding an adjustable factor λ to the Gaussian window function as the Equation (4) shown: |λ f | λ2 f 2 pt´τq2 2 wpt ´ τ, f , λq “ ? e´ 2π where λ is the adjustable factor and λ > 0. Then the generalized S-transform is obtained as:
(4)
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ż8 Gpτ, f , λq “
xptqwpt ´ τ, f , λqe´j2π f t dt
(5)
´8
In the generalized S-transform, λ < 1 corresponds to high frequency resolution, λ > 1 corresponds to high time resolution, λ = 1 corresponds to the standard S-transform window. In the application of generalized S-transform, an appropriate value of λ to the actual situation can be chosen to modify the time-frequency resolution. According to Equation (5), as f Ñn/NT, τÑiT, the discrete form of the generalized S-transform is expressed as: $ ´1 “ ‰ ’ GpiT, n , λq “ Nř `n ¨e´2π2 m2 {λ2 n2 ej2πmi{ N , n ‰ 0 ’ X mNT & NT m “0 (6) Nř ´1 ’ 1 m ’ % GpiT, 0q “ N xp NT q, n “ 0 m “0
where T is the sampling interval, N is the total number of sample points. The generalized S-transform of x(t) is a 2-dimensional complex matrix in time-frequency domain, where the columns correspond to time sampling points and the rows correspond to time frequency sampling points [25]. For simplifying the calculations, the MTFM can be obtained by modeling every element in the complex matrix. The time-frequency distribution based on the MTFM reflects the time-frequency features of UHF PD signal. These features contribute to the analysis of the periodic narrowband noise [26]. 3. De-Noising Method of Periodic Narrowband Noise 3.1. Simulated Signals Due to the noise interference in field-detected UHF PD signals, the original PD signal cannot be obtained for illustrating the procedure and verifying the feasibility of the proposed method. An UHF PD signal is a type of transient signal with a short time duration [11]. Therefore, the single exponential decay oscillating impulses (pulses 1 and 2) and double exponential decay oscillating impulses (pulses 3 and 4) are utilized to simulate four types of UHF PD signals [12,13]. Equations (7) and (8) give the corresponding mathematical models, respectively: Sensors 2016, 16, 941
Z1 ptq “ A1 e
´t τ
sin2π f c t
Table 1. Parameters of ´1.3t the simulated ´2.2t UHF PD signals.
Z2 ptq “ A2 pe
τ
´e
τ
qsin2π f c t
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(7)
(8)
PD Pulse Sequence 1 2 3 4 where A is the amplitude, τ is the decay coefficient τ/ns 0.5 and f c1 is the oscillation 0.5 1frequency. Table 1 gives the parameters of each PD pulse. fThe simulated sampling frequency pulse is 10 Gs/s. Figure 1a c/MHz 600 600 1000of each 1000 shows the four types of UHF PD signals. A/mV 10 40 10 40
Figure Figure 1. 1. Simulated Simulated UHF UHF PD PD signals: signals: (a) (a) Simulated Simulated PD PD signals; signals; (b) (b) Simulated Simulated noisy noisy PD PD signals. signals.
3.2. Calculation Method of Frequency and Amplitude 3.2.1. Sub-Matrix Maximum Module Value Method When the frequencies of the periodic narrowband noise and PD signals are partially overlapped, the PD pulse(s) are mixed with strong narrowband noise in a time-frequency distribution calculated by the S-transform, meaning that it is difficult to separate the narrowband
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Table 1. Parameters of the simulated UHF PD signals. PD Pulse Sequence 1 2 3 4 Table 1. Parameters of the simulated UHF PD signals. τ/ns 0.5 1 0.5 1 fc/MHz 600 600 1000 1000 PD Pulse Sequence 1 2 3 4 A/mV 10 40 10 40 τ/ns 0.5 1 0.5 1 f c /MHz 600 600 1000 1000 A/mV 10 40 10 40
The waveform of practical periodic narrowband noise is sinusoidal with constant amplitude [27]. Equation (9) gives the mathematical model of periodic narrowband noise, where A is the amplitude assumed to be 2 mV. Based on the frequencies of wireless communication signals in China, fi is the frequency assumed to be 470 MHz, 900 MHz, 1800 MHz, respectively. The simulated UHF PD signals with periodic narrowband noise and Gaussian white noise are given in Figure 1b. The SNR of these noisy PD signals is ´5.983 dB: n Figure 1. Simulated UHF PD signals: (a) Simulated PD signals; (b) Simulated noisy PD signals. ÿ Z3 ptq “ A sinp2π f i tq (9)
3.2. Calculation Method of Frequency and Amplitudei“1 3.2. Calculation Method of Frequency and Amplitude 3.2.1. Sub-Matrix Maximum Module Value Method 3.2.1.When Sub-Matrix Maximum Module Method the frequencies of the Value periodic narrowband noise and PD signals are partially overlapped, PD pulse(s) mixed with strong narrowband noise a time-frequency When thethe frequencies of theare periodic narrowband noise and PD signals arein partially overlapped, distribution calculated by the S-transform, meaning that it is difficult to separate the narrowband the PD pulse(s) are mixed with strong narrowband noise in a time-frequency distribution calculated noise the PD pulse(s) [26]. Theit MTFM of the PD signals calculated noise by thefrom S-transform by thefrom S-transform, meaning that is difficult to noisy separate the narrowband the PD and the generalized S-transform (λ = 0.3), respectively, are drawn as a time-frequency distribution pulse(s) [26]. The MTFM of the noisy PD signals calculated by the S-transform and the generalized contour map(λin Figure 2. The comparison results show that higher frequency resolution can be S-transform = 0.3), respectively, are drawn as a time-frequency distribution contour map in Figure 2. obtained by utilizing theshow generalized S-transform (λresolution < 1), which contribute extract the the The comparison results that higher frequency canmay be obtained bytoutilizing narrowbandS-transform noise features [26]. generalized (λ < 1), which may contribute to extract the narrowband noise features [26]. ×10-4 16
3
1.5
8 6
B
2 20
40
12
60 80 Time(ns) (a)
100
C
2
10 8
1.5
6
B
1
4
C
0.5
Frequency/GHz
Frequency/GHz
10
1
2.5
12
A
2
0 0
14
14
2.5
×10-4
3
120
140
0
0.5 0 0
4
A
2 20
40
60 80 Time(ns) (b)
100
120
140
0
2. The time-frequency distribution of the noisy PD signals calculated by (a) the S-transform; Figure 2. generalized S-transform. S-transform. and (b) the generalized
The time-frequency distribution of periodic narrowband noise is concentrated, with long The time-frequency distribution of periodic narrowband noise is concentrated, with long duration, duration, and the frequency distribution of a PD signal is discrete, with short duration. Therefore, and the frequency distribution of a PD signal is discrete, with short duration. Therefore, both types both types of signals can be recognized in the time-frequency distribution. Due to the influence of of signals can be recognized in the time-frequency distribution. Due to the influence of Gaussian Gaussian white noise, the accurate frequencies and amplitudes of periodic narrowband noise white noise, the accurate frequencies and amplitudes of periodic narrowband noise cannot be obtained cannot be obtained directly from the time-frequency distribution. In this paper, a novel sub-matrix directly from the time-frequency distribution. In this paper, a novel sub-matrix maximum module maximum module value (SMMV) method is proposed for calculating the frequencies and value (SMMV) method is proposed for calculating the frequencies and amplitudes of narrowband amplitudes of narrowband noise. The SMMV method is described as follows: noise. The SMMV method is described as follows:
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Step 1: As shown in Figure 2b, parts of the periodic narrowband noise are divided into Regions A, B, C, and the corresponding localized time-frequency sub-matrices are extracted; Step 2: Search the coordinate positions of the maximum module value points in each column of each sub-matrix; Step 3: Based on the MTFM calculated by the generalized S-transform (λ < 1), record the corresponding frequency values of the maximum module value points (f 1 , f 2 , f 3 , . . . , f n ) and the calculated time of each frequency value (t1 , t2 , t3 , . . . , tn ); Step 4: Calculate the frequencies f p of each periodic narrowband noise based on Equation (10): n ř
fp “
f k tk
k “1
(10)
n
Step 5: Record the module value of each sampling point (m1 , m2 , . . . , mn ) in time on the calculated frequencies in Step (4), and calculate the module value mp of each corresponding narrowband noise in the MTFM based on Equation (11): n ř
mp “
mk
k “1
(11)
n
Step 6: Calculate the amplitudes of periodic narrowband noise by utilizing the inverse generalized S-transform, and reconstruct time domain waveform based on Equation (3). 3.2.2. Selection of the Adjustable Factor λ In general, the generalized S-transform is a modified Fourier transform method, and the time-frequency resolution is still restricted by the Heisenberg uncertainty principle. This means that the time resolution decreases when the frequency resolution increases in a time-frequency distribution. Therefore, an appropriate value of λ contributes to the calculation of more accurate frequencies and amplitudes of periodic narrowband noise based on the SMMV method. Different values of λ were used to calculate the amplitudes and frequencies of each narrowband noise, with the results given in Table 2. Table 2. Calculated amplitudes and frequencies with changing values of λ. Narrowband Noise #A
Narrowband Noise #B
Narrowband Noise #C
λ
Frequency (MHz)
Amplitude (mV)
Frequency (MHz)
Amplitude (mV)
Frequency (MHz)
Amplitude (mV)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
470.085 470.085 470.085 470.312 470.476 470.851 470.994 471.265 471.325 471.785 472.046
1.949 1.954 1.986 1.984 1.979 1.974 1.976 1.968 1.968 1.962 1.953
900.032 900.011 900.076 899.391 901.328 898.398 901.852 902.347 903.481 903.911 904.056
1.919 1.925 1.931 1.933 1.926 1.915 1.929 1.924 1.919 1.918 1.911
1800.194 1800.194 1800.894 1802.140 1802.954 1803.823 1804.974 1805.293 1806.905 1807.488 1809.226
1.842 1.883 1.912 1.924 1.927 1.925 1.916 1.917 1.913 1.904 1.901
Compared with the original assumed value in the simulation, the relative errors of different calculated narrowband noise are given in Figure 3. Due to the improvement of frequency resolution, the relative errors of the calculated frequencies are reduced with the decrease in the value of λ.
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Sensors 2016, 16, 941 6 of 19 However, due to the reduction of time resolution, the relative errors of the calculated amplitudes are increased when the value of λ is lower than 0.3. The relative error is defined as:
O C Relative error |O ´ C| 100% O ˆ 100% Relativeerror “
(12)
O
(12)
where O and C are the amplitude (frequency) of the original and calculated periodical noise,
where O and C The are results the amplitude the be original and calculated periodical respectively. show that(frequency) the value of of λ can set between 0.3 and 0.4 to accuratelynoise, respectively. results show that the valueofofperiodic λ can be set between 0.3 and to of accurately calculateThe the frequencies and amplitudes narrowband noise. The 0.4 value λ is set tocalculate 0.3 in this paper. the frequencies and amplitudes of periodic narrowband noise. The value of λ is set to 0.3 in this paper.
Figure 3. Relative errorsbybyusing usingdifferent different value (b)(b) amplitude. Figure 3. Relative errors valueofofλ:λ:(a)(a)frequency; frequency; amplitude.
3.3. Method of Periodic Narrowband Noise Suppression
3.3. Method of Periodic Narrowband Noise Suppression
To suppress the periodic narrowband noise, the MTFM based on the generalized S-transform
To suppress the periodic narrowband noise, the MTFM based on the generalized S-transform is is applied as follows: applied as follows: Step 1:
Use the generalized S-transform to obtain the MTFM SM×N and time-frequency distribution of noisy UHF PD signals, where SM×N is the matrix with M rows and N columns ; Step 1: Use the generalized S-transform to obtain the MTFM SM ˆN and time-frequency distribution Step 2: Extract the localized matrices of periodic narrowband noise without PD signals based on of noisy UHF PD signals, where SMˆN is the matrix with M rows and N columns ; the time-frequency distribution; the localized matricesand of periodic narrowband noise without PD signals basedthe on the Step Step 2: Extract 3: Obtain the amplitudes frequencies of periodic narrowband noise utilizing time-frequency distribution; above-described SMMV method; 4: Usethe the amplitudes generalized and S-transform to calculate the MTFM P1, P2, …,noise Pn of utilizing periodic the Step Step 3: Obtain frequencies of periodic narrowband narrowband noise based on Equation (6) and the results of Step (3), where P k is the matrix above-described SMMV method; with M rows and N columns; Step 4: Use the generalized S-transform to calculate the MTFM P1 , P2 , . . . , Pn of periodic narrowband Step 5: Suppress the periodic narrowband noise by utilizing the reverse phase cancellation noise based on Equation (6) and the results of Step (3), where Pk is the matrix with M rows method is given in Equation (13), where WM×N is the MTFM without periodic narrowband and N columns; noise:
Step 5: Suppress the periodic narrowband noise by utilizing the reverse phase cancellation method is n given in Equation (13), where W M ˆ the periodic narrowband noise: WNM×is = SMMTFM - Pwithout (13) ×N N k k =1
Step 6:
n ÿ
Use the inverse generalized S-transform to obtain the de-noised UHF PD signals in time W Mˆ N “ S Mˆ N ´ Pk (13) domain of the WM×N. k “1
Figure 4 gives the de-noising results of simulated noisy UHF PD signals utilizing the proposed Step method. 6: Use the generalized S-transform to obtain the de-noised PD signals in time Theinverse de-noising results indicate that periodic narrowband noise UHF has been suppressed domainand of the MˆN . white noise exists in the UHF PD signals. The SNR of this signal is effectively onlyW Gaussian 7.085 dB.
Figure 4 gives the de-noising results of simulated noisy UHF PD signals utilizing the proposed method. The de-noising results indicate that periodic narrowband noise has been suppressed effectively and only Gaussian white noise exists in the UHF PD signals. The SNR of this signal is 7.085 dB.
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Figure 4. UHF PD signals with suppressed periodic narrowband noise: (a) in time domain; (b) in Figure 4. UHF PD signals with suppressed periodic narrowband noise: (a) in time domain; (b) in time-frequency domain. time-frequency domain.
4. De-Noising Method of Gaussian White Noise
4. De-Noising Method of Gaussian White Noise 4.1. Singular Value Decomposition De-Noising Method
4.1. Singular Value Decomposition De-Noising Method
Suppression of Gaussian white noise is vital to extract accurate UHF PD pulse waveforms. The
Suppression Gaussian white is vital to extract accurate UHF method, PD pulse waveforms. singular value of decomposition (SVD)noise de-noising method is a nonlinear filtering which was The singular value decomposition (SVD) de-noising method is a nonlinear filtering method, which shown to be capable of effectively suppressing white noise [17,28]. The conventional SVD was de-noising method given in Figure 5 andnoise is illustrated follows. shown to be capable ofprocedure effectivelyis suppressing white [17,28].as The conventional SVD de-noising method procedure is given in Figure 5 and is illustrated as follows. 16, 941 Step Sensors 1: For2016, a signal sequence x(i), i = 1, 2, . . . , N, a trajectory matrix of x(i) is defined as
fi xp1q xp2q ¨¨¨ xpnq — ffi xp2q xp3q ¨ ¨ ¨ xpn ` 1q — Figure 5. Procedure of the conventional SVD de-noising method. ffi — ffi A“— .. .. .. .. ffi – fl . . . . Step 1: For a signal sequence x(i), i = 1, 2, …, N, a trajectory matrix of x(i) is defined as xpN ´ n ` 1q xpN ´ n ` 2q ¨ ¨ ¨ xpNq x(1) x(2) x(n) and m = N – n + 1. where APRmˆn , 1 < n < N, x(2) x(3) x(n 1) A m ˆn is defined as: Step 2: The SVD of this real matrix APR x( N n 1) x( N n 2) T x( N ) A “ UΛV
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»
(14)
(14)
(15)
where A∈Rm×n, 1 < n < N, and m = N – n + 1. Step 2: TheU SVD A∈Rm×nmatrix, is defined as: m ˆn , VPRm ˆn . Λ = [diag(a1 , a2 , . . . , aq ), O] where andofVthis arereal thematrix orthogonal UPR
orFigure its transposition matrix, which is determined by m < n or(a)minětime n. domain; O is the(b)zero matrix, 4. UHF PD signals with suppressed periodic noise: in (15) A = U Λnarrowband VT q time-frequency = min(m, n), domain. a1 ě a2 ě . . . ě aq . ai , i = 1, 2, . . . , q, are the singular values of matrix. where U and V are the orthogonal matrix, U∈Rm×n, V∈Rm×n. Λ = [diag(a1, a2, …, aq), O] or its Step 3: In matrix Λ, the effective singular values corresponding to a PD signal are reserved while the transposition matrix, which isWhite determined 4. De-Noising Method of Gaussian Noise by m < n or m ≥ n. O is the zero matrix, q = min(m, corresponding white to the zero, and thus a new matrix Λre can be obtained. n), a1 ≥ a2 ≥…≥ aq. ai, noise i = 1, 2,are …, set q, are singular values of matrix. Step 4.1. 4: A new trajectory matrix A is reconstructed by utilizing Λ , the original U and Vwhile through resingular values StepSingular 3: In matrix the effectiveDe-Noising signal are reserved ValueΛ, Decomposition Methodcorresponding to arePD Equation (16), and the de-noised PD theΛtrajectory matrix Are : the corresponding white noise are setsignals to zero,are andrecovered thus a newfrom matrix re can be obtained. Suppression of Gaussian white noise is vital to extract accurate UHF PD pulse waveforms. The Step 4: A new trajectory matrix Are is reconstructed by utilizing Λre, the original U and V through singular value decomposition (SVD) de-noising method is a Tnonlinear filtering method, which was Equation (16), and the de-noised PD signals recovered from the trajectory matrix Are: Are “areUΛ (16) re V shown to be capable of effectively suppressing white noise [17,28]. The conventional SVD de-noising method procedure is given in Figure Are5and UΛreis VTillustrated as follows. (16)
Figure 5. Procedure of the conventional SVD de-noising method.
Figure 5. Procedure of the conventional SVD de-noising method.
Step 1: For a signal sequence x(i), i = 1, 2, …, N, a trajectory matrix of x(i) is defined as
A
x(1) x(2)
x(2) x(3)
x(n) x(n 1)
(14)
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4.2. Selection Method of Effective Singular Values Because the trajectory matrix is created by truncating the signal in the time domain, SVD results only contain time-amplitude information. It means that the recovered de-noised PD signal may be partially distorted [18]. The MTFM of the noisy PD signals is set as the trajectory matrix A in this paper. Eigenvectors of frequency and time are represented by the orthogonal matrices U and V, respectively. SVD results of the MTFM contain time-frequency-amplitude information, meaning that some characteristic quantities of another dimension can be obtained in the U, V and singular values. Therefore, the recovered PD signal after de-noising will be more accurate compared with the conventional algorithm ones. For the SVD de-noising method, the most important step is how to select the number of effective singular values, which largely determines the de-noising effect. The recovered PD signal is distorted when too small a number of effective singular values is selected, and residual white noise would remain when selecting over many effective singular values [17]. The methods proposed in [17,18] selected effective singular values by the hard thresholding technique, which may deteriorate the original UHF PD signal. In the MTFM, as Figure 4b showed, the module values corresponding to PD signal are greater with a concentrated distribution. On the contrary, the module values corresponding to Gaussian white noise are lower with a scattered distribution. Therefore, the magnitudes of singular values corresponding to PD signals are greater with a large difference. In this paper, a novel method for selecting effective singular values based on the fuzzy c-means (FCM) clustering algorithm is proposed. Features fe1 and fe2 are extracted as the features for clustering when selecting effective singular values. fe1 and fe2 are shown in Equation (17): #
fe1 “ a1 , a2 , ..., an fe2 “ b1 , b2 , ..., bn , bi “ ai ´ ai´1 , i “ 1, 2, ..., q ´ 1
(17)
The basic principle of FCM can be described briefly as follows. Given a dataset of PD features F = {fe1 , fe2 , ¨ ¨ ¨ , feN }, where fej = (fe1 , fe2 , ¨ ¨ ¨ , fem )T . FCM aims to minimize the c-means function, which is defined as: 2 c ÿ N ÿ m min f pUU, VVq “ pµij q ||fe j ´ vi || (18) i “1 j “1
$ ’ ’ ’ ’ ’ & s.t. ’ ’ ’ ’ ’ %
c ř
µij “ 1, p1 ď j ď Nq
i“1
0 ď µij ď 1, p1 ď i ď c, 1 ď j ď Nq N ř 0ď µik ď N, p1 ď i ď cq
(19)
j“1
where c is the pre-defined cluster number, VV indicates the cluster center matrix, vi is the cluster center of class i, 1 ď i ď c, UU = [µij ]c ˆN represents the fuzzy membership degree matrix, µij is the fuzzy membership degree of f j belong to class i, m is the weighting coefficient, and m is set as 2 in this paper, dij = //f j -vi //2 represents the Euclidean distance between sample f j and vi . In order to obtain the minimum of f (UU, VV) and find the optimal UU = [µij ]c ˆN and vi , a Lagrange multiplier is formed to solve the optimization problem: F“
c ÿ
pµij qm pdij q2 ´ λp
i “1
c ÿ
µij ´ 1q
(20)
i “1
The optimal conditions of Equation (20) are: $ ’ &
BF Bλ
’ % BF “ Bµ ij
c ř
µij ´ 1 “ 0 i“1 rmpµij qm´1 pdij q2 ´ λs “
(21) “0
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According to to Equation Equation (21), (21), μμijij isis calculated calculated uing uing Equation Equation (22). (22). Then ThenB fby by solving solving According pUU,VV q
“ 0, According Equation (21), µij is calculated uing Equation (22). Then by solving UU,VV B vi ff((UU,VV )) 0to ,vvi iisiscalculated calculatedas asin inEquation Equation(23). (23).More Moredetails detailsof ofthe theFCM FCMmethod methodcan can befound found , be 0 vi is calculated as in Equation (23). More details of the FCM method can be found in [29]: vvi i
in[29]: [29]: in
2
c ÿ dij 1´m µij “ p q 22 1mm c c d dd 1 k“1 (kj ijij ( )) μμijij d k k11 dkjkj N ř pµij qm ¨ fe j
vi “
j“1NN
vvi i
((μμ )) ··fefe N ř j j 11 NN
ijij
mm
jj
m
pµij q mm j“1 ((μμijij))
(22)
(22) (22)
(23) (23) (23)
j j 11
4.3. De-Noising Method of Gaussian White Noise 4.3.De-Noising De-NoisingMethod MethodofofGaussian GaussianWhite WhiteNoise Noise 4.3. The suppression procedure of Gaussian white noise based on the MTFM and FCM clustering The suppression suppression procedure procedure of of Gaussian Gaussian white white noise noise based based on on the the MTFM MTFM and and FCM FCM clustering clustering The involves the following steps: involvesthe thefollowing followingsteps: steps: involves Step 1: Step1: 1: Step Step 2: Step Step 2:2: Step 3: Step3: 3: Step
Calculate singular values ofof the MTFM ofof the noisy PD signals byby SVD; Calculate singular values of the MTFM of the noisy PD signals by SVD; Calculate singular values the MTFM the noisy PD signals SVD; Calculate features of singular values sequence based on Equation (17); Calculate features of singular values sequence based on Equation (17); Calculate features of singular values sequence based on Equation (17); Classify singular values intointo two groups utilizing FCM clustering algorithm, which represent Classify singular values into two groups groups utilizing FCM clustering clustering algorithm, which Classify singular values two utilizing FCM algorithm, which UHF PD signals and Gaussian white noise, respectively; represent UHFPD PD signals andGaussian Gaussian white noise,respectively; respectively; represent UHF signals and white noise, Step4: 4: Select Select the group whose maximum value greater as the effective effective singular values, the Step 4: the group whose maximum value is greater as theas effective singularsingular values, the number Step Select the group whose maximum value isis greater the values, the number ofeffective effective singular values defined askpaper; kin inthis thispaper; paper; of effective singularsingular values isvalues defined as k in this number of isisdefined as Step5: 5: Recover Recover the de-noised PD signal by the conventional SVD de-noising method. Step 5: the de-noised PD signal byby the conventional SVD de-noising method. Step Recover the de-noised PD signal the conventional SVD de-noising method.
Figure6. Calculatedsingular singularvalues valuesby bydecomposing decomposingthe theMTFM. MTFM. Figure Figure 6.6.Calculated Calculated singular values by decomposing the MTFM.
Figure7. 7.Selection Selectionresults resultsof ofeffective effectivesingular singularvalues valuesbased basedon onthe theFCM FCMclustering clusteringalgorithm. algorithm. Figure algorithm. Figure 7. Selection results of effective singular values based on the FCM clustering
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The The calculated calculated singular singular values of the the MTFM MTFM are given given in in Figure Figure 6, 6, and and the the classification classification result result The calculated singular values of the MTFM are given in Figure 6, and the classification result utilizing the FCM clustering algorithm is given in Figure 7. The value of k was set as 4 in this paper. utilizing value was set as in this utilizing the FCM clustering algorithm is given in Figure 7. The value of k was set as 4 in this paper. The areare given in Figure 8. The8.results Gaussian white noisewhite is suppressed The de-noising de-noisingresults results given in Figure The show resultsthat show that Gaussian noise is The de-noising results are given in Figure 8. The results show that Gaussian white noise is successfully by using the by proposed method. When the number effective singular values singular was set suppressed successfully using the proposed method. Whenofthe number of effective suppressed successfully by using the proposed method. When the number of effective singular to k ´ 1 and k + 1, the de-noised PD signals are given in the Figure 9. The first pulse of recovered values was set to k − 1 and k + 1, the de-noised PD signals are given in the Figure 9. The first pulse of values was set to k − 1 and k + 1, the de-noised PD signals are given in the Figure 9. The first pulse of PD signalsPD is distorted k = 3, and kthe the de-noised signals will be lowered recovered signals isseverely distortedwhen severely when = 3,SNR andof the SNR of the de-noised signals will be recovered PD signals is distorted severely when k = 3, and the SNR of the de-noised signals will be when k = when 5. Thekcomparison of results of indicates the proposed based on FCM clustering lowered = 5. The comparison results that indicates that the method proposed method based on FCM lowered when k = 5. The comparison of results indicates that the proposed method based on FCM algorithm capable of singular values. clustering is algorithm is selecting capable ofeffective selecting effective singular values. clustering algorithm is capable of selecting effective singular values.
Figure 8. De-noised De-noised PD signals: signals: (a) (a) time-domain time-domain signals; (b) (b) time-frequency distribution. distribution. Figure Figure 8. 8. De-noised PD PD signals: (a) time-domain signals; signals; (b) time-frequency time-frequency distribution.
Figure 9. De-noised PD signals employing SVD when (a) k = 3; and (b) k = 5. Figure 9. 9. De-noised De-noised PD PD signals signals employing employing SVD SVD when when (a) (a) kk == 3; 3; and and (b) (b) kk == 5. Figure 5.
5. Simulation De-Noising Results and Discussions 5. Simulation 5. SimulationDe-Noising De-NoisingResults Resultsand andDiscussions Discussions The procedure of the proposed de-noising method is summarized in Figure 10. For comparison, The procedure procedure of of the the proposed proposed de-noising de-noising method method is is summarized summarized in in Figure 10. 10. For comparison, comparison, four The conventional de-noising methods [9–13,17] were employed to suppressFigure the noisy For UHF PD signals four conventional conventional de-noising de-noising methods methods [9–13,17] were to suppress the noisy UHF UHF PD PD signals signals four were employed employed to suppress theinnoisy shown in Figure 1b. The types of noise[9–13,17] and de-noising techniques are shown Table 3. The db2 and shown in Figure 1b. The types of noise and de-noising techniques are shown in Table 3. The db2 and shown in Figure The types ofmother noise and de-noising techniques 3. The db2 level and db8 wavelets are1b. selected as the wavelet in Methods C andare D shown where in theTable decomposition db8 wavelets are selected as the mother wavelet in Methods C and D where the decomposition level db8 as PD the signals motherare wavelet and D where decomposition level was wavelets set as 10. are Theselected de-noised given in in Methods Figure 11.CTo compare thethe performance of various was set as 10. The de-noised PD signals are given in Figure 11. To compare the performance of various was set as 10. The de-noised PD signals are given in Figure To compare the performance of various de-noising methods, the following evaluation parameters are11. introduced [14,30,31]. de-noising methods, areare introduced [14,30,31]. de-noising methods,the thefollowing followingevaluation evaluationparameters parameters introduced [14,30,31].
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Figure de-noising method. method. Figure 10. 10. Procedure Procedure of of the the proposed proposed de-noising
(I) The SNR illustrates the effectiveness of noise suppression, which is defined as: (I) The SNR illustrates the effectiveness of noise suppression, which is defined as: N
SNR 10log10 ( SNR “ 10log10 p
s(i)2
2
N ř N
i 1|spiq|
i“1
2
)
r(i ) s(i ) q N ř 1 i |rpiq ´ spiq|2
(24) (24)
i “1
where s(i) is the original signal and r(i) is the de-noised signal. Higher SNR value means more where s(i)background is the original signal and r(i) is the de-noised signal. Higher SNR value means more effective effective noise suppression. background suppression. error (RMSE) is defined as: (II) The noise root-mean-square (II) The root-mean-square error (RMSE) is defined as: 2 1 N RMSEg (25) r(i) s(i) f NNi 1 f1ÿ |rpiq ´ spiq|2 RMSE “ e (25) RMSE is used to evaluate waveform distortion N of de-noised signal compared with the original i“1 signal. Lower RMSE value means less waveform distortion of UHF PD signal waveform. (III) The correlation coefficient (NCC) defined as: RMSE is normalized used to evaluate waveform distortion of is de-noised signal compared with the original signal. Lower RMSE value means less waveform distortion of UHF PD signal waveform. N s(is i)·rdefined (i) (III) The normalized correlation coefficient (NCC) as: i 1 NCC (26) N N N 2 ř 2 ( sspiq¨ (i))·(rpiq r ( i )) i 1
i 1
NCC “ d i“1 (26) N N ř ř NCC is used to evaluate waveforms similarity between the original and de-noised signal. The 2 2 p s piqq¨ p r piqq NCC value is between the negative one (−1) and (+1), −1 means fore-and-aft waveform i“1 positivei“one 1 reverse, zero means orthogonal, and +1 means almost the same. NCCThe is used to evaluate waveforms similarity between the originaltrend and parameter de-noised (RVTP) signal. (IV) variation trend parameter (VTP) is composed of rise variation The NCC value is between the negative one (´1) and positive one (+1), ´1 means fore-and-aft and fall variation trend parameter (FVTP). waveform reverse, zero means orthogonal, and +1 means almost the same.
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(IV) The variation trend parameter (VTP) is composed of rise variation trend parameter (RVTP) and fall variation trend parameter (FVTP). N ř
RVTP “
i “2 N ř
rrpiq ´ rpi ´ 1qs (27) rspiq ´ spi ´ 1qs
i “2
where s(i) > s(i ´ 1), r(i) > r(i ´ 1). N ř
FVTP “
i “2 N ř
rrpi ´ 1q ´ rpiqs (28) rspi ´ 1q ´ spiqs
i “2
where s(i) < s(i ´ 1), r(i) < r(i ´ 1). VTP is the mean value of RVTP and FVTP. VTP “
RVTP ` FVTP 2
(29)
VTP is used to describe the similarity of the wave variation tendency, which measures the oscillating situation of the waveform. When the VTP value is close to 1, the variation tendency of two waveforms is the most similar. Table 3. Conventional de-noising methods. Method
Noise Type
Technique
A B C D E
Narrowband noise + White noise Narrowband noise + White noise Narrowband noise + White noise Narrowband noise + White noise White noise
Proposed method in this paper Adaptive IIR filter Adaptive threshold WT de-noising (db2) Adaptive threshold WT de-noising (db8) Conventional SVD de-noising method
Calculated results of various evaluation parameters are given in Table 4. The comparison results of de-noised PD signals and evaluation parameters show that: (1)
(2)
(3)
(4)
The adaptive filtering method is capable of suppressing periodic narrowband noise. However, because the frequencies of narrowband noise and UHF PD signals are mixed, the de-noised PD signal waveforms are distorted. Due to the large variation in time-frequency characteristics of UHF PD signals, de-noising results are affected by the mother wavelet selection [11]. In this paper, the de-noised signals with lower SNR and larger waveform distortion are obtained by utilizing Method C (select db2 wavelet); the de-noised signals with higher SNR and less waveform distortion are obtained by utilizing Method D (select db8 wavelet). Therefore, the de-noising effectiveness when using wavelet decomposition may become worse when an unsuitable mother wavelet is selected. Because the effective singular values calculated from the MTFM represent more timefrequency-amplitude information, the proposed method in this paper is capable of obtaining the de-noised UHF PD signals with higher SNR and less waveform distortion compared with conventional SVD de-noising method. Compared with four conventional de-noising methods, the proposed method based on the MTFM is capable of suppressing the periodic narrowband noise and Gaussian white noise successfully. Hence, by the proposed method, the de-noised UHF PD signals are similar to the original signals.
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Figure 11. 11. De-noising De-noising results results by by employing employing each each method: method: (a) (a) Method Method B; B; (b) (b) Method Method C; C; (c) (c) Method Method D; D; Figure and (d) Method E. and (d) Method E. Table 4. De-noising evaluation parameters using each method. Table 4. De-noising evaluation parameters using each method. Evaluation Parameter Evaluation Parameter SNR
SNR
RMSE
RMSE
NCC
NCC
VTP
VTP
De-Noising Method De-Noising Method Method A Method MethodBA Method MethodCB Method MethodDC Method MethodED Method MethodAE Method B Method A Method MethodCB Method MethodDC Method MethodED Method MethodAE Method B Method A Method C Method B Method MethodDC Method MethodED Method MethodAE Method B Method A Method C Method B Method D Method C Method MethodED
Method E
Pulse 1 Pulse 1 14.445 2.875 14.445 −0.871 2.875 0.585 ´0.871 6.2548 0.585 0.0013 6.2548 0.0248 0.0013 0.0587 0.0248 0.0551 0.0587 0.0124 0.0551 0.9898 0.0124 0.7416 0.9898 0.3503 0.7416 0.4152 0.3503 0.8832 0.4152 1.0745 0.8832 0.7814 1.0745 1.7645 0.7814 1.1484 1.7645 1.0852 1.1484
1.0852
Pulse 2 Pulse 3 Pulse 4 Pulse 2 Pulse 3 Pulse 4 19.482 16.794 20.987 1.832 3.892 2.968 19.482 16.794 20.987 −0.362 −1.796 −1.704 1.832 3.892 2.968 0.315 1.607 1.861 ´0.362 ´1.796 ´1.704 7.629 8.662 9.029 0.315 1.607 1.861 −4 7.6 × 10 3.7 × 9.029 10−4 8.3 ×7.629 10−4 8.662 0.0418 ´4 0.0196 ´4 0.0327 ´4 8.3 ˆ 10 7.6 ˆ 10 3.7 ˆ 10 0.0697 0.0234 0.0212 0.0418 0.0196 0.0327 0.0497 0.0189 0.0212 0.0697 0.0234 0.0212 0.0133 0.0166 0.0390 0.0497 0.0189 0.0212 0.9942 0.9934 0.9970 0.0133 0.0166 0.0390 0.6446 0.8118 0.7256 0.9942 0.9934 0.9970 0.4506 0.5122 0.5418 0.6446 0.8118 0.7256 0.5738 0.6577 0.7479 0.4506 0.5122 0.5418 0.9112 0.8035 0.5389 0.5738 0.6577 0.7479 1.0382 1.0260 1.0515 0.9112 0.8035 0.5389 0.6767 0.7042 0.7237 1.0382 1.0260 1.0515 1.2887 1.8554 1.7592 0.6767 0.7042 0.7237 1.0606 1.4327 1.3084 1.2887 1.8554 1.7592 1.0790 0.7157 0.7880 1.0606 1.4327 1.3084 1.0790 0.7157 0.7880
All All 16.473 2.712 16.473 −1.647 2.712 0.941 ´1.647 8.342 0.941 3.4 8.342 × 10−4 0.0198 ´4 3.4 ˆ 10 0.0432 0.0198 0.0347 0.0432 0.0134 0.0347 0.9907 0.0134 0.5773 0.9907 0.4301 0.5773 0.6209 0.4301 0.8251 0.6209 1.0476 0.8251 0.7432 1.0476 1.6346 0.7432 1.2636 1.6346 0.8956 1.2636
0.8956
The pulse shapes of de-noised UHF PD signal utilizing Method A are given in Figure 12. The original and the de-noised PD pulses were drawn in the black dashed line and red solid line, The pulse shapes of de-noised UHF PD signal utilizing Method A are given in Figure 12. respectively. The results show that the de-noising method proposed in this paper can recover the The original and the de-noised PD pulses were drawn in the black dashed line and red solid line, original PD pulses successfully and thus can assist in feature parameters extraction of UHF PD respectively. The results show that the de-noising method proposed in this paper can recover the signals for insulation defect type recognition. original PD pulses successfully and thus can assist in feature parameters extraction of UHF PD signals for insulation defect type recognition.
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Figure 12. Comparisons originaland and the the de-noised de-noised PD shapes. (a) Pulse 1; (b)1;Pulse 2; Figure 12. Comparisons of of thethe original PDpulse pulse shapes. (a) Pulse Pulse (b) Pulse Pulse 2; Figure 12. Comparisons of the original and the de-noised PD pulse shapes. (a) 1; (b) 2; (c) Pulse 3; (d) Pulse 4. (c) Pulse 3; (d) Pulse 4. (c) Pulse 3; (d) Pulse 4.
6. De-Noising Results of Field Detected Signals
6. De-NoisingResults Resultsof ofField FieldDetected DetectedSignals Signals 6. De-Noising To verify the effectiveness of the proposed de-noising method for field detected PD signals,
To the effectiveness ofofthe proposed de-noising method for for fieldfield detected PD two To verify thesignals effectiveness the proposed de-noising method detected PD signals, twoverify UHF PD were detected from two 500 kV substations in Guangzhou, China. Thesignals, picture UHF PD werewere detected fromfrom two 500 500 kV An substations in Guangzhou, TheThe picture of two UHF PD signals detected two kVexternal substations in Guangzhou, China. picture of thesignals field detection setup is given in Figure 13. horn antenna sensorChina. (AInfo-LB530NF, the field detection setup is is given ininFigure 13. external horn sensor (AInfo-LB530NF, A-Info Technology Co., Ltd, Chengdu, China) was employed detect the PD (AInfo-LB530NF, signals. The of the field detection setup given Figure 13.An An external horntoantenna antenna A-Info Technology Co., Ltd, Chengdu, China) was GHz, employed tomaximum detect PD signals. The operation bandwidth ofLtd, antenna sensor was 0.5–3 the antenna gain was 11operation dBi. The Technology Co., Chengdu, China) was and employed to the detect the PD signals. The detected UHF PD signals were recorded using a digital oscilloscope (Lecory-WavePro740, bandwidth of antenna of sensor was sensor 0.5–3 GHz, maximum gainantenna was 11 dBi. operation bandwidth antenna was and 0.5–3the GHz, and theantenna maximum gainThe wasdetected 11 dBi. Teledyne LeCroy, New York,using USA),awhich has 4 GHz bandwidth, 20 GSamples/s sampling rate, UHF PD signals were recorded digital oscilloscope (Lecory-WavePro740, Teledyne LeCroy, The detected UHF PD signals were recorded using a digital oscilloscope (Lecory-WavePro740, central processing unit (CPU) with 2.6 GHz dominant frequency and random access (RAM) New York,LeCroy, NY, USA), which hasUSA), 4 GHz bandwidth, GSamples/s sampling rate,memory central processing Teledyne New York, which has 4 20 GHz bandwidth, 20 GSamples/s sampling rate, with 2 G memory. The phase resolved partial discharge (PRPD) patterns corresponding to two unit (CPU) with 2.6unit GHz dominant frequency and random accessand memory (RAM) with 2 G memory. central processing (CPU) with 2.6 GHz dominant frequency random access memory (RAM) detected UHF signals are given in Figure 14, and according to [20,31], the detected signals were The resolved partial discharge (PRPD) patterns corresponding to two detected UHF signalstwo are withphase 2 G memory. ThePD phase resolved partial discharge (PRPD) patterns corresponding originated from the sources. The time-domain waveforms and time-frequency distributions to of given in Figure 14,PD and according to [20,31], the detected signalsto were originated from PD sources. detected UHF signals are given in Figure 14, 15. and according [20,31], the detected signals were two detected signals are given in Figure It is obviously that narrowband noise the and white The time-domain waveforms and time-frequency distributions of two detected PD signals are given originated from the PD sources. The time-domain waveforms and time-frequency distributions of noise interfered with the detected PD signals severely. in Figure 15. It is obviously that narrowband noise and white noise interfered with the detected PD two detected PD signals are given in Figure 15. It is obviously that narrowband noise and white signals severely. with the detected PD signals severely. noise interfered
Figure 13. Field detection setup in substation.
Figure 13. Field substation. Figure 13. Field detection detection setup setup in in substation.
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Figure 14. 14. PRPD PRPD patterns patterns corresponding corresponding to to the the detected detected UHF UHF signals: signals: (a) (a) signal signal #1; #1; (b) (b) signal signal #2. #2. Figure
Figure 14. PRPD patterns corresponding to the detected UHF signals: (a) signal #1; (b) signal #2.
Figure 15. 15. Waveforms Waveforms and and time-frequency time-frequency distributions distributions detected detected UHF UHF PD PD signals: signals: (a) (a) signal signal #1; #1; (b) (b) Figure
Figure 15. Waveforms and time-frequency distributions detected UHF PD signals: (a) signal #1; signal #2. #2. signal (b) signal #2.
The calculated calculated frequencies frequencies and and amplitude amplitude of of periodic periodic narrowband narrowband noise noise employing employing the the SMMV SMMV The method are given in Table 5. method are given in Table 5. The calculated frequencies and amplitude of periodic narrowband noise employing the SMMV
method are given in Table 5.
Table 5. 5. Calculated Calculated frequencies frequencies and and amplitude amplitude of of narrowband narrowband noise noise in in field field detection. detection. Table
Table 5. Calculated frequencies and amplitude of narrowband noise in field detection. Signal Frequency (MHz) Amplitude (mV) Signal Frequency (MHz) Amplitude (mV) UHF signal signal 1# 1# UHF
Signal
UHF signal signal 2# 2# UHF
UHF signal 1#
875.9 875.9 941.1(MHz) 941.1 Frequency 876.4 876.4
875.9 961.3 961.3 941.1
1.03 1.03 2.16 2.16 Amplitude (mV) 3.53 3.53 1.03 1.24 1.24
2.16
3.53 communication Periodic narrowband narrowband noise originates originates in in 876.4 local mobile mobile phone phone wireless wireless communication signals. signals. Periodic noise local UHF signal 2# 961.3by The de-noising de-noising results results of of field field detected detected PD PD signals signals by the the methods methods in in1.24 Table 33 are are given given in in Figure Figure 16. 16. The Table
Periodic narrowband noise originates in local mobile phone wireless communication signals. The de-noising results of field detected PD signals by the methods in Table 3 are given in Figure 16.
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Because the original PD signal without noise is not obtained as a reference in field detection, the Because the originalintroduced PD signal without noise is not obtained as available. a referenceTherefore, in field detection, the evaluation parameters in Equations (24)–(29) are not noise reduction evaluation parameters introduced in Equations (24)–(29) are not available. Therefore, noise ratio (NRR) and the amplitude reduction ratio (ARR) are proposed to evaluate the de-noising reduction ratioare (NRR) and as: the amplitude reduction ratio (ARR) are proposed to evaluate the results [32], which defined de-noising results [32], which are defined as:
22 NRR “ 10plog σ22 q) NRR 10(log log10 1 1´log 1010σ 10 2 2
(30) (30)
Y´Z ARR “ Y Zˆ 100% (31) (31) ARR Y 100% Y where σ1 and σ2 are the standard deviation of detected signal and de-noised signal, Y and Z are the where σ1 and σ2 are the standard deviation of detected signal and de-noised signal, Y and Z are the maximum amplitude of detected noisy signal and de-noised signal. High NRR and low ARR means maximum amplitude of detected noisy signal and de-noised signal. High NRR and low ARR means effective de-noising results. The evaluation parameters using each de-noising method are given in effective de-noising results. The evaluation parameters using each de-noising method are given in TableTable 6. The result shows that the proposed method is capable of suppressing noise effectively with 6. The result shows that the proposed method is capable of suppressing noise effectively with less amplitude reduction. less amplitude reduction.
Figure 16. De-noised PD signals in field detection using each method: (a) signal #1; (b) signal #2.
Figure 16. De-noised PD signals in field detection using each method: (a) signal #1; (b) signal #2.
The computation of proposed de-noising method was carried out by MATLAB codes in computer platformof ofproposed oscilloscope. The calculating time of carried de-noising UHF PD signals s The computation de-noising method was outtwo by MATLAB codesisin4.751 computer and 4.897 s, respectively. means that time the proposed de-noising method feasibleisin4.751 field test, platform of oscilloscope. TheItcalculating of de-noising two UHF PDissignals s andand 4.897 s, the de-noised results canthe be proposed obtained inde-noising real time. method is feasible in field test, and the de-noised respectively. It means that results can be obtained in real time.
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Table 6. De-noising evaluation parameters of field detected signals. Signal
De-Noising Method
Evaluation Parameter NRR
ARR
PD Signal #1
Method A Method B Method C Method D
15.96 10.59 13.21 13.36
21.4% 34.6% 27.6% 41.3%
PD Signal #2
Method A Method B Method C Method D
19.23 17.25 16.68 17.66
32.5% 48.3% 43.4% 36.6%
7. Conclusions To suppress the periodic narrowband noise and Gaussian white noise in UHF PD signal, a novel de-noising method is proposed based on the generalized S-transform and MTFM in this paper. The results are concluded as follows: (1)
(2)
(3)
To suppress periodic narrowband noise, the SMMV method based on generalized S-transform and MTFM is employed. By calculating the frequencies and amplitudes of the narrowband noise, the corresponding MTFMs are obtained and removed. To suppress Gaussian white noise, singular values are calculated through decomposing the 2-dimensional MTFM based on the generalized S-transform. Effective singular values can be selected successfully by employing the FCM clustering method. De-noising results of simulated and field detected UHF PD signals validate the feasibility of this method. Compared with some conventional de-noising methods, the proposed method obtains de-noised signals with high SNR and less waveform distortion.
Acknowledgments: This research was supported by Guangzhou Power Supply Co. Ltd. (Guangzhou, China), the field tests were performed in the Guangnan and Ruibao 500 kV substations in Guangzhou, China. Author Contributions: Yushun Liu and Wenjun Zhou conceived the algorithm and wrote the manuscript. Yushun Liu, Pengfei Li and Shuai Yang designed and performed the experiment. Yan Tian provided the field detection equipment. All authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations The following abbreviations are used in this manuscript: PD UHF HV WT SNR MTFM FCM SMMV SVD PRPD CPU RAM RMSE NCC VTP RVTP FVTP NRR ARR
Partial discharge Ultrahigh frequency High voltage Wavelet transform Signal noise ratio Module time-frequency matrix Fuzzy c-means Sub-matrix maximum module value method Singular value decomposition Phase resolved partial discharge Central processing unit Random access memory Root mean square error Normalized correlation coefficient Variation trend parameter Rise variation trend parameter Fall variation trend parameter Noise reduction ratio Amplitude reduction ratio
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