A new wavelet selection method for partial discharge denoising

Electric Power Systems Research 125 (2015) 184–195

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A new wavelet selection method for partial discharge denoising Caio F.F.C. Cunha a,b , André T. Carvalho a,b , Mariane R. Petraglia a , Antonio C.S. Lima a,∗ a b

Federal University of Rio de Janeiro, COPPE/UFRJ, Rio de Janeiro, RJ, Brazil Electrical Energy Research Center, CEPEL/ELETROBRAS, Rio de Janeiro, RJ, Brazil

a r t i c l e

i n f o

Article history: Received 7 January 2015 Received in revised form 7 April 2015 Accepted 8 April 2015 Available online 25 May 2015 Keywords: Partial discharge Partial discharge measurement Dielectric measurement Wavelet transforms Signal Denoising Signal reconstruction

a b s t r a c t Partial discharge signals are used to evaluate the insulation condition in several power devices. Their measurements are often heavily contaminated by noise from different sources and thus the use of noise reduction techniques is required. Wavelet shrinkage denoising methods are frequently employed and several wavelet basis selection approaches have been discussed recently in the literature. This paper presents a new method for choosing the wavelet basis, comprising a novel approach for determining the number of wavelet decomposition levels based on the energy spectral density of the signals, and a scaledependent algorithm for selecting the wavelet functions based on signal to noise ratios computed from the wavelet coefficients. At each decomposition stage, the predominant band of the partial discharge pulse, called the signal band, is identified as the one that has the highest coefficient magnitude. The other band is assumed to contain predominantly noise, and at each level the method selects the wavelet that maximizes the signal to noise ratio. The proposed approach is compared to the correlation based wavelet selection (CBWS) and to the energy based wavelet selection (EBWS) methods for signals simulated and obtained from high voltage equipment, showing better filtering performance for over 70% of the tested signals and a smaller processing time. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In power systems there is a high demand for diagnosing and monitoring operating conditions of power equipment in an efficient and accurate way. In recent years this need has considerably grown, since any unexpected downtime results in very high costs. Furthermore, as power systems grow older, the number of aging devices becomes substantial, leading to a need to increase their lifetime, thereby reducing power outages and economic losses. The monitoring of partial discharges (PD) is an effective tool for the evaluation of insulating system degradation in high voltage equipment. A typical PD signal is often hampered by the presence of several noise sources. This contamination may overwhelm the PD signal and cause errors in the assessment of the insulation conditions. In this scenario, the use of digital filtering techniques is a critical step for the evaluation of PD signals. In particular, the wavelet transform (WT) has proved a powerful tool for PD denoising, in view of its time-frequency localization properties [1]. Among several noise reduction algorithms that have been

∗ Corresponding author. Tel.: +552139388599. E-mail addresses: [email protected] (C.F.F.C. Cunha), [email protected] (A.T. Carvalho), [email protected] (M.R. Petraglia), [email protected] (A.C.S. Lima). http://dx.doi.org/10.1016/j.epsr.2015.04.005 0378-7796/© 2015 Elsevier B.V. All rights reserved.

proposed [2], denoising by wavelet shrinkage is the most frequently employed approach to reduce wideband noise and discrete spectral interference [3–13]. For the extraction of pulse-shaped noise, additional denoising strategies have been successfully used after the wavelet shrinkage, such as the ones based on back-propagation neural networks [11] and support vector machines [14]. To effectively suppress pulse-shaped noise, the wavelet shrinkage filtering operation must minimize signal distortions to preserve as much as possible the PD waveform [10]. In [15], linear discriminant analysis was applied to multi-scale fractal dimensions and energy parameters of ultra-high PD signals in order to recognize defects. In [16], spectral power analyses of PD pulses was employed to classify PD sources and noise by means of a graphical representation. In [17], modern methods for robust measurement and monitoring of highvoltage apparatus are described, while in [18] and [19] the designs of Hilbert fractal antenna and stacked Hilbert antenna array to increase the frequency bandwidth and sensitivity of the PD measurement system are discussed. The discrete wavelet transform (DWT) decomposes discrete signals into a particular basis of the vector space of finite-energy complex series l2 (Z). The wavelet basis of the DWT is determined by the number of decomposition levels and by the wavelet filters adopted in each level. The separation of PD pulses immersed in wideband noise in the WT domain is possible because the noise energy is uniformly distributed among the wavelet coefficients,

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while the PD pulse energy is concentrated on a limited number of coefficients with amplitudes above those of the corresponding noise coefficients. This separation will be more efficient as the wavelet basis better represents the PD pulses with the smallest number of high amplitude coefficients [5]. Therefore, the problem of finding the best wavelet basis for PD denoising can be formulated as a search for the wavelet basis that maximizes the reference pulse wavelet coefficients that are above the wideband noise level. Despite the fundamental importance of determining an appropriate number of wavelet decomposition levels [6], very little attention has been given to this issue in the literature. Most articles have reported either the use of the maximum possible number or the trial and error approach [4,5,7,9,10]. Regarding the selection of the wavelet filters (associated to the wavelet function, also called wavelet mother), two important techniques have been proposed: the correlation based wavelet selection (CBWS) method [9], which searches the wavelet function that best correlates to the PD pulse and applies it in all decomposition levels, and the energy based wavelet selection (EBWS) method [10], which, for each level, selects the wavelet filters that maximize the energy of the approximation coefficients located in the lower frequency subband. According to [10], EBWS provides better denoising results than CBWS. In this work we propose a new approach for the selection of the number of wavelet decomposition levels based on the energy spectral density (ESD) of a reference pulse, resulting in the minimum number of levels necessary to represent the pulse mainly in the detail subbands. As the ESD is independent from the WT, the number of decomposition levels can be calculated beforehand and provides essential information to the wavelet filter selection method. We also propose a new scale-dependent wavelet selection method that identifies, at each level, the predominant band of the PD signal and selects the wavelet filters that maximize the signal to noise ratio (SNR) between the complementary decomposition bands considering their peak coefficient values. The proposed SNR based wavelet selection (SNRBWS) method tends to concentrate the information of the PD signal on fewer coefficients with higher values, producing better filtering results and reducing signal distortions when compared to other techniques. The remaining of this paper is organized as follows. Section 2 presents an overview of the wavelet-shrinkage denoising method, describing the most employed thresholding approaches. Section 3 describes the proposed method for the selection of the number of wavelet decomposition levels, hereafter denoted as NWDLS. Section 4 introduces the proposed wavelet basis selection technique, SNRBWS, as well as the previously proposed CBWS and EBWS methods. Section 5 shows the partial discharge signals used in the denoising evaluations. The outputs are discussed in Section 6 and concluding remarks are given in Section 7.

Fig. 1. Bank structure for wavelet decomposition (analysis).

that of the input signal. These two signals are denoted as approximation a1 and detail d 1 coefficients, respectively. In the next stage the coefficients a1 are applied to the low-pass and high-pass filters followed by decimators, to generate the new approximation a2 and detail d 2 coefficients. The decomposition process is repeated until the final approximation aJ and detail d J coefficients are obtained. The filter bank composed by the low-pass and high-pass filters, which are also referred to as quadrature mirror filters (QMF), are designed such that it is possible to obtain a perfect reconstruction of the original signal by applying the inverse digital wavelet transform (IDWT), which can be implemented by a synthesis filter bank with factor of two interpolators (up-samplers) at each level. The denoising of a signal S by the wavelet shrinkage technique can be summarized as follows: 1. Determine the number of decomposition levels J and choose the wavelet filters for each level j, where j = 1, 2, . . ., J; then perform the decomposition to obtain the DWT coefficients [21]. 2. Obtain the thresholded coefficient values, considering hard or soft approaches. 3. Reconstruct the signal by applying the IDWT to the thresholded coefficients to obtain the denoised signal. The selection of the thresholding function and its parameters determines which wavelet coefficients will be associated to noise and hence eliminated. The thresholding procedure, known as wavelet shrinkage, can be divided into three steps: 1. Calculate the threshold value, which is frequently performed in one of the following four different rules [7]: rigrsure, sqtwolog, heursure and minimax. 2. Apply a multiplicative threshold rescaling method [7], according to one of the following procedures: a. sln: estimates the noise level based on the first detail coefficients d1 only, since in most cases they are mainly formed by noise, with a standard deviation that can be estimated as =

2. Wavelet-based denoising technique for PD detection The decomposition process of a measured PD signal S through a wavelet transform allows us to separate in different coefficients most of the existing noise and the signal of interest, making it possible to mitigate or even eliminate the noise by the use of a threshold shrinkage function. The dyadic digital wavelet transform is the most frequently employed transform in data denoising techniques [5,6,20] due to its desirable time-frequency characteristics [1]. The practical implementation of a dyadic digital wavelet transform as an analysis filter bank is depicted in Fig. 1. In the first stage, the PD signal S passes through a two-channel filter bank composed of a low-pass filter G(n), corresponding to the scale function, and a high-pass filter H(n), corresponding to the wavelet function, followed by factor-of-two decimators (down-samplers). The outputs of the first stage are two signals whose lengths are equal to half of

185

MAD|d1 | q

(1)

where MAD|d1 | is the median absolute deviation of the detail coefficients, and q may vary between 0.4 and 1, with typical value equal to 0.6745. The combination of the threshold selection rule sqtwolog with the multiplicative threshold rescaling sln corresponds to the well-known global thresholding rule [22] given by thr =

MAD|d1 |  2 log(N) 0.6745

(2)

b. mln: similar to sln, but the estimation of the noise variance based on the coefficients of each level. The jth level noise variance is estimated by j =

MAD|dj | q

(3)

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The use of the sqtwolog rule with multiplicative threshold rescaling given by thr j =

MAD|dj |  0.6745

2 log(N)

(4)

is frequently employed since it tends to eliminate a smaller amount of the PD signal energy when compared to (2), and because the wavelet coefficients patterns of the PD pulse and of the noise are level dependent [5]. 3. Apply a threshold to the wavelet coefficients, which will determine how such coefficients will be attenuated or zeroed in order to eliminate or reduce the noise distributed among them. The most frequently used threshold functions are the hard and soft functions [23,24]. The hard threshold function, given by



fH (t, thr) =

t,

|t| ≥ thr

0,

otherwise

(5)

is commonly applied in denoising operations of PD signals, because it preserves the signal magnitude characteristics, which provides a better SNR [5,9]. Therefore, the hard function was used in this work to compare the performance of the different algorithms, rather than the soft function employed in [10]. 3. Number of wavelet decomposition levels selection (NWDLS) In the literature, a few procedures have been adopted for the automatic selection of the number of wavelet decomposition levels employed in the denoising of PD signals. In most cases [4,5,7,9,10], however, this selection is made at random or by verifying that, for a particular PD signal, a certain number of levels J is enough to provide good denoising results. Although pointed out in [6] that the number of levels must be related to the sampling rate and spectrum of the PD pulse, its maximum possible value was employed. Considering that the signal length at the highest level of decomposition must be at least equal to the length of the wavelet filter, the maximum number of decomposition levels is given by



J = fix log2



N NW − 1



(6)

where N is the signal length and NW is the length of the decomposition filter associated to the mother wavelet. However, such method cannot be applied to a scale-dependent wavelet selection algorithm, which employs different wavelet filters in each level, since a priori knowledge of all wavelet functions would be necessary to determine the number of decomposition levels. In this case, a wavelet independent maximum value could be calculated as Jmax = fix(log 2 N) [6]. In this work we propose that the number of decomposition levels be selected based on the value of the lowest representative frequency component present in the PD signal, obtained from its energy spectral density (ESD), as follows:



J = fix log2

 F  S Fmin

the signal, and the vertical dashed line indicates the lowest frequency component whose energy is greater than that indicated by the horizontal line, which in this case is Fmin = 5.4932 MHz. Substituting these values in (7) we found J = 8 decomposition levels, and by this procedure we ensured at least 85% of the pulse energy to be represented in detail subbands. Besides providing a good wavelet decomposition through the selection of an appropriate number of decomposition levels, this method avoids the need for decomposing the signal in its maximum number of levels, which would be Jmax = 13 in the above example. As a result, large process time savings is obtained in both the wavelet selection and the denoising process. 4. Wavelet filters selection In this section we briefly describe the previously proposed CBWS and EBWS methods [9,10]. Subsequently, we present a new approach for selecting the mother wavelet of each stage of the PD signal decomposition, named SNRBWS, which has shown itself an efficient alternative to the CBWS and EBWS methods. 4.1. CBWS and EBWS methods In the CBWS procedure [9,10], the wavelet selection is made with the purpose of maximizing the correlation  between the ideal PD signal and the wavelet function chosen from a library of wavelets. The selected wavelet is not scale-dependent, and hence the same mother wavelet is employed in all decomposition levels. The correlation is computed as

n−1

=



where Fmin is determined by the lowest frequency component, whose energy is greater than a certain percentage of the total signal energy. Such percentage is used with the purpose of discarding the frequency components that do not contain significant energy. If J exceeds the maximum allowable number of levels, then J = Jmax must be adopted. Consider, for example, the PD signal of length N =10,000 and sampling frequency FS =2500 MHz shown in Fig. 2a, and its energy spectral density in Fig. 2b. The horizontal dashed line in Fig. 2b indicates the level where the energy is 15% of the total energy of

¯ (xi − x¯ )(yi − y) − x¯ )2

(8)

n−1

¯ 2 (yi − y) i=0

where xi and yi represent the ith samples of the PD signal of interest and of the wavelet function, respectively, and the variables x and y are their average values. The main difficulty associated to the CBWS method, however, is the determination of the appropriate scale for the wavelet function in which the correlation with the PD pulse should be calculated. In [10] the authors proposed a heuristic determination of this scale, rescaling in time and aligning the wavelet function in order to coincide with the PD pulse in their maximum and in their first point of zero crossing after their maximum. This approach was also adopted in this work. An alternative procedure to select scale-depended wavelet filters is to apply the EBWS method presented in [10], which performs the search by an optimal wavelet at each level from a library of wavelet filters, by the maximization of the energy percentage in the approximation aj , which leads to a reduction of the energy percentage in d j . The energy percentage of the approximation coefficients Eaj is calculated by



Eaj =

 k

(7)

i=0

n−1 (xi i=0

k 2

(aj,k ) +

(aj,k )2

j  i=1

k

(di,k )

2

(9)

where j is the wavelet coefficient index. According to the results presented in [10], the EBWS method outperformed CBWS in both noise filtering and runtime. 4.2. SNRBWS method The proposed SNRBWS method stemmed from the fact that the EBWS approach, always seeking to maximize the energy of the approximation coefficients, not necessarily concentrates the energy of the PD signals in as few levels as possible, which is the main goal of the wavelet filters selection for PD denoising [9]. For example, consider again the ESD of the PD pulse in Fig. 2. Until the

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187

Fig. 2. Illustration of the NWDLS method: (a) PD signal, and (b) its energy spectral density (ESD). The vertical dashed line indicates the lowest frequency component whose energy is larger than 15% of the total pulse energy. With an 8-level decomposition, at least 85% of the pulse energy is in the detail subbands.

6th level, where the energy of the pulse was still concentrated on the approximation bands, the EBWS method chose the best filters in order to maximize the energy corresponding to the pulse. However, at level 7 the EBWS still tried to maximize the energy on the approximation band, while the energy of the PD pulse was concentrated on the detail coefficients. Consequently, the filters selected from this level on were not the best ones to concentrate the wavelet coefficients energy, but rather to spread it. To circumvent this difficulty, the proposed method identifies, at each decomposition level, the predominant band of the PD pulse (called the signal band) as the one that has the maximum absolute coefficient value. The complementary band, which is assumed to contain less relevant information, is then identified as the noise band. The wavelet filters for each level are then chosen in order to maximize the peak signal to noise ratio (SNR) between the two complementary decomposition subbands. Assuming that the band with the highest energy concentration corresponds to the band with the highest absolute coefficient value, the SNRBWS method maximizes, at each level, the energy difference between the adjacent approximation and detail subbands. By finding the wavelet filters that maximize the energy in those bands where supposedly the energy of the analyzed PD pulse is larger, this method concentrates the pulse energy in a smaller number of coefficients with larger amplitudes. Similar objective was applied in [9]. The proposed method is detailed next. At the jth stage, the signal decomposition is performed for each filter pair of a candidate wavelet, generating the approximation aj and the detail coefficients d j . The signal coefficients are identified as aj or d j , whichever has the highest magnitude. The remaining set of coefficients is identified as noise. Summarizing: if max(|dj |) ≥ max(|aj |) then signal = d j and noise = aj else signal = aj and noise = d j The SNR is then calculated by SNR = max(|signal|)/ max(|noise|)

(10)

The wavelet filters with the highest SNR are then selected for this stage.

Various wavelet filters suitable for signal decomposition are available in the literature [1,21]. Among the ones commonly applied to PD signals are the wavelets of Daubechies (db), Coiflets (coif) and Symlets (sym) families [9,10], because of properties such as compactness, orthogonality, speed and symmetry, that are desirable in the analysis of fast transient and irregular pulses as PD signals [25,26]. The wavelet library employed in this work comprised Daubechies wavelets of orders 1–40 (db1–db40), Symlets wavelets of orders 2–15 (sym2–sym15), and Coiflets wavelets of orders 1–5 (coif1–coif5). In summary, the selection process of the wavelet by the SNRBWS algorithm is composed of the following steps: 1. Create a library of wavelet function t,i , for each type of wavelet family (type t = 1, 2, ..., p) and selected wavelet orders (order i = 1, 2, ..., N); initialize j = 1; 2. Perform a single-level decomposition of the PD signal S with each wavelet of the library, generating the approximation aj and the detail d j coefficients; 3. Compare the peak values of aj and d j , assigning the coefficients with the highest peak to signal and the others to noise, and calculate the SNR(t,i) defined by (10); 4. Find the maximum SNR(t,i) , for t = 1, 2, . . ., p and i = 1, 2, . . ., N, and save the corresponding wavelet T,I as the best for that decomposition level; thus, T and I correspond to the best values of t and i; 5. Repeat steps 2–4, with j = j + 1, until the maximum number of levels J is reached, taking the approximation coefficients obtained with the best wavelet T,I at level j − 1 as the signal S to the jth level. The flowchart describing the whole procedure of SNRBWS algorithm is shown in Fig. 3, where the number of decomposition levels J is obtained by the method NDWLS. Since the wavelet filters selected by SNRBWS tend to concentrate components with higher amplitudes in the frequency bands that contain most of the desired PD signal information, there is a tendency that the reconstructed PD signals present less distortion with better denoising results when compared to the other methods, particularly in low SNR conditions. A large number of

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Fig. 3. Flowchart of SNRBWS.

experimental results confirming the above statement are presented in Section 6. As the calculation of the maxima of the complementary bands is computationally less expensive than the calculation of their energy, the SNRBWS method usually presents lower runtimes than does the EBWS. In [27], we employed the SNR metric in dB. In the present work, however, we adopted Eq. (10) that, being less expensive than the log equation, made the method even faster, while producing the same results. Simulation results presented in Section 6 confirm that the SNRBWS is faster than the other methods. A practical limitation of the three methods discussed above for the wavelet basis selection may be the previous determination of a typical PD pulse waveform, which can be difficult in very low SNR conditions [11,14,16]. Usually, any PD pulse with high amplitude can be used whenever available. When the noise level is too high, a reference PD pulse waveform can be calculated by the average of a set of normalized noisy pulses. When no PD pulse waveform is available, a calibration pulse can be adopted as a reference. Even without carrying any information about how the PD signal was

originated, the calibration pulse will be shaped by the transfer function of the measurement circuit, as would the real PD pulse.

5. Partial discharge signals In order to evaluate the performance of PD signals denoising, experiments were carried out with both simulated and measured PD signals.

Table 1 Parameters for S3 –S5 . Pulse

 (␮s)

S3 S4 S5

2 2.5 2

f0 (kHz) 1000 250 500

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Table 2 Parameters for S6 and S7 . Pulse

A1

A2

A3

A4

f1 (MHz)

f2 (MHz)

f3 (MHz)

f4 (MHz)

S6 S7

1.06 −1.09

0.70 −1.00

−1.06 −0.41

−0.01 0.01

30 60

20 40

15 30

10 20

Table 3 Sampling rates of simulated signals.

Table 4 Measured PD signals and sampling frequencies.

Signal

S1

S2

S3

S4

S5

S6

S7

Designation

Equipment

Fs (MHz)

Sampling rate (MHz)

60

60

10

10

10

1000

1000

S8 S9 S10 S11 S12 S13 S14

Stator bar Circuit breaker Generator GIS Surge arrester Transformer Current transformer

250 500 500 1000 2500 2500 1000

5.1. Simulated PD Signals Seven signals proposed in the literature were employed. The signals S1 and S2 , adopted in [9], are given by S1 = A(e−˛1 t − e−˛2 t ) S2 = A(e

−˛1 t

cos(ωt − )) − e

(11) −˛2 t

cos())

(12)

where A = 1.5, ˛1 = 1 ×106 s−1 , ˛2 = 1 ×107 s−1 , ω = 2 × 106 rad/s. The signals S3 , S4 and S5 , proposed in [20], are defined as

Sn =

⎧ ⎪ ⎨

A(e

−

⎪ ⎩ 0,

t − t0  cos(2f0 (t − t0 ))),

t ≥ t0

(13)

t < t0

where A = 1.0 and t0 = 1 ␮s. The values of  and f0 are shown in Table 1 for the signals S3 , S4 and S5 . Finally, signals S6 and S7 , proposed in [8], are

Sn = e−ıt

⎧ A sin(2f1 t), ⎪ ⎨ 1

t1 ≤t < t2

A2 sin(2f2 t),

t2 ≤t < t3

A3 sin(2f3 t) + A4 sin(2f4 t),

t3 ≤t < t4

⎪ ⎩

(14)

where ı = 7 ×106 , t1 = 200 ns, t2 = 300 ns, t3 = 500 ns, and t4 = 1000 ns. The remaining parameters are presented in Table 2. The sampling rates of all simulated pulses are given in Table 3. 5.2. Measured PD Signals The PD measuring circuit presented in [25] and shown in Fig. 4 was employed in our laboratory experiments. In this configuration, the object under test, Ca , is placed in parallel with a coupling capacitor Ck in series with a coupling device Z, from which the PD is detected and passed through a coaxial cable to the data acquisition system. A synchronization signal is obtained from the capacitive voltage divider composed by capacitors C1 and C2 , so that the system can identify the main phase where each PD pulse occurs. The measurement circuit is fed by a high voltage source in series with an inductor that acts as a filter to reduce the high frequency noise

Fig. 5. Impedance used for the transformer, generator, stator bar and breaker measurements.

components that may come from the source. This filter can only be used offline or in laboratory measurements, since in on-line measurements it would be impractical to insert a filter in the supply branch that comes to equipment, both for security reasons as not to interrupt the operation of the equipment under analysis. In this work, an inductor of 3 H was employed to evaluate a generator stator bar and a circuit breaker. For the generator, GIS, surge arrester, transformer and current transformer, the measurements were performed on site, and the arrangement was that of Fig. 4, except for the filter. The coupling device Z varied according to the equipment. The list of analyzed equipment is shown in Table 4, along with the corresponding sampling rates. In the transformer, generator, stator bar and circuit breaker measurements, the impedance Zm , shown in Fig. 5, was employed, where Cm , Rm and Lm are the measuring capacitance, resistance and inductance, respectively, which are accompanied by a gas current suppressor at both circuit input and output in order to protect the measuring system. The impedance Zm corresponds to a highpass filter and its component values are selected to provide a cutoff frequency of about 1.2 kHz, which is sufficient to remove the

Fig. 4. PD measurement circuit used in laboratory.

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Fig. 6. Photographs of evaluated generator, GIS and current transformer.

Table 5 Selected wavelets for obtaining the reference signals.

6. Results

Signal

S8

S9

S10

S11

S12

S13

S14

Levels Wavelet

13 coif5

7 sym11

8 coif2

13 db4

10 db10

6 sym12

8 coif3

The results of the SNRBWS method applied to the various PD reference signals described in Section 5 are next compared to those obtained with the EBWS and CBWS methods. 6.1. Wavelet basis selection

60 Hz component and its possible harmonics.1 . For the gas insulated switchgear (GIS) measurements, a capacitive strap was used as a coupling capacitor to collect the PD signal, without external impedance. On the evaluation of transformers, the coupling capacitor Ck is situated inside the bushing, and hence the impedance is connected directly to its tap output [28]. For the surge arrester and the current transformer, which have higher capacitances, a high frequency current transformer connected to the ground cable of the object under test was employed as the sensor, dismissing the use of Ck . Photographs of the measured generator, GIS and current transformer are shown in Fig. 6. The measurement system bandwidth is determined by the characteristics of the measurement equipment and cables [28]. Although the data were obtained at different sampling rates, each signal was analyzed considering 10,000 samples. In order to obtain filtered reference pulses from the measured signals, we applied a procedure similar to the one employed in [10]. The signals were filtered through the wavelet shrinkage algorithm with the hard thresholding function, with threshold values obtained by the sqtwolog method and mln rescaling approach. The number of decomposition levels employed in this process was selected by the NWDLS method described in Section 3, and the wavelets were selected by the CBWS method applied directly to the measured signals. Table 5 presents both parameters used to denoise each of the acquired signals. The reference pulses selected from the denoised signals, as well as the simulated pulses described in Section 5.1, are shown in Fig. 7.

1 In our experiments, the following values were adopted: Cm = 10 ␮F, Lm = 820 ␮H and Rm = 1 k .

The total number of decomposition levels for each reference signal was obtained from (7), using the NWDLS method described in Section 3. The frequency Fmin was calculated in order to concentrate 90% of the energy of the pulses in the details subbands. Fig. 8 shows the normalized ESD of each pulse and the decomposition bands generated with the calculated number of decomposition levels. Table 6 shows for each signal the optimal wavelets obtained with the CBWS, EBWS and SNRBWS methods in all decomposition levels. Note that for the EBWS and SNRBWS algorithms a different wavelet is obtained for each level, whereas for the CBWS algorithm a single wavelet is selected for all levels. As originally stated in [9], the objective of the wavelet basis selection is to concentrate the energy of the pulses in as few levels as possible in order to generate better denoising results. Fig. 9 presents the energy distributions by levels produced by each method for all signals. Note that the SNRBWS approach yielded the largest energy concentration for all the measured pulses, and for all the simulated pulses except for S1 and S4 . For signals S1 and S4 the methods EBWS and CBWS, respectively, resulted in better energy concentration. This fact confirms that no method can always produce the best energy concentration, as the result depends on the waveform and on the initial time of each pulse. 6.2. Runtime comparison Considering all decomposition levels and all signals, a total of 111 wavelet selections were performed by the SNRBWS and EBWS approaches, while only 14 wavelet selections were performed by the CBWS technique. The total processing time required for each method is shown in Table 7. The simulations were carried out on a processor Intel Core2 Duo 2.4 GHz personal computer. Although the number of wavelet selections was considerably smaller in the CBWS method, it should be noted that considering together all

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191

Fig. 7. Reference pulses: S1 –S7 (simulated) and S8 –S14 (filtered from measured signals).

analyzed signals, both EBWS and SNRBWS approaches spent less than half of the time required by the CBWS, confirming the result pointed in [10] that the CBWS method performs much slower than the multilevel techniques. However, as the processing time of EBWS (and also of SNRBWS) is proportional to the number of

decomposition levels, its difference to the CBWS processing time was not as large as found in [10]. The relative execution times of each method are shown in Fig. 10. It should be noted that for signal S11 the CBWS method required less time than the other two, indicating that the processing

Fig. 8. Normalized ESD and decomposition subbands for signals S1 –S14 with the NWDLS method for 90% of the pulses energy decomposed in the detail subbands (see Fig. 2).

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Table 6 Wavelets selected by: (A) CBWS, (B) EBWS and (C) SNRBWS. PD

S1

S2

S3

S4

S5

S6

S7

S8

S9

S1 0

S1 1

S1 2

S1 3

S1 4

Decomposition level

A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C

1

2

3

4

5

6

7

8

9

10

11

12

13

db3 coif1 coif2 sym6 coif2 coif2 db3 sym5 sym5 sym4 db1 db1 db3 sym5 db1 db10 sym5 db37 sym13 sym12 coif1 coif3 db2 coif5 sym11 sym5 sym4 coif2 coif2 sym4 db3 sym5 sym4 db10 sym5 db38 sym13 db11 db33 coif4 coif3 coif3

db3 db2 db1 sym6 db9 db39 db3 sym10 db40 sym4 sym5 db1 db3 sym4 sym13 db10 coif2 db8 sym13 coif2 db4 coif3 db9 db3 sym11 db7 db5 coif2 coif2 coif2 db3 db6 sym11 db10 db15 db22 sym13 db13 db18 coif4 db36 db35

db3 db2 db1 sym6 sym4 sym7 db3 db1 coif2 sym4 sym4 db8 db3 sym6 coif2 db10 coif2 sym5 sym13 coif2 db3 coif3 sym12 db5 sym11 db16 sym13 coif2 coif2 sym7 db3 db4 sym6 db10 sym9 sym12 sym13 db11 sym10 coif4 db34 db35

db3 db5 db1 sym6 db12 db11 db3 db2 db15 sym4 db2 sym7 db3 db3 db36 db10 db9 db39 sym13 coif3 sym7 coif3 sym11 sym12 sym11 db15 db12 coif2 coif2 db5 db3 sym5 db6 db10 db40 db37 sym13 db12 sym10 coif4 db36 db2

db3 db5 db4 sym6 db2 db2

db3 db27 db1 sym6 db34 sym4

db3 db5 db3

db3 db3 db1

db3 db6 db2

db3 sym7 db1

db3 db3 db1

db3 db4 db4

db3 db4 sym4

sym4 db4 sym5 db3 db2 coif1 db10 sym5 sym11 sym13 db9 db5 coif3 sym12 db12 sym11 db37 db28 coif2 coif3 sym6 db3 db3 coif3 db10 db3 db9 sym13 db33 db32 coif4 db1 db39

sym4 coif1 db1 db3 db2 db2 db10 sym12 db21 sym13 db34 db37 coif3 db4 sym8 sym11 db37 db22 coif2 coif2 db9 db3 db7 db3 db10 db37 db35 sym13 db1 sym12 coif4 db1 sym13

sym4 coif1 db1

sym4 coif1 db1

sym4 coif1 coif1

sym4 coif1 db1

sym4 db2 db1

sym4 coif1 sym11

sym4 coif1 db1

db10 db35 db28 sym13 db1 db28 coif3 db1 db1 sym11 db1 db19 coif2 db1 db13 db3 db39 coif1 db10 coif1 db39

db10 db1 db40 sym13 db3 db6

db10 db3 db2

Table 7 Total runtime required by each method to select the wavelet basis for all signals. Method

CBWS

EBWS

SNRBWS

Total runtime (s)

525.12

205.69

203.06

time also depends on the pulse waveform, and that there is no guarantee that any method always performs faster than the other ones. Table 7 shows that CBWS spent 155.29% more time than EBWS, and that EBWS spent 1.30% more time than SNRBWS. Although the execution times of SNRBWS and EBWS are similar, SNRBWS executed faster in 11 out of the 14 analyzed signals, as can be seen in Fig. 10.

db3 db5 db1

coif4 db1 db1

The shrinkage wavelet filtering was performed with the hard thresholding function, using the sqtwolog method with the mln scaling to find the threshold values. To reconstruct the signals, only the decomposition bands in which the percentage energy of the reference pulse was greater than 1% were used. This procedure presented better denoising results than the simple sqtwolog-mln approach, and was inspired by the visual band selection implementation previously published in [29]. To evaluate the distortion introduced in the denoised signals, we compared the resulting pulse waveforms with the reference ones before the addition of noise. The parameters employed to quantify such distortion were the correlation coefficient (CC) and the mean squared error (MSE), defined as

6.3. Denoising results

2 1

x1 (i) − x2 (i) N N

MSE = Noise is usually defined as an unwanted component of a measured signal that has no relationship with the signal of interest [2]. The most common sources of noise in PD measurements are described in [6]. Wavelet shrinkage denoising results are affected not only by the waveform of the pulse and its relative initial time, but also by the stochastic noise sample added to the pulse. To produce more general results, for each SNR, 50 independent noise signals were added to each PD pulse, and the average results of the 50 simulations are presented.

coif2 db1 db19 db3 db1 db4 db10 db4 db21

(15)

i=1

N CC =



x (i) x2 (i) i=1 1

N 2 x (i) i=1 1

N

(16) 2

x (i) i=1 1

The best filtering results correspond to the largest CC and/or lowest MSE.

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193

Fig. 9. Percent energy distributions among levels obtained by CBWS, EBWS and SNRBWS for signals S1 –S14 .

Fig. 10. Relative time of wavelet basis selection methods for signals S1 –S14 .

6.3.1. Wideband stochastic additive noise To evaluate the filtering performance of each basis selection method for each reference signal, 50 independent Gaussian white noise signals were generated with five different SNR values (0.25, 0.5, 1, 2 and 4), and each of such signals was added to each of the 14 PD pulses, performing a total of 10,500 filtering simulations. We present next the mean results obtained for each signal and SNR value. Fig. 11 shows the mean values of CC. It should be noted that, especially in low SNR conditions (Fig. 11a and b), the SNRWBS method produced higher CC for most of the signals. Fig. 12 presents the resulting MSE mean values for all signals and SNR conditions. For better visual comparison, we show the logarithm of the mean MSE. Again it can be seen that the basis produced by SNRBWS produced lower mean MSE values for all SNR values, particularly in low SNR conditions (Fig. 12a and b). Table 8 displays, for each value of SNR, the number of signals for which the basis selected by the proposed method produced the best denoising results when compared to CBWS and EBWS. Considering the universe of 10,500 filtering simulations, Table 9 shows the percentage of best filtering results obtained by each of the wavelet selection methods. Regarding both CC and MSE parameters, the SNRBWS method presented the best performance in over 70% of the cases.

Fig. 11. Mean value of the correlation coefficient for each method and signal with additive Gaussian white noise: (a) SNR = 0.25, (b) SNR = 0.5, (c) SNR = 1, (d) SNR = 2, (e) SNR = 4. Table 8 Number of signals, from S1 to S14 , for which the SNRBWS produced the best results for each SNR. SNR

Best CC value

Best MSE value

0.25 0.5 1 2 4

10 11 11 12 10

12 11 11 12 10

Fig. 13 illustrates the filtering results for signal S11 in presence of additive Gaussian white noise with SNR = 0.25. Note that the wavelet shrinkage filtering removes most of the wideband noise using any of the three proposed basis selection methods.

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Fig. 12. Logarithm of the MSE mean value for each method and signal with additive Gaussian white noise: (a) SNR = 0.25, (b) SNR = 0.5, (c) SNR = 1, (d) SNR = 2, (e) SNR = 4.

Fig. 14. (a) Mean value of the correlation coefficient for each method and signal with additive DSI noise of SNR = 1; (b) logarithm of the MSE mean value for each method and signal with additive DSI noise of SNR = 1.

Table 9 Percentage of best results obtained by each method in the universe of 10,500 simulations. Method

CBWS

EBWS

SNRBWS

Best CC (%) Best MSE (%)

4 8

25 19

71 73

However, this example clearly shows that the filtering operation using the SNRBWS basis yields better preservation of the waveform and amplitude of the original pulse than does the CBWS. The EBWS method, although performing better than CBWS in preserving the pulse shape, kept some noise power where no signal was expected. 6.3.2. Discrete spectral interference Although the wavelet shrinkage processing is theoretically more appropriate for wideband noise removal from pulsing signals, its performance was here evaluated in the presence of discrete spectral

Fig. 15. Filtering results (in blue) for signal S8 in presence of DSI noise with SNR = 1 and Gaussian white noise with SNR = 1. The original reference signal is shown (in red) for visual comparison: (a) noisy signal; (b) filtered signal using the CBWS basis; (c) filtered signal using the EBWS basis; and (d) filtered signal using the SNRBWS basis. (For interpretation of the references to color in this legend, the reader is referred to the web version of the article.)

interference (DSI). As suggested in [6], the AM noise was generated by nAM =

5

(c + m sin(2fm t)) sin(2fi t)

(17)

i=1

Fig. 13. Filtering results (in blue) for signal S11 in presence of additive Gaussian white noise with SNR = 0.25. The original reference signal is shown (in red) in all graphs for visual comparison: (a) noisy signal; (b) filtered signal using the CBWS basis; (c) filtered signal using the EBWS basis; (d) filtered signal using the SNRBWS basis. (For interpretation of the references to color in this legend, the reader is referred to the web version of the article.)

with c = 1, m = 0.4, fm = 1 kHz and fi varying from 600 kHz to 1400 kHz in steps of 200 kHz. We added to the reference signals the AM noise defined by (17) with SNR = 1 and a stochastic Gaussian white noise with SNR = 1, and performed for each signal 50 filtering simulations. The obtained CC and MSE mean values are shown in Fig. 14. The SNRBWS method again outperformed CBWS and EBWS, yielding higher CC in 10 out of 14 signals, and lower MSE in 11 out of 14 signals. Fig. 15 shows the filtering results of signal S8 in presence of additive AM noise with SNR = 1 and Gaussian white noise with SNR = 1. Again the wavelet shrinkage filtering process was able to remove

C.F.F.C. Cunha et al. / Electric Power Systems Research 125 (2015) 184–195

most of the wideband noise in all cases, but only the filtering with the SNRBWS basis preserved the waveform and the amplitude of the original pulse. In this case the EBWS approach was unable to filter all the DSI noise, and consequently presented worse results than did the other methods. 7. Conclusions The proposed methodology for obtaining the best wavelet basis for PD denoising is composed of two parts: a new approach for determining the wavelet decomposition level and a new scaledependent wavelet selection method based on a peak SNR measure between approximation and detail coefficients at each decomposition level. The proposed NWDLS method determines the minimum number of decomposition levels necessary to ensure the decomposition of a given percentage of the energy of a PD pulse into detail subbands. As the number of levels is calculated from the pulse ESD and the sampling rate, the method is independent from the wavelet functions adopted for the basis, and thus can be used with any other wavelet selection method. Avoiding the use of the maximum number of decomposition levels, the proposed method saves runtime on the selection of the wavelet functions, as well as on the wavelet shrinkage denoising processing. This contribution may fill a gap in determining wavelet basis for PD denoising, since the selection of WDL has not received as much attention in the literature as has the selection of the wavelet functions. The proposed SNRBWS method finds, at each level, the wavelet function that maximizes the difference between the peak amplitudes of complementary subbands. Simulation results indicated that the SNRBWS method outperformed the CBWS and the EBWS methods in over 70% of the cases, producing greater concentration of the energy of the pulses in the wavelet domain, better denoising regarding the minimization of pulse distortions, and smaller runtimes. The proposed NWDLS and the SNRBWS methods comprise therefore a more effective methodology to select the best wavelet decomposition to denoise PD signals, providing better denoising results and lower computational load. Its application is expected to greatly improve on-site PD measurements and, consequently, on-line diagnosis of high voltage electrical equipment. References [1] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, 1993. [2] I. Shim, J. Soraghan, W. Siew, Digital signal processing applied to the detection of partial discharge: an overview, IEEE Electr. Insul. Mag. 16 (3) (2000) 6–12. [3] I. Shin, J. Soraghan, W. Siew, Detection of PD utilizing digital signal processing methods: Part 3. Open-loop noise reduction, IEEE Electr. Insul. Mag. 17 (1) (2001) 6–13. [4] S. Sriram, S. Nitin, K. Prabhu, M. Bastiaans, Signal denoising techniques for partial discharge measurements, IEEE Trans. Dielectr. Electr. Insul. 12 (6) (2005) 1182–1191.

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