In this paper, a novel algorithm for PD localization in power transformers based on wavelet denoising technique and energy criterion is proposed. Partial discharge is one of the main failures in power transformers. The localization of which could be very useful for maintenance systems. Acoustic signals due to a PD event are transient, irregular and nonrepetitive. So wavelet transform is an efficient tool for this signal processing problem that gives a timefrequency demonstration. First, different wavelet based denoising methods are analyzed. Then, a reasonable structure for threshold value determining and applying manner on signals is presented. Evaluated errors are good evidences for choices. Next, applying the elimination low energy frequency bands is discussed and developed as a denoising method. Time differences between signals are used for PD localization. Different ways in time arrival detection are introduced and a novel approach in energy criterion method is presented. At the end, the quality of algorithm is verified through the different assays in lab.
1. Introduction
Transformers are important and expensive parts in power systems, and their interrupts are not affordable. So monitoring them to prevent wasting time and financial losses due to their shut down and maintenance is inevitable. During the life time of a power device, because of mechanical, thermal or electrical pressure, insulation may be damaged that may cause catastrophic destruction
[1

6]
.
Partial Discharge is one of the main failures in power transformers. PD starts its activity impalpable but can cause fundamental damages. PD occurs when electrical field change leads to produce the local stream. This stream indicates itself as an electrical pulse that is measureable in transformer output
[7]
.
Because PDs cause insulation breakdown and mechanism of further insulation damage, detection and localization are done to estimate condition and identify problems with insulation of power transformers. In recent years, several methods for detection and localization of PD in power transformers have been developed, which can be classified into four categories of Electrical, Chemical, Acoustic and Optical methods depending on detection quality
[8

13]
.
Acoustic method has high immunity against electromagnetic interferences. Because of better signal to noise ratio, this immunity makes the acoustic method ideal
[13]
. Another advantage of acoustic method is PD localizing using information obtained from ultrasonic sensors around the transformer tank
[14]
.There are two acoustic methods including acoustic systems with an electrical trigger and the allacoustic systems. All acoustic systems, because of sensors positions, can be divided into internal and external systems. The external system has another preference which is portable and invasive and can be implemented online in transformers
[15]
. This study focuses on external all acoustic PD measurement systems in oilimmersed power transformers.
In AE Based localization, denoising plays an important role. Noise is a serious problem through the acoustic method. In a factory, there is a lot of noise ,but many of this is not in frequency range of acoustic PD. Detecting the starting point of acoustic signal is very important for localization; therefore, every signal except PD acoustic wave is interpreted as a noise even an acoustic PD signal that is captured through the tank wall and not directly. So others have introduced some methods to denoise AE produced by partial discharge
[3
,
10
,
13]
and
[17

21]
. Also some methods for localization such as peak criterion are proposed in
[12
,
18
,
22]
and
[25]
.
The aim of this study is designing an algorithm for denoising of acoustic signals produced by partial discharge and localizing of it in oilimmersed transformers via external all acoustic measurement to implement practically in future for power transformers.
This paper first investigates wavelet based denoising methods. Then AE based localization methods is considered. Finally an effective algorithm for denoising and localizing is presented.
2. Wavelet Based Denoising
Noise is the fundamental limit in the partial discharge test. Methods are proposed by the existing standard
[16]
based on the time and frequency field that each is effective only on certain types of noise. Employing this method, Require knowledge of the presence of noise and be under consideration online PD is very difficult
[17]
.
The partial Discharge pulses are transient, irregular and nonrepetitive. Analyzing events in infinite range is the Fourier fundamental problem. A signal breaks to frequency harmonics which oscillate unlimited with fixed period and don’t have local features in time domain. As a result, information are carried by discharges pulse such as starting time aren’t clear
[17]
.
 2.1. Discrete Wavelet Transform (DWT) properties
Wavelet Transform (WT) is a mathematical tool that was designed to analyze the transient, irregular and noniterative signals in a phase space (time  scale or time  frequency). Due to the capability of wavelet Transform to remove noise from signals for example in the earthquake science and multimedia technology, this is an essential tool to extract partial discharge signals in noisy environments. Because the PD signals have transient, irregular and noniterative nature too.
 2.1.1. DWT filter pair structure
The DWT results is a series of coefficients including the approximation and several details that are presented by Ca
_{N}
, Cd
_{N}
, Ca
_{N1}
,···, Cd
_{2}
, Cd
_{1}
where N is the final decomposing level.
Fig. 1
illustrates this process.
DWT logarithm tree including filtering and downsampling
When a noisy signal is decomposed, the signal and noise in various levels of decomposition can be observed that present the possibility of separation. Using a threshold for each level, a denoised signal is reconstructed.
 2.1.2. Frequency bands
As described, using DWT is equivalent to filtering a signal with Quadrature Mirror Filters (QMF)
[8]
.
Fig. 2
illustrates the ideal wavelet filters frequency bands.
Filters frequency bands in various decomposition levels
 2.2. DWT denoising
DWT denoising method has three steps including
[18]
:

1. Desired signal decomposing and calculation of wavelet coefficients.

2. Inserting the threshold for the coefficients. For each step, a threshold is chosen and applied on the coefficients. Hence, denoising includes selecting the signal coefficients and elimination of the noise coefficients.

3. Signal reconstruction. Signal reconstruction is based on modified approximation and details coefficients.
 2.2.1. Threshold determining
Threshold selection plays an important role in the process of noise removal. Based on the noise variance, four common threshold estimating methods are presented
[19]
:
• General Threshold
Where n is the length of noisy signal.
• Minimaxi Threshold
Where n is the length and σ is the deviation of noisy signal.
• Heuristic Stein Unbiased Risk (HSUR) Threshold [20].
• Leveldependent Threshold
where λ
_{j}
is the threshold value at leverl j,
m_{j}
, is the mean value at level j and n is the length of signal. Leveldependent threshold method is widely used in recent papers
[21]
.
 2.2.2. Applied manner determining for threshold
After that threshold value was determined, threshold function needed to be selected. Hard thresholding keeps wavelet coefficients that are higher than threshold and removes the rest. Soft thresholding makes zero the wavelet coefficients that are below the threshold, keeping larger coefficients and then shrinking towards zero.
PD energy and its properties can be saved in hard threshold functions, but through this function a rough signal is obtained. Through soft threshold functions, a smooth signal is achieved, but the energy of PD is lost
[19]
. Threshold function Mark softens PD filter can provide the signal, but the energy is lost.
 2.2.3. Evaluation of noise rejection effect
Mean square error (MSE) and measure error (ME) are used to indicate the effect of noise removal, and are defined below
[19]
:
Where
f
is the original pulse signal,
r
is the denoised PD signal and n is the length of signal.
Where
A_{f}
is the original signal domain and
A_{r}
is the denoised PD signal domain.
3. AE based PD localization
A graphical view of a transformer tank with 4connected sensors, PD inside the transformer and the obtained distances Di are shown in
Fig. 3
. These variables are used to form geometric relations. PD event is modeled as a point which emits acoustic wave in a homogeneous environment (here in Oil). Corresponded nonlinear equations are characterized by spherical functions, which focuses on the PD source. A corner of tank is chose as center of coordinate.
Acoustic sensors Cartesian coordinate on tank wall of transformer
For making a relation between the introduced variables in
Fig. 3
and eventually establishing nonlinear equations for localization, there are several methods such as timedifference method, absolutetime method and pseudotime method. Because of using all acoustic systems, in this paper, timedifference method is used to form the equations.
 3.1.Timedifference approach
Time difference method is appropriate for all acoustic measuring in which an acoustic trigger is used for other channels .In fact, according to
Fig. 4
, in addition to the PD location, the exact time of PD happening (T) is unknown. Time arrival differences of signals in different sensors toward first sensor which receives signal sooner τ
_{li }
, are available.
Timedifferences between four sensors
In these equations, (x, y, z) is PD location coordinate, (x
_{si}
, y
_{si}
, z
_{si}
) is the sensor i coordinate, τ
_{li}
is the time arrival difference of acoustic signal in sensor i toward first sensor, T is the time arrival of acoustic signal is reached by first sensor and
v_{s}
is the velocity of sound in oil.
As is clear from equations above, solving the equations requires achieving the time arrival differences of acoustic signals due to partial discharge that is received by sensors.
 3.2. Time arrival determination
The First step into localization of partial discharge is determination of time arrival of signals for time differences calculation. Several different criteria for time arrival determination are surveyed.
 3.2.1. Peak criterion
One of the time arrival determination methods is the peak criterion. A threshold is applied on signal and time of first crossing through the threshold is determined. This time can be considered as PD pulse arrival time. In this method, the value of threshold according to Eq. (11) is 20 percent of signal’s max and liking two parallel horizontal lines is proposed
[18]
. The schematic is shown in
Fig. 5
.
Where MAX is the signal's extremum. This value is obtained after several experiments
[18]
.
One of the time arrival determination methods is the peak criterion. A threshold is applied on signal and time of first crossing through the threshold is determined. This time can be considered as PD pulse arrival time. In this method, the value of threshold according to Eq. (11) is 20 percent of signal’s max and liking two parallel horizontal lines is proposed
[18]
. The schematic is shown in
Fig. 5
.
Determination of signal's time arrival graphically
 3.2.2. Correlation criterion
Alternative method to find the time difference between two signals, is the correlation between them. When two signals received by two sensors are similar except for the delay time and intensity, the time delay can be determined via the time of the extremum of correlation diagram. This process is illustrated in
Fig. 6
. If two signals are the same, the peak occurs at zero.
Correlation criterion for time arrival determination
 3.2.3. Energy criterion
One of the other criteria for determining the arrival time, is energy criterion. This criterion is based on this fact that most of the time the acoustic signals are known by change of their energy component rather than their frequency components
[6]
.Energy curve
S_{i}
is a cumulative sum of signal samples values while partial energy at each points, according to Eq. (12), is subtracting energy till that point from a trend.
Where i is the loop variable which exchanges for all signal samples, δ is a negative trend that is related to the total signal energy
S_{N}
and length of signal
N
which is defined via Eq. (7).
Fig.7
illustrates the partial energy curve before reaching the energetic part of signal has a negative slope. But after that it rises up. This turning point introduces a min. The time of this min is assumed corresponding to the time of signal onset
[22]
.
Occurence time of the partial energy minimum as signal time arrival
4. Experimental Results
First, the qualifications and conditions of PD simulation in lab are explained. Next described techniques in the previous section is implemented. Finally, the results are presented
 4.1. Case study of PD
PDs are produced in a 0.3×0.3×0.6 m
^{3}
tank that is filled with oil and the acoustic signals are received by four ultrasonic sensors on the outer walls of the tank. The four sensors in terms of distance from the PD source are located so that signals are received via two sensors directly and via two sensors indirectly and through the tank wall. Because of the long distance from the PD source and this fact that velocity of sound in oil is not as much as in metal, two sensors receive signals through the wall first and then receive direct signal. The schematic is shown in
Figs. 8
and
9
. These signals are produced three times with changing the location of sensors and PD source.
Emission in various directions
A view of sensors arrangement on thank wall
Received signals are amplified 40dB, sampling frequency is 2.5 MHz and in each sampling, 150,000 samples have been captured. In
Fig. 10
, received signals by the four sensors are given in the first test.
Received signals to four sensors on thank wall
 4.2. Denoising results
There is a common equation to determine the numbers of levels in the wavelet decomposition:
Where
n
is the length of original signal and
n_{w}
is the length of mother wavelet function. According to the 150,000 samples, via Eq. (14), 14 levels is achieved that according to the
Table 1
and also PD frequency bands, 14 levels is unreasonable and 9 levels is adequate. Finally the truth of this selection is proved.
Signal processing is done in MATLAB software. In addition for signal decomposition, db3 is used as a mother wavelet function that following frequency bands is shown in
Table 1
.
Details frequency bands
After selecting the mother wavelet and number of decomposition levels, a threshold criterion and applying style are chosen. Threshold selection plays an important role in denoising process. Different results are given via soft and hard threshold.
Mixture threshold, is a combination of the general threshold and HSUR. Since the aim is selecting a wavelet denoising method and studying the signal's energy, hard threshold is selected because signal's energy is preserved although a rough signal is presented. In this step also the MEs and MSEs are observed.
Several models such as brigémassart model
[24]
, the model based on Gaussian white noise, the model based on nonGaussian white noise, automatic model and leveldependent model for threshold selecting are studied so that the value of threshold is obtained from above threshold estimating methods.
First 1000samples of signal which is received by means of sensor 3 are demonstrated by
Fig. 11
. The primary fluctuations of signal are surface signal that is received from the tank wall. Denoised signal is achieved via the model based on Gaussian white noise and hard threshold is shown in
Fig. 12
. The original signal is blue and denoised signal is red. Appropriate ME is shown in
Fig. 13
.
With comparison between
Figs. 11
and
14
, it’s obvious that this model not only destroys the primary parts of signal, but the fluctuations related to surface wave is also considered as noise and is omitted. Also general and mixture threshold present the best performance.
First 1000 samples of signal is received via sensor 3
Denoised signal with a model based on Gaussian white noise
Measure error in the model based on Gaussian white noise.
According to
Figs. 11

14
,
Tables 2

5
and the analyses, it seems the combination of the model based on Gaussian white noise and the calculated value via mixture threshold approach through the 9 level decomposition is the best choice. Dependence of wavelet based denoising method to decomposition levels is another finding. So in order to prevent the excessive details, it is better to choose Nyquis trate for sampling rate precisely.
First 1000 primary denoised signal via in the model based on Gaussian white noise
Mean square error of direct soft and hard thr
Mean square error of direct soft and hard thr
Mean square error of leveldependent hard and soft thresholding via mean and median
Mean square error of leveldependent hard and soft thresholding via mean and median
Mean square error of model based on Gaussian white noise thresholding and without it
Mean square error of model based on Gaussian white noise thresholding and without it
Mean square error of brigémassart and automatic thresholding in 7, 8 and 9 levels
Mean square error of brigémassart and automatic thresholding in 7, 8 and 9 levels
 4.2.1. Details energy distribution
In this section, the details energy is studied.
Where
EA_{k}
is the total energy of approximation coefficients in k levels,
EA_{k}
is the total energy of details coefficients in k levels and
N_{k}
is the length of signal in that level. In
Fig. 15
, details and approximation energy of signal is received via sensor 1 is demonstrated.
Details and approximation energy of signal is received via sensor 1
Reconstructed signal is obtained by means of Eq. (16):
Where
Ca
is approximation coefficient and
Cd
is detail coefficient.
According to
Fig. 15
, negligible energy is put in d1, d2, d3 and a9 which can be ignored as the unwanted signals via Eq. (16). Also the most part of energy is distributed in details 7 and 8 corresponding to 520 kHz frequency ranges. Useful information is provided via this fact. Also this energy distribution through the change of sampling rate or discharge type is varied.
According to
Figs. 15

17
and
Table 6
as expected, omission of d1, d2, d3 and a9 operate as an excellent system to denoise signal and primary fluctuations of signal produced by surface wave is omitted well. Mean square error and amplitude error demonstrate that the omission of a9 is not necessary.
First 800 primary samples of signal is received through d1, d2, d3 and a9.
Measure error of d1, d2, d3 and a9 omission
Mean square error of d1, d2, d3 and a9
Mean square error of d1, d2, d3 and a9
Finally in denosing, mother wavelet db3, 9 level of decomposition, hard thresholding through the model based on Gaussian white noise using value that is obtained from mixture threshold method and omitting of d1, d2 and d3 are the best selection
 4.3. Localization results
In this section, the described time arrival detection methods are compared. A criterion is needed to compare, so the time of first negative peak of acoustic signal is chosen. This process is illustrated in
Fig. 18
.
The time of first negative peak as time arrival graphically
In
Table 7
, this time is determined for signals which are received via four sensors in three assays.
Time arrival of signals which are received via sensors in three assays
Time arrival of signals which are received via sensors in three assays
In
Tables 8

10
, times arrival via correlation criterion, peak criterion and energy criterion are calculated respectively.
Time differences arrival of signals via correlation criterion in three assays
Time differences arrival of signals via correlation criterion in three assays
Time arrival of signals via peak criterion in three assays
Time arrival of signals via peak criterion in three assays
Time arrival of signals via energy criterion in three assays
Time arrival of signals via energy criterion in three assays
With comparison between
Tables 7
and
8
, it’s obvious that time differences are obtained from correlation criterion is different from absolute times completely because the signals are not same exactly.
Also comparison between
Tables 7
and
9
shows that peak criterion has a better performance rather than correlation, although an irrelevant time is obtained.
According to
Tables 7
and
10
and
Fig. 19
, in some signals which have low energy primary part, usage of energy criterion is not reasonable, for example signals which are received via sensor 4 in this experiment. It is because the low energy part can't change the slope of partial energy curve and make a minimum timely. Therefore a new criterion is introduced which is the time of local minimum in partial energy curve as time arrival (
Fig. 19
and Eq. (17)).
Partial energy in signals with low energy beginning
Where
S'_{n}
is the partial energy in sample n and
S"_{n}
is the differential of partial energy in that sample.
The reached times arrival through the differential partial energy criterion are given in
Table 11
. With comparison between
Tables 7
and
8

11
, it’s realized that the differential partial energy criterion is the best choice.
Time differences arrival of signals via differential energy criterion in three assays
Time differences arrival of signals via differential energy criterion in three assays
After determining the time arrival of signals, the localization equations are formed, and Time difference approach is chosen. In this approach the time arrival of signal is reached by first sensor T is not available but time differences between signals τ
_{li}
are valid. Also the sound velocity in oil Vs = 1413 m/s is assumed
[15]
. Results are shown in
Table 12
.
According to
Table 12
, the greatest difference between the coordinates calculated by the algorithm and PD coordinates is only 32 mm. So the error of algorithm is under 35 mm and about 7%. Therefore, the success of localization algorithm is clear.
Result of the localization algorithm (all coordinates are in mm)
Result of the localization algorithm (all coordinates are in mm)
5. Conclusion
Wavelet transform is a suitable method to denoise the acoustic signal due to partial discharge in power transformers. A novel algorithm based on wavelet for denoising and localizing acoustic signals is presented which according to the results, is successful. In this algorithm, there is aninelevel decomposition, and the threshold value is determined through the mixture threshold method and applied via the model based on Gaussian white noise. Omission of low energy frequency bands as unwanted signal is presented and developed. The number of decomposition levels can be verified through the frequency bands energy diagram. In addition, dependence of wavelet based denoising on the number of decomposition levels is demonstrated.
The PD source location is determined via this algorithm with very little error. So, it is clear that using of differential of partial energy criterion is the best method. In this paper, the algorithm for three assays with relocation of PD source and sensors is examined and the results are acceptable.
Acknowledgements
This work was supported by Mobarakeh Steel Company. Special thanks to Giscard Veloso and other friends who help us to write this paper.
BIO
Sina Mehdizadeh He was born in Masjedsoleiman, Iran, in 1983. He received the B.S. degree in biomedical engineering from University of Isfahan and M.S. degree in electronic engineeering from Islamic Azad University, Bushehr branch, Iran in 2007 and 2010 respectively. He is currently electrical engineer in the AGAHAN Engineering & Management Co. since 2011. His research interests include signal processing, wavelet transform, partial discharge, acoustic analyzing and power electronics.
Mohammadreza Yazdchi Mohammadreza Yazdchi obtained his B.Sc. in Electrical Engineering from the Isfahan University of Technology in 1997. He received his M.Sc. And Ph.D. in Biomedical Engineering from Amirkabir University of Technology 2000 and 2006 respectively. He is currently an Assistant Professor in the Department of Biomedical Engineering at the University of Isfahan since 2007. His research interests include Biomedical Signal Processing, Medical Image Processing, Biomedical Instrumentation, Biologically Inspired Computing and BioInspired Engineering.
Mehdi Niroomand He was born in Isfahan, Iran, in 1979. He received the B.S., M.S. and Ph.D. degree in electrical engineering from Isfahan University of Technology (IUT), Iran in 2001, 2004 and 2010 respectively. Since 2010, he has been with the Department of Electrical Engineering at University of Isfahan, where he is presently an Assistant Professor. His research interests include power electronics, uninterruptible power supplies, switching power supplies, control in power electronics and renewable energy.
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