Time-frequency Analysis for Power Transformer Fault Detection Using Vibration Method

Hong Kaixing, Huang Hai, Zheng Jing Department of Instrumentation Science and Engineering

Zhejiang University Hangzhou, China

hongkaixing@gmail.com

Zhou Jianping, Zhou Yangyang, Liu Jiangming State Grid Zhejiang Electric Power Company

Hangzhou, China

Abstract—The vibration methods are widely employed in the machinery condition monitoring and the time-frequency analysis has been applied in many fault detection applications. In this paper, the study of the vibrations using Wigner-Ville distribution is presented to assess the power transformer condition. First, the distribution contour plots of power transformer vibrations under different conditions are compared. Then, the pattern recognition based on the energy distribution similarity is presented. Finally, a health metric is proposed to represent the transformer health state. The results from more than 10 transformers are compared, and the preliminary study shows that the proposed method is effective to assess the power transformer condition.

Keywords—transformer vibration; fault detection; Wigner-Ville distribution;

I. INTRODUCTION Power transformers are important and expensive equipment

in the power system. More and more researches have been done to ensure their safety and reliability. Investigations show that the transformer failures can be categorized as electrical, mechanical, and thermal [1-2]. The mechanical failure is the toughest problem, and nothing except heavy maintenance should be taken when it happens. Among different mechanical failures, windings deformation is the most harmful one. Many methods have been used to diagnose transformers, but the vibration method has unique advantages. First, the vibration has a close relation with the mechanical structure. Second, the vibration method can be used in online monitoring, which means the diagnosis method based on the vibration analysis can be used when the transformer is running.

There are many vibration based methods [3-7] which are used in fault detection of power transformers. However, many of them are using classical signal processing techniques, such as discrete Fourier transform (DFT). The transformer vibration signals have some unique characteristics. Generally speaking, the vibration from an anomalous transformer may behave non-stationary, which also contains other vital information. Due to the DFT only shows the amplitude value in the frequency domain. So the observation of the signal magnitude using DFT lacks the ability to determine the instantaneous variation. More information can be obtained when the time-frequency analysis is performed [8]. The basic idea of the time-frequency analysis is to describe the signal distribution in a joint function

of time and frequency. The Wigner-Ville distribution (WVD) was the first to be introduced and has been widely used in practical applications. It was first developed in the quantum mechanics by Wigner and implemented in signal processing by Ville. The basic theory of the WVD can be found in [9]. Although the short time Fourier transform (STFT) provides a fixed-size window to solve the problem in the time-frequency dimension, the WVD provides a variety of real-time data rates and resolutions for signal processing. Meanwhile, the WVD has been widely used in machinery fault diagnosis, and the approach is clear to figure out the energy density in the time-frequency domain.

The WVD contour plot can be simply regarded as a two-dimensional image. Although the power transformer condition can be estimated by observing the energy distribution directly, the method using visual observation from the WVD contour plot has its limitation. The fault detection algorithm can be treated as a pattern recognition procedure [10]. There are many different pattern recognition methods. Statistical method [11] is one of the classical methods, which has been used widely in the fault detection. Knowledge-based model [12] focuses on the pattern recognition and decision-making which acts like human beings.

In this paper, the WVD is performed in the transformer condition assessment, and the proposed algorithm is based on the energy distribution similarity. Finally, the vibrations from plenty of transformers under different conditions are analyzed and conclusions are shown.

II. THEORY

A. Wigner-Ville distribution The Wigner distribution (WD) is a mathematical technique,

which has been widely used in the signal processing area to provide local frequency spectra. For a continuous-time signal x(t), the corresponding WD is defined as (1).

( ), ( / 2) ( / 2) jxW t x t x t e dωτω τ τ τ

+∞ ∗ −

−∞= + − (1)

where * stands for the complex conjugate, t is the time domain variable and is the frequency domain variable.

2110978-1-4799-4315-9/14/$31.00 c©2014 IEEE

Equation (1) can be considered as a Fourier transform of the signal instantaneous autocorrelation. The WD provides a high resolution of the instantaneous energy density both in time domain and in frequency domain. The Wigner distribution of an analytic signal is named Wigner-Ville distribution (WVD). The analytic signal is defined in (2).

( ) ( ) ( )= +ax t x t jx t (2) where is the Hilbert transform of the .

[ ] 1 1( ) ( ) ( )H x t x t x dt

τ τπ τ

+∞

−∞= =

− (3)

The WVD is treated as an energy distribution over the time-

frequency plane. But the interpretation cannot be limited in a pointwise sense, because the uncertainty principle prohibits the interpretation as a point time-frequency energy density

Equation (1) implies that the value of the whole signal must be known before the WVD result is evaluated. Besides, the other difficulty with applying the WVD is its quadratic nature which leads to the interference in the time-frequency domain. The limitation can be eliminated by using a truncating window. The windowed version of the WVD approach is called pseudo Wigner-Ville distribution (PWVD), which is shown in (4).

( ), ( ) ( / 2) ( / 2) ωτω τ τ τ τ+∞ ∗ −

−∞= + − j

xPW t w x t x t e d (4)

The function w(t) is a truncating window to the original

continuous-time signal. The effect of the window is to smear the WVD along the frequency axis. The frequency resolution can be increased by using a longer window without affecting the time resolution.

B. Numerical implementation The discrete version of the WVD is shown in (5).

( ) ( ) 2( , ) 2 ωω∞

∗ −

=−∞

= + − j kx

kW n x n k x n k e (5)

The Fourier transform should have a Nyquist sampling rate

limit to avoid any aliasing in the distribution. The periodicity of the discrete WVD is equal to , whereas the periodicity of the Fourier transform is equal to 2 . Therefore, the sampling frequency of the signal should be twice as high as that used for the Fourier transform. In some cases, it is difficult to raise the sampling frequency. The other way is to use an analytic signal, in this way, the distribution will never contain any negative frequency components which causing interference in the time-frequency domain.

For a discrete signal with limited length, the WVD still has two problems, one is the discretization of the angle frequency

, and the other is the range of the parameter k. A truncating window is imposed on the original signal, which is known as

the discrete version of the PWVD. The full expression of the discrete PWVD is shown in (6).

2 12

(2 1)

22 1

0

( , ) ( ) ( ) ( )

4 Re ( ) ( ) ( ) 2 ( ) ( )

ω

π

−∗ −

=− −

− −∗ ∗

=

= − +

= + − −

Lj k

xk L

kmL jL

k

PW n m w k x n k x n k e

w k x n k x n k e x n x n

(6)

where must be an even symmetric real window function, with the length of 4L-1.

Equation (6) shows that the signal x(n) should be defined beyond the interval between 0 and N-1 to obtain for n=0,1,2,…,N-1. Usually, the original signal with the length of three cycles is required to obtain one cycle result.

III. TIME-FREQUENCY ANALYSIS

Before using the WVD to analyze the transformer vibrations, the transformer monitoring system (TCMS) is introduced first. The TCMS can measure up to 8 channels of vibrations. The frequency of a transformer vibration covers the range from 0 to 2000 Hz, and 8192 Hz is regarded as an adequate sampling frequency. ICP (Integrated Circuits Piezoelectric) sensors are used to measure the vibration signals. Other signals such as current and voltage signals are collected synchronously. In the field tests, fixed interval sampling mode is used to collect all signals every one minute.

The vibration measured from a transformer tank is a mixed signal from different sources. The vibrations of an in-service transformer are mainly caused by the forces appearing in the windings and in the core. The main cause of the core vibration is the magnetostriction phenomenon. The windings vibration is caused by the electrodynamic force. The basic theory of the transformer vibration [5-7] is not intended to be described here, but one thing should be mentioned is the frequency of the fundamental vibration (100Hz), which is as twice as the power frequency (50Hz in China).

In the field tests, the TCMS is placed near the transformer. 8 vibration sensors are used to collect vibration signals for one transformer. Among these sensors, 6 of them are placed on the tank to assess the windings condition, and the other 2 sensors are placed in the bottom to assess the core condition. 6 sensors should be arranged in the tank surface close to the position of the internal windings to evaluate the condition of the whole transformer. For a three-phase transformer, the windings of each phase should be covered by more than one sensor. In the field tests, sensors are placed on the tank in the same manner, which is shown in Fig.1. One difficulty to study the transformer vibration is the lack of anomalous transformers, because most transformers are in good condition. In this paper, three typical transformers under different conditions are chosen. The first transformer is a new one. The second transformer is under critical condition. The third transformer is confirmed as windings damaged.

2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA) 2111

Fig.1 Vibration sensors in the field tests

During the time-frequency analysis, the hamming window is chosen to ensure the frequency resolution and precision. The parameter L is twice as long as the period of the fundamental frequency vibration. In the time domain, the whole signal is analyzed, but only a part of them are displayed in the contour plots.

First, the vibration from a healthy transformer is analyzed. The model of this transformer (QT#1) is SSZ10-180000 / 220, which has been running for only 2 years. Fig.2 (a) shows the time-domain and frequency-domain of the vibration signals. The corresponding WVD contour plot is shown in Fig.2 (b). It can be seen that the vibration has two obvious characteristics. First, the vibration doesn't contain any higher frequency components. Second, the energy of the vibration is distributed along the time axis uniformly.

Then the transformer vibration under critical condition is analyzed. The selected transformer (YJB#2) has been running for more than 19 years. The healthy parameter named short-circuit impedance exceeds the standard level and the rate of excess is 2%. The vibration details are shown in Fig.3. It can be seen from the WVD contour plot that the vibration contains higher frequency components.

At last, the vibrations from an anomalous transformer with deformed windings is analyzed. The model of this transformer (JN#1) is OSFPS7-150000 / 220. The damage to the windings is caused by a long time short-circuit current in an accident. The vibration details are shown in Fig.4. It’s obvious that the stationarity of the vibration signal is poor in the time domain, and higher frequency harmonics appears. It can be seen from the contour plot that the distribution is more disorganized than the healthy transformer.

(a)

(b)

Fig.2 vibrations under good condition: (a) time domain and frequency domain, (b) the WVD contour plot

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.1

-0.05

0

0.05

0.1

Time (s)

Vib

ratio

n (g

)

0 200 400 600 800 10000

0.01

0.02

0.03

Frequency (Hz)

Vib

ratio

n (g

)tim

e in

sam

ples

(s)

frequency (Hz)0 100 200 300 400 500 600 700 800 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.2

-0.1

0

0.1

0.2

Time (s)

Vib

ratio

n (g

)

0 200 400 600 800 10000

0.02

0.04

0.06

Frequency (Hz)

Vib

ratio

n (g

)

2112 2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA)

(b)

Fig.3 vibrations under critical condition: (a) time domain and frequency domain, (b) the WVD contour plot

(a)

(b) Fig.4 vibrations under anomalous condition: (a) time domain and frequency

domain, (b) the WVD contour plot

IV. PATTERN RECOGNITION

Getting the contour plots of the relevant vibrations using the WVD analysis is the first step. However, the key step of the fault detection is the pattern recognition stage. As mentioned above, the contour plots can be regarded as two-dimensional images. Then the diagnostic parameters can be derived from the image through the feature vector extraction and the fault classification.

A. Feature vector extraction It is impractical to use all points from the contour plot as a

series of feature vectors in the pattern recognition step, since a pixel image may yield K feature vectors, and each one

includes N elements. In this way, the feature matrix form the WVD is too large that a feature compression procedure should be applied.

From the comparison of the WVD contour plots mentioned above, it is known that the vibration energy of a healthy transformer is concentrated in the low frequency band. The mechanical faults, such as windings deformation, will change the energy distribution.

The vibration energy at the time t and the frequency can be defined as the measurement in the window between (t- t/2, t+ t/2) and ( - /2, + /2), which is shown as follows.

/2 /2

/2 /2

1( , ) ( , )2

t t

xt tE t W t dtd

ω ω

ω ωω ω ω

π+Δ +Δ

−Δ −Δ= (7)

The transformer vibration has a unique characteristics that the harmonics are concentrated in the frequencies multiple of the fundamental value (100Hz). Then a vector is defined to represent the energy distribution at the time index i, means E(t, ) at the time , and at the frequency of the k th harmonics, T is the corresponding time period (0.01s) of the fundamental frequency vibration. The equation (7) is an explicit definition, but there are many different forms of the numerical implementation. In this paper, the Gauss integration on a 2D square domain is used to get the vectors .

B. Fault classification After the feature vectors are extracted from the contour plot,

the fault classification step is used to assess the transformer condition. The fault classification is based on the numerical distance similarity and statistics.

The similarity is simply based on the Euclidean distance. The distance from the vector to the other vector is shown in (8).

( ) ( )( , ) Ti i id x x xμ μ μ= − − (8)

time

in s

ampl

es (

s)

frequency (Hz)0 100 200 300 400 500 600 700 800 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.2

-0.1

0

0.1

0.2

Time (s)

Vib

ratio

n (g

)

0 200 400 600 800 10000

0.02

0.04

0.06

Frequency (Hz)

Vib

ratio

n (g

)tim

e in

sam

ples

(s)

frequency (Hz)0 100 200 300 400 500 600 700 800 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA) 2113

where is the i th feature vector, is the mean vector of the group vectors. The unbiased estimator of is given by

1

1( ) ( )N

ii

k x kN

μ=

= (9)

The coefficient of the energy deviation (CED) is defined as

1

1 ( , )/ || ||μ μ=

=N

ii

CED d xN

(10)

The CED value is a dimensionless parameter, which is used

to represent the transformer condition. When the CED value becomes larger, the condition is supposed to become worse.

C. Applications Finally, the pattern recognition theory described above is

applied to several in-service transformers in the power stations. Among all measured transformers, most of them are in good condition, only a few of them are already known as windings deformed. The CED values of transformers under different conditions are compared in the table 1. It can be seen that the CED value increases monotonically when the structural fault happens. Therefore, the CED value can be used for condition assessment.

TABLE I

THE CED VALUES OF TRANSFORMERS UNDER DIFFERENT CONDITIONS

Transformer Fault type Level CED JN#1 Windings deformed Severe 15.7% ZJ#1 Windings deformed Moderate 2.8% YJB#2 Windings deformed Slight 1.1% QT#1 No fault 0.5% QT#2 No fault 0.5% QX#3 No fault 0.6% ZK#2 No fault 0.7% YA#1 No fault 0.8% XQ#1 No fault 0.6% XL#1 No fault 0.8% WD#1 No fault 0.7% WD#2 No fault 0.6%

V. CONCLUSIONS The Wigner-Ville distribution is a powerful tool to analyze

the vibration signals. In this paper, the WVD contour plots of the power transformer vibrations under different conditions are compared. The results show that the signal stationarity of the vibrations becomes worse and the vibration contains higher harmonics when the transformer condition deteriorates. After the WVD contour plots are obtained, the fault detection procedure is taken by calculating the similarities of the energy distribution. Then, a health parameter named CED is proposed to represent the transformer condition. At last, this theory is

applied to plenty of in-service transformers, and the results show that when the internal structure faults happen, the CED value becomes larger.

Although the algorithm proposed in this paper is proved to be an effective way to evaluate the transformer condition, the result cannot identify the problem belongs to the windings or the core. Therefore, if the result indicates that the transformer condition is suspicious, other standard procedures should be taken to verify the transformer condition.

ACKNOWLEDGMENTS This project is supported by the Science and Technology

Project of Zhejiang Province (2013C31008) and the Science and Technology Project of State Grid Zhejiang Electric Power Company (ZB13-026B-039).

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[7] B. García, J. C. Burgos, and Á. M. Alonso, “Transformer tank vibration modeling as a method of detecting winding deformations-part II: Experimental verification,” IEEE Trans. Power Del., vol. 21, no. 1, pp.164–169, Jan. 2006.

[8] L. Cohen, “Time-frequency distribution - A review," Proc. IEEE, vol. 77, pp.941 -981, Jul. 1989

[9] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, "The Wigner distribution - a tool for time-frequency analysis. Part 2: discrete time signals," Philips Journal of Research, vol. 35, pp. 276-300, 1980.

[10] W.J.Staszewski, K. Worden, G.R.Tomlinson, "The-frequency analysis in gearbox fault detection using the Wigner-Ville distribution and pattern recognition," Mechanical Systems and Signal Processing, vol.11, no.5, pp.673-692, 1997.

[11] A.K. Jain, R.P.W. Duin and J. Mao, "Statistical Pattern Recognition: A Review," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 1, pp. 4-37, Jan. 2000.

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2114 2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA)