Comparison of spectral estimation techniques applied to induction ...

SDEMPED 2003 Symposium on Diagnostics for Electric Machines, Power Electronics and Drives Atlanta, CA, USA, 24-26 AugusI2003

Comparison of Spectral Estimation Techniques Applied to Induction Motor Broken Bars Detection F. Cupertino, E. de Vanna, L. Salvatore, S. Stasi DEE - Politecnico di B a r i Via. Orabona, 4 - 70125 Bari - Italy

Abstract- This paper presents a performance comparison among some of the most effective spectral estimation techniques applied to the fault diagnosis o f induction machines. The diagnostic test is based on the analysis ofthe current space vector during motor starling via short-time analysis, using a sliding window and different spectral estimation algorithms. Differently f r o m most o f the diagnostic techniques already proposed in the technical literature, the approach, presented in this work, i s effective regardless the load condition of the machine. Algorithms based on the FFT o r optimal band-pass filters (nonparametric methods), on the estimation o f a linear time-invariant model of the signal (parametric mefhods), and on the eigenanalysis of the autocorrelation matrix (high-resolution methods) have been used to process the motor current spare-vector. Experiments prove that both parametric and high-resolution methods overcame the FFT-based approaches, keep only the principal frequency components o f the signal and decrease the noise influence, thus permitting a better interpretation of the current vector spectrum and an automatic fault detection procedure.

1. INTRODUCTION Broken rotor bars can be a serious problem when induction motors (IM) have to perform hard duty cycles. Broken rotor bars do not initially cause an IM to fail, but they can cause serious mechanical damage to the stator windings i f they are left undetected [I]. Moreover an IM with broken rotor bars cannot operate i n dangerous environments due to sparking at the fault site. For these reasons much research has been devoted to this topic i n the last years. The techniques more efficient i n identifying broken rotor bars are mainly based on the steady-state analysis of stator voltages and currents via FFT. In 121, comparison and performance evaluation of different diagnostic procedures, to detect and quantify broken rotor bars o f an induction machine supplied by the mains, are reported. The authors o f [2] compare steady-state analysis methods based on FFT of onephase stator current, current space-vector modulus, instantaneous power, and torque. The information inside onephase current and current space-vector make these signals preferable to detect broken rotor bars. A l l the diagnostic tests

proposed i n [2], as well as i n most o f the scientific literature, need the machine to be loaded and at steady-state to detect the rotor bar failure. In this work the current space-vector w i l l be adopted as medium to detect IM faults, because its spectral analysis permits one to separate negative and positive sequence components. This fact greatly helps to detect the fault by time-frequency representations, as it w i l l be shown later. Furthermore we will pay attention to start-up test, which can be also performed without load. Recently the application o f signal processing techniques different from FFT has been proposed to diagnose 1M faults. This i s justified by the possibility o f better distinguishing sinusoidal signals, even if they are heavily corrupted by noise, as i t happens when voltages and currents are measured in electric motor drives. In [3], for example, the authors use high-resolution spectral-analysis techniques to diagnose faults i n induction machines. MUSIC algorithm has been used both to distinguish the fundamental frequency and the twice slip frequency side bands due to broken rotor bars, and to identify stator voltage unbalance. The high-resolution property o f a frequency estimator based on eigen-analysis i n the former case, and its ability to evidence the main frequency components o f a signal i n the latter case, prove the superiority o f MUSIC algorithm over FFT to reduce noise influence and extract useful information. A i m of this paper i s to compare different spectral estimation techniques to diagnose broken rotor bars i n induction motors. Our approach consists in analyzing the current space-vector via short-time analysis, i.e. using a sliding window, during motor starting. Unlike many other condition monitoring techniques, the new diagnostic method is effective regardless the load condition o f the motor. This means that the test can be executed without disconnecting the motor from its load when the machine i s used i n industrial applications, and without a load when the test is realized in a laboratory. The shon-time analyses, using non-parametric, parametric, and high-resolution spectral estimation algorithms, have been adopted to obtain a time-frequency representation of the motor current space-vector during starting. Experimental results have been reported to prove the superiority of parametric and

high-resolution spectral estimation algorithms over traditional FFT in fault diagnosis. A technique to automatically detect the fault has also been developed and its results presented.

11. SPECTRAL ESTIMATION ALGORITHMS OVERVIEW

optimal trade-off between variance and frequency resolution by choosing an adequate time-bandwidth product. This method i s computationally more intensive than previous approaches.

B. Paramelric algorilhms

FFT-based algorithms have been adopted, in the last decades, in a wide range of applications, including condition monitoring of electric motors. Although FFT oflen produces results accurate enough, it i s possible to improve frequency resolution and reduce variance using more advanced methods of spectral estimation. The algorithms considered in this paper to estimate the power spectrum of an induction motor current space vector can be mainly divided in three categories: nonparametric, parametric and high-resolution algorithms [4].

A. Non-paramelric algorithms Among the non-parametric algorithms we have considered the classical periodogram, the Welch method and the multitaper method. Let us consider a signal i(1) as a sum o f p complex sinusoids and white noise:

Parametric algorithms are based on the estimation o f a linear time-invariant model that has white noise as input and the signal i as output. A linear time-invariant model can be described by the following difference equation: P

i(r) = - c a . ; ( I - n) ?,=I

+c

b,w( I - n)

(3)

,,=O

where w i s the white noise sequence with unit variance, a, and b, are constant parameters. Equation (3) i s often referred as auto-regressive-moving-average (ARMA) model, and its discrete-time transfer function is:

The power spectrum estimate is: where I h , f h , and 4% are the amplitude, the frequency and the phase o f the h'h current space vector respectively, and W ( I ) i s white noise. I t has to be noted thatf, can assume both positive and negative values, and a harmonic space-vector having negative frequency corresponds to harmonic phase currents o f negative sequence. A n N-sample vector

i = [i(I),i(/+

I ) ,...,i ( / + N -I)]

of

the noisy signal, sampled at frequency 6,can be written as i = i, + w . Classical spectral estimation methods are based on -periodogram, defined as follows:

where F

=h/fT, h

= ! & I N , k = O , l , .....,N - I .

To reduce the variance o f power spectrum estimate, the Npoint sequence is subdivided into nonoverlapping (Bartlen method) or overlapping (Welch method) segments, and the periodogram o f each segment is calculated. The obtained periodograms are then averaged to give the power spectrum estimate. These methods permit one to reduce the variance of the estimate at the expense of a decreased frequency resolution. The signal segments 'can be windowed before calculating FFT (Blackman-Tukey method) to increase the quality o f power spectrum estimates. However the differences in performance are relatively small. , The multitaper method uses a bank of optimal bandpass filters to estimate the power spectrum of a signal. l h e filters are derived from the discrete prolate spheroidal sequences, also known as Slepian sequences. I t i s possible to find the

p-,2nl:")2 L - , v -

FM(F)=

In=0

I

I

2

I

I +Cane-,2nl:n P "=I

(5)

There are power spectrum estimate methods that use models without zeros (AR) and models without poles (MA). Autoregressive models lead to power spectrum estimated sharply peaked, and better suited for electric motors fault detection. Moreover the linear equations, to find the coefficients o f AR models, are simple to he solved. The AR methods tested in this work are the Yule-Walker, Burg, covariance, and modified covariance. The Yule-Walker and covariance methods solve the set o f linear equations by minimising the fonvard prediction error in the least squares sense. The Burg and modified covariance methods solve the set o f linear equations by minimising the forward and backward prediction errors i n the least squares sense. The Yule-Walker and Burg approache: always guarantee a stable model. Unfortunately, the performance o f the YuleWalker approach degrades when the number o f samples decreases. The covariance-based approaches perform well also when p i s chosen smaller than the number o f sinusoids actually present in the analyzed signal. This fact will be exploited in the fault diagnostic procedure described later.

C. High resolulion algorirhms The high-resolution spectral estimation is based on subspace eigen-analysis. We have considered the multiple signal classification (MUSIC) method and the eigenvector (EV) method. These approaches are best suited to analyse

sinusoidal signals in white noise, especially when the signallnoise ratio i s low. T h e autocorrelation matrix o f the noisy signal i i s the sum of the autocorrelation matrices ofthe signal is and the noise w :

h=l

where the exponent H denotes Hermitian transpose, I i s the identity matrix, Fh = fh sf =

[1

/f, ,and _ _ _ e-'

'2nFh(N-')l

e-j2nfi

(7)

i s the vector o f complex sinusoids. The eigenvectors and the eigenvalues of the correlation matrix R; can be divided in two disjoint groups. The first group of eigenvectors associated with the largest p eigenvalues span the signal subspace and the second group, having o', a$ eigenvalue, the noise one. As a consequence the signal vectors can he expressed as a linear combination of the first group of eigenvectors (principal eigenvectors). The signal subspace i s orthogonal to the noise one [5]; therefore the sinusoidal signal vectors s,, (h=l,,., p ) are orthogonal to the

form of the Fourier transform i s called Short-Time Fast Fourier Transform (STFFT), and gives a time-frequency representation of a signal. The STFFT represents a sort of compromise between the time- and frequency-based representationso f a signal. It provides some information about both when and at what frequencies a signal event occurs. In this work we propose to window the signal and then to compute the spectrum o f each segment of the signal, using different spectral estimation algorithms. In this way it i s possible to compare the ability o f the spectral estimation algorithms to evidence the main frequency components of a signal, reduce noise influence and extract useful information. As a maner of fact the result o f the ST analysis can be represented using surfaces in the time-frequency-amplitude space, as i t i s usual in STFFT applications. When experimental data measured at electric motor terminals are analysed, the interpretation o f the STFFT surface i s complicated by the high number o f harmonics and by the presence o f noise. As it w i l l be shown later, both parametric and high-resolution algorithms give more regular surfaces, mitigate the effect o f noise, and evidence only the larger frequency components making fault diagnosis simpler.

nuist. subspace:

zw,

Iv.

N-l

DIAGNOSTIC STARTUPTEST

( ~ ) e - ' ~ ~=' 0' "V h = 1,___, p; k = p+ I ,...,N (8)

It i s well known that broken rotor bars result in current

where q is the Kh eigenvector o f the matrix RiT . The MUSIC

components being induced in stator windings at frequencies given by:

sfw, =

"4

pseudo-spectrum of the current space-vector i s defined as follows: jWs" ( F ) = I

i

(9)

ISH (+kI2

k=p+l

and has large peaks at the frequencies o f the principal sinusoidal components where sfw, = 0 : the frequencies of its peaks are taken as the MUSlC estimates. The EV method slightly modifies (9) by weighting the summation with the eigenvalues hk o f the matrix Ri, :

.f, = f ( l + 2 k s ) k = l , 2 ,_._, m (11) wheref is the supply frequency [ 6 ] .For kl the classical.hvice slip frequency side-bands due to broken rotor bars are obtained. These components have been widely used in scientific literature to diagnose broken bars. For b 3 the sidebands are usually too small to be detected. Deleroi [7] has demonstrated that, when the rotor o f an induction motor contains an asymmetry (i.e. due to a broken rotor bar), there are frequency components in the supply current at the following frequencies:

where np is the pole pairs number, s the slip o f the motor, and Y

k=p+l

111. SHORTTIMEANALYSIS When the Fourier analysis i s adopted to obtain a frequency representation of a signal, time information i s lost. This i s not a problem if the frequency components o f the signal do not change their amplitude and frequency during the period analysed via FFT (window amplitude), otherwise the FFT result will be not accurate and lead to misunderstand the nature o f the physical phenomenon to he analysed. To overcome this limitation it is possible to analyse only a small section o f the signal at a time, by using a sliding window. This

k = l,2, ...m . If the motor is supplied directly by the mains, at startup the slip w i l l change from one to almost zero, depending on the load. This implies that the components characterised by (11) and (12) w i l l change their frequency during motor starting. For example the twice slip frequency side-bands obtained from ( I I ) with kl w i l l change from -50Hz and +I50 Hz to about +50 Hz during starting as slip goes from one to about zero. The diagnostic startup test consists in detecting the presence o f the larger frequency mmponents among those given by ( I 1) and ( 1 2) in the time-frequency spectrum o f the current space-vector.

V. AUTOMATIC FAULT DETECTION

I f we indicate with A ( 1 , F ) the power spectral density, at the time instant I and frequency F, AI the time resolution of the time-frequency representation, and N , the number of windows used for the short time analysis, it i s possible to calculate the following index:

1=0 F=l;Smr,

where Fs,on and Fed identify a suitable frequency range chosen so that no main frequencies but those given by (11) and (12) are inside. The short time analysis executes N , times each spectral estimation algorithm, giving N,, spectra between

- A /2 and f,/2.

We calculate the absolute value of

the difference between consecutive spectra so that the index c returns a small number for a healthy motor and a large number for a motor with broken rotor bars.

VI. EXPERIMENTALRESULTS A 1.1 k W induction motor was used to test both the diagnostic technique proposed in this work and performance o f non-parametric, parametric, and high-resolution algorithms. The rated motor parameters are reported in table 1. The motor was supplied by the mains via a variac, with no load attached on the shaft. Two line voltages and G o phase currents-were measured and stored in the computer memory, during the motor starting, using a National Instrument DAQPad 6020e acquisition board and the Matlab-Data Acquisition Toolbox software. The block diagram o f the test bench i s reported in Fig.1. ST analyses were performed on both one-phase current and current space vector. The latter results were preferred because separating negative and positive sequence components greatly helps to detect the fault using timefrequency representations. T h e sampling frequency was chosen equal to IS00 Hz for all the tests. The motor starting duration was about 0.7 s. Several tests were performed using different sliding-window amplitudes, equal to 26/1S00 s, Z7/ISOO s, and 2*/1500 s respectively. The overlap time o f two consecutive windows was always chosen equal to 16/1SOO s. This resulted in about 65, 60, and 50 consecutive sliding windows respectively, during motor startup. To apply Welch method, the N windowed points were further divided in iVl2 length sections, with N / 8 points 01 overlap. The time-bandwidth product, used to compute the Slepian sequences and implement the multitaper method, was chosen equal to N . BFV//, = 0.001 . Some more parameters I

set to implement each spectral estimation algorithm, such as the order o f the linear model or the number o f principal frequency components o f the high resolution methods, are detailed in Table II. The frequency range o f a l l the figures has been chosen from

-300 Hz to 100 Hz to stress the attention in the range between the fundamental frequency and the fifth harmonic. Twodimensional images, plotted with a greyscale colormap and defined by the time-frequency-amplitude results o f the ST analyses, are reported. When the time-frequency spectra obtained with the healthy motor are considered, it i s possible to notice that there are mainly constant frequency components (see Fig. 3). Among them the fundamental components o f positive and negative sequence (+SO Hz and -SO Hz), the fifth harmonic (-250 Hz) and the third harmonic (ISOHz) are easily located on the surfaces. When the machines have broken bars, some components, having variable frequency during motor starting, appear beside the constant frequency components (see Fig. 4). The variable frequency components move depending on the motor slip. Note that the motor is supplied by the mains, with constant frequency. Thence, when the motor starts, at time t = 0 s the speed i s zero and the slip equals I , and at the end of the starting the slip depends on the load torque, but i s always close to zero. As already stated in a previous section, there are two components starting from -SO Hz and 150 Hz and moving toward SO Hz during the startup. I t i s possible to see these components in the reported figures, obtained with the faulty motor data. I t i s also possible to detect a component that starts at -SO Hz and moves in the direction o f the fifth harmonic. T h i s component i s predictable from (12) with suitable value o f

k. For sake o f brevity only the results obtained using FFT with Hamming window, multitaper method, modified covariance, and MUSIC algorithms are reported. From a glance to the reported figures there i s evidence that both modified covariance and MUSIC algorithms are more effective to extract useful information by incrementing the signallnoise ratio. This fact is even clearer when the proposed automatic fault detection technique is used. To implement (13) it is necessary to select the frequency range F,. I t has been selected equal to the union o f different frequency intervals between 250 Hz and 0 Hz, so to avoid the presence o f large harmonic components: F, =[-240 - 2 1 0 ] ~ [ - 1 9 0 - 1 6 0 ] ~ [ - 1 4 0 - I l O ] u (14) U[-90 -6O]u['40 -IO]

If the motor i s healthy there are not large frequency components in this range, and the absolute difference between two consecutive spectra i s small. I f the motor has broken rotor bars there are time instants in which the Deleroi components cross the considered frequency range. This will contribute to make the c coefficient larger. The average ratio between the coefficients calculated for the faulty motor (cd and for the healthy motor (ch) has been reported in Table I1 for all the tested spectral estimation algorithms, and for different sliding windows width. Table II demonstrates the proposed coefficient 'c ' clearly distinguishesbetween healthy and faulty IMs when the spectral estimation algorithm i s accurate enough. Moreover, using the modified covariance and MUSIC

algorithms c,/ch ratio increases about 20 times and 100 times respectively, with respect of the periodogram result, when a 256 points sliding window is employed. When the sliding window becomes shorter only the MUSIC algorithm gives satisfactory results: it is able to clearly distinguish a healthy motor from a faulty one even if the sliding window has only 64 points.

141 151 161

171

VII. CONCLUSIONS This paper has presented a performance comparison among different spectral estimation techniques applied to diagnose broken rotor ban in induction motors. The new diagnostic test consists in the ST analysis of the stator current space-vector during motor startup. This technique has several advantages over most of the fault detection techniques already presented in the scientific literature. It can be recommended because: - is effective regardless the load condition of the machine; - is easily adoptable when 1Ms have to perform many starlups during hard duty cycles, which is the most critical condition for bar breakage; - permits one to separate negative and positive sequence components, greatly simplifying the fault and unbalance detection. An original condition-monitoring coefficient able to automatically detect the fault has been developed and its results also used to compare the performances of different spectral estimation algorithms. It results: - windowing the signal is essential to extract useful fault information when periodogram is used, despite the lost of frequency resolution; - Welch method reduces the variance of the estimated power spectrum, but the lost of frequency resolution makes it effective as periodogram; - the multitaper approach solves the problem of windowing the signal and performs slightly better than periodogram; - the covariance-based and high-resolution approaches perform well also when the model order and the number of principal frequencies are chosen smaller than the number of sinusoids actually present in the analysed signal and clearly overcome the other tested algorithms; - the MUSIC algorithm is the best one in reducing noise influence and extracting useful information. Experimental results have been shown lo prove the abovementioned results.

Fig I -Test bench block diagram

Fig 2 -Stator current amplitude during motor slarling TABLE I l.lkW 220 v 4.8 A 280Orpm

Power

Voltage Current Speed

TABLE I1

~ J ~ : ~ n n n o u s STFFT r w ANDSTMUSIC ~ 1 . w R l l t l M S . Spectral estimation

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Processing Techniques", IEEE Tram Power Electronics, ~01.14, n~l, pp14-22, 1999. J.G. Proatis, D.G. Manolakis, "Digital Signal Processing, Principles, Algorithms, and Applications'', Prentice-Hall, Inc. 1996. S. Vaseghi, "Advanced Signal Processing and Noise Reduction", John Wiley & Sons Idt, 2000. F. Filippetti, G. Francerchini, C. Tassoni, and P Vas,"AI Techniques in Induction Machines Diagnosis Including the Speed Ripple Effect", IEEE Trans. Ind. Applications: "01.34, n I.pp98-108, 1998. W. Deleroi, "Broken bars in squirrel cage rotor of an induction motor, Pan I:Desenption by superimposed fault currents," (in German), Arch fur Elektrotechnik, vol. 67, pp. 91-99, 1984.

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I33

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