1. Introduction
Induction motors are widely used in electrical machines in the electrical and processing industries. There are various induction motor failures from stator, rotor, bearing, or other faults [1, 2]. Induction motor failure results in severe damage to the motor itself, as well as to the motorrelated processes in the industrial plant.
Many studies based on the frequency domain analysis using the fast Fourier transform (FFT) have been performed in order to detect broken rotor bars (BRBs), stator windings, and mechanical faults [3-6]. Using a motor current signature analysis (MCSA), which is based on the FFT method, the spectrum of stator currents can be used to diagnose BRBs faults and stator winding faults with various faulty motor conditions [3]. The MCSA based on the FFT analysis of the sideband components provides the information to diagnose accurately and quantitatively the broken rotor bars in induction machines [4]. A FFT-based MCSA with advanced signal-and-dataprocessing algorithms is proposed for an online induction motor diagnosis system [5]. A detection method is proposed for stator winding faults in the low-frequencyrange of the flux spectrum with low resolution that exploits an external stray flux sensor [6]. However, the frequency resolution of FFT-based methods is limited to the number of data points.
Based on the FFT, zoom-FFT (ZFFT) techniques with window functions have been proposed to increase the frequency resolution [7-9]. ZFFT techniques reduce the computational time, save the memory size, and increase the accuracy in a specified frequency range [7-9]. However, the ZFFT techniques for the high-frequency resolution requires a long acquisition time [8, 9].
In the literature, time-frequency domain analyses based on wavelet-transform techniques have been investigated to overcome the resolution problem encountered in fault diagnostic analysis [10-12]. Wavelet-transform techniques, which are effective for stationary as well as nonstationary signal processing, provide a multiresolution signal analysis with high resolution in the time and frequency domains [10-12]. An automatic online diagnosis algorithm for broken-rotor-bar detection has been proposed based on the wavelet-transform application to the start-up current transient [10]. By using the wavelet decomposition, fault diagnosis has been performed in both time and frequency domains [11]. Based on a continuous wavelet technique with frequency B-spline functions, a diagnosis method identifies as many BRB fault-related frequency components as possible during large transients [12]. Wavelet-transform techniques can be applied to detect faults during stationary and non-stationary conditions [10-12]. However, in stationary conditions, wavelet-transform techniques have the disadvantage in the frequency resolution compared to the subspace methods.
Subspace methods such as a multiple signal classification (MUSIC) algorithm and an estimation of signal parameters via rotational invariance techniques (ESPRIT) have been applied to diagnose induction motor faults while overcoming the frequency resolution limitation of the FFT method in low signal-to-noise ratios (SNRs) [13-17]. The MUSIC and ESPRIT compute an autocorrelation matrix of the signals, estimate a frequency signal dimension order (FSDO), use signal and noise subspaces, and convert the fault detection problem into a generalized eigenvalue problem [13-17]. The MUSIC was applied to estimate the harmonic components in order to improve induction motor fault detection [13]. The MUSIC and short-time MUSIC were used for the analysis of the space-vector of voltages induced in stator windings, after supply disconnection, to detect BRBs faults in induction machines [14]. The MUSIC algorithm with zooming methods was proposed to detect fault sensitive frequencies based on a frequency analysis of the stator current [15]. Using the motor leakage flux spectral analysis, the MUSIC is applied to identify BRB faults in induction motors [16]. The ESPRIT with the zooming technique has been proposed to estimate both the frequencies and amplitudes of BRB fault harmonics using the stator current analysis [17]. However, the difference in the spectrum of healthy motors and those with BRB faults determined using the zoom MUSIC (ZMUSIC) [15] or zoom ESPRIT (ZESPRIT) [17] is not obvious when the load level is close to no-load condition. To overcome this limitation, this paper proposes another fault diagnostic index using the result of the FSDO estimator, which is needed to perform the ZMUSIC [15] or ZESPRIT [17].
In experiments, the small slip induces small current flowing in the rotor even though under no-load condition [22, 23]. To detect BRBs faults under no-load condition, the wavelet transform is proposed using transient current analysis to overcome the resolution problem of FFT-based methods [22]. A detection method based on particle swarm optimization can estimate BRBs fault-related frequencies using stator current under no-load condition [23]. However, performance analysis using detection and false alarm probabilities is required in order to provide a stable diagnosis for BRBs faults [22, 23].
In this paper, a robust fault diagnosis algorithm is proposed to detect BRB faults in induction motors. In this approach, a FSDO estimator, and a fault decision module are proposed. For the FSDO estimator, we use the minimum description length (MDL) criterion [18] to improve the performance of BRB fault detection for motors operating with no load or a finite load. On the basis of the FSDO estimator, we propose a fault decision module that uses the FSDO estimate as a fault diagnosis index. In this module, the predetermined threshold values depending on the load conditions are optimized using the detection and false alarm probabilities from the experimental results. Experimental results show that the proposed diagnosis algorithm achieves the optimal performance for BRBs faults in various load conditions.
2. Detection of Broken Rotor Bar
In this section, frequencies in the current spectrum are analyzed in order to diagnose BRB faults. Then, a data model for BRB faults is presented using the stator current.
2.1 Frequency analysis of broken rotor bar
When an induction motor rotor bar is broken, the stator steady-state current has harmonic frequencies given by [15, 17]
where s is the per-unit motor slip, fs is the electrical supply frequency, and k=1,2,3,… is the harmonic frequency index. In (1), the frequency of interest for BRB fault detection, f BRB , exists near the electrical supply frequency fs , Therefore, BRB faults are verified with the presence of frequency components of interest, fBRB , near the electrical supply frequency, fs , in the current spectrum.
The characteristic frequency for BRB fault detection, fBRB , is a function of the slip s , where the slip s depends on the load condition [15, 17]. When the relative mechanical speed of the motor is almost the same as the motor synchronous speed, which occurs when the motor operates with no load, the slip s is very small and it is very hard to identify frequencies arising from BRB faults in (1) using the FFT method or the MUSIC algorithm [15, 17].
2.2 Data model
Discrete time-domain measurement data for the stator current can be represented as follows:
where s[n] consists of sinusoids, e[n] represents white noise with a zero mean and a variance of σ2, K is the number of pure sinusoids, and ai, fi, and ϕi are the amplitude, the frequency, and the phase of the i th pure sinusoid, respectively. Using cos(2πfin + ϕi ) = from Euler’s formula [19], (2) can also be expressed in complex exponential form as
where , for 1 ≤ i ≤ K , for 1 ≤ i ≤ K , for 1 ≤ i ≤ K , for 1 ≤ i ≤ K , and M = 2K . Using L > M serial discrete-time samples {x[n], x[n +1],⋯, x[n + L −1]} we can rewrite (3) in a compact matrix format as follows:
where x(n) = [x[n] x [n+ 1] ⋯x [ n + L −1]]T, with , i = 1, 2, ⋯, M , and e(n) = [e [n] e [n +1] ⋯ e [n+L–1]]T. Here , F is a L×M Vandermonde matrix of rank M and f() is the mode vector with frequency , where i = 1, 2,⋯, M .
3. BRB Fault Diagnosis Algorithm
In this section, a robust diagnosis scheme is proposed to detect BRB faults. The proposed diagnosis algorithm consists of a FSDO estimator, and a fault decision module. The FSDO estimator provides the number of frequency components in the measured data using the MDL criterion. A fault decision module uses the number of frequency components as the fault index to detect BRB faults.
3.1 FSDO estimator
The autocorrelation matrix Rx of the measurement data x(n) can be expressed as
where E[⋅] denotes the expected value, denotes the signal correlation matrix, and IL denotes an L× L identity matrix.
To implement the FSDO estimator, we need an estimate, , of Rx in (5). By using N > L serial measured time samples {x[n], x[n +1], ⋯, x[n + N −1]} , a spatial smoothing (SS) method is applied to calculate as follows [20]:
where Ps= N − L + 1 . In (6), the eigenvalues of is defined as { λ1 ≥ λ2 ≥⋯≥ λL } and can be obtained using the eigenvalue decomposition. When has the full rank of M , we obtain
Information theoretic criteria, such as the MDL or Akaike information criterion (AIC), have been used to obtain the number of frequencies [18]. The MDL criterion for the estimation of M is used in this paper, and is given as follows [17, 18]:
The estimate of M can then be expressed as
where k = 0,1,⋯, L −1 .
This estimate of M can be used to make a decision for the BRB fault detection. For example, when the no-load condition is satisfied, the slip s is 0.00022 and therefore the frequency estimation for harmonic frequencies in (1) cannot be performed precisely. However, when a faulty induction motor with a BRB fault operates in low load conditions, harmonic frequencies are introduced into the stator current due to the air-gap flux. Therefore, even though the difference between fBRB and fs is smaller than the frequency resolution of the frequency estimation, harmonic frequencies are present in the stator current in (1) for BRB faulty motors. In the proposed method, is defined as a fault diagnostic index to detect BRB faults. The procedure for BRB fault detection is described in the following subsection.
3.2 Fault decision module
A functional block diagram of the proposed diagnosis scheme for BRB faults in induction motors is shown in Fig. 1. First, the measured data vector x(n) in (4) is used to calculate the autocorrelation matrix in (6). The eigenvalues of the correlation matrix are then computed. Through the analysis of the eigenvalues by the MDL criterion, the FSDO estimator calculates and supplies the estimate of the number of frequencies in (9) to the fault decision module.
Fig. 1.Block diagram of the BRB fault diagnosis algorithm
In the fault decision module, we use the property that for BRB faulty motors is larger than that in healthy motors, because the BRB fault introduces harmonic frequencies in (1) in different load conditions. The fault decision module uses the motor speed Rs and from the FSDO estimator. During the initial stage, the motor speed Rs can be estimated using the rotor-slot harmonics [5]. Let Mth be the threshold parameter predetermined from experimental results depending on the motor speed. At different load conditions, in (9) is used for the BRB fault decision and is compared to Mth . This is because increases even when the frequency estimation cannot be performed exactly due to the low load condition [15, 17]. The proposed diagnosis algorithm detects BRB faulty motors with the optimal performance even at no load condition using Mth .
4. Experimental Results
In this section, we present our experimental setup and analyze the proposed BRB fault diagnosis algorithm with different motors and load conditions. For the fault decision module of the proposed method, the cumulative distribution functions (CDFs) for detection and false alarm probabilities obtained from experimental results. The performance of the proposed BRB fault diagnosis algorithm is compared with those of the ZMUSIC and ZESPRIT algorithms for different motors and under different load conditions.
4.1 Experimental setup
The experiment tests were performed with 75-kW squirrel-cage induction motors. Fig. 2 shows the induction motor test system, which is composed of a test motor, a load motor, an inverter, and a data acquisition system (DAS). The specifications of the test and load motors are described in Table 1. The inverter is connected to the load motor to control the load condition of the test motor. Three load conditions were considered: no-load, 50%, and 80% of the full load. Under the no-load, 50% load, and 80% load conditions, the slips are almost 0.00022, 0.0054, and 0.0091, respectively, and the speeds of the motor are 1799.6 rpm, 1791.3 rpm, and 1783.2 rpm, respectively. For motor conditions, three types of motors were used: a healthy motor, a motor with one BRB, and a motor with two BRBs. As shown in Fig. 3, BRBs faults were simulated by cutting the end of the rotor bar.
Fig. 2.Experimental setup
Table 1.Characteristics of the induction motor used in experiments
Fig. 3.BRB faults: (a) one BRB and (b) two BRBs
The experimental data was gathered with a sampling rate fd of 2 kHz. This can be achieved by downsampling 200 kHz sampled data from the DAS by an integer factor 100. This is because the BRB fault can be detected by the analysis of the frequency band near the electrical supply frequency fs [15, 17]. For the FSDO estimator, the size of the correlation matrix, L = 2048 , and the size of the measured time samples for the SS method, N = 4096 , are used. The measurement data acquisition time is 2.048 seconds and the measurement data size is N = 4096 . In order to verify the proposed diagnosis algorithm, the results are analyzed for all possible experimental cases.
4.2 Proposed BRB diagnosis algorithm
In the fault decision module, the fault diagnostic index is exploited in order to detect BRB faults. Fig. 4 shows CDFs of the FSDO estimate for healthy and faulty motors at the different load conditions. For the no-load condition, as shown in Fig. 4(a), there is a significant difference between the CDF of a healthy motor and the CDFs of faulty ones. Using the threshold value with Mth = 163 obtained from the CDF-based experimental results shown in Fig. 4(a), we can easily detect BRB faults in motors operating under the no-load with the false alarm probability for , i.e., = 0 , and the detection probability for i.e., = 1 , which are defined as follows [21]:
Fig. 4.CDFs of for the (a) no-load, (b) 50% load, and (c) 80% load conditions. Threshold values are Mth = 163 , Mth = 128 , and Mth = 124 for no-load, 50% load, and 80% load conditions, respectively.
and
where H0 is the healthy motor hypothesis, and H1 is the BRB faulty motor hypothesis. As shown in Figs. 4(b) and 4(c), the threshold values are determined to be Mth =128 for 50% load condition and to be Mth =124 for 80% load condition, respectively, using experimental results based on the CDFs of healthy and faulty motors. Using the threshold values, BRB faults under 50% and 80% load conditions can be easily detected using with = 0 and =1. Therefore, for the fault decision module, the fault index can be used for the no-load and finite load conditions.
Fig. 5 shows the threshold values as a function of load rates, D, using experimental results. As predicted by the experimental results as shown in Fig. 4, the threshold value Mth decreases as the load rate D increases. This is because when the load rate D increases, the motor slip s increases and the BRBs fault-related frequencies fBRB in (1) are getting away from the electrical supply frequency, therefore the number of BRBs fault-related frequencies from the autocorrelation matrix of measured data in (5) decreases. Moreover, lines with two different slopes can be calculated with a small set of samples. The absolute value of slope between no-load and 50 % load conditions is larger than that between 50 % and 80 % load conditions.
Fig. 5.Threshold values as a function of load rate D.
For comparison purposes, the ZMUSIC [15] and ZESPRIT [17] have been analyzed at no-load condition. For the ZMUSIC, the frequency shift was -60 Hz, the acquisition time was 3 s, and the 100 Hz sampling frequency data, which can be obtained by downsampling 200 kHz sampled data from the DAS by an integer factor 2000. For the ZESPRIT, the acquisition time was 3 s, and the 200 Hz sampling frequency data, which can be obtained by downsampling 200 kHz sampled data from the DAS by an integer factor 1000. The fault index is defined as the amplitude of lower side band harmonic (LSH) in bandwidth BW1 of the pseudospectrum for the ZMUSIC, where the fault index is computed with respect to the noise level [15]. The fault index for the ZEPSRIT is defined as the sum of amplitudes of LSH in bandwidth BW1 and upper sideband harmonic (USH) in bandwidth BW2 [17].
Fig. 6 shows the CDFs of fault indices and for ZMUSIC [15] and ZESPRIT [17], respectively, for no-load condition where the slip s is 0.00022 and the motor speed is 1799.6 rpm. For fault indices, the bandwidth BW1 is between 57 and 60 Hz, and the bandwidth BW2 is between 60 and 63 Hz. For the ZMUSIC as shown in Fig. 6(a), the CDF of for the healthy motor is between those for the faulty motor with one BRB and two BRBs.
Fig. 6.CDFs at no-load condition. (a) Fault index in BW1 = [57, 60 Hz] for the ZMUSIC [15], (b) Fault index in BW1 = [57, 60 Hz] and BW2 = [60, 63 Hz] for the ZEPSRIT [17].
For the ZESPRIT as shown in Fig. 6(b), the CDF of for the healthy motor is close to those for faulty motors with one BRB and two BRBs. Compared to Fig. 4(a), the ZMUSIC and ZESPRIT cannot detect the BRB fault frequencies at the no-load condition due to the fault-related frequency fBRB is almost the same with the electrical supply frequency fs .
Furthermore, under the 50% and 80% load conditions as shown in Figs. 4(b) and 4(c), the proposed BRB fault diagnosis algorithm has better performance for the detection probability and false alarm probability compared to those of ZMUSIC and ZESPRIT [17]. Therefore, compared to the ZMUSIC and ZESPRIT [17], the proposed BRB fault diagnosis algorithm accurately detects BRB faulty motors under no load and finite load conditions.
5. Conclusion
In this paper, an accurate and efficient diagnosis algorithm is proposed for BRB fault detection in threephase squirrel-cage induction motors under no-load and finite load conditions. Fault diagnostic index from the FSDO estimator is proposed considering the stator current signature based on all fault sensitive frequencies. Experimental results show that the proposed diagnosis algorithm provides the optimal performance under no-load and finite load conditions of induction motor.
The main advantages of the proposed diagnosis algorithm are that it can correctly identify BRB faults using a few samples and can be easily implemented using software. The main drawback, which is also found in other high-resolution spectral analysis methods such as the MUSIC, ZMUSIC and ZESPRIT, is that the FSDO estimator is limited to the steady-state operating condition of the induction motor.
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