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energies
Article
Time-Frequency Analysis Based on Minimum-Norm
Spectral Estimation to Detect Induction Motor Faults
Tomas A. Garcia-Calva 1 , Daniel Morinigo-Sotelo 2, ∗^ , Oscar Duque-Perez 2 , Arturo Garcia-Perez 1 and Rene de J. Romero-Troncoso 3 (^1) HSPdigital-Electronics Department, University of Guanajuato, Salamanca 36700, Mexico; ta.garciacalva@ugto.mx (T.A.G.-C.); agarcia@hspdigital.org (A.G.-P.) (^2) HSPdigital-ADIRE, ITAP, University of Valladolid, 47011 Valladolid, Spain; oscar.duque@eii.uva.es (^3) HSPdigital-Mechatronics Department, Autonomous University of Querétaro, San Juan del Río 76806, Mexico; troncoso@hspdigital.org ***** Correspondence: dmorinigo@hspdigital.org
Received: 13 July 2020; Accepted: 4 August 2020; Published: 7 August 2020
Abstract: In this work, a new time-frequency tool based on minimum-norm spectral estimation is introduced for multiple fault detection in induction motors. Several diagnostic techniques are available to identify certain faults in induction machines; however, they generally give acceptable results only for machines operating under stationary conditions. Induction motors rarely operate under stationary conditions as they are constantly affected by load oscillations, speed waves, unbalanced voltages, and other external conditions. To overcome this issue, different time-frequency analysis techniques have been proposed for fault detection in induction motors under non-stationary regimes. However, most of them have low-resolution, low-accuracy or both. The proposed method employs the minimum-norm spectral estimation to provide high frequency resolution and accuracy in the time-frequency domain. This technique exploits the advantages of non-stationary conditions, where mechanical and electrical stresses in the machine are higher than in stationary conditions, improving the detectability of fault components. Numerical simulation and experimental results are provided to validate the effectiveness of the method in starting current analysis of induction motors.
Keywords: fault detection; induction motors; signal processing; spectrogram; spectral analysis; stator current; transient regime; time-frequency analysis
1. Introduction Condition monitoring has become a critical issue in rotatory machines, and the induction machines are not an exception. Induction motors (IMs) are widely used in industrial systems due to their excellent performance, high robustness and low-cost. However, they are susceptible to many different types of electrical and mechanical faults including, stator winding faults, rotor faults, bearing faults, etc. [ 1 – 5 ]. These faults can result in a wide range of detrimental and sometimes dangerous consequences, such as reduced efficiency of the machine, increased wear on pulleys or belts, excessive energy consumption, increased noise and vibration. Several approaches based on the analysis of the stator current in a steady-state motor operation have been studied for fault detection in induction machines. However, the monitoring of a steady-state operation can produce false fault alarms due to superposition of different frequency components in the stator current spectrum. To overcome this issue, different time-frequency analysis techniques have been proposed for time-varying spectrum analysis, including Fourier-based schemes [ 6 – 9 ], application of adaptive windows [ 10 ], adaptive scales [ 11 ], quadratic time-frequency (t, f ) distributions such as the Wigner–Ville distribution [ 12 – 14 ], the use of (t, f ) atoms [ 15 ], continuous time-frequency
Energies 2020 , 13 , 4102; doi:10.3390/en13164102 www.mdpi.com/journal/energies
tools [ 16 ], and parametric methods for power spectrum estimation [ 17 – 19 ]. Most of these methods operate adequately only for long transients and at moderate noise levels. The time-frequency representations based on the Fourier transform (FT) suffer from the major problem of poor time-frequency resolution because of some important assumptions (stationarity of the signal, integer digital frequencies, noise-free, and infinite observation data length) made in these methods. The estimated time-varying spectrum results in a very noisy and diffuse version of the actual spectrum [ 20 , 21 ]. (t, f ) decompositions based on quadratic operations of the observed data, such as the Wigner–Ville distribution, are often able to accomplish a better resolution than FT-based decompositions of approximately a factor of 2 in the (t, f ) sharpening of the distribution. However, they are not suited for multi-component signals since false "cross-terms" appear between true signal components [ 22 ]. The most recent methods of parametric power spectrum estimation are based on statistical properties of the signal. Their resolution is high in frequency and independent of the signal-to-noise ratio (SNR). However, this improvement is achieved in the presence of additional false or “spurious” frequency components in the spectrum that may lead to an inaccurate fault diagnostic of the machine. In this paper, a time-frequency tool is proposed that can make an accurate and reliable assessment of the presence of specific fault signatures in an induction motor starting current. To avoid spurious signatures and obtain high resolution, the spectral estimator called Minimum Norm (Min-Norm) is proposed to jointly estimate both frequency and magnitude of the time-varying stator current components. Simulation and experimental results show the potentiality of the methodology as a time-frequency decomposition technique that is suited for transient and stationary conditions.
2. Stator-Current Spectral Content
2.1. Induction Motor Faults Most fault detection methods used in condition monitoring are based on frequency analysis because induction motor’s behaviour is cyclic, and the most common faults encountered in the machine, such as bearing faults, broken rotor bars, and misalignments, produce predictable frequencies in the line current spectrum [23–25]. This work focuses on the detection of two of these failures.
2.1.1. Misalignment Fault An air-gap eccentricity arises in a motor when its shaft and the shaft of the mechanical load do not rotate on a common axis. Like shaft misalignment, belt misalignment results in an air-gap eccentricity. This eccentricity causes a non-uniform distance between the stator and the rotor, giving rise to unbalanced magnetic flux within the air gap. As a consequence, a group of harmonic components arise in the stator current whose frequencies are given by:
fpsh(nd ) =
[
(kR ± nd) ( 1 − s) P ± v
]
fs, (1)
where nd = 0 for static eccentricity, nd = 1, 2, 3... the order of the dynamic eccentricity, s is the motor slip, R is the number of rotor slots, k is an integer, P is the number of pole pair, v represents the harmonics present in the power supply, and fs is the fundamental supply frequency [ 26 – 28 ]. Shaft misalignment also introduces more spectral components in the low-bandwidth at frequencies:
fl = fs ± k ρ fr, fr = 1 − s P fs (2)
where fr is the machine rotating frequency, k is an integer and ρ is a scale factor when a reduction coupling is present in the transmission chain.
d = [d 1 d 2 ... dN ]T^ (10)
that has the property that, v H^ ( ψ i) d = 0, i = 1, 2, · · · , D, (11) or equivalently, Φ Hi d = 0, i = 1, 2, · · · , D, (12)
Then, the polynomial,
D(z) =
N
k= 1
dkz−(k−^1 ), (13)
has zeros at, zk = exp(j ψ k), i = 1, 2, · · · , D, (14) v H^ ( ψ i) d = 0, i = 1, 2, · · · , D, (15) finding d as a liner a combination of the N − D, the power spectral density will have zeros at ψ 1 , ψ 2 , ..., ψ D. It is required that d 1 equal unity and minimize
Q( ψ ) =
N
k= 1
|dk|^2 , d 1 = 1. (16)
Minimizing Q( ψ ), the Min-Norm null spectra [33] can be written as
Q̂ MN ( ψ ) = | v H^ ( ψ ) d |^2 = v H^ ( ψ ) U ̂ n W U ̂ nH v ( ψ ), (17)
where W = w (^12) w H 2 and d = U ̂ n w (^12) interpreted as a weighted eigenspace algorithm. A pseudo-code representation for practical implementation is given in Algorithm 1.
Algorithm 1: Min-Norm Spectral Estimation Algorithm Data: Ss, K, p Result: QˆMN begin nm = [(K + 1 ), · · · , (p + 1 )] T p = Toeplitz(Ss) F b =
[
T p T ∗ p J m
]T
USV = SVD( F b) g = V (1, mn) G = V (2 : p + 1, nm) d = [ 1
g∗^ gT^
G∗^ gT^
]T
Q^ ˆMN = 20log 10 | v H^ ( ψ ) d |^2
where Tp denotes the Toeplitz matrix of p order [ 32 ] and J m is a reflection matrix. It is possible to compute the (t, f ) decomposition of Ss(l) by sectioning the measured signal into overlapping windows and applying the Min-Norm spectral estimator to each frame.
4. Simulation Results In this section, simulation experiments will illustrate the performance of the Min-Norm strategy in terms of spectrum estimation accuracy when applied to the analysis of the signature of a motor stator current. Some synthetic signals were created in the time-domain within the bandwidth [0, λ s/ 2 ), to simulate an IM stator current in a constant-speed regime. The sequence contains five sinusoids
located at [ fl 1 − = 40 Hz, fbb− = 54 Hz, fs = 60 Hz, fbb+ = 66 Hz, fl 1 + = 80 Hz], with amplitudes of [−26 dB, −31 dB, 0 dB, −31 dB, −26 dB] respectively. The Min-Norm is compared with two benchmark methods, the fast Fourier transform (FFT) and the MUSIC algorithm. The window observation considered has a length of N = 256, and the noise is modeled as an additive zero-mean white noise with variance σ^2 = 0.08. Figure 1 illustrates the comparison of the true spectrum values and the spectral estimation performance of the FFT algorithm, the MUSIC algorithm, and the Min-Norm method. Spectra have been normalized to 0 dB at its maximum estimation value to facilitate the comparison.
0 10 20 30 40 50 60 70 80 90 100 110 120 Frequency (Hz)
0
PSD (dB)
MUSIC Min-Norm FFT True spurious spurious
Figure 1. Comparison of spectra obtained from simulated data (IM stator current in white noise data record length of N = 256), using: FFT, MUSIC, Min-Norm, and the true values.
Figure 1 shows that all methods obtain a good estimate of fs since its amplitude is the highest of the time-sequence. However, it is not easy to distinguish in the FFT spectrum additional peaks that would prove the existence of fbb− or fbb+. This is explained by the small observation window length that reduces the FFT frequency resolution ∆ f = λ s/N, resulting in a poor spectral estimate. On the other hand, the MUSIC method produces a better spectral estimation (its frequency resolution is better than FFT), which allows the observation of spectral peaks located at the sinusoidal frequencies fl 1 −, fbb−, fs, fbb+, fl 1 +. However, it should be noted that their amplitudes differ significantly from their true values and the noise floor is altered; also, spurious or fake frequencies appear in the spectrum at 4, 48, 73, 95 and 112 Hz. On the contrary, the results reveal that the Min-Norm method achieves not only a high-frequency resolution but also a good estimation accuracy of the amplitudes from relatively small length data. This result confirms that this tool is suitable for the transient analysis of signals from induction motors.
Time-Frequency Analysis of Stator Current Broken rotor bar and misalignment faults can be detected in IM under stationary conditions by observing the fault components fbb and fl. However, for certain transmission chain configurations, the two frequency components can overlap between them in steady state producing false positive or negative fault alarm because of constructive or destructive superposition of two or more components located at the same frequency. By monitoring the fault components at the startup transient, it is possible to accurate evaluate the fault components before they overlap with another frequency components and avoid false fault alarms. Classical estimation techniques are not adequate to analyze the transient current since its amplitude, frequency, and phase are time-variant. When the induction motor is started, the frequency components that indicate a fault varies as the motor accelerates to its final operation speed. The behavior of motor speed and slip are illustrated in Figure 2a,b for a startup transient with a motor fed from line and by an inverter, respectively. It can be seen that for the inverter-fed IM, the slip reaches smaller values faster than line-fed IM.
RAM memory and a microprocessor Intel (R) Core (TM) i7. All the digital sequences of the stator current were decimated from its original sampling frequency λ s to a new sampling frequency λ Ds for selecting the frequency band analysis, and to reduce computational burden. The decimation process was implemented with a cascade multi-stage approach to avoid spectral aliasing at the sample rate conversion stage. The startup transient current of the stator was measured for the line-fed IM and ASD-fed IM as shown in Figure 4. It should be noted that when the IM was started from the line at constant rated frequency, an ovecurrent happens 7 times greater than rated current. However, when the ASD was used, the starting current is limited and slightly higher than the rated current.
0 1 2 3 4 5 6 7 Time (s) (a)
Current (A)
time-varying operation
0 1 2 3 4 5 6 7 Time (s) (b)
0
2
4
6
Current (A)
Figure 4. Stator current experimental waveform during startup transient of induction motor feds by: ( a ) Power grid. ( b ) Adjustable speed drive.
The Min-Norm-based (t, f ) decompositions of the stator current under ASD startup for: healthy, shaft misalignment, and broken rotor bar are shown in Figure 5a–c, respectively. On the other hand, in Figure 6a,c and Figure 4 are presented the (t, f ) analyses for the stator current when the motor is started directly from the line for the same cases. The proposed methodology is implemented in PC software. Each time-frequency decomposition has a size of 512 × 1155 points and the analyzed bandwidth is [0, 128) Hz by 8 s.
f (^) s
f (^) l1-
PSHs (^) f (^) l2+
f (^) l2-
f (^) bb
f (^) l1+
Figure 5. Min-Norm (t, f ) analysis of the starting current with the adjustable speed drive (ASD) induction motor driving a rated load, for the case of: ( a ) Healthy motor. ( b ) Mechanical misalignment. ( c ) Motor with one broken rotor bar. 6.1. ASD-Fed Startup Transient Analysis (0 to 60 Hz) The (t, f ) analysis for the healthy IM case is shown in Figure 5a, where the principal components inside the stator current spectrum are observed. It can be seen the main harmonic following a linear profile provided by the ASD supply fs, the space-harmonics for the first eccentricity orders k = ±1, ± 2 in (2). The Min-Norm-based method also enables the observation of the Principal Slot Harmonics PSHs
for v = ±3 and v = ±5 in (1). Notice that after the startup transient (next to the second 5), a frequency signature appears parallel and near the fs component, which is a fact that may lead to a false-positive indication of rotor fault if only the steady-state operation is analyzed. Figure 5b shows the (t, f ) analysis for the induction motor under shaft misalignment condition. It can be observed a significant increase in the magnitude of the PSH components due to the asymmetry induced by the misalignment. Moreover, it can be observed additional spectral lines around the fl for k = ±2 trajectories.
f (^) l2-
f (^) l2+ PSHs
f (^) bb
Figure 6. Min-Norm (t, f ) analysis of the starting current with the line-fed induction motor driving a rated load, for the case of: ( a ) Healthy motor. ( b ) Mechanical misalignment. ( c ) Motor with one broken rotor bar.
The rotor bar defect is detected in the ASD-fed IM with the clear identification of the fault harmonic located at fbb, as predicted in Section 2.1.2 by (3). Figure 5c shows the Min-Norm-based decomposition, where the fault-pattern is clearly recognized evolving parallel and close to the fs component. In this case, the ( 1 + 2 s) fs known as Right side harmonic (RSH) oscillation is not present due to inertia attenuation and its amplitude is so small that is submerged by the measurement noise.
6.2. Line-Fed Startup Transient Analysis This section shows the ability of the Min-Norm-Based (t, f ) analysis to detect the presence of fault indicators from short-duration transients, such as when a small motor is started from the line. The results are presented in Figure 6. In these starting current spectra, the dominant component is the power grid frequency, with a constant value of 60 Hz. Figure 6a illustrates the result of Min-Norm analysis of a startup transient of the induction motor under healthy contion. As a result of the short-time speed transient, all the space harmonics vary rapidly in frequency. The principal slot harmonics reach high-frequency levels in less than a second, being difficult to notice its trajectories. On the other hand, the evolution of the fl components is clearly visible and they are symmetrically located at a distance from the fs, which is equal to ρ k fr. In Figure 6b, it is shown that when the transmission system is misaligned, the asymmetry produces (t, f ) oscillating components in the vicinity of fl , for k = ±2. It is important to note that, during the steady-state IM operation, the spectrum has a frequency component apparently located at ( 1 − 2 s) fs, even when the rotor is in perfect health condition. In contrast with the (t, f ) analysis of the induction motor under healthy condition in Figure 6a, in Figure 6c the rotor failure is manifested by the clear appearance of a “V” pattern trajectory, which corresponds to ( 1 − 2 s) fs in (3). Furthermore, Min-Norm-based (t, f ) decomposition permits instant detection of a change in the mechanical load during the steady-state at second 5. The high (t, f ) resolution of the proposed method enables the instant visualization of the sudden change in frequency
Funding: This research was partially supported by CONACyT scholarship 487058; and in part by the University of Guanajuato, DAIP, under grand 248495/2019. Conflicts of Interest: The authors declare no conflict of interest.The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Abbreviations
The following abbreviations are used in this manuscript:
ASD Adjustable Speed Drive fbb Broken Bar frequency FFT Fast Fourier Transform algorithm fl Load frequency fr Rotor frequency fs Fundamental supply frequency FT Fourier Transform IM Induction Motors LSH Left Side Harmonic Min-Norm Minimum Norm MUSIC Multiple Signal Classification PSH Principal Slot Harmonics ρ Reduction relation RSH Right Side Harmonic s Motor slip SNR Signal to Noise Ratio STFT Short Time Fourier Transform STM Short Time MUSIC STMN Short Time Minimum Norm (t, f ) time-frequency
Appendix A Three-phase induction motor rated characteristics: Rated power = 0.74 kW, Rated voltage = 230/460 V, Rated current = 2.9/1.4, Rated speed = 3355 r.p.m., Power factor = 0.87, Efficiency = 75.5%.
Appendix B Adjustable speed drive characteristics: Rated output voltage: 220/240 V, Rated output current: Start-up mode = Linear (v/ f ), Control mode: pulse width modulation (16 bits).
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