Short Circuit Fault Detection in PMSM by means of Empirical Mode Decomposition (EMD) and Wigner Ville Distribution (WVD) 1
J. Rosero1, L. Romeral1, J. A. Ortega1, E. Rosero2 Motion Control and Industrial Applications Group, Technical University of Catalonia, C/ Colon 1 Tr 2-225. 08222 Terrassa. Catalonia. Spain, e-mail:
[email protected] 2 Research Group of Industrial Control (GICI), University of Valle, Calle 13 N° 100-00, Cali – Colombia, e-mail:
[email protected] include either saturation or control interactions. From years ago, the failure detection in a motor is studied by analyzing the stator current harmonic by means of FFT [9]. Hovewer FFT cannot be applied in no-stationary signals [10, 11]. Fortunately, signal processing theories provide several algorithms for applications with no-stationary signals [12]. Joint time-frequency analysis is a novel approach in the motor diagnosis applications. Successful use of these techniques requires understanding of their respective properties and limitations. The selection of a suitable temporal window size is required when computing the ShortTime Fourier Transform (STFT) to match with the specific frequency content of the signal, which is generally not known a priori [13]. A very appealing feature of the continuous wavelet analysis (CWT) is that it provides a predetermined resolution for each scale [14]. Moreover, an important limitation of the wavelet analysis is its non-adaptive nature. Once the basic wavelet is selected, one will have to use it to analyze all the data [15]. The Wigner-Ville distribution (WVD) is a time-frequency representation, which is part of the Cohen class of distribution [16]. The drawback of this transform is the presence of cross terms as indicated by the existence of negative power for some frequency ranges. In addition, the WVD of discrete time signals suffers from the aliasing problem, which may be overcame by employing various approaches [15]. The Hilbert–Huang transform (HHT) is based on the instantaneous frequencies resulting from the intrinsic mode functions (IMF) of the signal being analyzed; thus, it is not constrained by the uncertainty limitations with respect to the time and frequency resolutions to which other time-frequency techniques are subject [13]. The result is a three-dimensional energy-frequency-time spectrum designated as Hilbert spectrum that provides an effective way to get the local information which is vital to the non-stationary signals. In recent years, HHT has been used to characterize the time evolution of non-stationary power system oscillations following large perturbations [17]. Xu [18] has applied the EMD method to multiscale decomposition and raw textile defect detection over synthetic and actual texture images. In this work, the HHT is applied to stator currents for analysis of PMSM under failures [19]. This paper shows that
Abstract – This paper presents and analyzes short circuit failures for Permanent Magnet Synchronous Motor (PMSM). The study includes stable condition and speed transients in simulation and realistic experimental conditions. The stator current is analyzed by the empirical mode decomposition (EMD) method, which will generate a collection of intrinsic mode functions (IMF). Finally, the Hilbert–Huang transform (HHT) is used to compute the instantaneous frequencies resulting from the IMFs obtained from the stator currents. Moreover, the IMF 1 and IMF 2 have been analyzed by means of Wigner Ville distribution (WVD). Experimental laboratory results validate the analysis and demonstrate that this kind of time-frequency analysis can be applied to detect and identify short circuit failures in synchronous machines. Index Terms – PMSM drive, motor fault, short circuit, Hilbert Huang Transform, HHT, empirical mode decomposition, EMD, intrinsic mode functions, IMF, Wigner Ville, WVD.
I. INTRODUCTION In many applications the failure of a drive has a serious impact on the operation of a system. In some cases the failure results in lost production, whilst in others it may jeopardize human safety. In such applications it is advantageous to use a continuous operation in the presence of any single point failure. Such a drive is termed fault tolerant and the development of a fault tolerant drive is the aim of the research presented here [1, 2]. Stator or Armature faults are usually related to insulation failure. In common parlance, they are generally known as phase-to-ground or phase-to-phase faults. It is believed that these faults start as undetected turn-to-turn faults that finally grow and culminate into major ones [3, 4]. Short circuit between turns is the most critical fault in the machine, and is quite difficult to detect and almost impossible to remove. Stator winding faults might have a destructive effect on the stator coils [5]. There are many techniques to detect turn-to-turn faults, the majority of them based on the analysis of stator voltages and currents, axial flux and d-q current and voltage component. Toliyat [6, 7] and Penman [8] have described the operation of induction motor drives under the conditions of loss of one phase, broken bars and shorted stator turns using winding functions. However, they considered models that did not 978-1-4244-1874-9/08/$25.00 ©2008 IEEE
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is possible to identify short circuits in the windings of the PMSM, even in an early stage, by means of empirical mode decomposition (EMD) and Wigner Ville Distribution (WVD). Healthy and faulty PMSM motors were simulated and experimentally tested. Faulty conditions were generated with a short circuit in fourth, eight or twelfth stator winding turns. PMSM motors were operated at nominal torque and with different speeds between 1500 rpm and 6000 rpm. Simulation and experimental results obtained from healthy and faulty machines were compared, and finally conclusions are also presented. II. HILBERT HUANG TRANSFORM (HHT) Fig. 1. Empirical mode decomposition (EMD) algorithm
The traditional time–frequency analysis techniques have their own limitations [20, 21]. The consequence is the misleading energy-frequency distribution for nonlinear and non-stationary data analysis. The Hilbert Huang transform (HHT) represents the characteristic oscillations of the original signal from time scales contained in the IMF functions. The local energy and the instantaneous frequency derived from the intrinsic mode function (IMF) through the Hilbert transform can give us a full frequency-time energy distribution of the signal. IMF is the decomposition of signal at different frequency ranges.
IMF 5
IMF 4
IMF 3
IMF 2
IMF 1
1
A) Empirical Mode Decomposition (EMD) The EMD method was motivated by computation of instantaneous frequency defined in terms of Hilbert transform. As is well known, for a real-valued signal x(t); the Hilbert transform is defined by the principal value (PV) integral.
y (t ) =
1
π
+∞
x (t ' ) ∂t ' −∞ t − t '
P∫
[
-1 5 0 -5 10 0
-10 0.2 0 -0.2 0.2 0 -0.2 0
0.05
0.1 Time(s)
0.15
0.2
Fig. 2. IMF of stator current in PMSM with 12 short circuit turns running at 6000 rpm. Simulation results
2.
(1)
3.
As the cubic spline interpolate of its local minima. Compute the mean envelope mn, k(t) as the average of the upper and lower envelopes. Compute hn , k (t ) = hn , k −1 (t ) − mn , k (t ) . Considering
hn, 0 (t)=rn-1 (t). If hn, ,k (t) fulfill the condition expressed in equation (3), IMF was obtained and then stop. Otherwise, treat hn,k (t) as the signal and iterate to hn, k (t) through Steps 1–4. The stopping condition is:
Where P indicates the Cauchy principal value (constant). This leads to the definition of an analytic signal, z(t). (2) z (t ) = x (t ) + j. y (t ) = a (t )e jθ ( t )
4.
]
, θ = arctan( y (t ) / x(t ) ) Where a(t) is amplitude and θ(t) is phase. a(t ) = x 2 (t ) + y 2 (t )
0
1/ 2
∑
The instantaneous frequency is then defined by ω(t)=dθ(t)/dt. In the above equation, both the amplitude and instantaneous frequency are function of time. One would therefore hope to construct a time–frequency representation based on Hilbert transform. An intrinsic mode function (IMF) is a function that satisfies two conditions: First, in the whole signal set, the number of extremes and the number of zero crossings must be either equal or differ at most by one; and second, at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima has to be zero [21]. The EMD extracts the IMFs by the following iterating process [20], as can been seen in Fig. 1. Considering r0(t)= x(t): 1. Find the upper envelope of rn(t) as the cubic spline interpolate of its local maxima, and the lower envelope.
t =0
[h
(t ) − hn ,k (t )]
2
n , k −1
hn2, k −1 (t )
< SD
(3)
where hn,k (t) is the sifting result in the kth iteration for calculus of IMF n, and SD is standard deviation, typically set between 0.2 and 0.3. 5. The EMD extracts the next IMF (see Fig. 2) by applying the above procedure to the residue: (4) rn (t ) = rn −1 (t ) − Cn (t ) Where Cn (t) denotes the IMF n. This process is repeated until the last residue rn (t) has at most one local extremum. III. WIGNER VILLE DISTRIBUTION The best way to analyze the content frequency of non stationary stator current is by means of a joint time-frequency analysis as the Wigner Ville distribution (WVD). Furthermore, it possesses a great number of good 99
WVD (n, k ) =
∑ R[n, p]e N −1
⎛ ⎜− ⎝
600
⎡ ⎣
p⎤ f 2 ⎥⎦
*⎡
n− ⎢ ⎣
j 2πkp ⎞ ⎟ N ⎠
∑
N −1
g 60
50
450 40
400 30
350 300
20
250 10
200
p⎤ 2 ⎥⎦
150 0
(5)
w[ p ]R[n, p ]e
⎛ ⎜− ⎝
j 2 πkp ⎞ ⎟ N ⎠
0.05
0.1
0.15 Time (S)
0.2
0.25
0.3
Fig. 3. SPWVD of IMF 1 & 2 in healthy PMSM. Speed change from 1500 to 1000 rpm
where R[n,p] is the instantaneous correlation, f[n] is the discrete signal, p is an integer. This calculation requires knowing the value of f at half integers. These values are computed by interpolating f, with an addition of zeros to its Fourier transform. If the analyzed signal contains more than one frequency components, the WVD method suffers from cross term interference. Assigning different weights to R[n,p] suppresses the less important parts and enhances the fundamental parts of the signal. Two traditional methods: the Pseudo Wigner Ville Distribution (SPWVD) or Zao-Atlas-Marks distribution (ZAM) are usually applied as the weighting function to the instantaneous correlation. The first is in the time domain and known as the Pseudo Wigner-Ville distribution (SPWVD), it can be computed by [23]:
WVD ( n, k ) =
p
500
p=− N
R [n , p ] = f ⎢ n +
g
550
Frequency (Hz)
properties and it has wide interest for non stationary signal analysis [22]. WVD is a time – frequency energy density computed by correlating f(t) with a time and frequency translation of itself. This avoids any loss of time – frequency resolution. The discrete Wigner-Ville distribution is possible for a discrete signal f(n) defined over 0≤n
