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Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy
Accepted Manuscript Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy
Xuejun Chen, Yongming Yang, Zhixin Cui, Jun Shen PII:
S0360-5442(19)30461-X
DOI:
10.1016/j.energy.2019.03.057
Reference:
EGY 14893
To appear in:
Energy
Received Date:
25 October 2018
Accepted Date:
09 March 2019
Please cite this article as: Xuejun Chen, Yongming Yang, Zhixin Cui, Jun Shen, Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy, Energy (2019), doi: 10.1016/j.energy.2019.03.057
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ACCEPTED MANUSCRIPT
Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy
Xuejun Chena,c,*, Yongming Yang b, Zhixin Cui c, Jun Shen c a
Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China b State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044, P.R. China; c School of Mechanical Engineering and Automation, Fuzhou University, 350108, P.R. China
Correspondence information*: Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China; E-mail address:[email protected]
1
ACCEPTED MANUSCRIPT Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy Xuejun Chen a,c, , Yongming Yang b, Zhixin Cui c, Jun Shen c Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044, P.R. China c School of Mechanical Engineering and Automation, Fuzhou University, 350108, P.R. China a
b
Abstract: The bearing vibration of wind turbines is nonlinear and non-stationary. To effectively extract bearing vibration signal features for fault diagnosis, a method of feature vector extraction based on variational mode decomposition (VMD) and energy entropy is proposed. In addition, the support vector machine (SVM) classifier is used to identify the types of vibration faults. VMD transforms the constrained variational objective function into the unconstrained one to optimize solution. Compared with the VMD and empirical mode decomposition (EMD), the modal decomposition layer of VMD is less than EMD, no found false modality, and truly reflecting signal components. After the Hilbert transformation, double logarithmic coordinates show that the VMD-based spectral characteristics are significant. VMD is performed on the different types of vibration signals of wind turbines. Therefore, VMD is not that affected by noise and has few decomposition levels. The energy entropy of the normalized four modal components is considered the eigenvalue and classified by SVM, and compared with EMD-based and wavelet db4based energy entropy eigenvalue extraction methods. Experimental results indicate that the accuracy of the method is higher than those based on EMD and wavelet db4, under the limited sample condition. Thus, a referential diagnostic method is provided for practical applications. Keywords: variational mode decomposition; energy entropy; vibration signal; wind turbine; fault 1. Introduction Traditional energy is increasingly becoming scarce, and the environment and climate are getting worse. Hence, the utilization and development of new energy has become the focus of all countries. Wind energy is a kind of renewable green energy, which has been vigorously developed, researched, and utilized by many countries [1−3]. However, with the large-scale construction, operation and production of wind turbines, a series of new technical and environmental problems have arisen. It is urgent to study the related wind power monitoring and fault diagnosis technology [4−6]. An effective equipment monitoring and fault diagnosis can continuously monitor various parameters of wind turbine operation. It can track all kinds of state information in real time, analyse wind power operation status and diagnose faults. Finally, the appropriate maintenance plan is arranged according to the
Corresponding author. Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China. E-mail address: [email protected]
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ACCEPTED MANUSCRIPT diagnosis results. These factors reduce the equipment accident rate as well as operation and maintenance costs [7-11]. At present, monitoring and fault diagnosis are mainly used in wind turbine bearings and gearboxes and other major components [12-14]. Vibration signal is the most important carrier of much diagnostic information. More and more attention has been paid to the diagnosis methods based on vibration signals. [15−19]. In view of the relatively wide distribution of vibration frequency, time-frequency transformation of vibration signals can be performed to obtain the special quantity of frequency domain [20]. Thus, the vibration characteristics of different faults are distinguished. However, this method sometimes cannot display local information. In addition, wavelet transformation has multi-scale characteristics and "mathematical microscopy," which has been used by many researchers in the time-frequency analysis of vibration signals. Moreover, wavelet transformation can provide localized information about time and frequency domains. However, wavelet has certain problems, such as selection of basis function, sensitivity of parameters, and inaccurate description of frequency-to-time transformation [21-24]. For non-linear unsteady-state signals, Huang et al. proposed a Hilbert-Huang Transform, which can obtain the signal Hilbert and time-frequency energy spectrums based on empirical mode decomposition (EMD). The signal time-frequency localization analysis can also be achieved [25−26]. However, certain signals have modal aliasing. To solve this problem, Konstantin Dragomiretskiy and Dominique Zosso (2014) proposed variational mode decomposition (VMD) [27]. By redefining the modal function, if each modal component is a finite bandwidth with different center frequencies, then the alternating direction multiplier method is used to iteratively update the modal and its center frequency. Using this method solves the variational problem. Each mode is demodulated onto a corresponding baseband, and each modal component is extracted. Compared with the EMD recursive “screening” mode, VMD transforms the signal into non-recursive, variational mode decomposition modes, showing good robustness and stability [28−29]. The adaptive decomposition of VMD to non-stationary signals has high time-frequency resolution. Therefore, a method based on VMD and energy entropy is proposed to extract vibration signal features for bearing of wind turbines. Support vector machine (SVM) is also used to diagnose faults, providing a new method for monitoring and fault diagnosis of wind turbine safety production. 2. Vibration signal feature analysis method based on VMD 2.1. VMD theory
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ACCEPTED MANUSCRIPT VMD is based on classical Wiener filtering, Hilbert transform, and frequency mixing. Original signal f (t ) can be decomposed into k modal components uk (t ) with center frequency k (t ) , so that the estimated bandwidth of each mode The sum is the smallest [27−28]. Note that k is the preset scale. The fractional modal decomposition algorithm constructs variational objective functions and solves them. 2.1.1. Constructing variational objective functions First, the analytical signal of each modal function uk (t ) is obtained by Hilbert transform to obtain its unilateral spectrum, as shown in Formula (1). j ) uk (t ) (1) t In Formula (1), (t ) is an impact function. The estimated center frequency e jk t of each modal ( (t )
analysis signal is mixed, and each mode is modulated to the corresponding baseband, as in Equation (2). j ) uk (t )]e jk t (2) t The square L2 norm of the given demodulation signal gradient is calculated, and the bandwidth of [( (t )
each mode signal is estimated. Formula (3) shows the constrained variational objective functions. 2 j jk t { t [( (t ) ) uk (t )]e } {umin },{ } t k k k s.t. k uk f
(3)
K
In Formula (3), {uk }: {u1 , , uK } ， {k }: {1 , , K } ， : . k
k 1
2.1.2. Solving the variational objective function The quadratic penalty factor and Lagrangian multiplication operator (t ) are introduced to transform the constrained variational objective function into a non-binding variational objective function. The quadratic penalty term is a classical method to guarantee the fidelity of the reconstructed signal. The method also has good convergence, especially in the presence of Gaussian noise. The Lagrangian multiplier keeps the constraints strictly, and the extended Lagrangian function is given by Equation (4).
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ACCEPTED MANUSCRIPT L({uk },{k }, ) : t [( (t ) k
j ) uk (t )]e jk t t
2
2
(4)
2
f (t ) uk (t ) (t ), f (t ) uk (t ) k
k
2
Therefore, the minimization problem of variational objective functions is transformed into the iterative search for the extended Lagrangian function "saddle point" problem, which is the alternate direction method of multipliers. The Lagrangian function "saddle point" is obtained by alternately updating ukn 1 ， kn 1 , and kn 1 . The iterative problem in solving Equation (4) is expressed below. j u kn 1 arg min t [( (t ) ) uk (t )]e jk t t uk X 2 (t ) f (t ) ui (t ) 2 2 i
Among them, k equals kn 1 and
u (t ) i
equals
u (t ) ik
i
n 1
i
2
2
(5)
. Formula (5) is converted to the
frequency domain by Parseval/Plancherel Fourier equidistant transformation.
2
uˆ kn 1 arg min j[(1 sgn( k ))uˆk ( k )] 2 uˆk ,uk X
fˆ ( ) uˆi ( ) i
(6)
2 ˆ ( )
2
2
Replace in the first term of Formula (6) with k .
2
uˆ kn 1 arg min j ( k )[(1 sgn( ))uˆk ( )] 2 uˆk ,uk X
2 ˆ ( )
fˆ ( ) uˆi ( )
2
i
2
(7)
Transforming Formula (7) into a half-space integral form with non-negative frequencies, we obtain Formula (8). uˆ kn 1 arg min uˆk ,uk X
4 ( )
2
k
0
2 fˆ ( ) uˆi ( ) i
uˆk ( )
ˆ ( ) 2
2
d
(8)
By making the first variable of positive frequency disappear, the solution of the two optimization problems can be solved as follows: ˆ ( ) fˆ ( ) uˆi ( ) 2 . ik uˆ kn 1 ( ) 2 1 2 ( k )
(9)
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ACCEPTED MANUSCRIPT Using the same solution process, the central frequency solution problem is converted to frequency domain. n 1 arg min k
k
( )
2
k
0
2
uˆk ( ) d
(10)
The update formula of center frequency can be obtained through the following:
n 1 k
0 0
2
uˆk ( ) d
.
(11)
2
uˆk ( ) d
Formula (11) enables the new k to be in the center of gravity of the power spectrum of the corresponding modal function. 2.2. VDM algorithm Through the given construction and solving process, a complete VMD algorithm can be obtained by optimizing in the appropriate Fourier domain.
① Initialize uˆ1k ， k1 ， ˆ1 ， n 0 ； ② n n 1 ； ③ Update uˆk and ˆ k using Equations (9) and (10); ④ For all 0 , update the double rising step length by using Equation (12) ˆ n 1 ( ) ˆ n ( ) fˆ ( ) uˆkn 1 ( ) .
Among them, is a noise tolerance parameter.
k
(12)
⑤ If convergence condition Formula (13) is satisfied, then iteration is stopped; otherwise it returns to step ② to continue.
uˆ k
n 1 k
n
uˆ k
uˆ kn 2
2 2
(13)
2
Among them, 0 of Formula (13). 2.3. Vibration signal feature extraction and fault diagnosis Vibration signals vary when wind turbines either fail or are in a normal state. The energy of the vibration signal can also be different in different frequency bands. Vibration signal can be decomposed into k modal components uk (t ) by VMD algorithm. Then, the energy distribution of vibration signal can be obtained by calculating energy entropy [30−32]. When the working state of
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ACCEPTED MANUSCRIPT wind turbines changes, the energy entropy of modal component uk (t ) changes. Therefore, wind turbine faults can be diagnosed on the basis of their energy entropy. The energy entropy feature extraction steps based on the modal component uk (t ) , which are as follows: ① The original vibration signal is decomposed by VMD, and former n modal component uk (t ) containing the main fault information is selected. ② Energy Ek of each modal component uk (t ) is obtained.
Ek
2
k 0,1, 2,..., n
uk (t ) dt
(14)
③ The characteristic vector T is constructed by the energy Ek of each modal component uk (t ) .
T E1
E2 En
(15)
Then, normalize feature vector T using
T ' E1 / E
E2 / E En / E .
(16)
1/2
n 2 In Formula (16), T ' is the vector T after normalization, and E Ei . i 1 SVM is first proposed by Vapnik. Similar to the multi-layer perceptron and radial basis function networks, SVM can be used for pattern classification and nonlinear regression. Its basic idea is that input space is transformed into high-dimensional space by nonlinear transformation defined by kernel function. Thus, samples are linearly separable, and the optimal classification surface is constructed in high-dimensional space, leading to the formation of the decision rules of sample classification [33−36]. Let vibration signal training sample set ( xi , yi ) ， i 1, , n ， x R d ， y {1, 1} ， n be the number of training samples, and d be the dimension of the feature vector of vibration signal. Under the constraint of yi [( , xi ) b] 1 0 ， i 1, , n , function ( )
1 2
2
is made to have a minimum
value. Use the Lagrangian function to find the partial derivatives of and b and make them equal to 0. Then, find the minimum value of the Lagrangian function. Doing so determines the classification rules only by few support vectors right on the edge of the optimal classification surface. Note that such rules have nothing to do with the other samples. According to the nonlinear characteristics of the four kinds of wind turbines' working state vibration signals, the VMD method is used to decompose various state signals, Moreover, the
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ACCEPTED MANUSCRIPT energy entropy characteristics of the first few modal components containing most information in each working state signal are calculated. Therefore, the feature vector of each working state is composed and integrated into the SVM model for fault diagnosis. Fig.1 illustrates the fault feature diagnosis process. Original signal f(t)
Calculating the energy entropy of modal component
VMD decomposition
Fault identification
SVM
Fig.1. Flow chart of wind turbine fault diagnosis based on VMD energy entropy and SVM
3. Simulation experiment and analysis To illustrate the VMD performance and robustness with respect to the input signal noise, a threeharmonic signal affected by noise is used to test VMD, as shown in Equation (17). Among them, is
(0, ) Gaussian noise, and the standard deviation is considered 1 to control the noise level. Here, the amplitude of noise is quite important and relative to the amplitude of the highest and weakest harmonics. In addition, = 0.1 is selected. x1 (t ) cos(2 5 t ) 0.2 cos(2 20 t ) 0.1cos(2 80 t )
(17)
Although EMD can automatically estimate the mode decomposition layer on the basis of the signal, VMD can set the number of decomposition layers according to different signals. For simulation signal x1 (t ) ,
K = 4. Fig.2 (a) and (b) show the time domain waveform and simulated signal spectrum and the
detected boundaries supported by each filter, respectively.
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（a）Time domain waveform of simulation signal
x1 (t )
50 40 30 20 10 10
1
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（b）Spectrum and supporting boundary of simulation signal Fig.2. Simulation signal
x1 (t )
x1 (t )
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ACCEPTED MANUSCRIPT The simulation signal is decomposed by VMD, in which 5, 20, and 80 Hz signals are completely separated from each other. The maximum amplitude of 80 Hz signal is 0.1, while the standard deviation of Gaussian noise is considered 1, and = 0.1 in Equation (17). So the 80HZ signal is annihilated by noise. It can also be seen from Fig.3 (a) that “V3”is the 80 HZ signal and most of the amplitude is less than “V4” noise. However, after VMD, the center frequency of the 80 Hz signal is adjusted to this harmonic. Fig.3 (a) illustrates the VDM. To compare and analyze the EMD, Fig.3 (b) provides the EMD result of simulation signal x1 (t ) . Fig.3 (b) shows that the simulation signal x1 (t ) is decomposed into four modal function components and one remainder. Simulation signal x1 (t ) is composed of four empirical mode components by the VMD. Although EMD can be adaptively decomposed according to the signal, the decomposition is greatly affected by noise and is prone to over-decomposition, thereby reducing operational efficiency. Moreover, if too many modal components are separated, then false modal components may appear. Such components can accurately analyze unfavorable signals. Furthermore, this excessive decomposition of the iterative computation can inevitably result in increased computation. For most signals, VMD computation is less than that of EMD. v1
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ACCEPTED MANUSCRIPT Empirical Mode Decomposition
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（b）EMD of simulation signal 1 Fig.3. Decomposition result of the simulation signal
In Fig.3 (a), the four modal components decomposed by the VMD well reflect the components of the original simulation signal. In addition, noise is considered an independent modal component. Fig.3 (b) illustrates that the EMD only approximately and correctly decomposes a high-frequency modal component intrinsic mode function 1 (IMF1) and a residual. The EMD also decomposes several false modal components IMF2, IMF3, and IMF4, thus causing distortion and loss of 20 Hz and 80 Hz components in the original simulated signal. Moreover, the VMD can decompose the different modal components of the signal. The VMD can also analyze the modal components of different frequencies in the signal, and each modal component has substantial characteristic energy of the original signal. In theory, the IMF of each resolution obtained by the EMD cannot guarantee strict orthogonality but can only be approximate orthogonality. To date, no complete and rigorous theoretical support has been found. Therefore, problems of modal aliasing arise in decomposition. VMD is based on classical Wiener filter and Hilbert transform, which has a good theoretical support. Fig.3 (a) and (b) are transformed by Hilbert transform respectively, and then the energy spectrum of Hilbert can be obtained by integrating the square of amplitude with time. The energy spectrum represents the energy accumulated by each frequency over the whole time period. The results of the time-frequencyenergy amplitude spectrum, as shown in Fig.4 (a) and (b), can clearly reflect the distribution of signal energy with frequency and time (the darker the color, the greater the energy; conversely, the smaller the energy).
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ACCEPTED MANUSCRIPT 100
frequency/Hz
80 60 40 20
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（b）Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) based on EMD Fig.4. Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) .
Fig.4 (a) is a Hilbert two-dimensional time-frequency energy amplitude spectrum of a simulated signal based on VMD. There are three frequency components on the graph, which are 5 Hz, 20 Hz, and 80 Hz. The wave energy is clearly distributed at frequencies of 5 Hz, 20 Hz and 80 Hz, respectively. And the smaller the frequency component, the darker the color is, and the maximum amplitude of the 5Hz component of the original simulation signal is the same. Fig.4 (b) is a Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) based on EMD. It can be seen that there are 5 Hz and 20 Hz frequency components in the diagram, but no change of energy amplitude spectrum of 80 Hz components can be seen. There is only scattered energy amplitude spectrum between 40 Hz and 160 Hz, which indicates that there is too much energy of frequency components. This is completely consistent with the result of EMD decomposition.
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ACCEPTED MANUSCRIPT Comparing Fig.4 (a) and (b), it can be seen that the Hilbert time-frequency energy amplitude spectrum based on VMD better characterizes the time-frequency energy distribution characteristics of the original signal than that based on EMD. To distinguish the display effect, double logarithmic coordinates are used to obtain the corresponding spectrum as shown in Fig.5 (a) and (b). The Fig.5 shows that the angular frequency centers of the three modal components of the original simulation signal are 31.4159, 125.6637, and 502.6548 rad/s, respectively. The frequency centers of the three main signals of the original simulation signal are identical, and the energy amplitudes are basically the same. Among them, the high-frequency part is disturbed by noise, but VMD still separates each segment of frequency.
A/V
10
10
2
0
2πf =31.4159
10
2πf =125.6637 2πf =502.6548
-2
10
1
2
10 2πf/(rad/s)
10
3
（a）Hilbert spectrum of original simulation signal
A/v
10
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10
2
0
-2
10
1
2
10 2πf/(rad/s)
10
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（b）VMD-Hilbert spectrum Fig.5. Time-frequency spectrogram for simulation signal
x1 (t )
4. Vibration fault diagnosis of wind turbines Although the types of motor vibration signals vary, along with the frequency and energy distribution, these signals remain comparable. In the past decade, the data of the Case Western Reserve University (CWRU) Bearing Data Center website Loparo has become the standard reference for many researchers to test various algorithms[37-42]. Therefore, the proposed method in this paper is entirely tested based on the
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ACCEPTED MANUSCRIPT vibration data obtained from the CWRU. The experimental database from CWRU simulates all kinds of fault vibration, including the bearing inner ring, outer ring, and rolling element. Different parts and sizes of single point defects are set on the bearing with electrical discharge machining (EDM) technology. The data analyzed by the authors were based on the experimental end bearing 6205-2RS JEM SKF. When no fault setting is available, the motor has a load of 3 Hp, a speed of 1730 rpm, and a sampling frequency of 12 kHz. Fig.6 (a) provides the diagram of the normal vibration signal time domain. After VMD, four empirical modal components are obtained, as shown in Fig.6 (b). From these four components, the first one reflects the main vibration characteristic components. Although the three other components also decompose such components, high-frequency vibration noise is included and annihilated.
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0.4 0.2 0 -0.2 -0.4
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（a）Waveform of normal vibration signal
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（b）Figure (a) waveform after VMD
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ACCEPTED MANUSCRIPT
res.
imf9 imf8 imf7 imf6 imf5 imf4 imf3 imf2 imf1 signal
Empirical Mode Decomposition
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（c）Figure (a) waveform after EMD
2
Input signal spectrum
10
0
v1
A/(m.s -2)
v2 10
10
-2
v3 -4
v4 10
-6
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f/Hz
10
3
（d）Hilbert spectrum of Figure (b) Fig.6. Analysis charts of the experimental vibration signals
Similarly, to compare the effects of the VMD and EMD on the actual vibration signal analysis, the EMD was also performed on Fig.6 (a). This was done to obtain an empirical modal function, as shown in Fig.6 (c). This figure illustrates that the vibration signal is decomposed into nine IMF components and one remainder. The second and third IMF components separate the main vibration signal components and contain noise. Then, the excessive IMF component is decomposed. Although certain components reflect vibration information, most IMF components have no vibration characteristics and cannot be explained. This condition may be referred to as false mode, which is not conducive to the further decomposition of vibration signal to obtain feature quantity. Fig.6 (b) is transformed by Hilbert transform, and the spectrum of Fig.6 (d) is obtained by using double logarithmic coordinates. The dotted line in Fig.6 (d) is the original signal spectrum, which is just the sum of the spectrums of the four decomposed VMD components. It can be seen that the energy of the signal is mainly concentrated in the first modal component, and the other three modal components have less energy,
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ACCEPTED MANUSCRIPT which mainly reflects various noises during normal operation. For the same motor, the fault defect diameters of bearing inner ring, outer ring, and rolling element are set at 21 mils, whereas bearing outer ring is set at the six o'clock direction (orthogonal to the load area). The motor is also loaded at 3 Hp, the speed is 1730 rpm, and the sampling frequency is 12 kHz. Fig.7 (a) presents the time domain diagram of bearing outer ring’s fault vibration signal, whereas Fig.7 (b) shows its modal component diagram decomposed by the VMD.
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（c）Hilbert spectrum of Figure (b) Fig.7. Analysis of vibration signal under bearing outer ring failure
Compared with Fig.7 (a) and Fig.6 (a), vibration signal has evident characteristics and periodicity in the case of faults. Fig.7 (b) and Fig.6 (b) are decomposed into four variational modal components, which
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ACCEPTED MANUSCRIPT contain the same high and low frequency information content of vibration signal, but the intensity is different. Hilbert transform is applied to Fig.7 (b), and double logarithmic coordinates are used to obtain the spectrogram as shown in Fig.7 (c). Compared with Fig.6 (d), the first and fourth mode components change greatly, reflecting the amount of fault information. The second and third mode components slightly change, especially in the high frequency part, and the noise content is the majority. Similarly, the time-domain diagrams of fault vibration signal of the bearing inner ring and the rolling element can be obtained, as shown in Fig.8 (a) and Fig.9 (a), respectively. The spectrum diagrams of the bearing inner ring and the rolling element after VMD and Hilbert transformation are shown in Fig.8 (b) and Fig.9 (b), respectively. Fig.7 (c), Fig.8 (b), and Fig.9 (b) illustrate the significant differences in vibration time-frequency maps of the same type of faults at different locations. Whether the spectrum energy value is different in the low frequency band or the center frequency is different in the high frequency band, fault identification is much improved based on VMD and energy entropy.
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f/Hz
（b）Spectrum by VMD and Hilbert transform of the failure of bearing inner ring Fig.8. Analysis of the vibration signal under bearing inner ring failure
A/(m.s -2)
0.5
0
-0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
t/s
（a）Vibration signal waveform when rolling element is faulty
16
ACCEPTED MANUSCRIPT 10
2
Input signal spectrum
A/(m.s -2)
10
10
0
v1
-2
v2
10
-4
v3 v4
10
-6
10
1
10
2
f/Hz
10
3
（b）Spectrum by VMD and Hilbert transform of the failure of rolling element Fig.9. Analysis of vibration signal under rolling element failure
Vibration signal acquisition is performed for the inner ring, outer ring, rolling element faults and normal state, and 40 groups are measured for each state. Each state of vibration signal in the motor load is 0 Hp, 1 Hp, 2 Hp, and 3 Hp. Motor speed is between 1730 rpm and 1797 rpm. Sampling frequency is 12 kHz, and the number of data signals per group is 2,160. The vibration signals of the three fault types and normal conditions are extracted into VMD component energy features, and Table 1 reveals the results of each four series. Table 1 Normalized vector elements of each VMD energy for different vibration signals Status
normal
Inner ring fault
Outer ring fault
Category label
1
2
3
Rolling element fault
4
Sample sequence number
V1
1
0.29339
2
0.66328
3
Feature vector V2
V3
V4
0.95248
0.08183
0.00013
0.67764
0.31759
0.00019
0.46557
0.68641
0.55865
0.00022
4
0.59969
0.46197
0.65342
0.00021
1
0.00582
0.01023
0.80092
0.59866
2
0.07105
0.01151
0.80317
0.59139
3
0.03655
0.00807
0.97639
0.21273
4
0.04727
0.01802
0.96027
0.27446
1
0.18010
0.02640
0.55957
0.80855
2
0.19333
0.03476
0.60249
0.77359
3
0.10852
0.02634
0.23788
0.96486
4
0.07255
0.03029
0.25408
0.96398
1
0.17217
0.08619
0.97727
0.08871
2
0.01271
0.01359
0.99978
0.00910
3
0.03528
0.18481
0.98202
0.01517
4
0.02266
0.46274
0.88616
0.00789
From Table 1, each state vibration signals selected under the load of 3Hp are used for analytical comparisons. The VMD components energy features are extracted and normalized, the energy distribution map of each type of vibration signals containing four VMD components are shown in Fig.10. It can be
17
ACCEPTED MANUSCRIPT seen from Fig.10 that the energy distribution of each type of vibration signals are significantly different, and the energy of the same order component varies greatly under different operating states. This is very helpful for the diagnosis and identification of vibration faults of wind turbines. 1 0.9 0.8
Normal Inner ring fault Outer ring fault Rolling element fault
Normalized energy
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
2 3 Decomposed components
4
Fig.10 Energy spectrum characteristics of VMD-based decomposition components of vibration signals in various operating states under load of 3HP.
From Table 1, each type of vibration signal contains four VMD components to form a four-dimensional vector. Let X V1 V2 V3 V4 be the input feature sample of the SVM model for motor vibration failure. Digital labels 1−4 are assigned to the normal condition, inner ring, outer ring, rolling element faults, respectively. In this experiment, 20 groups are selected for normal motor condition, inner ring, outer ring, and rolling element failures. A total of 80 samples are used for training, and the remaining 80 samples are used for testing. Fig.11 (a) displays the sample data distribution. The training and test samples are integrated into the LIBSVM model respectively for fault recognition in classification. To achieve satisfactory results, relevant parameters (c [penalty parameter] and g [function parameter]) must be adjusted. Cross-validation is used to avoid under-learning and avoid over-learning. Select parameter c = 2, g = 1. Eighty types of training samples are integrated into the SVM model for the four types of characteristic training. After training, the remaining 80 groups of test samples are input into the trained SVM model to verify classification. Fig.11 (b) illustrates the training classification effect. "○" represents the actual training sample fault type classification, and "*" represents the prediction sample fault type classification. The figure also reveals that only one fault identification error is found.
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ACCEPTED MANUSCRIPT class
V1
2 0
100 200 Sample V3
Feature Energy
1 0.8 0.6 0.4 0.2 0
0
100 200 Samples
V2
1 0.8
Feature Energy
3
1
Feature Energy
Feature Energy
Class Label
4
0.6 0.4 0.2 0
0
100 200 Samples V4
0
100 200 Samples
1 0.8 0.6 0.4 0.2 0
0
100 200 Samples
1 0.8 0.6 0.4 0.2 0
（a）Sample data
Actual Classifition and Predicted Classifition of Testing Samples 4
Actual Classifition Predicted Classifition
Class Label
3.5 3 2.5 2 1.5 1
0
10
20
30
40
50
60
70
80
Testing Samples
（b）Classification results Fig.11. Vibration fault classification of wind turbines based on SVM
The experimental results show that all 80 test samples of normal condition, inner ring fault, and outer ring fault are correctly identified, whereas rolling element fault correctly identifies 79 test samples. Only one group is classified incorrectly. Thus, the SVM can classify the validation samples with an accuracy of 98.75%. The reasons for the recognition errors may be that few training samples are available. In addition, a large difference between individual feature vector data and training samples is found. If the number of training samples is increased, then the accuracy of the recognition results can be further improved. In order to compare and verify the diagnostic recognition effect of the VMD energy entropy method, EMD and wavelet db4 are used to decompose the four operating states of the vibration signal, and the energy entropy of the modal state is calculated and decomposed. These three methods are used to train and classify the SVM, and Table 2 is the comparisons of the fault recognition rates of the three feature extraction methods.
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ACCEPTED MANUSCRIPT Table 2 Comparisons of recognition rate of different fault feature extraction methods. Order number 1 2 3
Methods VMD energy entropy EMD energy entropy db4 energy entropy
Number of training samples
Number of test samples
Number of misjudgement
Recognition rate /%
80
80
1
98.75
80
80
25
68.75
80
80
9
88.75
It can be seen from Table 2 that the correct recognition rate of the feature vector based on the VMD energy entropy method is higher than that of the feature vector obtained based on EMD and wavelet db4. This is because VMD has better robustness than other decomposition algorithms, and can effectively extract modal components with different center frequencies. It has good decomposition characteristics, and the energy entropy eigenvectors of the composition are significantly different. Therefore, it has a higher state recognition rate. The results show that the fault diagnosis method based on VMD energy entropy eigenvalue extraction and SVM can accurately classify various fault states of wind turbines. 5. Conclusions The actual vibration signal of wind turbines is weak and is seriously affected by environmental noise. Thus, fault identification is relatively difficult. A vibration fault diagnosis method based on the VMD and energy entropy is proposed in this study. VMD converts vibration signal decomposition into non-recursive and variational mode decomposition. This conversion is a function optimization problem under constrained conditions. It updates the modal functions and corresponding central frequencies with the objective of minimizing the bandwidth of modal estimation. The VMD also comprises a set of adaptive Wiener filter banks. The simulation results show that the VMD can effectively avoid the mode aliasing problem in the EMD when dealing with modal components with multiple frequencies. For the actual vibration signal, the EMD also has serious modal aliasing problems due to the influence of noise interference. The VMD can still accurately decompose and extract vibration fault information with respect to EMD. The energy entropy of each modal component decomposed by VMD is input into SVM as an eigenvalue. One part of the eigenvalue is used as the training sample of SVM, whereas the other part is used as the test sample of SVM. And it was compared with SVM classification and recognition by energy entropy eigenvalue extraction methods based on EMD and wavelet db4. The experimental results show that the classification accuracy of the proposed method is higher than those of EMD-based method and wavelet db4-based method.
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ACCEPTED MANUSCRIPT Although the SVM model has a limited training sample, the accuracy of the classified verification sample can reach 98.75%. The next step is to increase the vibration fault sample library. The accuracy of classification and recognition can be further improved by increasing the sample base of SVM model, which provides a reference diagnosis method for practical application. In addition, the next step is to study the application of the proposed method to vibration fault diagnosis of other components of wind turbines. 6. Acknowledgements This work is supported by National Natural Science Foundation of China (51477015), the Visiting Scholarship of State Key Laboratory of Power Transmission Equipment & System Security and New Technology (Chongqing University) (2007DA10512714406) , the Natural Science Foundation of Fujian Province of China (2018J01511), and the Program for New Century Excellent Talents in Fujian Province University (2018047). 7. References [1] Chu S, Majumdar A. Opportunities and challenges for a sustainable energy future. Nature 2012; 488: 294. [2] Solangi K H, Islam M R, Saidur R, Rahim N A , Fayaz H. A review on global solar energy policy. Renew Sust Energ Rev 2011; 15: 2149-2163. [3] Lewis J I, Wiser R H. Fostering a renewable energy technology industry: An international comparison of wind industry policy support mechanisms. Energ policy 2007; 35: 1844-1857. [4] Qiao W, Lu D. A survey on wind turbine condition monitoring and fault diagnosis—Part I: Components and subsystems. IEEE T Ind Electron 2015; 62: 6536-6545. [5] Nie M, Wang L. Review of condition monitoring and fault diagnosis technologies for wind turbine gearbox. Procedia Cirp 2013; 11: 287-290. [6] Amirat Y, Benbouzid M E H, Al-Ahmar E, Bensaker B, Turri S. A brief status on condition monitoring and fault diagnosis in wind energy conversion systems. Renew Sust Energ Rev 2009; 13: 2629-2636. [7]Yang W, Tavner P J, Crabtree C J, Feng Y, Qiu Y. Wind turbine condition monitoring: technical and commercial challenges. Wind Energy 2014; 17: 673-693. [8] Hang J, Zhang J, Cheng M, Wang W, Zhang M. An overview of condition monitoring and fault diagnostic for wind energy conversion system. T China Electrotech Soc 2013; 28:261-272. [9] Ogidi O O, Barendse P S, Khan M A. Fault diagnosis and condition monitoring of axial-flux permanent magnet wind generators. Electr Pow Syst Res 2016; 136: 1-7. [10]Liu W Y, Tang B P, Han J G, Lu X N, Hu N N, He Z Z. The structure healthy condition monitoring and fault diagnosis methods in wind turbines: A review. Renew Sust Energ Rev 2015; 44: 466-472.
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ACCEPTED MANUSCRIPT [11] Tchakoua P, Wamkeue R, Ouhrouche M, Slaoui-Hasnaoui F, Tameghe T A, Ekemb G. Wind turbine condition monitoring: State-of-the-art review, new trends, and future challenges. Energies 2014; 7: 2595-2630. [12] Lei Y, Lin J, Zuo M J, He Z. Condition monitoring and fault diagnosis of planetary gearboxes: A review. Measurement 2014; 48: 292-305. [13] Chen J, Pan J, Li Z, Zi Y, Chen X. Generator bearing fault diagnosis for wind turbine via empirical wavelet transform using measured vibration signals. Renew Energ 2016; 89: 80-92. [14] Feng Z, Qin S, Liang M. Time–frequency analysis based on Vold-Kalman filter and higher order energy separation for fault diagnosis of wind turbine planetary gearbox under nonstationary conditions. Renew Energ 2016; 85: 45-56. [15] Jablonski A, Barszcz T. Validation of vibration measurements for heavy duty machinery diagnostics. Mech Syst Signal Pr 2013; 38: 248-263. [16] Márquez F P G, Tobias A M, Pérez J M P, Papaelias M. Condition monitoring of wind turbines: Techniques and methods. Renew Energ 2012; 46: 169-178. [17] An X, Jiang D, Chen J, Liu C. Application of the intrinsic time-scale decomposition method to fault diagnosis of wind turbine bearing. J Vib Control 2012; 18: 240-245. [18] Feng Z, Liang M. Complex signal analysis for wind turbine planetary gearbox fault diagnosis via iterative atomic decomposition thresholding. J Sound Vib 2014; 333: 5196-5211. [19] Wang J, Peng Y, Qiao W. Current-aided order tracking of vibration signals for bearing fault diagnosis of directdrive wind turbines. IEEE T Ind Electron 2016; 63: 6336-6346. [20] de Jesus Rangel-Magdaleno J, de Jesus Romero-Troncoso R, Osornio-Rios R A, Cabal-Yepez E, DominguezGonzalez A. FPGA- based vibration analyzer for continuous CNC machinery monitoring with fused FFT-DWT signal processing. IEEE T Instrum Meas 2010; 59: 3184-3194. [21] Khanam S, Tandon N, Dutt J K. Fault size estimation in the outer race of ball bearing using discrete wavelet transform of the vibration signal. Procedia Technology 2014; 14: 12-19. [22] Yan R, Gao R X, Chen X. Wavelets for fault diagnosis of rotary machines: A review with applications. Signal Process 2014; 96: 1-15. [23] Chen J, Li Z, Pan J, Chen G, Zi Y, Yuan J, Chen B, He Z. Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review. Mech Syst Signal Pr 2016; 70: 1-35. [24] Yaqub M F, Gondal I, Kamruzzaman J. Inchoate fault detection framework: Adaptive selection of wavelet nodes and cumulant orders. IEEE T Instrum Meas 2012; 61: 685-695. [25] Huang N E, Wu Z, Long S R. Hilbert-Huang transform. Scholarpedia 2008; 3: 2544. [26] Huang N E, Wu Z. A review on Hilbert-Huang transform: Method and its applications to geophysical studies. Rev Geophys 2008; 46: RG2006. [27] Dragomiretskiy K, Zosso D. Variational mode decomposition. IEEE T Signal Proces 2014; 62: 531-544.
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ACCEPTED MANUSCRIPT [28] Dragomiretskiy K, Zosso D. Two-dimensional variational mode decomposition. In: International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition; 2015. P.197-208. [29] Lahmiri S. Comparative study of ECG signal denoising by wavelet thresholding in empirical and variational mode decomposition domains. Healthcare technol lett 2014; 1: 104-109. [30] Bafroui H H, Ohadi A. Application of wavelet energy and Shannon entropy for feature extraction in gearbox fault detection under varying speed conditions. Neurocomputing 2014; 133: 437-445. [31] Jurado S, Nebot À, Mugica F, Avellana N. Hybrid methodologies for electricity load forecasting: Entropybased feature selection with machine learning and soft computing techniques. Energy 2015; 86: 276-291. [32] Dasgupta A, Nath S, Das A. Transmission line fault classification and location using wavelet entropy and neural network. Electr Pow Compo Sys 2012; 40: 1676-1689. [33] Chang C C, Lin C J. LIBSVM: a library for support vector machines. ACM T Intel Syst Tec 2011; 2: 1-27. [34] Gryllias K C, Antoniadis I A. A Support Vector Machine approach based on physical model training for rolling element bearing fault detection in industrial environments. Eng Appl Artif Intel 2012; 25: 326-344. [35] Al-Shammari E T, Keivani A, Shamshirband S, Mostafaeipour Ali,Yee P L,Petković D, Ch S. Prediction of heat load in district heating systems by Support Vector Machine with Firefly searching algorithm. Energy 2016; 95: 266-273. [36] Deka P C. Support vector machine applications in the field of hydrology: a review. Appl Soft Comput 2014; 19: 372-386. [37]
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ACCEPTED MANUSCRIPT Highlights
A method of feature vector extraction based on variational mode decomposition (VMD) and energy entropy is proposed. The time-frequency energy spectrum obtained by VMD is compared with that by empirical mode decomposition (EMD). Different types of bearing vibration signals of wind turbine are analyzed. The fault diagnostic identification of the proposed method is compared with EMD-based and wavelet db4-based methods.
Xuejun Chen, Yongming Yang, Zhixin Cui, Jun Shen PII:
S0360-5442(19)30461-X
DOI:
10.1016/j.energy.2019.03.057
Reference:
EGY 14893
To appear in:
Energy
Received Date:
25 October 2018
Accepted Date:
09 March 2019
Please cite this article as: Xuejun Chen, Yongming Yang, Zhixin Cui, Jun Shen, Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy, Energy (2019), doi: 10.1016/j.energy.2019.03.057
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy
Xuejun Chena,c,*, Yongming Yang b, Zhixin Cui c, Jun Shen c a
Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China b State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044, P.R. China; c School of Mechanical Engineering and Automation, Fuzhou University, 350108, P.R. China
Correspondence information*: Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China; E-mail address:[email protected]
1
ACCEPTED MANUSCRIPT Vibration fault diagnosis of wind turbines based on variational mode decomposition and energy entropy Xuejun Chen a,c, , Yongming Yang b, Zhixin Cui c, Jun Shen c Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044, P.R. China c School of Mechanical Engineering and Automation, Fuzhou University, 350108, P.R. China a
b
Abstract: The bearing vibration of wind turbines is nonlinear and non-stationary. To effectively extract bearing vibration signal features for fault diagnosis, a method of feature vector extraction based on variational mode decomposition (VMD) and energy entropy is proposed. In addition, the support vector machine (SVM) classifier is used to identify the types of vibration faults. VMD transforms the constrained variational objective function into the unconstrained one to optimize solution. Compared with the VMD and empirical mode decomposition (EMD), the modal decomposition layer of VMD is less than EMD, no found false modality, and truly reflecting signal components. After the Hilbert transformation, double logarithmic coordinates show that the VMD-based spectral characteristics are significant. VMD is performed on the different types of vibration signals of wind turbines. Therefore, VMD is not that affected by noise and has few decomposition levels. The energy entropy of the normalized four modal components is considered the eigenvalue and classified by SVM, and compared with EMD-based and wavelet db4based energy entropy eigenvalue extraction methods. Experimental results indicate that the accuracy of the method is higher than those based on EMD and wavelet db4, under the limited sample condition. Thus, a referential diagnostic method is provided for practical applications. Keywords: variational mode decomposition; energy entropy; vibration signal; wind turbine; fault 1. Introduction Traditional energy is increasingly becoming scarce, and the environment and climate are getting worse. Hence, the utilization and development of new energy has become the focus of all countries. Wind energy is a kind of renewable green energy, which has been vigorously developed, researched, and utilized by many countries [1−3]. However, with the large-scale construction, operation and production of wind turbines, a series of new technical and environmental problems have arisen. It is urgent to study the related wind power monitoring and fault diagnosis technology [4−6]. An effective equipment monitoring and fault diagnosis can continuously monitor various parameters of wind turbine operation. It can track all kinds of state information in real time, analyse wind power operation status and diagnose faults. Finally, the appropriate maintenance plan is arranged according to the
Corresponding author. Department of Mechanical and Electrical Engineering, Putian University, Putian, 351100, P.R. China. E-mail address: [email protected]
2
ACCEPTED MANUSCRIPT diagnosis results. These factors reduce the equipment accident rate as well as operation and maintenance costs [7-11]. At present, monitoring and fault diagnosis are mainly used in wind turbine bearings and gearboxes and other major components [12-14]. Vibration signal is the most important carrier of much diagnostic information. More and more attention has been paid to the diagnosis methods based on vibration signals. [15−19]. In view of the relatively wide distribution of vibration frequency, time-frequency transformation of vibration signals can be performed to obtain the special quantity of frequency domain [20]. Thus, the vibration characteristics of different faults are distinguished. However, this method sometimes cannot display local information. In addition, wavelet transformation has multi-scale characteristics and "mathematical microscopy," which has been used by many researchers in the time-frequency analysis of vibration signals. Moreover, wavelet transformation can provide localized information about time and frequency domains. However, wavelet has certain problems, such as selection of basis function, sensitivity of parameters, and inaccurate description of frequency-to-time transformation [21-24]. For non-linear unsteady-state signals, Huang et al. proposed a Hilbert-Huang Transform, which can obtain the signal Hilbert and time-frequency energy spectrums based on empirical mode decomposition (EMD). The signal time-frequency localization analysis can also be achieved [25−26]. However, certain signals have modal aliasing. To solve this problem, Konstantin Dragomiretskiy and Dominique Zosso (2014) proposed variational mode decomposition (VMD) [27]. By redefining the modal function, if each modal component is a finite bandwidth with different center frequencies, then the alternating direction multiplier method is used to iteratively update the modal and its center frequency. Using this method solves the variational problem. Each mode is demodulated onto a corresponding baseband, and each modal component is extracted. Compared with the EMD recursive “screening” mode, VMD transforms the signal into non-recursive, variational mode decomposition modes, showing good robustness and stability [28−29]. The adaptive decomposition of VMD to non-stationary signals has high time-frequency resolution. Therefore, a method based on VMD and energy entropy is proposed to extract vibration signal features for bearing of wind turbines. Support vector machine (SVM) is also used to diagnose faults, providing a new method for monitoring and fault diagnosis of wind turbine safety production. 2. Vibration signal feature analysis method based on VMD 2.1. VMD theory
3
ACCEPTED MANUSCRIPT VMD is based on classical Wiener filtering, Hilbert transform, and frequency mixing. Original signal f (t ) can be decomposed into k modal components uk (t ) with center frequency k (t ) , so that the estimated bandwidth of each mode The sum is the smallest [27−28]. Note that k is the preset scale. The fractional modal decomposition algorithm constructs variational objective functions and solves them. 2.1.1. Constructing variational objective functions First, the analytical signal of each modal function uk (t ) is obtained by Hilbert transform to obtain its unilateral spectrum, as shown in Formula (1). j ) uk (t ) (1) t In Formula (1), (t ) is an impact function. The estimated center frequency e jk t of each modal ( (t )
analysis signal is mixed, and each mode is modulated to the corresponding baseband, as in Equation (2). j ) uk (t )]e jk t (2) t The square L2 norm of the given demodulation signal gradient is calculated, and the bandwidth of [( (t )
each mode signal is estimated. Formula (3) shows the constrained variational objective functions. 2 j jk t { t [( (t ) ) uk (t )]e } {umin },{ } t k k k s.t. k uk f
(3)
K
In Formula (3), {uk }: {u1 , , uK } ， {k }: {1 , , K } ， : . k
k 1
2.1.2. Solving the variational objective function The quadratic penalty factor and Lagrangian multiplication operator (t ) are introduced to transform the constrained variational objective function into a non-binding variational objective function. The quadratic penalty term is a classical method to guarantee the fidelity of the reconstructed signal. The method also has good convergence, especially in the presence of Gaussian noise. The Lagrangian multiplier keeps the constraints strictly, and the extended Lagrangian function is given by Equation (4).
4
ACCEPTED MANUSCRIPT L({uk },{k }, ) : t [( (t ) k
j ) uk (t )]e jk t t
2
2
(4)
2
f (t ) uk (t ) (t ), f (t ) uk (t ) k
k
2
Therefore, the minimization problem of variational objective functions is transformed into the iterative search for the extended Lagrangian function "saddle point" problem, which is the alternate direction method of multipliers. The Lagrangian function "saddle point" is obtained by alternately updating ukn 1 ， kn 1 , and kn 1 . The iterative problem in solving Equation (4) is expressed below. j u kn 1 arg min t [( (t ) ) uk (t )]e jk t t uk X 2 (t ) f (t ) ui (t ) 2 2 i
Among them, k equals kn 1 and
u (t ) i
equals
u (t ) ik
i
n 1
i
2
2
(5)
. Formula (5) is converted to the
frequency domain by Parseval/Plancherel Fourier equidistant transformation.
2
uˆ kn 1 arg min j[(1 sgn( k ))uˆk ( k )] 2 uˆk ,uk X
fˆ ( ) uˆi ( ) i
(6)
2 ˆ ( )
2
2
Replace in the first term of Formula (6) with k .
2
uˆ kn 1 arg min j ( k )[(1 sgn( ))uˆk ( )] 2 uˆk ,uk X
2 ˆ ( )
fˆ ( ) uˆi ( )
2
i
2
(7)
Transforming Formula (7) into a half-space integral form with non-negative frequencies, we obtain Formula (8). uˆ kn 1 arg min uˆk ,uk X
4 ( )
2
k
0
2 fˆ ( ) uˆi ( ) i
uˆk ( )
ˆ ( ) 2
2
d
(8)
By making the first variable of positive frequency disappear, the solution of the two optimization problems can be solved as follows: ˆ ( ) fˆ ( ) uˆi ( ) 2 . ik uˆ kn 1 ( ) 2 1 2 ( k )
(9)
5
ACCEPTED MANUSCRIPT Using the same solution process, the central frequency solution problem is converted to frequency domain. n 1 arg min k
k
( )
2
k
0
2
uˆk ( ) d
(10)
The update formula of center frequency can be obtained through the following:
n 1 k
0 0
2
uˆk ( ) d
.
(11)
2
uˆk ( ) d
Formula (11) enables the new k to be in the center of gravity of the power spectrum of the corresponding modal function. 2.2. VDM algorithm Through the given construction and solving process, a complete VMD algorithm can be obtained by optimizing in the appropriate Fourier domain.
① Initialize uˆ1k ， k1 ， ˆ1 ， n 0 ； ② n n 1 ； ③ Update uˆk and ˆ k using Equations (9) and (10); ④ For all 0 , update the double rising step length by using Equation (12) ˆ n 1 ( ) ˆ n ( ) fˆ ( ) uˆkn 1 ( ) .
Among them, is a noise tolerance parameter.
k
(12)
⑤ If convergence condition Formula (13) is satisfied, then iteration is stopped; otherwise it returns to step ② to continue.
uˆ k
n 1 k
n
uˆ k
uˆ kn 2
2 2
(13)
2
Among them, 0 of Formula (13). 2.3. Vibration signal feature extraction and fault diagnosis Vibration signals vary when wind turbines either fail or are in a normal state. The energy of the vibration signal can also be different in different frequency bands. Vibration signal can be decomposed into k modal components uk (t ) by VMD algorithm. Then, the energy distribution of vibration signal can be obtained by calculating energy entropy [30−32]. When the working state of
6
ACCEPTED MANUSCRIPT wind turbines changes, the energy entropy of modal component uk (t ) changes. Therefore, wind turbine faults can be diagnosed on the basis of their energy entropy. The energy entropy feature extraction steps based on the modal component uk (t ) , which are as follows: ① The original vibration signal is decomposed by VMD, and former n modal component uk (t ) containing the main fault information is selected. ② Energy Ek of each modal component uk (t ) is obtained.
Ek
2
k 0,1, 2,..., n
uk (t ) dt
(14)
③ The characteristic vector T is constructed by the energy Ek of each modal component uk (t ) .
T E1
E2 En
(15)
Then, normalize feature vector T using
T ' E1 / E
E2 / E En / E .
(16)
1/2
n 2 In Formula (16), T ' is the vector T after normalization, and E Ei . i 1 SVM is first proposed by Vapnik. Similar to the multi-layer perceptron and radial basis function networks, SVM can be used for pattern classification and nonlinear regression. Its basic idea is that input space is transformed into high-dimensional space by nonlinear transformation defined by kernel function. Thus, samples are linearly separable, and the optimal classification surface is constructed in high-dimensional space, leading to the formation of the decision rules of sample classification [33−36]. Let vibration signal training sample set ( xi , yi ) ， i 1, , n ， x R d ， y {1, 1} ， n be the number of training samples, and d be the dimension of the feature vector of vibration signal. Under the constraint of yi [( , xi ) b] 1 0 ， i 1, , n , function ( )
1 2
2
is made to have a minimum
value. Use the Lagrangian function to find the partial derivatives of and b and make them equal to 0. Then, find the minimum value of the Lagrangian function. Doing so determines the classification rules only by few support vectors right on the edge of the optimal classification surface. Note that such rules have nothing to do with the other samples. According to the nonlinear characteristics of the four kinds of wind turbines' working state vibration signals, the VMD method is used to decompose various state signals, Moreover, the
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ACCEPTED MANUSCRIPT energy entropy characteristics of the first few modal components containing most information in each working state signal are calculated. Therefore, the feature vector of each working state is composed and integrated into the SVM model for fault diagnosis. Fig.1 illustrates the fault feature diagnosis process. Original signal f(t)
Calculating the energy entropy of modal component
VMD decomposition
Fault identification
SVM
Fig.1. Flow chart of wind turbine fault diagnosis based on VMD energy entropy and SVM
3. Simulation experiment and analysis To illustrate the VMD performance and robustness with respect to the input signal noise, a threeharmonic signal affected by noise is used to test VMD, as shown in Equation (17). Among them, is
(0, ) Gaussian noise, and the standard deviation is considered 1 to control the noise level. Here, the amplitude of noise is quite important and relative to the amplitude of the highest and weakest harmonics. In addition, = 0.1 is selected. x1 (t ) cos(2 5 t ) 0.2 cos(2 20 t ) 0.1cos(2 80 t )
(17)
Although EMD can automatically estimate the mode decomposition layer on the basis of the signal, VMD can set the number of decomposition layers according to different signals. For simulation signal x1 (t ) ,
K = 4. Fig.2 (a) and (b) show the time domain waveform and simulated signal spectrum and the
detected boundaries supported by each filter, respectively.
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x1 (t )
x1 (t )
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ACCEPTED MANUSCRIPT The simulation signal is decomposed by VMD, in which 5, 20, and 80 Hz signals are completely separated from each other. The maximum amplitude of 80 Hz signal is 0.1, while the standard deviation of Gaussian noise is considered 1, and = 0.1 in Equation (17). So the 80HZ signal is annihilated by noise. It can also be seen from Fig.3 (a) that “V3”is the 80 HZ signal and most of the amplitude is less than “V4” noise. However, after VMD, the center frequency of the 80 Hz signal is adjusted to this harmonic. Fig.3 (a) illustrates the VDM. To compare and analyze the EMD, Fig.3 (b) provides the EMD result of simulation signal x1 (t ) . Fig.3 (b) shows that the simulation signal x1 (t ) is decomposed into four modal function components and one remainder. Simulation signal x1 (t ) is composed of four empirical mode components by the VMD. Although EMD can be adaptively decomposed according to the signal, the decomposition is greatly affected by noise and is prone to over-decomposition, thereby reducing operational efficiency. Moreover, if too many modal components are separated, then false modal components may appear. Such components can accurately analyze unfavorable signals. Furthermore, this excessive decomposition of the iterative computation can inevitably result in increased computation. For most signals, VMD computation is less than that of EMD. v1
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ACCEPTED MANUSCRIPT Empirical Mode Decomposition
signal
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（b）EMD of simulation signal 1 Fig.3. Decomposition result of the simulation signal
In Fig.3 (a), the four modal components decomposed by the VMD well reflect the components of the original simulation signal. In addition, noise is considered an independent modal component. Fig.3 (b) illustrates that the EMD only approximately and correctly decomposes a high-frequency modal component intrinsic mode function 1 (IMF1) and a residual. The EMD also decomposes several false modal components IMF2, IMF3, and IMF4, thus causing distortion and loss of 20 Hz and 80 Hz components in the original simulated signal. Moreover, the VMD can decompose the different modal components of the signal. The VMD can also analyze the modal components of different frequencies in the signal, and each modal component has substantial characteristic energy of the original signal. In theory, the IMF of each resolution obtained by the EMD cannot guarantee strict orthogonality but can only be approximate orthogonality. To date, no complete and rigorous theoretical support has been found. Therefore, problems of modal aliasing arise in decomposition. VMD is based on classical Wiener filter and Hilbert transform, which has a good theoretical support. Fig.3 (a) and (b) are transformed by Hilbert transform respectively, and then the energy spectrum of Hilbert can be obtained by integrating the square of amplitude with time. The energy spectrum represents the energy accumulated by each frequency over the whole time period. The results of the time-frequencyenergy amplitude spectrum, as shown in Fig.4 (a) and (b), can clearly reflect the distribution of signal energy with frequency and time (the darker the color, the greater the energy; conversely, the smaller the energy).
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ACCEPTED MANUSCRIPT 100
frequency/Hz
80 60 40 20
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0
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time/s
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0.1 energy 0.2 0.3 0.4 of the 0.5 simulation 0.6 0.7 0.9 （a）Hilbert time-frequency spectrum signal0.8x1 (t ) based on1 VMD Time/s
160 140
frequency/Hz
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（b）Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) based on EMD Fig.4. Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) .
Fig.4 (a) is a Hilbert two-dimensional time-frequency energy amplitude spectrum of a simulated signal based on VMD. There are three frequency components on the graph, which are 5 Hz, 20 Hz, and 80 Hz. The wave energy is clearly distributed at frequencies of 5 Hz, 20 Hz and 80 Hz, respectively. And the smaller the frequency component, the darker the color is, and the maximum amplitude of the 5Hz component of the original simulation signal is the same. Fig.4 (b) is a Hilbert time-frequency energy spectrum of the simulation signal x1 (t ) based on EMD. It can be seen that there are 5 Hz and 20 Hz frequency components in the diagram, but no change of energy amplitude spectrum of 80 Hz components can be seen. There is only scattered energy amplitude spectrum between 40 Hz and 160 Hz, which indicates that there is too much energy of frequency components. This is completely consistent with the result of EMD decomposition.
11
ACCEPTED MANUSCRIPT Comparing Fig.4 (a) and (b), it can be seen that the Hilbert time-frequency energy amplitude spectrum based on VMD better characterizes the time-frequency energy distribution characteristics of the original signal than that based on EMD. To distinguish the display effect, double logarithmic coordinates are used to obtain the corresponding spectrum as shown in Fig.5 (a) and (b). The Fig.5 shows that the angular frequency centers of the three modal components of the original simulation signal are 31.4159, 125.6637, and 502.6548 rad/s, respectively. The frequency centers of the three main signals of the original simulation signal are identical, and the energy amplitudes are basically the same. Among them, the high-frequency part is disturbed by noise, but VMD still separates each segment of frequency.
A/V
10
10
2
0
2πf =31.4159
10
2πf =125.6637 2πf =502.6548
-2
10
1
2
10 2πf/(rad/s)
10
3
（a）Hilbert spectrum of original simulation signal
A/v
10
10
10
2
0
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10
1
2
10 2πf/(rad/s)
10
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（b）VMD-Hilbert spectrum Fig.5. Time-frequency spectrogram for simulation signal
x1 (t )
4. Vibration fault diagnosis of wind turbines Although the types of motor vibration signals vary, along with the frequency and energy distribution, these signals remain comparable. In the past decade, the data of the Case Western Reserve University (CWRU) Bearing Data Center website Loparo has become the standard reference for many researchers to test various algorithms[37-42]. Therefore, the proposed method in this paper is entirely tested based on the
12
ACCEPTED MANUSCRIPT vibration data obtained from the CWRU. The experimental database from CWRU simulates all kinds of fault vibration, including the bearing inner ring, outer ring, and rolling element. Different parts and sizes of single point defects are set on the bearing with electrical discharge machining (EDM) technology. The data analyzed by the authors were based on the experimental end bearing 6205-2RS JEM SKF. When no fault setting is available, the motor has a load of 3 Hp, a speed of 1730 rpm, and a sampling frequency of 12 kHz. Fig.6 (a) provides the diagram of the normal vibration signal time domain. After VMD, four empirical modal components are obtained, as shown in Fig.6 (b). From these four components, the first one reflects the main vibration characteristic components. Although the three other components also decompose such components, high-frequency vibration noise is included and annihilated.
A/(m.s -2)
0.4 0.2 0 -0.2 -0.4
0
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0.08 0.1 t/s
0.12
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（a）Waveform of normal vibration signal
0.1
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0 0.02 -3 x 10
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0.08 0.1 t/s
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0.1
v2
0
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v3
0.1 0
-0.1
v4
2 0 -2
0
0.02
（b）Figure (a) waveform after VMD
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ACCEPTED MANUSCRIPT
res.
imf9 imf8 imf7 imf6 imf5 imf4 imf3 imf2 imf1 signal
Empirical Mode Decomposition
0
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t/s
10
（c）Figure (a) waveform after EMD
2
Input signal spectrum
10
0
v1
A/(m.s -2)
v2 10
10
-2
v3 -4
v4 10
-6
10
1
10
2
f/Hz
10
3
（d）Hilbert spectrum of Figure (b) Fig.6. Analysis charts of the experimental vibration signals
Similarly, to compare the effects of the VMD and EMD on the actual vibration signal analysis, the EMD was also performed on Fig.6 (a). This was done to obtain an empirical modal function, as shown in Fig.6 (c). This figure illustrates that the vibration signal is decomposed into nine IMF components and one remainder. The second and third IMF components separate the main vibration signal components and contain noise. Then, the excessive IMF component is decomposed. Although certain components reflect vibration information, most IMF components have no vibration characteristics and cannot be explained. This condition may be referred to as false mode, which is not conducive to the further decomposition of vibration signal to obtain feature quantity. Fig.6 (b) is transformed by Hilbert transform, and the spectrum of Fig.6 (d) is obtained by using double logarithmic coordinates. The dotted line in Fig.6 (d) is the original signal spectrum, which is just the sum of the spectrums of the four decomposed VMD components. It can be seen that the energy of the signal is mainly concentrated in the first modal component, and the other three modal components have less energy,
14
ACCEPTED MANUSCRIPT which mainly reflects various noises during normal operation. For the same motor, the fault defect diameters of bearing inner ring, outer ring, and rolling element are set at 21 mils, whereas bearing outer ring is set at the six o'clock direction (orthogonal to the load area). The motor is also loaded at 3 Hp, the speed is 1730 rpm, and the sampling frequency is 12 kHz. Fig.7 (a) presents the time domain diagram of bearing outer ring’s fault vibration signal, whereas Fig.7 (b) shows its modal component diagram decomposed by the VMD.
A/(m.s -2)
5 0
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（a）Vibration signal waveform when bearing outer ring is faulty
v1
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（b）Figure (a) waveform after VMD 10
10
4
2
A/(m.s -2)
Input signal spectrum 10
0
v1 10
-2
v2 v3
10
v4
-4
10
1
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2
f/Hz
10
3
（c）Hilbert spectrum of Figure (b) Fig.7. Analysis of vibration signal under bearing outer ring failure
Compared with Fig.7 (a) and Fig.6 (a), vibration signal has evident characteristics and periodicity in the case of faults. Fig.7 (b) and Fig.6 (b) are decomposed into four variational modal components, which
15
ACCEPTED MANUSCRIPT contain the same high and low frequency information content of vibration signal, but the intensity is different. Hilbert transform is applied to Fig.7 (b), and double logarithmic coordinates are used to obtain the spectrogram as shown in Fig.7 (c). Compared with Fig.6 (d), the first and fourth mode components change greatly, reflecting the amount of fault information. The second and third mode components slightly change, especially in the high frequency part, and the noise content is the majority. Similarly, the time-domain diagrams of fault vibration signal of the bearing inner ring and the rolling element can be obtained, as shown in Fig.8 (a) and Fig.9 (a), respectively. The spectrum diagrams of the bearing inner ring and the rolling element after VMD and Hilbert transformation are shown in Fig.8 (b) and Fig.9 (b), respectively. Fig.7 (c), Fig.8 (b), and Fig.9 (b) illustrate the significant differences in vibration time-frequency maps of the same type of faults at different locations. Whether the spectrum energy value is different in the low frequency band or the center frequency is different in the high frequency band, fault identification is much improved based on VMD and energy entropy.
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（b）Spectrum by VMD and Hilbert transform of the failure of bearing inner ring Fig.8. Analysis of the vibration signal under bearing inner ring failure
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ACCEPTED MANUSCRIPT 10
2
Input signal spectrum
A/(m.s -2)
10
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（b）Spectrum by VMD and Hilbert transform of the failure of rolling element Fig.9. Analysis of vibration signal under rolling element failure
Vibration signal acquisition is performed for the inner ring, outer ring, rolling element faults and normal state, and 40 groups are measured for each state. Each state of vibration signal in the motor load is 0 Hp, 1 Hp, 2 Hp, and 3 Hp. Motor speed is between 1730 rpm and 1797 rpm. Sampling frequency is 12 kHz, and the number of data signals per group is 2,160. The vibration signals of the three fault types and normal conditions are extracted into VMD component energy features, and Table 1 reveals the results of each four series. Table 1 Normalized vector elements of each VMD energy for different vibration signals Status
normal
Inner ring fault
Outer ring fault
Category label
1
2
3
Rolling element fault
4
Sample sequence number
V1
1
0.29339
2
0.66328
3
Feature vector V2
V3
V4
0.95248
0.08183
0.00013
0.67764
0.31759
0.00019
0.46557
0.68641
0.55865
0.00022
4
0.59969
0.46197
0.65342
0.00021
1
0.00582
0.01023
0.80092
0.59866
2
0.07105
0.01151
0.80317
0.59139
3
0.03655
0.00807
0.97639
0.21273
4
0.04727
0.01802
0.96027
0.27446
1
0.18010
0.02640
0.55957
0.80855
2
0.19333
0.03476
0.60249
0.77359
3
0.10852
0.02634
0.23788
0.96486
4
0.07255
0.03029
0.25408
0.96398
1
0.17217
0.08619
0.97727
0.08871
2
0.01271
0.01359
0.99978
0.00910
3
0.03528
0.18481
0.98202
0.01517
4
0.02266
0.46274
0.88616
0.00789
From Table 1, each state vibration signals selected under the load of 3Hp are used for analytical comparisons. The VMD components energy features are extracted and normalized, the energy distribution map of each type of vibration signals containing four VMD components are shown in Fig.10. It can be
17
ACCEPTED MANUSCRIPT seen from Fig.10 that the energy distribution of each type of vibration signals are significantly different, and the energy of the same order component varies greatly under different operating states. This is very helpful for the diagnosis and identification of vibration faults of wind turbines. 1 0.9 0.8
Normal Inner ring fault Outer ring fault Rolling element fault
Normalized energy
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
2 3 Decomposed components
4
Fig.10 Energy spectrum characteristics of VMD-based decomposition components of vibration signals in various operating states under load of 3HP.
From Table 1, each type of vibration signal contains four VMD components to form a four-dimensional vector. Let X V1 V2 V3 V4 be the input feature sample of the SVM model for motor vibration failure. Digital labels 1−4 are assigned to the normal condition, inner ring, outer ring, rolling element faults, respectively. In this experiment, 20 groups are selected for normal motor condition, inner ring, outer ring, and rolling element failures. A total of 80 samples are used for training, and the remaining 80 samples are used for testing. Fig.11 (a) displays the sample data distribution. The training and test samples are integrated into the LIBSVM model respectively for fault recognition in classification. To achieve satisfactory results, relevant parameters (c [penalty parameter] and g [function parameter]) must be adjusted. Cross-validation is used to avoid under-learning and avoid over-learning. Select parameter c = 2, g = 1. Eighty types of training samples are integrated into the SVM model for the four types of characteristic training. After training, the remaining 80 groups of test samples are input into the trained SVM model to verify classification. Fig.11 (b) illustrates the training classification effect. "○" represents the actual training sample fault type classification, and "*" represents the prediction sample fault type classification. The figure also reveals that only one fault identification error is found.
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ACCEPTED MANUSCRIPT class
V1
2 0
100 200 Sample V3
Feature Energy
1 0.8 0.6 0.4 0.2 0
0
100 200 Samples
V2
1 0.8
Feature Energy
3
1
Feature Energy
Feature Energy
Class Label
4
0.6 0.4 0.2 0
0
100 200 Samples V4
0
100 200 Samples
1 0.8 0.6 0.4 0.2 0
0
100 200 Samples
1 0.8 0.6 0.4 0.2 0
（a）Sample data
Actual Classifition and Predicted Classifition of Testing Samples 4
Actual Classifition Predicted Classifition
Class Label
3.5 3 2.5 2 1.5 1
0
10
20
30
40
50
60
70
80
Testing Samples
（b）Classification results Fig.11. Vibration fault classification of wind turbines based on SVM
The experimental results show that all 80 test samples of normal condition, inner ring fault, and outer ring fault are correctly identified, whereas rolling element fault correctly identifies 79 test samples. Only one group is classified incorrectly. Thus, the SVM can classify the validation samples with an accuracy of 98.75%. The reasons for the recognition errors may be that few training samples are available. In addition, a large difference between individual feature vector data and training samples is found. If the number of training samples is increased, then the accuracy of the recognition results can be further improved. In order to compare and verify the diagnostic recognition effect of the VMD energy entropy method, EMD and wavelet db4 are used to decompose the four operating states of the vibration signal, and the energy entropy of the modal state is calculated and decomposed. These three methods are used to train and classify the SVM, and Table 2 is the comparisons of the fault recognition rates of the three feature extraction methods.
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ACCEPTED MANUSCRIPT Table 2 Comparisons of recognition rate of different fault feature extraction methods. Order number 1 2 3
Methods VMD energy entropy EMD energy entropy db4 energy entropy
Number of training samples
Number of test samples
Number of misjudgement
Recognition rate /%
80
80
1
98.75
80
80
25
68.75
80
80
9
88.75
It can be seen from Table 2 that the correct recognition rate of the feature vector based on the VMD energy entropy method is higher than that of the feature vector obtained based on EMD and wavelet db4. This is because VMD has better robustness than other decomposition algorithms, and can effectively extract modal components with different center frequencies. It has good decomposition characteristics, and the energy entropy eigenvectors of the composition are significantly different. Therefore, it has a higher state recognition rate. The results show that the fault diagnosis method based on VMD energy entropy eigenvalue extraction and SVM can accurately classify various fault states of wind turbines. 5. Conclusions The actual vibration signal of wind turbines is weak and is seriously affected by environmental noise. Thus, fault identification is relatively difficult. A vibration fault diagnosis method based on the VMD and energy entropy is proposed in this study. VMD converts vibration signal decomposition into non-recursive and variational mode decomposition. This conversion is a function optimization problem under constrained conditions. It updates the modal functions and corresponding central frequencies with the objective of minimizing the bandwidth of modal estimation. The VMD also comprises a set of adaptive Wiener filter banks. The simulation results show that the VMD can effectively avoid the mode aliasing problem in the EMD when dealing with modal components with multiple frequencies. For the actual vibration signal, the EMD also has serious modal aliasing problems due to the influence of noise interference. The VMD can still accurately decompose and extract vibration fault information with respect to EMD. The energy entropy of each modal component decomposed by VMD is input into SVM as an eigenvalue. One part of the eigenvalue is used as the training sample of SVM, whereas the other part is used as the test sample of SVM. And it was compared with SVM classification and recognition by energy entropy eigenvalue extraction methods based on EMD and wavelet db4. The experimental results show that the classification accuracy of the proposed method is higher than those of EMD-based method and wavelet db4-based method.
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ACCEPTED MANUSCRIPT Although the SVM model has a limited training sample, the accuracy of the classified verification sample can reach 98.75%. The next step is to increase the vibration fault sample library. The accuracy of classification and recognition can be further improved by increasing the sample base of SVM model, which provides a reference diagnosis method for practical application. In addition, the next step is to study the application of the proposed method to vibration fault diagnosis of other components of wind turbines. 6. Acknowledgements This work is supported by National Natural Science Foundation of China (51477015), the Visiting Scholarship of State Key Laboratory of Power Transmission Equipment & System Security and New Technology (Chongqing University) (2007DA10512714406) , the Natural Science Foundation of Fujian Province of China (2018J01511), and the Program for New Century Excellent Talents in Fujian Province University (2018047). 7. References [1] Chu S, Majumdar A. Opportunities and challenges for a sustainable energy future. Nature 2012; 488: 294. [2] Solangi K H, Islam M R, Saidur R, Rahim N A , Fayaz H. A review on global solar energy policy. Renew Sust Energ Rev 2011; 15: 2149-2163. [3] Lewis J I, Wiser R H. Fostering a renewable energy technology industry: An international comparison of wind industry policy support mechanisms. Energ policy 2007; 35: 1844-1857. [4] Qiao W, Lu D. A survey on wind turbine condition monitoring and fault diagnosis—Part I: Components and subsystems. IEEE T Ind Electron 2015; 62: 6536-6545. [5] Nie M, Wang L. Review of condition monitoring and fault diagnosis technologies for wind turbine gearbox. Procedia Cirp 2013; 11: 287-290. [6] Amirat Y, Benbouzid M E H, Al-Ahmar E, Bensaker B, Turri S. A brief status on condition monitoring and fault diagnosis in wind energy conversion systems. Renew Sust Energ Rev 2009; 13: 2629-2636. [7]Yang W, Tavner P J, Crabtree C J, Feng Y, Qiu Y. Wind turbine condition monitoring: technical and commercial challenges. Wind Energy 2014; 17: 673-693. [8] Hang J, Zhang J, Cheng M, Wang W, Zhang M. An overview of condition monitoring and fault diagnostic for wind energy conversion system. T China Electrotech Soc 2013; 28:261-272. [9] Ogidi O O, Barendse P S, Khan M A. Fault diagnosis and condition monitoring of axial-flux permanent magnet wind generators. Electr Pow Syst Res 2016; 136: 1-7. [10]Liu W Y, Tang B P, Han J G, Lu X N, Hu N N, He Z Z. The structure healthy condition monitoring and fault diagnosis methods in wind turbines: A review. Renew Sust Energ Rev 2015; 44: 466-472.
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ACCEPTED MANUSCRIPT Highlights
A method of feature vector extraction based on variational mode decomposition (VMD) and energy entropy is proposed. The time-frequency energy spectrum obtained by VMD is compared with that by empirical mode decomposition (EMD). Different types of bearing vibration signals of wind turbine are analyzed. The fault diagnostic identification of the proposed method is compared with EMD-based and wavelet db4-based methods.