Advanced Eccentricity Fault Recognition in Permanent ... - CiteSeerX

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 4, APRIL 2014

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Advanced Eccentricity Fault Recognition in Permanent Magnet Synchronous Motors Using Stator Current Signature Analysis Bashir Mahdi Ebrahimi, Mehrsan Javan Roshtkhari, Jawad Faiz, Senior Member, IEEE, and Seyed Vahid Khatami

Abstract—In this paper, a novel index is introduced for static and dynamic eccentricity fault diagnosis in permanent magnet synchronous motors (PMSMs). The proposed index is a linear combination of the energy, shape factor, peak, head angle of the peak, area below the peak, gradient of the peak of the detail signals in wavelet decomposition, and coefficients of the autoregressive model, which are extracted from the stator current signature analysis. Principal component analysis is applied to the features as the linear transform for dimension reduction and elimination of linear dependence between the features. In order to demonstrate the capability of these indexes to estimate eccentricity type and degree, the fuzzy support vector machine is employed as a classifier. Classification of the results indicates that the nominated index can be utilized to detect eccentricity occurrence, recognize its type, and determine its degree precisely. Since extraction of efficient indexes closely depends on precise computation of necessary signals, the time-stepping finite element method is utilized to model the PMSM under eccentricity fault and calculate the stator currents as a proper signal for processing. Simulation results are verified by the experimental results. Index Terms—Eccentricity fault diagnosis, feature extraction, finite element (FE) method, permanent magnet (PM) motor, support vector machine (SVM), wavelet transform (WT).

I. I NTRODUCTION

R

ELIABILITY of permanent magnet (PM) synchronous motors (PMSMs) is an essential requirement for their applications in industries. Faults in PMSMs are classified into magnetic, electrical, and mechanical faults. Eccentricity fault as a mechanical fault is categorized to static eccentricity (SE), dynamic eccentricity (DE), and mixed eccentricity (ME). Since online access to the rotor is not easily possible, accurate eccentricity fault detection is difficult. Expert fault diagnosis systems enable precise on-time prediction of faults. Introducing a competent index is one of the most important stages of any reliable eccentricity fault recognition procedure. In [1], different analytical, artificial intelligence, and finite element (FE) modeling methods developed for fault analysis and detection in induction and PM motors have been reviewed. It has been concluded that motor current is the most

Manuscript received November 6, 2012; revised February 3, 2013; accepted April 16, 2013. Date of publication June 4, 2013; date of current version September 19, 2013. The authors are with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran 14174, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2263777

commonly monitored signal for fault diagnosis because it is easily monitored without sensors. Wavelet transform (WT) and other time–frequency analysis techniques are found to be very efficient in extracting fault-related features even under nonstationary operating conditions of the drive. It is noticeable that the necessary time to detect a fault depends heavily on the feature extraction and fault decision processes. Algorithms used in these two steps have to be chosen and coordinated well in order to develop a quick and reliable fault diagnosis procedure. Albeit, many indexes have been nominated for eccentricity detection in induction motors [2], and only two indexes have been so far proposed for eccentricity diagnosis in PMSMs [3], [4]. The magnitude of the sideband components at frequency (2fs /P ), where fs is the frequency supply and P is the number of pole pairs, has been used to determine eccentricity fault [3]. In [4], the amplitude of sideband components (ASBCs) at frequencies (1 ± (2k − 1)/P )fs , where k is an integer number, has been employed for SE diagnosis in PMSMs. Despite the fact that the PMSMs under eccentricity fault have been investigated in a few papers and the performance of the faulty motor has been analyzed, no criterion has been so far recommended for eccentricity fault diagnosis. In [5], the wavelet packet transform (WPT) and artificial neural network have been utilized for fault identification in interior PM (IPM) motors. The line currents of different faulty and healthy IPM motors have been preprocessed by the WPT. The secondlevel WPT coefficients of line currents have been employed as inputs of a three-layer feedforward neural network. In [6], short circuit fault has been recognized in the steady-state and transient modes of the PMSM by decomposing the stator currents. For this, empirical mode decomposition has been used to generate a set of intrinsic mode functions (IMFs). Then, the quadratic time–frequency distributions, such as smoothed pseudo-Wigner–Ville and Zhao–Atlas–Marks, have been applied to the more significant IMFs for fault detection. In [7], the short-time Fourier transform, wavelet analysis, and Wigner and Choi–Williams distributions of the field-oriented currents have been used to diagnose electrical faults in the PMSM. Different fault types have been classified using the linear discrimination classifier and k-means classification. Comparison between the different methods has been based on the number of correct classifications and Fisher’s discrimination ratio. In [8] and [9], the magnetic flux densities of a faulty surface-mounted PM (SPM) motor and an IPM motor have been compared and it is concluded that the variation rate of the faulty IPM motors under eccentricity is more than that of SPM motors. In addition, it

0278-0046 © 2013 IEEE

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was demonstrated that eccentricity effects on the SPM motors’ performance are less than those of IPM motors. In [10] and [11], distribution of the air-gap magnetic flux density of the PMSM under eccentricity with and without considering slot effects has been calculated. In [12]–[14], unbalanced magnetic pull (UMP), radial force, cogging torque, electromotive force, and developed torque of the healthy and faulty PMSMs under eccentricity have been evaluated. It has been revealed that UMP, radial force, torque ripples, and cogging torque increase due to the eccentricity. In [15], stator currents, cogging torque, and air-gap flux density have been analytically calculated in the brushless PM motor under eccentricity. It has been presented that eccentricity raises the variation rate and magnitude of the cogging torque. Since performance of the faulty machines is very important in industries, proposing applicable methods for fault detection in PMSMs can be utilized to design and analyze fault-tolerant PM machines [16]–[18]. In [19], impacts of eccentricity fault on the generated electromagnetic noise in the PMSM have been investigated. It has been shown that eccentricity increases noise and vibration. Previous findings demonstrate the lack of the competent index to predict occurrence, type, and degree of eccentricity fault in PMSMs. Therefore, the main contribution of this study is to propose competent indexes for eccentricity fault diagnosis, which are capable of estimating the severity of eccentricity, which has been always ignored in the previous studies. In this paper we note the following. 1) Since extraction of appropriate indexes closely depends on precise computation of the required signals, the 2-D time-stepping FE method (TSFEM) is employed to model the PMSM under SE and DE faults. Then, stator currents as a proper signal are calculated for processing. 2) Wavelet decomposition is used to analyze the stator currents. Hence, selection of particular frequency bonds that should be analyzed for feature extraction is essential. These bonds are determined analytically by monitoring air-gap magnetic field and stator currents. Furthermore, spectra of the stator current in the selected bond are demonstrated for the healthy and different faulty cases. 3) The linear combination of the energy, peak, head angle of the peak, the area below the peak, the gradient of the peak of the detail signals in wavelet decomposition, and coefficients of the autoregressive (AR) model, which are extracted as features, is utilized to introduce a novel index for fault diagnosis. 4) Principal component analysis (PCA) is used for dimension reduction and elimination of linear dependence between the features. Then, the linear combination of the aforementioned features is nominated as a proper index. 5) The fuzzy support vector machine (FSVM) is employed as a classifier to evaluate the ability of the proposed index for accurate eccentricity fault diagnosis. II. G ENERALIZED A LGORITHM FOR E CCENTRICITY FAULT R ECOGNITION IN PMSM S Accurate eccentricity fault detection depends on the implementation of the proposed algorithm stages in Fig. 1. According

Fig. 1.

Eccentricity fault diagnosis algorithm.

to Fig. 1, precise modeling of the faulty PMSM is the first stage of any fault diagnosis strategy, which has considerable impacts on the results accuracy. This is due to the close dependence of the next stages to the calculated outputs from this stage [20], [21]. This stage has been presented in Section III. Since the second stage defines the type of fault recognition technique as invasive or noninvasive, selection of appropriate signals for processing is very important. Here, in order to implement a noninvasive fault diagnosis approach, the stator current has been chosen for processing. Hence, the stator current of the healthy and different faulty cases has been calculated in Section III. The third stage is selecting a proper processor to analyze the calculated signal of the previous stage. Here, WT has been selected for processing the computed stator current. Albeit, the WT can be applied to stationary or nonstationary signals; fast Fourier transform (FFT) as a well-known processor in this area is only employed in the steady state. However, it should be noted that specifying particular bonds, instead investigation of all bonds, of the basic wavelet is necessary. Hence, these bonds are analytically determined in Section V, and therefore, in the

EBRAHIMI et al.: ECCENTRICITY FAULT RECOGNITION IN PMSMs USING CURRENT SIGNATURE ANALYSIS

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Fig. 2. Geometric configurations of the modeled motor using 3-D TSFEM. (Left) Stator. (Right) Rotor.

other sections, only determined bonds in Section V are analyzed for feature extraction. The final stage presents the efficiency of the fault detection strategy. According to Fig. 1, the introduced index must detect eccentricity occurrence precisely in order to prevent the motor from fault expansion. The minimum eccentricity degree that must be detected is 10% [22]. Eccentricity below 10%, which has been named inherent eccentricity, happens during manufacturing and installing procedures and can be ignored. Moreover, the introduced index should be able to recognize type and degree of the occurred eccentricity. Determining the eccentricity degree is so important to make a decision to permit motor working or motor fixing. Therefore, in order to avoid touching the stator and rotor, estimating 50% or even 60% eccentricity is requisite. These characteristics are so important for the PMSM fault-tolerant working or fixing. This stage, which includes three states, is presented in Sections VI–VIII. III. M ODELING OF PMSM W ITH E CCENTRICITY U SING TSFEM A 3-D scheme of the simulated PMSM has been displayed in Fig. 2. Since, in this approach, spatial distribution of the stator windings, nonlinear characteristics of the ferromagnetic and PM materials, and geometrical and physical characteristics of the stator slots and PMs are taken into account, it has been used to model and analyze the healthy and faulty PMSMs under different types of eccentricity. Transient analysis of rotating machines (RMs) has been employed for modeling and analyzing the PMSM with mechanical coupling. The RM Program is a transient eddy current solver, which is extended to include the effects of rigid body (rotating) motion [23]. The solver also provides for the use of external circuits and coupling to mechanical equations. In this simulation, electrical equations of the external circuits, which exhibit supply, and electrical circuits are combined with the magnetic field equation in FEM and motion equations due to the mechanical coupling. Modeling of a PMSM has the following three basic parts. A. Modeling of Motor Elements Accuracy of the fault diagnosis approaches depends on considering materials physical characteristics. The reason is the noticeable impacts of the materials characteristics on the fault diagnosis criteria [24]. In this modeling, nonlinear character-

Fig. 3. Time variation of PMSM stator current simulated by 2-D TSFEM. (Top) Healthy. (Bottom) 50% DE.

istics of the PMs, stator and rotor cores, spatial harmonics due to the stator slots, and nonuniformity of the air gap due to eccentricity fault are taken into account. The stator and rotor consist of laminated M-19 sheets; 36 slots in the stator filled with copper. The B-H curve, magnetic orientation, and permeability (μr = 1.145) of the magnets (Nd–Fe–B) have been taken into account in the simulation procedure. B. Modeling of Motor Supply Winding configuration has considerable impact on the fault diagnosis precision [25]. The reason is related to the windings distribution impacts on the harmonics of line currents and voltages, which affect the aforementioned signals spectra. In this modeling, two-layer stator windings and their spatial distribution are reckoned. A three-phase balanced voltage source is used in the simulations. C. Modeling of Eccentricity Fault In this modeling, nonuniformity of the air gap due to the SE and DE faults in different degrees is modeled. In the SE case, the rotor is displaced from the center of the stator, and the rotor is rotated around its own center. Albeit, in the DE fault, the rotor is moved from the center of the stator; the rotor is rotated around the center of the stator. In the ME fault, the stator and rotor are displaced from the rotor rotation center. Fig. 3 depicts the stator current profile of the healthy and faulty PMSMs with 50% DE. It is seen that eccentricity distorts stator currents. This distortion indicates the need to increase particular sideband components in the stator current spectra, which can be utilized for feature extraction and pattern recognition.

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cylinder axis (DCA) of the rings and the air-gap length (AGL) of the motor is computed by SED = DCA of inner eccentric ring/AGL DED = DCA of eccentric ring around shaft/AGL. The experimental setup has been used to sample line currents, line voltages, and speed signal (by an angular speed sensor). The motor windings are Δ-connected, and the third line current (Ic ) and line voltage (Vca ) can be easily estimated from the sampled currents and voltages, respectively. The test rig and block diagram of the PMSM drive have been shown in Fig. 4, which indicates the fault detection system. This system is independent of the control system and only has the currents as inputs. The drive has an outer speed control loop, two inner current-controlled loops, and a PWM inverter feeding the PMSM. An encoder is used to provide the rotor position for the transformation to the q-d-o-rotor rotating reference frame. For diagnosis purposes, it is assumed that only the stator voltage and current are measured. Stator currents and torque were sampled while the motor was in steady state and FFTs were performed on the signals. A signal conditioning board, including voltage, current, and torque sensors, is designed and connected to the data acquisition (DAQ) system through voltage amplifiers to scale the magnitude and low-pass filters in order to obtain raw current, voltage, and torque data. A 12-bit A/D at a 100-kHz PCI-1710HG FFT spectrum analyzer is used to monitor the real time current and voltage spectra. The main parts of the experimental setup with numbers shown in the figure are as follows.

Fig. 4. Experimental setup. (a) Left to right: outer ring, new bearing, and inner ring. (b) Left to right: new bearing with rings assembled on it and main bearing. (c) Used devices. (d) Control strategy diagram.

IV. E XPERIMENTAL S ETUP Static, dynamic, and mixed eccentricities in PMSMs are created by a ball bearing, where its inner diameter, enclosing the shaft, is slightly larger than that of the shaft and its outer diameter, placed in a specific room on the motor bung, is smaller than the mentioned inner diameter of the room. If this ball bearing is changed by the main ball bearing of the motor, there will be a gap between the shaft and the ball-bearing inner circumference and also between the outer circumference and the ball-bearing room on the bung. These two gaps can be filled by two rings with suitable dimensions. If these rings are eccentric, the ME is generated in the motor. The smaller ring placed around the shaft generates DE, if the ring fixed on the shaft rotates with it [see Fig. 4(a)]. The other ring that is placed on the bung around the ball bearing creates SE. In this case, the rotor rotation axis coincides with its symmetrical axis but is displaced from the stator symmetrical axes. In this case, distribution of the air gap around the rotor loses its uniformity, but it is time invariant [see Fig. 4(a)]. If these two eccentricities coexist, ME is created. In order to determine the SE degree (SED) and DE degree (DED), the ratio between the distance of

1) A three-phase 3.5-kW PMSM with specifications given in the Appendix. 2) A dc generator coupled to the PMSM to provide load variation. 3) A tachogenerator coupled to the shaft of the generator as an angular speed sensor to measure and record the time variation of the speed. 4) Mechanical coupling between PMSM and dc generator. 5) A variable resistor bank as a variable load of the generator; the load of the generator, and consequently PMSM, can be adjusted by varying this resistance and/or regulating the excitation current of the generator by relevant variable resistor. 6) A PMSM drive type ACS800 from ABB with rating values corresponding to the rated values of the motor. 7) A three-phase change-over switch for exchanging the motor connection from the main to the drive output and vice versa. 8) A PC equipped with a DAQ card of type PCI-1710HG from Advantech on its PCI bus for sampling the electrical data at certain adjustable frequency and storing them in the memory. 9) Signal conditioning and terminal box: Since card PCI1710HG accepts only voltage-type signals with a maximum amplitude of ±10 V at its analog inputs, the type and range of the proposed signals for sampling must be prepared before connecting to this card.


It is noticeable that fault detection using only stator currents is rather difficult with tightly controlled currents and slow dynamic speed control. However, monitoring and processing other motor signals have different difficulties and limitations. For instance, processing air-gap flux density can be used for fault diagnosis. However, the search coil, which is employed to monitor flux density, is under influence of electromagnetic interference and electromagnetic compatibility. Analyzing torque and speed is another applicable approach. However, technical and economic limitations are reasons to avoid utilizing these signals for processing and fault identifying. V. A NALYSIS OF E CCENTRICITY E FFECTS ON S TATOR C URRENTS FOR F EATURE E XTRACTION One of the well-known methods for practically monitoring and processing air-gap magnetic field is utilizing search coils. A search coil is usually put in the stator slots. The induced air-gap flux in the search coil, which includes effects of eccentricity, has complete information of occurred fault that can be processed for eccentricity diagnosis. Hence, the voltage due to this flux is processed by different tools such as FFT, WT, WPT, or other applicable processors. Then, eccentricity fault is detected using extracted features via applying the aforementioned tools to the induced voltage. In order to monitor air-gap magnetic field analytically, the AGL or permeance should be calculated. In the healthy PMSM, the AGL is constant (g0 ), in which SE and DE make the air gap nonuniform. Thus, motor air gap under DE fault can be expressed as follows [26]: g(ϕ, t) = g0 (1 − δde cos(ωr t − ϕ))

(1)

where δde is the DE degree, ωr is the rotor angular velocity, ϕ is the space variable in the stator frame reference, and δse is the SE degree. The inverse of the AGL, which is named Λ, is calculated by 1 1 = g(φ, t) g0 (1 − δde cos(ωr t − φ)) (δde cos(ωr t − φ))2 1 1 + δde cos(ωr t − φ) − = g0 2 (δde cos(ωr t − φ))3 + ... . (2) + 6

Λ(φ, t) =

Since δde and cos(ωr t − ϕ) are between −1 and 1, the third, fourth, and later terms can be ignored, and (2) is simplified as follows: Λ(ϕ, t) =

1 (1 + δde cos(ωr t − ϕ)) . g0

(3)

Due to ωr = ωs /p, where p is the number of pole pairs, ωs (2πfs ) is the angular supply frequency, and fs is the supply frequency, (3) is rewritten as follows: ωs 1 t−ϕ . (4) Λ(ϕ, t) = 1 + δde cos g0 p

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Assuming sinusoidal variation of the magnetomotive force wave, the current density (js ) on the stator inner surface can be written as js (ϕ, t) = Js sin(ωs t − pϕ)

(5)

where Js is the peak of the current density. Based on Ampere’s circuital law, stator air-gap flux density is defined by (6) Bs (ϕ, t) = Λ(ϕ, t) μ0 js (ϕ, t)dϕ. Substituting (3) and (5) into (6) yields ωs 1 Bs (ϕ, t) = t−ϕ μ0 Js sin(ωs t−pϕ)dϕ. 1+δde cos g0 p (7) Equation (7) is simplified as follows: Bs (ϕ, t) =

μ0 J s μ0 Js δde cos(ωs t − pϕ) + g0p 2g0 p 1 × cos 1+ ωs t − (p + 1)ϕ p 1 + cos 1− . ωs t − (p − 1) ϕ p

(8)

Similarly, rotor air-gap flux density is determined as follows: μ0 Jr δde 1 Br (ϕ, t) = cos 1+ ωs t − pϕ 2g0 P p 1 + cos 1− . (9) ωs t − pϕ p As these harmonic fluxes move relative to the stator, they induce corresponding current harmonics in the stationary stator windings [27]. Thus, stator current spectra can be used for eccentricity prediction. The vital stage in fault diagnosis procedure is determining frequency bonds, which these harmonics induce [28]. Based on (8) and (9), eccentricity generates sideband components at frequencies (1 ± 1/p)fs . As mentioned for (2), the third term and later terms due to their small values were ignored. By considering these terms, a frequency pattern is extracted for fault detection in PMSMs as follows: k (10) feccentricity = 1 ± fs . p Equation (10) shows sideband components due to eccentricity generated around the fundamental harmonic. These sideband frequencies are 12.5, 25, and 37.5 Hz, . . ., and (1 ± k/p)fs . The current spectra of the healthy and faulty PMSMs under different SE and DE degrees have been depicted in Figs. 5 and 6, respectively. It is noticeable that harmonic components are not competent indexes for fault recognition. Because, in addition to fault impacts, different factors such as design characteristics, the winding configuration, load level variation, and even the manufacturing procedure have noticeable effects on the harmonic components. Therefore, in order to remove harmonic component frequencies from the aforementioned

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Fig. 5. Stator current spectra of the PMSM under different SE levels.

Fig. 6. Stator current spectra of the faulty PMSM under different DE levels. TABLE I ASBC S D UE TO DE IN D ECIBELS

TABLE II ASBC S D UE TO SE IN D ECIBELS

frequency pattern, the following frequency pattern is proposed for eccentricity fault recognition: 2k − 1 feccentricity = 1 ± (11) fs . p The correlation between different SE and DE degrees with the ASBCs at aforementioned frequencies has been presented in Tables I and II. According to Figs. 5 and 6 and also Tables I and II, fault occurrence and its extension increase ASBCs at these frequencies, which can be utilized for eccentricity detection.

Fig. 7. Amplitude variation of sideband component versus load variation and different percentages of (top) DE and (bottom) SE.

One of the most important notes, which should be discussed here, is impacts of load level variation on this index. It is clearly seen that number of poles and supply frequency only change the frequency pattern. Moreover, based on the aforementioned proved formula, ASBCs at these frequencies are not under influence of load variation. Therefore, it is expected that fault percentage is the only factor which changes this index that is a very important characteristic of this index, because most of the proposed indexes are under influence of load variation. Indeed, except fault, there are some factors which affect indexes which make difficult accurate fault detection. Fig. 7 presents the variation of the ASBC at frequency (1 − 1/P )fs versus different SE and DE percentages and load level variation. As shown in these figures and tables, the eccentricity fault and its extension increase ASBCs at frequencies 12.5, 25, and 37.5 Hz, . . ., and (1 ± k/p)fs considerably, which can be utilized as a competent criterion for eccentricity fault recognition. Therefore, it is concluded from aforementioned results that analysis of frequency bonds around the fundamental harmonic is a proper attitude for feature extraction via wavelet decomposition. VI. F EATURE E XTRACTION U SING WAVELET Fault diagnosis based on the frequency spectrum can be performed as long as the signals are stationary. The stator current signal is stationary when the PMSMs operate in the steadystate mode without speed and load variation [29]. In general, motor currents and voltages are nonstationary signals whose properties vary with the time-variant operating conditions of


Fig. 8.

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Feature extraction procedure.

the motors, such as load variation, power supply fluctuation, noise, motor geometries, and fault conditions. Such variations generate features similar to those of certain faults, resulting in the improper classification of the machine conditions [30]. Using wavelet for feature extraction is based on two reasons: 1) Many of PMSMs are supplied by power electronic drives which generate many harmonics, and it is not possible to easily extract frequency spectrum features. 2) For offline detection and the diagnostic system of large PMSMs, it is not possible to run the motor at full-load conditions, and the features obtained from low-load conditions of the frequency spectrum of stator current normally cannot result in accurate detection. In order to have a reliable fault diagnosis system, it is necessary to extract the most relevant features to the fault. The data set consists of 100 current signals for healthy, SE, and DE conditions with different degrees. The length of signals is 5 s with a sampling frequency of 800 Hz. Fig. 8 demonstrates the feature extraction procedure. The features are extracted from the detail signals at different levels of discrete wavelet decomposition of the stator current using Daubechies 6 (db6) as a base function. Selection of the mother wavelet function has been based on [31], in which the basis that is best adapted to the global signal properties, among all bases of the wavelet, will be selected. In order to select the decomposed signals for feature extraction, the signals that have more relation to the eccentricity have been selected. Based on the used sampling frequency (800 Hz), mentioned detail signals contain the frequency range of 200–400 Hz, 100–200 Hz, 50–100 Hz, and 25–50 Hz, respectively. According to Section III (nominated frequency bonds) and used sampling frequency, the detail signals at levels 1–4 of discrete wavelet decomposition have more relation to the eccentricity. Moreover, this relation is calculated between the entropy of the decomposed signals at various levels and the eccentricity degrees using mutual information as a proper criterion [32]. These values have been shown in Fig. 9. Based on the calculated mutual information, detail signals of the decomposed current at levels 1–4 have high relation to the eccentricity degree. As the relation of the approximation signal at the fourth level of decomposition (a4) has the least information about eccentricity, further decomposition is not necessary. Therefore, the aforementioned levels are employed for feature extraction. Heuristically, some features in a signal are considered as its characteristic for classification, which are energy, shape factor, peak, head angle of the peak, the area below the peak, and the gradient of the peak. Figs. 10 and 11 depict the peak of some detail signals at levels 1 and 2 of wavelet decomposition for

Fig. 9. Mutual information between entropy of the decomposed signals and eccentricity degree.

various SE and DE degrees. It is seen that these peak values have a linear relationship to both SE and DE degrees, i.e., any increase in the eccentricity degree leads to the rise in the magnitude of those peaks. In addition to the mentioned features, some additional features can be extracted from a signal for classification. These features are based on modeling the detail signals by linear dynamic models [33]. First, the AR model is selected as the appropriate model for these signals, and then, the coefficients of the model are used as the features, in addition to the previously mentioned features. To decrease dependence of these features on the load, the features must be normalized with respect to the load. As the RMS value of the stator current is related to load, the stator current is first divided to its RMS value, and then, the features are extracted from the normalized decomposed current. VII. R EDUCING F EATURE D IMENSIONS U SING PCA It is clear that the extracted features are appropriate features, some may be unrelated to the fault, some may be related to each other, and there might be redundancy. To eliminate the linear dependence among features, the dimension reduction is carried out by PCA [33]. It is noticeable that we did not achieve good classification result for healthy/faulty machines by employing the linear discriminant analysis (LDA) instead of PCA for feature selection. This is due to the fact that the dimension of the projected feature space is equal to 1 in the LDA. As the healthy and faulty machines are not linearly separable in the feature space, it is required to keep more features to improve classification results. PCA is only able to eliminate the linear relation between features and is a powerful tool for feature extraction or feature selection. Consider a set of N -dimensional sample data vectors: x1 , x2 , . . . , x5 . Then, x is an N × M data matrix whose columns are made of the sample vectors xi , i = 1, 2, . . . , M , and ST = xxT is defined as the total scatter matrix of the sample vectors. The aim of PCA is to find the transform matrix of a subspace whose basis vectors correspond to the maximum-scatter directions in the original N -dimensional vector space. Therefore, the WPCA transform matrix is chosen to maximize the determinant of the total scatter matrix of the projected samples and expressed as follows: WPCA = arg max |W St W T |. W

(12)

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Fig. 11. Peak of detail signal at second level of wavelet decomposition of stator current. (a) Simulated DE degrees. (b) Experimented DE degrees. (c) Simulated SE degrees. (d) Experimented SE degrees.

Fig. 10. Peak of detail signal at first level of wavelet decomposition of stator current. (a) Simulated DE degrees. (b) Experimented DE degrees. (c) Simulated SE degrees. (d) Experimented SE degrees.

Solution to this optimization equation is the transform matrix W constructed, so that its row vectors are the eigenvectors of the scatter matrix St arranged in the order of decreasing magnitude of the corresponding eigenvalues λ, i.e., St w j = λ j w j ,

j = 1, 2, . . . , Q.

(13)

Here, λj is the nonzero eigenvalues associated with the eigenvectors wj , and Q denotes the rank of St and cannot exceed the

lesser of N and M . Because of the maximum-scatter projection, PCA provides an optimal transformation for reconstructing the original data vector from a lower dimensional subspace in terms of minimal mean-square error (MSE) [34]. Let WPCA be the R × N matrix (R < N ) formed by discarding the lower , then the transformed R × 1 vector y is N − R rows of WPCA given by y = WPCA x. The x vector can still be reconstructed T y , with approximation error given by as x = WPCA MSE =

N

λk .

(14)

k=R+1

By applying PCA to the feature vector, a new feature vector for each class is formed with eight elements. In Fig. 12, the scatter plots of the first three features obtained by PCA have been represented for various SE and DE. It can be seen that these features can separate the SE and DE fault profiles. As it can be seen in Fig. 12, the first three features can discriminate various


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TABLE III I DENTIFYING E CCENTRICITY T YPE BY P ROPOSED I NDEXES

Fig. 12. Scatter plot of the three features (S/N = 40 dB).

noise, which is the most important parameter in the fault diagnosis systems. In order to evaluate the robustness of the proposed indexes, white Gaussian noise is added to the current, and the variance of the noise is gradually increased until the recognition rate becomes unacceptable. A. Identifying Eccentricity Type

Fig. 13. Correlation coefficients of the index and eccentricity degree (S/N = 40 dB).

DE degrees completely, but they may not separate various SE degrees as well as DE degrees. Moreover, it can be seen that these features can completely separate SE and DE faults, because they form separate clusters in the 3-D feature spaces. In Fig. 13, the correlation coefficients between the proposed indexes and both SE and DE degrees have been shown. It is clear that the proposed indexes have a higher relation to the DE degrees than that of the SE degrees. VIII. E STIMATION OF E CCENTRICITY T YPE AND D EGREE It seems that the proposed indexes are capable of discriminating between different eccentricities. To measure this ability, a classification system is employed to detect the eccentricity degree and define its type. To estimate the eccentricity degree, first its type must be identified, and then, the degree should be calculated. Hence, a hierarchical scheme is employed [35]. For better analysis of the results, a k-nearest neighbor (k-NN) classifier is employed for identifying the eccentricity type, and FSVMs are used to estimate eccentricity severity. Moreover, a good index needs to have high recognition rate along with robust performance to environmental changes, e.g., measured

A common classification scheme (k-NN) is employed to estimate the eccentricity type. The k-NNs are nonparametric classifiers based on nonparametric estimation of the class densities [35]. The aim is to find the nearest neighbors of an undefined test pattern within a hypersphere of predefined radius in order to determine its true class. In other words, k-NN classifiers find k-nearest samples in some reference set, by taking a majority vote among the classes of these k samples. Provided that the number of training samples is large enough, this simple rule exhibits good performance. This classifier also is well known for two class problems in electrical machine fault detection (healthy or faulty machine) [36] with impressive results. The reason for choosing k-NN for classification of the eccentricity type is that the clusters of the data for SE and DE faults form mass-type cluster in the feature space, and hence, using the k-NN classifier yields accurate results with a simple structure of the classifier. Classification has been carried out by selecting k = 5 at various noise levels, and the results have been presented in Table III. It is seen that the proposed features are able to completely classify the two eccentricity types. Moreover, the indexes show a robust performance in the presence of measurement noise and even at S/N = 20 dB, the correct classification rate is completely satisfactory. The indexes can identify the type of eccentricity without any noticeable degradation in their performance in the presence of noise. B. Estimating Eccentricity Degree One of the objectives in a fault diagnosis system is to estimate the fault severity in addition to detect its occurrence. Hence, the ability of these indexes to estimate the eccentricity degree must be evaluated by a classification system. The FSVM is utilized to estimate eccentricity severity. Here, it is assumed that the eccentricity type is correctly detected, and we want to evaluate their ability for estimation of eccentricity degree in the case of pure SE or DE. Support vector machine (SVM) is a relatively new and powerful technique for solving supervised classification problems and is very useful due to its generalization ability [34], [37], but it has some limitations. FSVMs are modified SVMs with less sensitivity to the outline data and more generalization ability (due to larger margins) [38].

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TABLE IV C LASSIFICATION R ESULT OF U SING I NDEXES (U PPER ROW: E STIMATION OF DE D EGREES , L OWER ROW: E STIMATION OF SE D EGREES )

The basic idea is the same as the SVM, i.e., maximizing the margin between the training data and the decision boundary, which can be formed by hyperplanes. In other words, input points are mapped to a high-dimensional feature space, where a separating hyperplane can be found, and then, the algorithm is chosen in such a way as to maximize the distance between the decision boundary and the closest subset patterns, which are called support vectors. The main difference between the FSVM and the SVM is that the different training samples have different contributions to their own class; the training set is fuzzified, and each data sample has been assigned a membership value according to its relative importance in the classification. Given training vectors xi , i = 1, . . . , m in two classes, and a vector of labels di such that di ∈ {1, −1}, the FSVM solves a quadratic optimization problem 1 ξi min (W T W ) + C w,ξ 2 i=1 N

s.t. di W T Φ(xi ) + b ≥ 1 − ξi ξi ≥ 0,

i = 1, . . . , N.

(15)

The only difference with SVM is using a weighting parameter μi in (13). Like original SVM, training data are mapped to a higher dimensional space by the function Φ, and C is a penalty parameter on the training error. To find μi , a similar approach reported in [34] is used, in which μi is estimated as the probability distribution of the data. For any testing instance x, the decision function is (16) f (x) = sign W T Φ(x) + b . Practically, we need only K(x, xi ) = ΦT (x)Φ(xi ), the kernel function, to train the SVM. The radial basis function kernel is used in our experiments as follows: x − xi 2 i . (17) K(x, x ) = exp − 2σ 2 As seen, the basic scheme of FSVM can be only used for the two class problems. For multiclass problem, we used the one-versus-all methods. It assembles classifiers that distinguish one from all the other classes [37]. Since the number of product samples was limited, it was important to obtain the best generalization performance and reduce the overfitting problem.

Practical implementation is to partition these data samples into two sets of training and testing data. In this paper, repeated random subsampling was used to determine the optimal parameters. The whole training samples were randomly divided into two subsets for testing and training. The data set consists of 100 current vectors for each pure SE and DE values, 240 feature vectors are used as the training set (about 60% of total data) randomly and the other employed to test the classification performance. The procedure of randomly dividing the data set into the two sets is performed 20 times, and results of classification are presented in Table IV, including the average success on the training and testing sets. The standard deviation and the best and worst performance of the classification results are also reported in Table IV. Table IV shows that the proposed features can accurately estimate the eccentricity degree. The correct recognition rate in both test and train sets is very high. Moreover, it is clear that the proposed features can estimate the DE degree more accurately than the SE. Meanwhile, higher noise has less effect on the estimation of DE degree, and the proposed features show more robust performance in recognition of the DE degree than that of the SE. When the noise level is high (e.g., S/N = 20 dB and less), the degradation in performance of estimating the eccentricity degree becomes unacceptable; as a result, the indexes can be used only when the S/N is not less than 30 dB. IX. C ONCLUSION New indexes based on the decomposition WT of the stator current have been proposed. The ability of the proposed indexes for eccentricity detection has been analyzed in terms of their relations to SE and DE and capability for identifying the eccentricity type and estimating its degree. The type of eccentricity was determined using a k-NN classifier, an FSVM was employed to estimate the eccentricity degree, and the result showed that they can identify the type and degree of eccentricity correctly. Moreover, their robustness was analyzed with respect to additive measurement noise. The results show that the proposed indexes can be used for eccentricity severity estimation in the presence of high noise (S/N = 20 dB). This makes them capable of diagnosing SE and DE faults in the PMSMs. Due to the high capability of eccentricity classification, the proposed indexes can be used to develop the high-performance fault detection systems.


TABLE V S PECIFICATIONS OF P ROPOSED T HREE -P HASE PMSM

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Bashir Mahdi Ebrahimi received the B.Sc. and M.Sc. (Hons.) degrees from the University of Tabriz, Tabriz, Iran, in 2004 and 2006, respectively, and the Ph.D. degree in electrical engineering from the University of Tehran, Tehran, Iran. He is with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran. He has published more than 50 papers in international journals and 55 papers in international conference proceedings. His research interests include design, modeling, control, and fault diagnosis of electrical machines. Dr. Ebrahimi has received a number of awards, including Top Student Researcher in 2008, Top Student in 2009 and 2010, and Top Thesis in 2011, all from the University of Tehran. Moreover, he has received the Top Thesis Award in Iran from the IEEE-Iran section in 2012.

Mehrsan Javan Roshtkhari received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Tehran, Tehran, Iran, in 2006 and 2009, respectively. He is currently working toward the Ph.D. degree in electrical engineering at McGill University, Montreal, QC, Canada. He is with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran. His research interests include computer vision, signal processing, and pattern recognition.

Jawad Faiz (M’90–SM’97) received the Master’s (Hons.) degree in electrical engineering from the University of Tabriz, Tabriz, Iran, in 1975 and the Ph.D. degree in electrical engineering from the University of Newcastle upon Tyne, Tyne, U.K., in 1988. Since February 1999, he has been a Professor with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran, where he is currently the Director of the Center of Excellence on Applied Electromagnetic Systems. He is the author of 170 papers in international journals and 172 papers in international conference proceedings. His teaching and research interests include switched reluctance and VR motor design, design and modeling of electrical machines and drives, transformer modeling and design, and fault diagnosis in electrical machinery. Dr. Faiz has received a number of awards, including the first basic research award from the Kharazmi International Festival in 2007; the silver Einstein medal for academic research from the UNESCO; the first rank medal in research from the University of Tehran in 2006; and the Elite Professor Award from the Iran Ministry of Science, Research, and Technology in 2004. He is a member of the Iran Academy of Science. He has been a member of the Iran Academy of Sciences since 1999.

Seyed Vahid Khatami received the B.Sc. degree in electrical engineering from the University of Zanjan, Zanjan, Iran, in 2008 and the M.S. degree from the Department of Electrical and Computer Engineering, University of Tehran, Tehran, Iran, in 2011. He is with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran. His current research interests include electrical machines, fault diagnosis distribution generation, renewable energy, and wind turbines.