312 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 16, NO. 4, DECEMBER 2001 Pattern Recognition—A Technique for Induction Machines Rotor Broken Bar Detection Masoud Haji, Student Member, IEEE and Hamid A. Toliyat, Senior Member, IEEE Abstract—A pattern recognition technique based on Bayes minimum error classifier is developed to detect broken rotor bar faults in induction motors at the steady state. The proposed algorithm uses only stator currents as input without the need for any other variables. First rotor speed is estimated from the stator currents, then appropriate features are extracted. The produced feature vector is normalized and fed to the trained classifier to see if motor is healthy or has broken bar faults. Only number of poles and rotor slots are needed as pre-knowledge information. Theoretical approach together with experimental results derived from a 3 hp ac induction motor show the strength of the proposed method. In order to cover many different motor load conditions data are obtained from 10% to 130% of the rated load for both a healthy induction motor and an induction motor with a rotor having 4 broken bars. Index Terms—Broken bars fault, fault diagnosis, induction motor, speed estimation, statistical classifier. I. INTRODUCTION M OST electric motor failures interrupt a process, reduce production, and may damage other related machinery. In some factories, a very expensive scheduled maintenance is performed in order to prevent sudden motor failures. Therefore, there is a considerable demand to reduce maintenance costs and prevent unscheduled downtimes for electrical drive systems, especially ac induction motors. In the past two decades, there has been a substantial amount of research to provide new condition monitoring techniques for ac induction motors based on analyzing vibration signals –, or signals other than currents . Vibration transducer is expensive and care should be taken into account for mechanical installation and transmitting the signal . Similar problems exist while working with other sensors like speed and temperature. Interestingly, signatures of all signals are available on electrical terminals (currents) of electric machines including the vibration signals . Current signals can easily be monitored for condition monitoring and control purposes. The objective is how to extract different features from the current signal and discriminate among various machine conditions. Noise together with nonlinear behavior of machine with or without faults make this task very difficult. Most of the works on motor current signature analysis (MCSA) use second order based techniques like FFT analysis – or Eigenanalysis-based frequency estimation (high Manuscript received April 19, 2001. Paper was approved for presenting at the IEMDC01, Cambridge, MA USA, June 2001. The authors are with the Electric Machines and Power Electronic Laboratory, Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128 (e-mail: [email protected]). Publisher Item Identifier S 0885-8969(01)10054-9. resolution spectral analysis)  and a few time-frequency analysis such as wavelet , . Yazici and Kliman  have presented a second order time frequency statistical method for detection of broken bars and bearing faults for induction motors. They have estimated torque, then knowing machine nameplate information speed has been linearly estimated, finally a spectral feature vector is produced which can be used for classification. Recent developments in hardware and software make it possible to produce a system for condition monitoring of induction machines if we utilize signal processing and classification techniques for fault diagnosis. In the following section, a brief transformation approach review of faults is presented. The comes in Section III. Section IV deals with estimating speed from current signal as a necessary element. In Section V, a brief review for statistical pattern recognition and Bayes classifiers are discussed. Feature extraction and the classifier are covered in Section VI. A block diagram of the proposed method plus the details of the developed algorithm and experimental results are presented in Sections VII and VIII. II. FAULTS IN AC INDUCTION MACHINES—A BRIEF REVIEW  Faults in electric machines produce one or more of the following symptoms: a) Unbalanced air-gap voltages and line currents, b) Increased torque pulsation, c) Decreased average torque, d) Increased losses and reduction in efficiency, e) Excessive heating. The most prevalent faults in AC induction machines are briefly described in the following four categories: 1) Bearing Faults: Though almost 40–50% of all motor failures are bearing related, very little has been reported in the literature regarding bearing related fault detection using motor current techniques. Bearing faults might manifest themselves as rotor asymmetry faults from the category of eccentricity related faults. 2) Stator or Armature Faults: These faults start as undetected turn-to-turn faults, which grow and culminate into major ones. Almost 30–40% of all reported faults of induction motor failures falls in this category. Toliyat and Lipo have shown through both modeling and experimentation that these faults result in asymmetry in the machine impedance causing the machine to draw unbalance phase currents . 3) Broken Rotor Bar and End Ring Faults: Rotor failures now account for 5–10% of total induction motor failures. 0885–8969/01$10.00 © 2001 IEEE HAJI AND TOLIYAT: PATTERN RECOGNITION—A TECHNIQUE FOR INDUCTION MACHINES ROTOR BROKEN BAR DETECTION Broken rotor bars give rise to a sequence of side-bands given by: 313 Equation (3) under ideal condition has physical meaning and in steady state reduces to: (1) where is the supply frequency and is the slip. Frequency domain analysis (second order) and parameter estimation techniques have been widely used to detect this type of faults. As suggested in , presence of interbar currents in uninsulated rotor cages, where the contact between the rotor core and the bars are good, might make broken bar detection difficult. In practice, the current side bands around fundamental may exist even when the machine is healthy . Also rotor asymmetry, resulting from rotor ellipticity, misalignment of the shaft with the cage, magnetic anisotropy, etc., shows up at the same frequency components as the broken bars . Therefore, other features of this fault need to be investigated. 4) Eccentricity Related Faults: is the condition of unequal air-gap between the stator and rotor. It is called static air-gap eccentricity when the position of the minimal radial air-gap length is fixed in the space. This maybe caused by the ovality of the stator core or by the incorrect positioning of the rotor or stator at the commissioning stage. In case of dynamic eccentricity, the center of rotor is not at the center of rotation, so the position of minimum air-gap rotates with the rotor. This maybe caused by a bent rotor shaft, bearing wear or misalignment, mechanical resonance at critical speed, etc. In practice, an air-gap eccentricity of up to 10% is permissible. Both static and dynamic eccentricities tend to exist in practice. Using Motor Current Signature Analysis (MCSA) the equation describing the frequency components of interest is: (2) for static eccentricity, and (1, 2, 3, …) for dywhere namic eccentricity. is the number of rotor slots and is the 1, 3, 5, … . number of poles, and Other equations are also presented in the literature as low frequency components for mixed eccentricity . As it is obvious, sometimes different faults produce nearly the same frequency components or behave like healthy machine, which make the diagnosis impossible. Specially, if it is only based on second order frequency analysis. This is the reason why new techniques must also be considered to reach a unique policy for distinguishing among faults. (4) Under abnormal conditions, such as the broken rotor bars, (4) is no longer valid, however, without loss of information we can still work with and in (3). Benbouzid and Nejjari have shown that – pattern differs from each other in healthy machine and under open phase or stator unbalance faults . Cardoso et al.have shown different – pattern for faults in rectifier diodes and power switches for ac drive systems and for broken rotor bars in ac induction motors . transformation by itself is not enough for Use of Park’s fault diagnosis for a number of reasons; first, it is not obvious if patterns are unique for different faults, secondly classification is very difficult if noise and practical problems are considered. transformation keeps all However, as mentioned above the information in the currents while reduces the number of variables from three to two. In this paper, and variables are used in the proposed approach. IV. SPEED ESTIMATION—RSH APPROACH Effective sensorless speed estimation is desirable for both on-line condition monitoring of induction motor and sensorless adjustable speed ac drive application . The mechanical speed information of an induction machine is embedded in the stator currents. The slots produce a continuous variation of the air-gap permeance in squirrel cage induction motors. During operation, the rotor slot MMF harmonics will interact with the fundamental component of the air-gap flux because of the rotor currents. Therefore, the air-gap flux will be modulated by the passing rotor slots, producing rotor slot harmonics (RSH). A number of researches have been performed to extract rotor speed from RSH –. The induction motor speed, , can be calculated using RSH at any slip condition from the following expression: (5) III. DQ TRANSFORMATION In a 3-phase induction motor, the sum of stator currents is zero. Therefore, only two currents are sufficient for processing and the third one can be obtained from the other two phases. A suitable representation is the use of Park’s transformation given by: (3) where and are as in (2). Since the rotor slot harmonics are related to the rotor currents, their magnitude reduce with decreasing load making it difficult to detect RSH. In the absence of detectable RSH, the use of eccentricity-related harmonics has been proposed . In the proposed approach, the rotor speed is needed first as a normalizing factor for features and secondly for windowing broken bars harmonics from current spectrum. It will be shown later that the power of signal in this window plus phase at desired harmonics, will be other features. 314 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 16, NO. 4, DECEMBER 2001 V. PATTERN RECOGNITION AND BAYES MINIMUM ERROR CLASSIFIER—A BRIEF REVIEW  A pattern recognition system contains three parts, a transducer, a feature extractor and a classifier. The transducer senses the input and converts it into a form suitable for machine processing. The feature extractor extracts presumably relevant information from the input data. The classifier uses this information to assign the input data to one of a finite numbers of category . The fundamental theory and the necessary formulas of the statistical pattern recognition technique are presented next. More details can be found in . A. Bayes Decision Theory Bayes decision theory is a fundamental statistical approach to the problem of pattern classification. Let be the finite set of states of nabe the probability of each state (or class). ture (class) and Having an observed vector (or feature vector), the Bayes theory is based on the formula: Fig. 1. Power spectral density of stator currents. normal density is completely specified by two parameters, mean vector and covariance matrix: (8) (6) where (9) is the state-conditional probability density funcwhere tion for , i.e., the probability density function for given that is the probability of selected class state of nature is . , given the feature vector . The strength of the above formula is that it relates our observation and priori probability, , to a posteriori probability, . . It is For simplicity it is often abbreviated as shown that by using normal distribution in our analysis, which in most cases is a fair assumption, we can have a very simple form for the discriminant function for the Bayes minimum error classifier: (10) B. Bayes Minimum Error Classifier The problem of classification comes with the optimization problem. We want to have a threshold or boundary conditions in the space of feature vectors in order to discriminate different classes. This boundary condition is called discriminant function or decision surface. Different decision surfaces have different properties. If the decided class is but the true class is , then and in error if not. If errors are to the decision is correct if be avoided, it is natural to seek a decision rule that minimizes the average probability of error, i.e., the error rate. It is proved that in (6) as the discriminant function if we use (or decision surface), then we will minimize the probability of error by: Decide if (7) This is called Bayes Minimum Error Classifier. Another important classifier is Bayes Minimum Risk where different errors have different weights. C. Simplified Formula for Normal Distribution Primarily the conditional densities determine the structure of a Bayes classifier. Of the various density functions that have been investigated, none has received more attention than the multivariate normal density function. The general multivariate where is the feature vector and and are the covariance matrix and the mean vector for each class. VI. FEATURE EXTRACTION AND THE CLASSIFIER Based on the discussion in the previous sections, torque developed at the broken rotor bar frequency (Fig. 1) in and are assumed to be features. Another feature is phase at that frequency. In this paper, experiments were performed on a broken rotor bar and a healthy rotor induction motors. Without loss of generality, the appropriate features can be extracted from eccentricity related harmonicas as well. All features are normalized according to the running speed and load to have comparable data for different conditions. The classifier is Bayes minimum error assuming that features have normal distribution with equal likely classes. This is a fair assumption at this level but in case of multiple classes and online monitoring, different probability should be assigned to different classes. For example: in online monitoring techniques, statistical rate of occurrence of main faults can be used as a probability measure (refer to Section II). The utilized discriminant function has the form in (10), where is the feature vector and and are the covariance matrix , respectively. and the mean vector for each class Further details plus the block diagram of the proposed method HAJI AND TOLIYAT: PATTERN RECOGNITION—A TECHNIQUE FOR INDUCTION MACHINES ROTOR BROKEN BAR DETECTION 315 Fig. 3. Normalized torque and phase produced by broken bar harmonics (Id current). Horizontal axis shows number of samples in each class. number of columns (features) and refers to each sample. The normalized feature vector is calculated by: Fig. 2. Block diagram of the proposed algorithm. and experimental results derived from both a healthy induction motor and the one having four broken bars are covered in the next two sections. VII. BLOCK DIAGRAM Fig. 2 shows the proposed algorithm. Stator currents are meatransformation they are consured first, and then using the verted to – form. For normalization and getting features, rotor speed is important. Therefore, the rotor speed is estimated first by using RSH technique. Knowing the number of rotor bars and stator poles, rotor speed is estimated from the power spectral density of the – currents. Then, necessary features are extracted, normalized and fed to the classifier. Applying the discriminant function (10) on the produced vector, the classifier decides whether it comes from a healthy motor or a motor with broken rotor bars. Normalization has a two step procedure: 1) For the 1st and 3rd features; first the integral of power spectral density (PSD) in a window around broken rotor bar harmonics and fundamental harmonics are calculated (Fig. 1). Then, division of these two gives a normalized , if power of the broken bar harmonics. Since we divide the result by (1-s), we will have a normalized torque produced by the broken bar harmonics (the desired feature). 2) In order to give the same weight to different features, all produced data from healthy and broken bar faults are put together and normalized according to (11). (11) where equation for , and 1 in the is for the unbiased estimation . is the (12) is the dimension of our feature space and the feawhere . To be able to classify, acture vector is: cording to Bayes minimum error classifier, we should find mean , and covariance matrix of each class (healthy vector and rotor broken bar fault) which is called offline training. For simplicity, we assumed that features have normal distribution. Therefore, the maximum likelihood estimate for the mean and covariance of each class are : (13) (14) and in (11) with and One should not confuse in (13) and (14). The first two are used for normalizing the ex, while the last two are used for classitracted feature vector fication in discriminant function (10). VIII. EXPERIMENTAL RESULTS The proposed algorithm was implemented on a 3 hp, 3-phase induction motor with a rotor having 44 bars. The induction motor was run from 10% to 130% of rated load with a healthy rotor and a 4 broken bar rotor. Therefore, 47 sets of different data were collected and processed to cover all different torque-speed machine conditions. Figs. 3 and 4 show features derived from these data. First and third features in these figures are normalized power in a window around broken bar transformation harmonics in and , respectively. Use of will be more useful in case of stator unbalances or eccentricity. As can be seen, there is not much difference between first and third features. But, the algorithm remains more robust if we consider both features. The second and fourth features are the 316 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 16, NO. 4, DECEMBER 2001 steady state. Only number of poles and rotor slots are needed as pre-knowledge information. Stator currents are the only inputs to this algorithm. For normalization and getting features, rotor speed is important. Therefore, the rotor speed is estimated first by using rotor slot harmonics method, then features are extracted. Once normalized mean and variance plus mean and covariance of each class are determined for an ac induction motor, the technique can be used in online condition monitoring of the motor. Theoretical approach plus experimental results from a 3 hp induction motor show the strength of the proposed method. Without loss of generality, the algorithm can be revised to include other faults such as eccentricity and phase unbalance. Also, if appropriate features are derived, this method can be applied for fault classification in other electric machines like DC machines. Fig. 4. Normalized torque and phase produced by broken bar harmonics ( current). Horizontal axis shows number of samples in each class. Iq ACKNOWLEDGMENT The authors would like to express their gratitude to Prof N. Kehtarnavaz, and B. N. Araabi from Texas A&M University for their remarks. REFERENCES Fig. 5. Classification results. Horizontal axes 1 corresponds to healthy class and 2 to rotor broken bar faults. Only 1 out of 47 samples is misclassified (error 2.1%). = normalized phase information, which as the results show, do not carry much information for this case. Since not very many data for training and test were available, one suitable approach is to take one sample out from the data pool as a test sample and use the remaining for training the classifier. This procedure was iterated 47 times and results were compared with the true classes. 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