Diagnostics of mechanical and electrical faults in induction motors using wavelet-based features of vibration and current through support vector machine algorithms for various operating conditions

Abstract

Fault diagnosis of induction motors (IMs) is always a challenging task in the practical industrial field, and it is even more challenging in the case of inadequate information of IM working conditions. In this paper, a new methodology for fault detection has been proposed for IMs to detect various electrical and mechanical faults as well as their severities, where the data are unavailable at required operating conditions (i.e., speed and load) based on wavelet and support vector machine (SVM). For this, the radial, axial and tangential vibrations, and three-phase current signals are acquired from IMs having different faults. The acquired time domain signal is then transformed to time–frequency signals using continuous wavelet transform (CWT). Ten different base wavelets are used to investigate the impact of different wavelet function on the fault diagnosis of IMs. Statistical features are extracted based on the CWT, and then appropriate feature(s) are selected using the wrapper model. These features are fed to the SVM to detect whether a defect has occurred. The fault detection is performed for identical speed and load case using a number of mother wavelets. To analyze the robustness of the present system, diagnosis is attempted for various operational conditions of IMs. The result showed that the feature(s) selected using the Shannon wavelet diagnose, the fault categories of IM more accurately as compared to other wavelets, and remarkably found to be robust at all working conditions of IMs. The work is finally extended to perform the fault diagnosis when limited information is available for the training. From the results, it is observed that the proposed methodology does not only take care of the practical problem of unavailability of data at different operating conditions, but also shows good performance and takes low computation time, which are vital requirements of a condition monitoring and diagnostic system.

Appendix 1: Support vector machine

The basic SVM is a binary classification problem. The classification can be done by mapping first the input vectors into one or more feature spaces either linearly or nonlinearly; then, SVM constructs a hyperplane or set of hyperplanes between the two classes in feature space [41]. Then, the optimal hyperplane is selected which maximizes the margin that exists between the boundary and the nearest data point in each class of the two classes. The margin is chosen as a trade-off between the margin level and the generalization error. The optimal hyperplane is selected as it has low generalization error. SVM basic principle is demonstrated in a two-dimensional plane as shown in Fig. 1. It shows the separation of data from two different classes: class A (rectangle) and class B (circle). The SVM aims to place a linear boundary between the data of two classes in such a manner that it maximizes the margin. The nearest samples that are used to define the margin are called as support vectors (SVs) [42].

The input vector from two classes can be separated by,

$$p = \left\{ {(x_{1} ,y_{1} ),(x_{2} ,y_{2} ), \ldots ,(x_{n} ,y_{n} )} \right\},\quad x \in R^{m} ,\quad y \in \left( { - 1,1} \right)$$

(6)

The SVM classifier function is,

$$f(x) = w\phi (x) + b,$$

(7)

where $x_{i}$ represent the training vectors, $y_{i}$ represents label of $x_{i}$, m represents the no. of samples, w represents the normal vector, and parameter b represents the offset of the hyperplane. The function, $\phi (x)$, is the mapping function that is used to transfer the data from input space to higher-dimensional space. The $f(x)$ is basically a nonlinear function for original data; however, it is linear for the transformed data.

To find optimal hyperplane, the optimization problem has to be solved:

$$\mathop {\hbox{min} }\limits_{w,b,\xi } \, \frac{1}{2}\left\| w \right\|^{2} + C\sum\limits_{i = 1}^{m} {\xi_{i} }$$

(8)

Subjected to

$$y_{i} f(x_{i} ) = y_{i} (wx_{i} + b) \ge 1 - \xi_{i} ,\quad i = 1, \ldots ,m$$

(9)

where $\xi \ge 0,$ represents slack variables that permit misclassification for some data points so that the calculation complexity can be decreased. The C is a penalty parameter, which trade-off between the misclassification and boundary complexity. This is called as the soft-margin SVM. For nonlinearly separable condition, the original data have to be mapped from input space to a feature space to produce an optimal hyperplane with a nonlinear kernel function, $k\left( {x,x^{\prime}} \right)$. Vapnik suggested some kernel functions, though the Gaussian radial basis function (RBF) kernel, i.e., $k\left( {x,x_{i} } \right) = \exp ( - \gamma \left\| {x - x_{i} } \right\|^{2} );$$\gamma = 1/2\sigma^{2} > 0;$ is always preferred. Herein, $\sigma$ represents width of the RBF kernel and $\, \gamma$ represents kernel parameter.