# Fault diagnosis of wind turbine bearing based on stochastic subspace identification and multi-kernel support vector machine

## Abstract

In order to accurately identify a bearing fault on a wind turbine, a novel fault diagnosis method based on stochastic subspace identification (SSI) and multi-kernel support vector machine (MSVM) is proposed. First, the collected vibration signal of the wind turbine bearing is processed by the SSI method to extract fault feature vectors. Then, the MSVM is constructed based on Gauss kernel support vector machine (SVM) and polynomial kernel SVM. Finally, fault feature vectors which indicate the condition of the wind turbine bearing are inputted to the MSVM for fault pattern recognition. The results indicate that the SSI-MSVM method is effective in fault diagnosis for a wind turbine bearing and can successfully identify fault types of bearing and achieve higher diagnostic accuracy than that of *K*-means clustering, fuzzy means clustering and traditional SVM.

## Keywords

Wind turbine Bearing Fault diagnosis Stochastic subspace identification (SSI) Multi-kernel support vector machine (MSVM)## 1 Introduction

Recent decades renewable energy sources have received increasingly wide attention. As one of the most promising new clean renewable energy sources, wind power generation is in large-scale development around the world [1, 2, 3, 4]. However, wind turbines are prone to various failures due to long-term operation under tough conditions, complex alternating loads and variable speeds [5]. The bearing is a critical component of a wind turbine and bearing failures form a significant proportion of all failures in wind turbines. These, can lead to outage of the unit and a high maintenance cost [6, 7]. Hence, the development of an accurate fault diagnosis method for wind turbine bearing would be extremely valuable for improving safety and economy.

Vibration analysis is an effective condition monitoring method, especially suitable for rotating machinery. So far, a vast number of vibration signal processing methods have been used in fault detection of the gear box and bearing for wind turbines, such as spectrum analysis [8], wavelet transform [9], Wigner-Vile distribution [10] and empirical mode decomposition (EMD) [11]. Compared with the former three methods, EMD performs better in processing the vibration signal but it has the drawback of being time-consuming [12]. Variational mode decomposition (VMD) [13] and empirical wavelet transform (EWT) [14] achieve better signal processing performance than EMD and avoid mode aliasing, and they have strong noise robustness. In the past 20 years, the application of the stochastic subspace identification (SSI) method in vibration signal analysis has also developed very fast [15, 16], especially in the fault diagnosis field related to buildings and rotating equipment [17, 18]. However, the SSI method has so far seldom been used to diagnose a bearing fault for a wind turbine. The SSI method directly constructs a model based on time-domain data and can identify the mode parameters. This is suitable for mining the most essential fault information [19]. In this paper, SSI is employed to extract fault features by processing the collected vibration signal of the wind turbine.

In recent years, intelligent diagnosis methods have been widely applied in diagnosing a bearing fault in a wind turbine. As one of the artificial intelligence methods, the support vector machine (SVM) has many special advantages in solving with small samples, nonlinear and high dimensional pattern recognition. Reference [20] used SVM to classify different states of a rolling bearing and conducted experiments to validate the effectiveness of the proposed method. Reference [21] proposed a method based on wavelet packet and a locally linear embedding algorithm to extract fault features, and then intelligently classified the different fault degrees of a rolling bearing using SVM. Reference [22] combined EMD and SVM to identify different fault states of a rolling bearing. A single kernel function is used in the traditional SVM method to solve the classification problem of simple data. However, traditional SVM cannot effectively solve a complex classification problem, especially for a heterogeneous and imbalanced data classification problem. In order to improve the performance of SVM, a multi-kernel support vector machine (MSVM) is adopted for pattern recognition in this paper. MSVM not only integrates the generalization ability but also the self-learning ability of the traditional single kernel SVM [23, 24, 25], thus having better adaptability and robustness.

We use wind turbine bearing vibration data to construct the subspace model and to realize the bearing feature extraction, then use the MSVM algorithm to classify the feature parameters for bearing fault diagnosis. The paper is organized as follows. The principles of SSI and MSVM are respectively introduced in Section 2 and Section 3. The procedure of fault diagnosis method based on SSI-MSVM is described in Section 4. Diagnostic performance is tested by applying the SSI-MSVM method to a bearing experimental signal of a wind turbine in Section 5. Finally, conclusions are drawn in Section 6.

## 2 Principle of SSI

### 2.1 Stochastic state-space model

**X**_{ k }∈

**R**

^{ n }and

**Y**_{ k }∈

**R**

^{ l }are the state variable and output of the system in discrete time

*k*, respectively;

*∈*

**A****R**

^{ n×n }is the system matrix, which describes the dynamic behaviour of the system;

*∈*

**C****R**

^{ l×n }is the output matrix of the system;

**w**_{ k }∈

**R**

^{ n }and

**v**_{ k }∈

**R**

^{ l }are system noise and measurement noise, respectively.

### 2.2 SSI method

The SSI method can be completed in three steps: orthogonal projection, singular value decomposition and system parameter estimation.

#### 2.2.1 Orthogonal projection

*composed of the measurement signal is defined as follows:where*

**Y***y*

_{ k }(

*k=*1, 2, …,

*i*+

*j*+

*N*) is the collected signal;

**Y**_{ p }is the past part in matrix

*and its row number is*

**Y***i*;

**Y**_{ f }is the future part in matrix

*and its row number is*

**Y***j+*1; both

*i*and

*j*are generally not less than the maximum order of the system model, namely, \(\hbox{min} \{ i,j + 1\} \ge n\); and the column number of the block Hankel matrix

*is generally far greater than*

**N***i*and

*j*+1, namely, \(N \gg \hbox{max} \{ i,j + 1\}\).

*is newly divided into two blocks as follows:where \({\varvec{Y}}_{p}^{ + }\) is the new past part generated by shifting the first row of*

**Y**

**Y**_{ f }into the last row of

**Y**_{ p }; \({\varvec{Y}}_{f}^{ - }\) is the new future part generated by deleting the last row of

**Y**_{ f }.

**Y**_{ f }is orthogonally projected to the space of

**Y**_{ p }by the following equation:

#### 2.2.2 Singular value decomposition

**P**_{ m }as follows:

*σ*

_{ i }is the

*i*th singular value of

**S**_{1};

**U**_{0},

**V**_{0},

**S**_{0}are null matrices.

**P**_{ m }can also be expressed as the product of the observable matrix

**Г**_{ m }and the Kalman filter sequence \(\hat{\varvec{X}}_{m}\).

**Г**_{ m }.

**Г**_{ m }and the state variable \(\hat{\varvec{X}}_{m}\) can be inferred.

#### 2.2.3 System parameter estimation

**ρ**_{ w }and

**ρ**_{ v }are residual vectors and not related to \(\hat{\varvec{X}}_{m}\).

*and output matrix*

**A***can be estimated by applying the least squares method as follows:*

**C***and*

**A***, respectively.*

**C**In (13), the matrix * A* contains the feature information of the system model built by the bearing vibration data. That is, the eigenvalues of matrix

*correspond to different fault modes.*

**A***is given by:*

**A***is the left singular matrix of the system matrix*

**U***;*

**A**

**V**^{T}is the transpose of the right singular matrix of the system matrix

*; \(\varvec{\varSigma}= {\text{diag}}\{ \lambda_{1} ,\lambda_{2} , \cdots ,\lambda_{n} \}\) is a diagonal matrix and*

**A***λ*

_{ i }is the

*i*th singular value of the system matrix

*.*

**A**## 3 Principles of standard SVM and MSVM

### 3.1 Standard SVM

**x**_{ i }∈

**R**

^{ n }is the input samples and

*y*

_{ i }∈{−1, +1} is the output samples, SVM maps the input samples to the

*n*dimensional feature space using the kernel function, in which the optimal classification hyper plane \(\sum\limits_{i = 1}^{n} {w_{i} k(\varvec{x},\varvec{x}_{i} )}\) is constructed, where

*w*

_{ i }is the

*i*th element of the coefficient vector

*. The classification interval in optimal hyper plane 2/||*

**w***|| is expected to achieve a maximum in the SVM method. By introducing slack variables, the optimal hyper plane is transformed into the following constrained optimization problem:*

**w***e*

_{ i }is the

*i*th element of the slack variable vector

*;*

**e***C*is penalty coefficient;

*b*denotes the distance from the hyper plane to the origin; \(\varvec{\phi}( \cdot )\) is mapping function.

### 3.2 MSVM

*σ*is the kernel parameter.

*d*is the kernel parameter.

*λ*(0<

*λ*<1) is the tuning parameter.

According to (18), it can be readily seen that the multi-kernel function degenerates into the Gauss kernel function under the condition that *λ=*0 and into the polynomial kernel function under the condition that *λ=*1. The multi-kernel function can adapt to different input samples by adjusting the tune parameter *λ*. The MSVM integrates prior knowledge of specific problems in the process of selecting the kernel function, and thus it has the ability of learning and generalization combined.

## 4 Fault diagnosis model based on SSI method and MSVM

- 1)
Vibration signal acquisition. Setting the mounting position of vibration sensors and sampling rate, the vibration signal of the wind turbine bearing is acquired through a signal acquisition system.

- 2)
SSI analysis. First, the collected vibration signal is used to construct a stochastic state-space model of a wind turbine bearing as shown in (1). Then, the SSI method is applied to estimate the system matrix

, eigenvalues of which are extracted as fault feature vectors.**A** - 3)
Training MSVM model. Samples in different states are taken as training samples to train the MSVM, establishing the fault diagnosis model. The trained model can be used to distinguish different patterns.

- 4)
Fault diagnosis. The test samples are input to the trained fault diagnosis model for classification, and the working state and fault type of the wind turbine bearing can be determined according to the output of the MSVM model.

## 5 Experiment analysis

*g*(

*g*= 9.8 m/s

^{2}), sensitivity of 500 mV per

*g*, frequency range of 0.3-12000 Hz and resolution of 0.004

*g*. To simulate the typical faults of a wind turbine bearing, a spark erosion technique is used to seed a pit in the inner race, roller and outer race.

**Y**_{ p }and

**Y**_{ f }are 9×8991 dimensional matrices. The projection matrix is analysed by singular value decomposition and the results indicate that the order of the stochastic subspace model is 5. Then the SSI method is applied to process the collected signal and eigenvalues of the system matrix are obtained. Figure 5 shows the features of the bearing vibration signal in different conditions extracted by the SSI model. It can be seen that all the eigenvalues are distributed in the unit circle, which indicates that the identified bearing stochastic subspace model is stable and effective. At the same time, the distributed locations of eigenvalue corresponding to normal mode and different fault modes are non-coincident which illustrates that these different eigenvalues can be discriminated by a clustering algorithm.

For strong faults and weak faults of bearing in this study, 320 samples gained from an experimental roller bearing are respectively adopted to verify the superior diagnostic performance of the SSI-MSVM method compared with that of *K*-means clustering, fuzzy means clustering (FCM) and traditional SVM. All data are divided into two data sets: the training and test, in which the training data sets including 120 samples are used to calculate the fitness function and train the diagnosis model, and the test data sets are used to examine the classification accuracy of each model.

Fault diagnosis result comparison among *K*-means clustering, FCM, SVM and MSVM

No. | Total samples | Classification accuracy (%) | |||
---|---|---|---|---|---|

| FCM | SVM | MSVM | ||

1 | 40 | 82.53 | 86.35 | 89.25 | 90.27 |

2 | 80 | 86.27 | 88.05 | 91.03 | 91.46 |

3 | 120 | 87.73 | 89.02 | 90.75 | 92.72 |

4 | 160 | 88.06 | 89.03 | 90.81 | 92.73 |

5 | 200 | 87.71 | 89.02 | 90.76 | 92.47 |

6 | 240 | 87.75 | 89.32 | 89.82 | 92.35 |

From Table 1 and Fig. 6, it can be seen that the fault diagnosis accuracy of the SSI-MSVM method is in the range of 90.27%-92.73%, while that of *K*-means clustering, FCM and SVM are in the range of 82.53%-88.06%, 86.35%-89.32% and 89.25%-91.03%, respectively. Overall, the SSI-MSVM method has a higher diagnostic accuracy than that of *K*-means clustering, FCM and SVM, which indicates that the proposed diagnosis model is clearly superior to the traditional diagnosis method.

## 6 Conclusion

This paper presents a novel fault diagnosis method for a wind turbine bearing based on SSI and MSVM. The SSI method directly constructs a model based on time-domain data and can identify the mode parameters. This is suitable for extracting the fault features. The MSVM is an improved mode recognition method which combines the Gauss kernel SVM and the polynomial kernel SVM, so it can identify the fault types of bearing more accurately.

The results indicate that the SSI-MSVM method is an effective fault diagnosis method for a wind turbine bearing, and can successfully identify fault types of bearing and achieve higher diagnostic accuracy than that of *K*-means clustering, FCM and traditional SVM.

## Notes

### Acknowledgment

This work was supported by National Key Technology Research and Development Program (No. 2015BAA06B03).

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