# Wayside Detection of Wheel Minor Defects in High-Speed Trains by a Bayesian Blind Source Separation Method

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## Abstract

**:**

## 1. Introduction

## 2. FBG-Based Wayside Wheel Defect Detection System

#### 2.1. FBG Sensing System in Wayside Detection

- Assurance of immunity to electromagnetic field: most of the conventional wheel condition monitoring systems, either resistance strain gauge- or accelerometer-based, are vulnerable to EMI induced by high voltage power supply system of modern HSR [23];
- Massive multiplexing capability: HSR always has strict requirements on clearance, which can be problematic for conventional sensing systems when considerable measuring points are needed. In contrast, FBG-based sensing system allows the use of hundreds of sensing points (FBGs) in a single fiber cable. This ability facilitates easy installation on HSR tracks with light-weight trackside equipment;
- High reliability and durability: the FBG-based sensing system can operate for more than 20 years without losses in performance even in extreme climate, such as heavy rains and snows, strong winds, or extremely hot summer days, and corrosion environment and large shocks caused by track maintenance work [22];
- Long conduction distance: the FBG-based sensing system can offer up to 100 km distant detection [23], because the optical fiber has a salient advantage in long-distance transmission with much lower signal attenuation. This allows the monitoring equipment to be installed far away from the instrumented rail section where the sensors are deployed and both the sensors and connecting fibers at the instrumented zone require no power supply.

#### 2.2. FBG-Based Wayside Wheel Defect Detector

## 3. Wheel Defect Identification

#### 3.1. Strain Response Extraction

#### 3.2. Defect-Sensitive Feature Extraction Based on Bayesian BSS

#### 3.2.1. Bayesian BSS

#### 3.2.2. Assumptions and Model Establishment

**X**(t) = [x

_{1}(t), x

_{2}(t), ..., x

_{m}(t)]

^{T}is a vector of size m standing for noisy observations,

**S**(t) = [s

_{1}(t), s

_{2}(t), ..., s

_{n}(t)]

^{T}is a vector of size n containing the hidden sources mixing in the observation signals;

**A**is called ‘mixing matrix’ representing the transfer function from the sources to the sensors;

**Z**(t) = [z

_{1}(t), z

_{2}(t), ..., z

_{m}(t)]

^{T}is the noise vector of size m;

**Y**(t) is observation signals without noise contamination. Unlike traditional BSS techniques (e.g., SOBI method) which assume that different noise sequences have a same variance, the present study considers a diagonal covariance matrix Σ

_{Z}to model the noise sequences

**Z**(t). Thus,

**X**,

**S**, and

**Z**; and $\mathcal{N}\left(Z(t);0,{\sum}_{Z}\right)$ represents normal distribution with the mean μ and variance σ

^{2}. The diagonal elements in the matrix are equal to different noise power σ

_{i}

^{2}of the ith observation point (i.e., ${\sum}_{{Z}_{ii}}={\sigma}_{i}^{2}$). After modeling the noise, the likelihood function of the observation

**X**can be expressed as

**S**

_{j}= [s

_{j}(t

_{1}), s

_{j}(t

_{2}), ..., s

_{j}(t

_{L})] is the jth source signal; K

_{j}is the covariation matrix with a GP kernel expression of any two times t and t’

_{j}is the hyperparameter of the jth source signal.

**A**, we consider discriminative inferences for different measuring points (FBGs) in modeling, and it is written as

_{ij}is the element in the ith row and the jth column of the mixing matrix; ɛ

_{ij}is the variance of a

_{ij}and it can be considered as a hyperparameter of the mixing matrix prior.

_{Z}) and mixing matrix (ɛ), the conjugate prior for the variance in Gaussian likelihood is used

_{Z}, β

_{Z}, α

_{a}, and β

_{a}are known parameters in inverse-gamma distribution.

_{S}and β

_{S}are two known parameters in the above gamma distribution.

**A**,

**S**, Σ

_{Z}, h, and ɛ. The procedure, which was detailed in our previous research [40], consists of: (i) generating samples of the source, mixing matrix, noise covariance matrix and mixing matrix hyperparameter from the corresponding conditional posteriors p(

**S**|

**X**,

**A**,Σ

_{Z},h), p(

**A**|

**S**,

**X**,Σ

_{Z},h,ɛ), p(Σ

_{Z}|

**A**,

**S**,

**X**), and p(ɛ|

**A**) by Gibbs sampling; (ii) deriving the expression of these conditional posteriors; and (iii) deriving the posterior of the source hyperparameter p(h|

**S**), which does not belong to a standard conjugate family by the M-H algorithm.

#### 3.2.3. Defect-Sensitive Feature Extraction

#### 3.3. Defect Identification

_{u}and x

_{l}are the upper and lower limits of the probability band, F

^{−1}is the normal inverse function, and N is the sample size. Given the lower and upper limits, the anomalies on the time history of normalized strain data can then be easily detected, as shown in Figure 7. Note that in this study, anomalies are the data points that are beyond lower or upper limits and the adjacent data points within a certain range in time series.

- The speed variation of passing trains: The process of train passage lasts from seconds to dozens of seconds, so it is possible that the train is speeding up or slowing down during this process and the speed is not constant. However, as described in the proposed method, the condition of wheels is assessed individually, that is, the detection of each wheel is free from the interference of other wheels. Since the instrumented rail section with FBG array is only about 3 m long, the speed of each wheel is unlikely to change dramatically during its passage across the instrumented section. In addition, it has been proven that the dynamic strain monitoring data of rail obtained under different constant running speeds of a train give rise to consistent wheel defect detection results as long as the running speed is instantly measured and enough large dynamic strain of the rail is excited by the passing wheel. It is observed that when the train’s running speed is lower than 20 km/h, the anomaly stemming from minor wheel defect is difficult to perceive in the measured rail dynamic strain response.
- The temperature effect on strain measurement: For strain measurement using FBG sensors, the temperature effect usually should not be ignored, since the output wavelength of FBG sensors can shift with temperature variation. However, temperature-induced wavelength change would not influence the performance of the proposed method. This is because the wavelength change caused by temperature variation mainly results in the change of baseline of the output signal. The influence of temperature can be easily eliminated by deducting the mean value of wavelength before or after train passage. Particularly in the proposed method, after pursuing BSS, the change of temperature will be reflected in the first component (source) rather than the second component (source), the latter being used for wheel defect detection. Also, the temperature variation during the short time of the wheel’s passage across the instrumented section is ignorable.
- Different locations of FBGs with respect to sleepers: In this study, the FBG sensors on the array have different locations with respect to sleepers. These FBGs measure the rail strain due to bending, and the measurement result may be influenced by the distance of the sensor from the sleeper. Therefore, it is necessary to compare the signals generated by different FBGs on the array. As shown in Figure 4, under the excitation of the same wheel, the waveforms of the rail strain responses at different locations are similar. Even if there are slight differences in the amplitude of response peak, this kind of difference is mainly reflected in the first component after signal processing using BSS, rather than in the second component. Therefore, the detection results would not be affected by this issue.

## 4. In-Situ Verification

#### 4.1. Implementation of Online Detector

#### 4.2. Blind Test

#### 4.3. Test Results and Validation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Procedure of rail response extraction.

^{1}The user-defined threshold Th is based on the knowledge of passing trains (e.g., heavy wagons, locomotives, metro trains, or high-speed trains). For high-speed trains concerned in this study, for example, the variance of the peak values is relatively small, the value of Th can be greater than 0.5.

^{2}In this step, the maximum strain value may not be the peak point considering that the noise in observation data may generate false peaks.

**Figure 5.**Raw signal of rail response to the excitation of a healthy wheel and its decomposed components: (

**a**) plotted in different panels; (

**b**) plotted in the same panel.

**Figure 6.**Raw signal of rail response to the excitation of a defective wheel and its decomposed components: (

**a**) plotted in different panels; (

**b**) plotted in the same panel.

**Figure 7.**Detection of localized anomalies from the normalized strain data – an example of two strain response datasets: blue and green curves—normalized strain time histories from two different FBGs; black straight lines—the upper and lower thresholds specified by the Chauvenet’s criterion; red curves—the anomalies identified using the Chauvenet’s criterion.

**Figure 9.**(

**a**) The test train of an eight-car high-speed EMU; (

**b**) In-depot offline wheel inspection by radius deviation measurement.

**Figure 10.**Defect detection results of the right wheel of wheelset no. 1: (

**a**) online detection result; (

**b**) offline wheel radius deviation measurement.

**Figure 11.**Defect detection results of the right wheel of wheelset no. 6: (

**a**) online detection result; (

**b**) offline wheel radius deviation measurement.

**Figure 12.**Defect detection results of the right wheel of wheelset no. 24: (

**a**) online detection result; (

**b**) offline wheel radius deviation measurement.

**Figure 13.**Defect detection results of the left wheel of wheelset no. 27: (

**a**) online detection result; (

**b**) offline wheel radius deviation measurement.

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## Share and Cite

**MDPI and ACS Style**

Liu, X.-Z.; Xu, C.; Ni, Y.-Q. Wayside Detection of Wheel Minor Defects in High-Speed Trains by a Bayesian Blind Source Separation Method. *Sensors* **2019**, *19*, 3981.
https://doi.org/10.3390/s19183981

**AMA Style**

Liu X-Z, Xu C, Ni Y-Q. Wayside Detection of Wheel Minor Defects in High-Speed Trains by a Bayesian Blind Source Separation Method. *Sensors*. 2019; 19(18):3981.
https://doi.org/10.3390/s19183981

**Chicago/Turabian Style**

Liu, Xiao-Zhou, Chi Xu, and Yi-Qing Ni. 2019. "Wayside Detection of Wheel Minor Defects in High-Speed Trains by a Bayesian Blind Source Separation Method" *Sensors* 19, no. 18: 3981.
https://doi.org/10.3390/s19183981