Open AccessArticle

Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM)

College of Automation, Jiangsu University of Science and Technology, Zhenjiang 212100, China

School of Computer Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212100, China

Author to whom correspondence should be addressed.

Machines 2024, 12(12), 832; https://doi.org/10.3390/machines12120832

Submission received: 19 October 2024 / Revised: 11 November 2024 / Accepted: 20 November 2024 / Published: 21 November 2024

(This article belongs to the Section Automation and Control Systems)

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Figure 1
Choice of probability-relevant matrix W. "> Figure 2
The flow chart of incipient fault diagnosis scheme based on PRPCA and SVM. "> Figure 3
Body properties. "> Figure 4
Primitive properties. "> Figure 5
Joint properties. "> Figure 6
Rail properties. "> Figure 7
The wheelset model. "> Figure 8
The bogie model. "> Figure 9
The vehicle model. "> Figure 10
The incipient fault detection comparisons for sensors. "> Figure 11
The incipient fault detection comparisons for actuators. "> Figure 12
The incipient fault detection comparisons for secondary suspension dampers. "> Figure 13
The incipient fault detection comparisons for secondary suspension springs. "> Figure 14
The fault classification comparisons between PCA-SVM and PRPCA-SVM with two fault types. "> Figure 15
The fault classification comparisons between PCA-SVM and PRPCA-SVM with three fault types. "> Figure 16
The fault classification comparisons between PCA-SVM and PRPCA-SVM with four fault types. ">

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Abstract

As a crucial component of CRH (China Railway High-speed) trains, the safety and stability of the suspension system are of paramount importance to the overall vehicle system. Based on the framework of probabilistic relevant principal component analysis (PRPCA), this paper proposes a novel method for incipient fault diagnosis in the CRH suspension system using PRPCA and support vector machine (SVM). Firstly, simulation data containing multiple types of fault information are obtained from the Simpack2018.1-Matlab2016a/Simulink co-simulation platform. Secondly, the nonlinear PRPCA approach, based on the Wasserstein distance, is employed for fault detection and data preprocessing in the suspension system. Furthermore, SVM is used for fault recognition, and the F1-Measure index is utilized for a comprehensive evaluation to assess the fault diagnosis performance more intuitively. Finally, based on the comparison results with traditional principal component analysis (PCA) and SVM-based methods, the proposed incipient fault diagnosis method demonstrates superior efficiency in fault detection and recognition. However, the proposed method is not very sensitive to sensor faults, and the performance of sensor fault diagnosis needs to be further improved in subsequent research.

Keywords:

PRPCA; SVM; F1-measure; incipient fault diagnosis

1. Introduction

The suspension system is a critical component of China Railway High-speed (CRH) trains. Accidents involving CRH trains can cause significant economic losses and negative social impacts. Therefore, it is essential to continuously monitor the condition of the suspension system to promptly identify abnormalities and determine their causes. Simultaneously, timely detection at the early stage of fault is imperative to reduce suspension damage. In summary, research on the incipient fault detection and diagnosis (FDD) of high-speed train suspension devices, including actuators and sensors, is particularly important.

Generally, the FDD methods are divided into three categories: quantitative model-based methods, qualitative model-based methods, and process history-based methods. The quantitative model-based methods, such as Kalman filtering design [1,2] and observer design [3,4], use mathematical modeling to generate residuals by comparing the characteristics of the actual dynamic system with those predicted by the system’s mathematical model. These residuals are then analyzed and processed to enable fault detection and diagnosis. The qualitative model-based (knowledge-based) methods, such as qualitative simulation theory [5], symbolic directed graphs [6] and expert systems [7], utilize incomplete prior knowledge of the system to develop a qualitative model that describes the system structure, predicts system behavior, and compares this predicted behavior with the actual behavior to detect faults. Subsequently, it identifies the cause of the system faults through logical deduction. Obviously, all these methods heavily rely on mathematical models and prior knowledge. Contrary to the previous two categories, the third category, which includes methods such as partial least squares (PLS) [8,9], fuzzy logic [10,11], principal component analysis (PCA), artificial neural networks (ANNs) [12,13], wavelet packet transform (WPT) [14,15] and support vector machines (SVMs), is primarily based on process history and does not require accurate system modeling. Among these, PCA is the most commonly used multi-variate statistical analysis method, while SVM is widely used as a classification tool.

Given its simplicity and efficiency in handling large amounts of process data, PCA is widely recognized as a powerful tool for statistical process analysis. Therefore, data-driven FDD methods based on PCA have been extensively studied in Refs. [16,17]. However, traditional PCA methods, typically applicable only to linear processes, are not suitable for analyzing nonlinear process data. To address this limitation, Refs. [18,19] proposed an FDD approach based on KPCA (kernel principal component analysis). By utilizing kernel functions, the KPCA method transforms a low-dimensional, nonlinearly indivisible data set into a high-dimensional, linearly divisible data set. Nevertheless, this approach focuses solely on the global structure and disregards the importance of local structural features in data processing. For example, Ref. [20] proposed an improved KPCA method (Multi-way Kernel Principal Component Analysis, MKPCA), incorporating the concept of popular learning into the objective function of KPCA to preserve the local structure. This method ensures that the resulting feature space not only preserves the global structure of the original sample space but also maintains its local nearest-neighbor structure simultaneously. Refs. [16,21] investigated the probabilistic relevant principal component analysis (PRPCA) method, which is better suited for nonlinear and non-Gaussian systems. However, recognizing that the

T^{2}

and

S P E

statistics lack sensitivity to incipient fault, researchers proposed an FDD method in Ref. [22] based on Kullback–Leibler divergence and PCA to enhance incipient fault detection. According to the aforementioned, PCA plays a significant role in fault detection. However, its application in fault identification is limited. Therefore, support vector machines (SVMs) are often needed as an auxiliary method to achieve fault identification. SVM is an effective binary classification technique that is also commonly used for multi-classification problems. There are primarily three combinations: directed acyclic graph (DAG) [23,24], one-against-one (1-v-1) [25,26], and one-against-rest (1-v-r) [27,28] methods. The 1-v-1 method requires designing an SVM for every pair of categories, while the 1-v-r method classifies one category of samples into one class and all other samples into another class. In contrast, DAG-SVM is a multi-class classifier that uses an SVM classifier as its base classifier and employs a directed acyclic graph as its topological structure. Due to the multi-variate and nonlinear characteristics of monitoring data, SVMs are rarely used directly for condition monitoring and fault diagnosis. Therefore, some research works used PCA as a data preprocessing method, followed by SVM for fault diagnosis [29]. Furthermore, to improve the performance of data preprocessing, Ref. [30] proposed a FDD method that combines KPCA and SVM. However, traditional KPCA suffers from poor adaptability. To address this limitation, Ref. [31] proposed a real-time fault diagnosis method for high-voltage circuit breaker (HVCB) based on adaptive KPCA and SVM.

In this paper, we focus on incipient fault detection in the suspension systems of high-speed trains. Initially, we use an improved PRPCA method for fault detection and data preprocessing. Subsequently, we employ SVMs for fault classification. The primary contributions of this study are as follows:

(1) The proposed improved PRPCA method not only enhances fault characteristics but also improves the accuracy of fault detection. This improvement further enhances the sensitivity of the proposed combined scheme to incipient faults.

(2) Considering the error accumulation issue in DAG-SVM, this paper opts for 1-v-1 SVM as the fault classification method. Furthermore, the performance evaluation index F1-Measure is used to comprehensively assess both fault detection and classification accuracy.

2. PRPCA-Based Incipient Fault Detection

To achieve incipient fault diagnosis in the suspension system of high-speed trains, it is essential to detect faults in the samples accurately and promptly. Additionally, effective dimensionality reduction in fault data during the data preprocessing stage is crucial. This paper proposes a novel method based on nonlinear PRPCA and SVM, consisting of fault detection, data preprocessing, and fault classification. For the fault detection section, this paper constructs a nonlinear PRPCA-based model using raw data for fault detection. In the data preprocessing stage, this paper uses the nonlinear PRPCA algorithm to reduce the dimensionality of the original fault data, while also grouping large-scale sample data and using variance to measure the deviation degree of each group. In the fault classification stage, optimized data from the previous steps are used for fault recognition using the SVM method. The detailed steps are shown in the following sections.

2.1. Nonlinear PRPCA Method

The nonlinear PRPCA method extends PCA into the nonlinear domain by nonlinearly projecting measurement data into a new space, thereby effectively separating the initial fault data from the normal data:

\hat{p} (t_{j}) = \frac{1}{N h} \sum_{i = 1}^{N} K (\frac{t_{j} - t_{i \cdot j}}{h})

(1)

Let

x_{k} \in R^{N \times m}

(

k = 1, 2, \dots, N

) be the N training samples in the row data X, then

t_{j}

is the j-th column of the score vector T, where

T = X P

, and P is the loading matrix.

t_{i \cdot j}

is the i-th element of

t_{j}

, h is the bandwidth, and K is a kernel function.

The probability density function distribution is shown in Figure 1. W is the probability-relevant matrix with the form

W = d i a g (w_{1}, \dots, w_{m})

, and the selection of

w_{i}, i = 1, 2, \dots, m

is depicted in Algorithm 1 shown as follows, where the boundary

β

and parameters

P_{c}

and

P_{d}

should ensure that all potential incipient faults are taken into account.

α

is the tuning factor, and

θ

represents the steepness of the logistic curve.

Algorithm 1: The selection of w.

1 . Set β

;

2 . Determine P_{c} and P_{d} based on probability density

function (PDF)

;

3 . Calculate w by using the logistic function as

w = 1 + \frac{α}{1} + exp (- θ (t - (p_{c} + p_{d}) / 2))

(2)

To distinguish between the variables of PRPCA and PCA, the subscript “New” is added to the variables of the PRPCA algorithm. Furthermore, the overload matrix in the PRPCA algorithm can be obtained as follows:

P_{N e w} = W^{- 1 / 2} P

(3)

Due to the limitations inherent in engineering applications, the optimal principal component selection based on the cumulative variance contribution ratio (CVR) is adopted in this paper as follows:

C V R (r) = \frac{\sum_{i = 1}^{r} λ_{i}}{\sum_{i = 1}^{m} λ_{i}} \times 100

(4)

where

λ_{i}

denotes the eigenvalues from the eigenvalue decomposition of the covariance matrix of the original data X. If v is the percentage of variance, the optimal number of principal components r can be determined as follows:

C V R (r) \geq v

(5)

The r-dimensional PRPCA model can be represented as

X_{f c} = X P_{r N e w} P_{r N e w}^{T}

(6)

where

P_{r N e w}

is the r eigenvectors retained in matrix

P_{N e w}

, and

X_{f c}

can be regarded as the initial data for fault classification. Further, the PRPCA-based residuals can be represented as

R = X - X_{f c}

(7)

In this paper, the Wasserstein distance (WD) serves as a measure of dissimilarity between two continuous distributions. R and

R_{i f}

represent the model residuals of the normal data and the fault data, respectively. The WD between two probability distributions R and

R_{i f}

is shown as follows with

k \geq 1

W_{k} (R, R_{i f}) = {(inf_{γ \in Π (R, R_{i f})} \int || x - {y ||}^{k} d γ (x, y))}^{1 / k}

(8)

Assuming that the distributions of the n-dimensional variables x and y are both Gaussian distributions, with mean vectors

μ

and

μ_{i f}

, and covariance matrices

Σ

and

Σ_{i f}

, respectively, the closed-form expression for the 2-Wasserstein distance between x and y can be formulated as

\{\begin{matrix} W_{2} (R, R_{i f}) = {\{μ_{1}^{2} + (Σ + Σ_{i f} - 2 T r {(Σ_{1})}^{1 / 2})\}}^{1 / 2} \\ Σ_{1} = Σ^{1 / 2} Σ_{i} f Σ^{1 / 2} \\ μ_{1} = {∥μ - μ_{i f}∥}_{2} \end{matrix}

(9)

2.2. The Fault Detection Accuracy

In this paper, the fault detection accuracy refers to the percentage of correctly detected samples. Specifically, it represents the proportion of correctly detected normal samples and faulty samples. The formula for calculating detection accuracy is

\hat{R} = \frac{T P + T N}{T P + T N + F P + F N}

(10)

where

T P

(true positive) represents the number of fault samples correctly detected,

T N

(true negative) represents the number of normal samples correctly identified,

F P

(false positive) represents the number of normal samples incorrectly identified as faults, and

F N

(false negative) represents the number of fault samples incorrectly identified as normal.

3. SVM-Based Incipient Fault Recognition

This research involves separating the fault data

X_{f c}

obtained from Equation (6) into k groups and then calculating the variance of each group. Consequently, a new data set can be obtained for each fault type. These data sets for each fault type are then merged to form

{\hat{X}}_{f c}

. This data set serves as the initial training set for one-versus-one SVMs to classify multiple faults.

3.1. Multi-SVMs Scheme

SVM provides significant advantages when dealing with small sample sizes and high-dimensional problems, making it an excellent machine learning algorithm. The fundamental principles of SVM are described as follows.

For the given training data

x_{i} \in R^{m}

y_{i} \in {1, 2, 3, 4}

(i = 1, 2, \dots, N)

, where m is the dimension of the input data and N is the number of the training data. SVM needs to address the fundamental issue of quadratic programming (QP) as follows:

\begin{matrix} min f (w) = \frac{{∥ w ∥}^{2}}{2} \\ s . t g_{i} (w, b) = y_{i} * (w {x_{i}}^{T} + b) - 1 \geq 0; \\ i = 1, 2, 3, 4 \dots N \end{matrix}

(11)

where w is the normalized weight vector of the same dimension as x, and b is the normalized bias of the hyperplane. Furthermore, convert the inequality involving

g_{i}

into an equation:

\begin{matrix} g_{i} (w, b) = y_{i} * (w {x_{i}}^{T} + b) - 1 = {p_{i}}^{2}; \end{matrix}

(12)

Thus, the Karush–Kuhn–Tucker (KKT) conditions can be derived using the Lagrange multiplier approach:

\{\begin{matrix} w - \sum_{i = 1}^{N} λ_{i} * y_{i} * x_{i} = 0 \\ - \sum_{i = 1}^{N} λ_{i} * y_{i} = 0 \\ y_{i} * (w x_{i} + b) - 1 \geq 0 \\ λ_{i} * [y_{i} * (w x_{i} + b) - 1] = 0 \\ λ_{i} \geq 0 \end{matrix}

(13)

where the KKT condition problem can be transformed into an equivalent Lagrangian dual problem. The formula is shown as

\begin{matrix} max q (λ_{i}) = (\sum_{i = 1} λ_{i} - \frac{1}{2} \sum_{i = 1} \sum_{j = 1} λ_{i} λ_{j} y_{i} y_{j} {\vec{x}}_{i} \cdot {\vec{x}}_{j}) \\ s . t . λ_{i} \geq 0, i = 1, 2, 3, 4 \dots N \end{matrix}

(14)

In the equation above,

λ_{i}

represents the Lagrange multiplier of the i-th sample. Utilizing dual problems has two advantages. Firstly, it results in more concise expressions and constraints. Secondly, the optimal solution, denoted as

{\vec{x}}_{i} \cdot {\vec{x}}_{j}

, is determined solely based on the dot product results of the support vectors. This can also be understood as being determined only by the spatial similarity of the support vectors.

The basic type of SVM is a hard-margin classifier that requires all samples to satisfy the hard-margin constraint. However, this constraint can reduce the geometric margin of the decision boundary, leading to decreased generalization of the model in the presence of outliers. Moreover, hard-margin classifiers can also result in overfitting. To address these issues, SVM introduces a soft-margin classifier by incorporating slack variables. A penalty factor is then added before the slack variables to represent the tolerance for loss caused by outliers. As C approaches infinity, the decision hyperplane is forced to classify all samples correctly, degenerating into a hard-margin classifier. The specific formula is as follows:

\begin{matrix} min f (w) = \frac{{∥ w ∥}^{2}}{2} + C \sum_{i = 1}^{N} ξ_{i} \\ s . t g_{i} (w, b) = y_{i} * (w {x_{i}}^{T} + b) - 1 \geq 0; \\ i = 1, 2, 3, 4 \dots N \end{matrix}

(15)

where

ξ_{i}

is the slack variables, and C is the penalty factor.

From the above theories, it is not difficult to discover that SVM is initially applied to linearly separable cases. However, when faced with nonlinearly inseparable data, a method called the kernel trick is introduced. Kernel functions nonlinearly map a nonlinearly inseparable model in a low-dimensional space into a high-dimensional space, thereby transforming it into a linearly separable model. Consequently, a linear decision function in dual form can be obtained as follows:

f (x) = sign (\sum_{i, j = 1}^{N} λ_{i} y_{i} k^{'} (x_{i}, x_{j}) + b)

(16)

where

s i g n (\cdot)

is the signum function, and

k^{'}

is the kernel function that returns the dot product of the feature space map of low-dimensional spatial data points. In this paper, the Radial Basis Function (RBF) is employed as the kernel function of the SVM, defined as follows:

k^{'} (x_{i}, x_{j}) = exp (- \frac{{∥x_{i} - x_{j}∥}^{2}}{2 γ^{2}})

(17)

where

γ

is the width of the RBF kernel which is crucial in SVM. When

γ

is small, there may still be pronounced similarity values despite a large distance between the sample points and the decision boundary. This indicates an amplification of the similarity between data points, facilitating their segmentation by simple hyperplanes. Conversely, when

γ

is large, most data points, except those in close proximity, lack similarity to other points, rendering them more susceptible to overfitting. Therefore, when calculating the decision boundary, the spatial characteristics of these data points must be taken into account. Consequently, to achieve the best classification effectiveness of the SVM classifier, it is imperative to find the optimal combination of

γ

and C. This article employs the cross-validation method to determine the optimal values.

3.2. The Comprehensive Evaluation Metrics

This study employs the F1-Measure to comprehensively assess the accuracy of fault detection and classification. The specific formula for the F1-Measure is as follows:

F = \frac{2^{*} {\hat{P}}^{*} \hat{R}}{\hat{P} + \hat{R}}

(18)

where

\hat{R}

represents the fault detection accuracy and

\hat{P}

represents the fault classification accuracy.

3.3. The Fault Diagnosis Scheme

The incipient fault diagnosis scheme based on PRPCA and SVM is shown in Figure 2. The main steps involve data preprocessing, fault detection, and fault classification.

Data preprocessing:

(1): The data sets are obtained from Simpack2018.1-Matlab2016a/Simulink co-simulation platform under both normal and faulty operating conditions.
(2): The data sets are standardized and normalized, and their respective covariance matrices underwent singular value decomposition to obtain the loading matrix P.
(3): The probability correlation matrix W, corresponding to the score vectors of each data set, can be obtained based on Equation (1) and Algorithm 1.
(4): The new loading matrix $P_{n e w}$ is reconstructed by scaling the loading matrix P using the probability correlation matrix W.
(5): Dimensionality reduction is conducted on $P_{n e w}$ based on the variance percentage v in Equation (5), resulting in a new load matrix $P_{r n e w}$ .
(6): The dimensionally reduced PCA model is established based on Equation (6) and $P_{r n e w}$ .

Fault detection:

(1): According to the PCA model in data preprocessing, the model residuals $R_{1}$ and $R_{2}$ under normal working conditions, and the model residual $F_{i}$ under fault conditions are calculated separately.
(2): Set $R_{1}$ as $x_{n}$ .
(3): Design a moving window of sample size m on both $R_{2}$ and $F_{i}$ , and set the corresponding residual data sets as $y_{n}$ and $y_{f}$ , respectively.
(4): Calculate the $W D_{n}$ between $x_{n}$ and $y_{n}$ for each sample according to Equation (9).
(5): Set the threshold $h = μ_{W D_{n}} + σ_{W D_{n}}$ , where $μ_{W D_{n}}$ and $σ_{W D_{n}}$ are the mean and standard deviation of $W D_{n}$ under normal operating conditions, respectively.
(6): Similarly, calculate the $W D$ between $x_{n}$ and $y_{f}$ for each sample according to Equation (9).
(7): If $W D > h$ , it indicates a fault case; otherwise, it indicates a normal case.

Fault classification:

(1): The PCA model $X_{f c}$ on the faulty data in the data preprocessing part will be divided into k groups, and the variance of each group of data will be calculated to obtain a new data set ${\hat{X}}_{f c}$ . This data set will be used to train a multi-classification model later.
(2): Since SVM needs to consider the following two factors when determining the hyperplane, SVM classification can be formulated as a QP problem, as shown in Equation (15).
(3): According to the Lagrange multiplier method, the KKT conditions can be obtained. Using these conditions, the QP problem can be transformed into a dual problem as shown in Equation (14).
(4): Given the significant influence of the slack variable $ξ_{i}$ and penalty factor C on the classification performance, cross-validation is employed to identify the optimal combinations of $ξ_{i}$ and C.
(5): By substituting the obtained $ξ_{i}$ and C in the classification hyperplane, the final decision function of the SVM can be obtained as shown in Equation (16).
(6): Finally, the fault type of each sample can be determined using the voting method.

4. Experimental Verification

4.1. Simulation Experiment Platform

The simulation data of the CRH suspension system include incipient damper and spring faults, actuator faults, and sensor faults in the secondary suspension. These data are derived from the Simpack2018.1-Matlab2016a/Simulink co-simulation platform. The wheelset model, train bogie, and vehicle body are constructed step by step using the dynamics simulation software Simpack2018.1.

Since the ‘Rail_Wheelset’ model included in the Simpack2018.1 software is a nonindependent wheelset model, and in most cases, the bogie of the car body utilizes an independent wheelset, engineers are required to establish an independent wheelset model. The following are the relevant steps with parameter settings shown as in Figure 3, Figure 4, Figure 5 and Figure 6 to create an independent wheelset model.

Step 1: Start by selecting the Rail-Track model when building the simulation. Subsequently, define the mass characteristics and moment of inertia of the car body based on the actual parameters.

Step 2: Proceed to build and visualize the axles. Set the geometry type to cylinder (type #2) and define the height, outer diameter, and number of planes using axis-dependent variables. The smoothness of the cylinder is affected by the number of planes; more planes result in a smoother cylinder. Next, create mark points on both the left and right sides of the axle to ensure accurate axle positioning and wheel–rail interaction.

Step 3: Adjust the joint status. In this article, the wheelset joint employs the General Rail Track Joint (type #7), and the z-position of the initial jointed state should correspond to the specified wheel radius in the wheel–rail interaction.

Step 4: Set up the track elements. Create track elements and configure them for dual rail mode. Then, adjust the left and right track profiles, track slopes, discrete steps, and lateral rail positions. Furthermore, the calculation of tangential forces and moments at the wheel–rail contact points is determined by the wheel–rail contact elements.

Step 5: Set up the wheel–rail pair. Please note that only objects defined using Track Joint #07 and #09 Joint can be selected here. Start by creating a wheel–rail pair and selecting the wheel frame object previously defined within the General tab. Next, use the parameters under the Wheel label to specify the wheel face, nominal wheel radius, and wheel lateral distance. In this model, wheel flat scars are disregarded. Navigate to the Rail tab, select the track element previously defined, and choose the default track mounting method, which is the inertia-fixed track method. Under the Contact Normal Force tab, employ the default contact search method (Equivalent elastic). Define the distance between the wheels and select the Hertzian method to calculate the normal force. Subsequently, define the material parameters necessary for calculating both the normal and tangential forces based on the actual parameters. The contact reference damping value is defined using the corresponding variable, which varies with the square root of the contact stiffness to maintain the natural damping constant. Under the Tangential Forces tab, select the wheel–rail contact element defined earlier and use the corresponding variables to set the coefficient of friction.

Once all the parameters are set, visualize the settings and apply the same principles to the other wheel–rail interaction. Then, create wheelset elements to connect the two wheel–rail interactions.

In this study, the cartographic approach is utilized due to the unavailability of measure files. During inward turns on the railway, the curved track is typically tilted around the x-axis. This tilt is designed to counteract the radial outward centrifugal force experienced during the turn by increasing the force acting perpendicular to the track. As a result, it enhances vehicle stability and reduces the risk of rollover. At the same time, the line tilt is defined as ‘ultra-high’. In Chinese railways, inner rail ultra-high is commonly employed. In addition to setting the ultra-high for the railway line, it is necessary to establish horizontal and vertical curves. Ramps and transitions are crucial for ensuring a smooth connection between different sections of the line, such as between two ultra-high curves or from a straight section to a curved one. Furthermore, the established model is shown in Figure 7.

When creating a new model, select the ‘Rail_Track’ component. Define the mass and shape of the bogie based on the train parameters. Import the pre-established wheelset model as a substructure and adjust its position accordingly. Subsequently, establish the bearing based on the marker points of the wheelset and determine the bearing marker point. The bearing can be simulated using the axle positioning force element. In this study, Force Element #86 represents the primary springs, and Force Element #6 represents the single-stage vertical shock absorbers. When modeling the secondary suspension, a virtual car body is introduced, which has no length or mass, as the actual automobile body model has not yet been constructed. This virtual car body is used to mimic the presence of the actual car body and facilitate the modeling process of the secondary suspension. In the secondary suspension modeling, Force Element #79 represents the air springs, Force Element #6 represents the lateral shock bumpers and anti-roll bars, Force Element #13 represents the vertical shock absorbers, and Force Element #5 represents the shock absorbers. The established bogie model is shown in Figure 8.

When creating a new model, select the ‘Rail_Track’ option. Define the overall look, length and mass of the trailer bodies according to the precise parameters of the trains. To connect the virtual bodies of the front and rear bogies, create marker points at the front and rear ends. Next, load the pre-established bogie model and adjust its position accordingly. The established vehicle model is shown in Figure 9.

The details of the fault injections in this paper are shown in Table 1. Each simulation process involves nine monitored variables. When a fault occurs, all these variables are affected, causing corresponding changes in their values. Additionally, the sampling time for each fault process is set to 30 s and is further divided into 30 intervals. A statistic is then calculated for each interval, generating 30 data points for each fault type, which aids in the multi-classification of faults.

4.2. Simulation Results and Analysis

Figure 10, Figure 11, Figure 12 and Figure 13 illustrate the performance of the PRPCA and PCA methods in incipient fault detection for sensors, actuators, secondary dampers, and secondary springs. The solid blue lines and the red dashed lines represent statistical values and corresponding thresholds, respectively. Furthermore, a detailed comparison of the fault detection accuracy between the PRPCA and PCA methods is presented in Table 2, where the detection accuracy of PRPCA for the actuator fault is 0.9093, while the traditional PCA method is 0.8789. Similarly, PRPCA demonstrates higher detection accuracy for the secondary damper and secondary spring faults. The detection accuracy of PRPCA for the sensor fault is 0.6991, while the traditional PCA method is 0.7178. Generally, the PRPCA method shows superior performance in fault detection for the actuator, secondary damper, and secondary spring, while the PCA method performs better in sensor fault detection.

Figure 14, Figure 15 and Figure 16 illustrate the fault recognition results of PCA-SVM and PRPCA-SVM for different fault categories. As shown in Figure 14, both PCA-SVM and PRPCA-SVM methods can achieve a classification accuracy of 100% for fault binary classification. Figure 15 shows that the PRPCA-SVM method achieves a classification accuracy of 95.8% for fault three-classification, while the PCA-SVM method achieves an accuracy of 83.3%. Figure 16 indicates that the PRPCA-SVM method achieves a classification accuracy of 93.1% for fault three-classification, while the PCA-SVM method can only achieve an accuracy of 68.9%. According to the comparison of the evaluation metrics (F1-Measure, as shown in Formula (18)) between PCA-SVM and PRPCA-SVM in Table 3, it can be concluded that the fault diagnosis scheme proposed in this paper exhibits superior performance. Both methods demonstrate low sensitivity to sensor fault detection, leading to a decrease in detection accuracy. However, the algorithm proposed in this paper outperforms the traditional PCA-SVM algorithm in locating the other three typical faults. In multi-fault classification, the proposed incipient fault diagnosis strategy using PRPCA and the “one-vs-one” multi-class SVM classifier with a Gaussian RBF kernel offers better overall performance. It is worth noting that its performance does not degrade as the number of fault types increases.

5. Conclusions

This paper proposes an incipient fault diagnosis scheme based on PRPCA and SVM for application in the CRH suspension system. Through experiments conducted in the Simpack2018.1-Matlab2016a/Simulink co-simulation environment, and by comparing with traditional PCA and SVM-based methods, the results demonstrate that the proposed scheme can effectively detect and recognize actuator fault, sensor fault, secondary spring fault, and secondary damper fault, especially in comprehensive evaluation metrics represented by accuracy and robustness. In summary, the PRPCA-SVM based strategy outperforms traditional methods in incipient fault diagnosis and maintains high recognition accuracy, even when the number of fault types increases. However, the proposed method is not very sensitive to sensor faults, and the performance of sensor fault diagnosis needs to be further improved in our future work.

Author Contributions

Conceptualization, Y.W. and K.F.; methodology, K.F., Y.Z. (Yang Zhou) and Y.Z. (Yijin Zhou); software, K.F.; validation, Y.W. and K.F.; formal analysis, Y.W.; investigation, K.F.; resources, Y.W.; data curation, Y.W. and K.F.; writing—original draft preparation, K.F.; writing—review and editing, Y.W.; visualization, Y.Z. (Yang Zhou) and Y.Z. (Yijin Zhou); supervision, Y.Z. (Yang Zhou) and Y.Z. (Yijin Zhou); project administration, Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grants (62173164, 62203192), in part by the 6th regular meeting exchange program (6-3) of China-North Macedonian science & technology cooperation committee.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

CRH	China Railway High-speed
PRPCA	Probabilistic Relevant Principal Component Analysis
SVM	Support Vector Machine
PCA	Principal Component Analysis
FDD	Fault Detection and Diagnosis
PLS	Partial Least Squares
ANN	Artificial Neural Network
WPT	Wavelet Packet Transform
KPCA	Kernel Principal Component Analysis
MKPCA	Multi-way Kernel Principal Component Analysis
HVCB	High-Voltage Circuit Breaker

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Figure 1. Choice of probability-relevant matrix W.

Figure 2. The flow chart of incipient fault diagnosis scheme based on PRPCA and SVM.

Figure 3. Body properties.

Figure 4. Primitive properties.

Figure 5. Joint properties.

Figure 6. Rail properties.

Figure 7. The wheelset model.

Figure 8. The bogie model.

Figure 9. The vehicle model.

Figure 10. The incipient fault detection comparisons for sensors.

Figure 11. The incipient fault detection comparisons for actuators.

Figure 12. The incipient fault detection comparisons for secondary suspension dampers.

Figure 13. The incipient fault detection comparisons for secondary suspension springs.

Figure 14. The fault classification comparisons between PCA-SVM and PRPCA-SVM with two fault types.

Figure 15. The fault classification comparisons between PCA-SVM and PRPCA-SVM with three fault types.

Figure 16. The fault classification comparisons between PCA-SVM and PRPCA-SVM with four fault types.

Table 1. Fault injections.

Fault Number	Fault Location Description
1	Actuator
2	Secondary suspension damper
3	Secondary suspension spring
4	Sensor

Table 2. The comparisons of fault detection accuracy.

Fault Types	Detection Accuracy
Fault Types	PRPCA	PCA
Actuator fault	0.9093	0.8789
Secondary suspension damper fault	0.9061	0.8693
Secondary suspension spring fault	0.9164	0.8694
Sensor fault	0.6991	0.7178

Table 3. The comparisons of evaluation metrics.

Method	Fault Number	$\hat{R}$	$\hat{P}$	F
PCA-SVM	2	0.7936	1.0	0.8849
	3	0.8220	0.8333	0.8276
	4	0.8338	0.6896	0.7549
PRPCA-SVM	2	0.8076	1.0	0.8935
	3	0.8416	0.9583	0.8961
	4	0.8577	0.9310	0.8928

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Feng, K.; Wu, Y.; Zhou, Y.; Zhou, Y. Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM). Machines 2024, 12, 832. https://doi.org/10.3390/machines12120832

AMA Style

Feng K, Wu Y, Zhou Y, Zhou Y. Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM). Machines. 2024; 12(12):832. https://doi.org/10.3390/machines12120832

Chicago/Turabian Style

Feng, Kang, Yunkai Wu, Yang Zhou, and Yijin Zhou. 2024. "Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM)" Machines 12, no. 12: 832. https://doi.org/10.3390/machines12120832

APA Style

Feng, K., Wu, Y., Zhou, Y., & Zhou, Y. (2024). Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM). Machines, 12(12), 832. https://doi.org/10.3390/machines12120832

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Incipient Fault Detection and Recognition of China Railway High-Speed (CRH) Suspension System Based on Probabilistic Relevant Principal Component Analysis (PRPCA) and Support Vector Machine (SVM)

Abstract

1. Introduction

2. PRPCA-Based Incipient Fault Detection

2.1. Nonlinear PRPCA Method

2.2. The Fault Detection Accuracy

3. SVM-Based Incipient Fault Recognition

3.1. Multi-SVMs Scheme

3.2. The Comprehensive Evaluation Metrics

3.3. The Fault Diagnosis Scheme

4. Experimental Verification

4.1. Simulation Experiment Platform

4.2. Simulation Results and Analysis

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

References

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