An Evidential Reasoning Assessment Method Based on Multidimensional Fault Conclusion

Abstract

The running gear mechanism is a critical component of high-speed trains, essential for maintaining safety and stability. Malfunctions in the running gear can have severe consequences, making it imperative to assess its condition accurately. Such assessments provide insights into the current operational status, facilitating timely maintenance and ensuring the reliable and safe operation of high-speed trains. Traditional evidential reasoning models for assessing the health of running gear typically require the integration of multiple characteristic indicators, which are often challenging to obtain and may lack comprehensiveness. To address these challenges, this paper introduces a novel assessment model that combines evidential reasoning with multidimensional fault conclusions. This model synthesizes results from various fault diagnoses to establish a comprehensive health indicator system for the running gear. The diagnostic outcomes serve as inputs to the model, which then assesses the overall health status of the running gear system. To address potential inaccuracies in initial model parameters, the covariance matrix adaptation evolution strategy (CMA-ES) algorithm is utilized for parameter optimization. Comparative experiments with alternative methods demonstrate that the proposed model offers superior accuracy and reliability in assessing the health status of high-speed train running gear.

1. Introduction

With the swift advancement of modern industrial technology, China’s high-speed trains have experienced six significant upgrades. They now boast the fastest development speed, the strongest integration capabilities, comprehensive system technologies, the longest operating mileage, the largest scale under construction, and the highest operating mileage globally. The running gear, being the central element that influences the performance and velocity of high-speed trains, is vital for guaranteeing their smooth functioning. With the increasing mileage of high-speed trains, the components of the running gear are prone to damage and aging. If not promptly identified and addressed, these issues can cause train stoppages and, in severe cases, lead to major traffic accidents. Therefore, it is essential to monitor the running gear mechanism’s health condition in a timely manner, conduct necessary maintenance and repairs, and ensure the optimal performance of all its components.

A health status assessment involves using reliable and effective assessment methods and comprehensive data to judge the system’s health status and understand its condition in real time. Existing health status assessment methods are categorized into qualitative knowledge-based, data-driven, and semi-quantitative information-based methods.

(1) Methods Based on Qualitative Knowledge

This category of methods primarily relies on expert experience, domain knowledge, and other non-quantitative functions to perform health status assessments. Examples include expert knowledge systems [1], Petri nets [2], and fault tree analysis [3]. Hou et al. [4] addressed the issue of performance degradation in quasi-Monte Carlo methods when dealing with high-dimensional problems by proposing a quasi-Monte Carlo reliability assessment method based on the importance of dimensions. These methods require only a small amount of data to conduct system assessments and possess the ability to handle uncertain information with a highly interpretable reasoning process.

However, these methods depend heavily on expert experience and knowledge. The system’s status cannot be effectively evaluated when expert knowledge is limited. Additionally, the lack of comprehensive expert knowledge can ultimately affect the assessment’s accuracy when dealing with a completely new system. Monitoring data are crucial for directly reflecting the system’s health status. Therefore, a combination of qualitative knowledge and appropriate data is necessary to assess the system’s health status accurately.

(2) Methods Based on Data-Driven Approaches

There are many data-driven assessment methods, including principal component analysis (PCA) [5,6], support vector machines (SVMs) [7,8], neural networks [9,10,11], and so on. This category of methods primarily utilizes extensive monitoring data from complex engineering systems to establish assessment models. For instance, Zhang et al. [12] addressed the issues of data imbalance and a lack of labeled data in the operational condition of high-speed train suspension systems. They introduced a hybrid approach combining deep transfer learning and graph neural networks, integrating information from multiple sensors. Zhang et al. [13] proposed a support vector machine optimized through a combination of the whale optimization algorithm and adaptive weight adjustment to improve the accuracy of network security situational assessments. Similarly, Xiao et al. [14] tackled the issues of high labor costs, inefficiency, and accuracy challenges in the health management of industrial robot mechanical axes by introducing a health assessment and status prediction model, leveraging the hidden Markov model and temporal convolutional network. Ana et al. [15] introduce a novel approach to constructing health indicators using the latent reconstruction error from a deep autoencoder. By leveraging the latent space’s compact and disentangled data representations, this method offers a more robust and accurate measure of machine health compared to traditional reconstruction errors. Although some success has been achieved, autoencoders are considered black-box models, which lack sufficient interpretability when handling complex systems, particularly in applications involving intricate systems like high-speed train running gear.

Traditionally, these methods require extensive monitoring data to achieve accurate health status assessments. However, obtaining such large datasets can be challenging and may limit the applicability of these approaches. Nonetheless, there are emerging methods designed to perform well even with limited data [16,17,18]. For example, recent work by Magadán et al. [18] introduced TabPFN, an approach specifically aimed at early fault classification in rotating machinery with limited data. This method shows that effective assessments can be made with a minimal amount of data, challenging the notion that extensive data are always necessary. As such, it is crucial to consider both data-rich and data-limited scenarios when evaluating the effectiveness of data-driven methods.

(3) Methods Based on Semi-Quantitative Information

These methods successfully combine quantitative data with qualitative knowledge, enabling the representation of various uncertainties. Zhang et al. [19] tackled the challenge of inaccurate expert-provided parameters in traditional Belief Rule-Based (BRB) models and the possible decline in model interpretability post-optimization. A new approach is presented for assessing the health condition of intricate systems. Yin [20] highlighted the absence of adequate methods to improve interpretability during the optimization stage of BRB models and the inclination of BRBs to rely on black-box models that prioritize accuracy alone. A new interpretable health status assessment method was proposed. Zhang et al. [21] tackled the reliability assessment problem in train control and management systems by proposing an evidential reasoning (ER) rule-based model. This model boasts strong reliability modeling capabilities, interpretable assessment processes, and the ability to describe assessment results under probabilistic and ignorance uncertainties, demonstrating robust performance. Wang et al. [22] noted that existing evidential reasoning models typically set the evidence reference point as a quantitative value or interval value, making it challenging to handle both fuzzy and probabilistic uncertainties simultaneously. A distributed reference point aerospace relay performance assessment model based on evidential reasoning rules is proposed.

Evidential reasoning is a typical semi-quantitative information method that leverages various quantified uncertain information and expert knowledge. It has performed well in non-linear health status assessments of complex electromechanical systems. Assessing the running gear system requires monitoring multiple indicators to reflect the system’s status accurately. Evidential reasoning can effectively integrate multi-source information indicators, achieving a thorough assessment of the running gear system’s health condition. Its unique feature lies in fully considering the weight and reliability of each evidence, enabling the handling of fuzziness, uncertainty, and incompleteness while effectively utilizing qualitative knowledge. This is particularly beneficial for welding robots, which often lack sufficient data.

Despite significant achievements, the complex and highly interconnected structure of high-speed train running gear systems poses challenges in obtaining comprehensive health indicator data. Additionally, such data often lack sufficient detail to reveal the system’s failure modes, complicating the assessment process. Traditional ER assessment methods typically integrate multiple indicators [23]. However, certain indicators, such as fault indicators, may more objectively reflect the running gear’s overall health status than the system’s own indicators. This approach addresses the challenge of obtaining comprehensive health indicators, making the assessment objective and simple. Therefore, a multidimensional fault conclusion-based ER rule model is introduced, utilizing fault diagnosis outcomes as model inputs to perform a thorough assessment of high-speed train running gear.

The primary contributions of this paper are as follows:

• The proposed method includes both the running gear system’s indicators and those related to fault conclusions. This approach enables a more objective assessment of the system’s overall health status.
• Applying the ER method in engineering is extended by proposing a new system health status assessment approach. This method enables health status assessment not only through health indicators but also based on fault diagnosis results, making the process both convenient and objective.

The rest of this paper is structured as follows: Section 2 discusses the basic concepts of ER and the challenges in assessing running gear health status. Section 3 outlines the ER assessment method based on fault conclusions. Section 4 showcases practical case studies to demonstrate the model’s effectiveness. Section 5 concludes with a summary of the research.

2. ER Basic Knowledge and the Health Status Assessment Problem Description

To fully understand and apply the ER method for assessing the health status of high-speed train running gear, this section provides a detailed description of the foundational concepts of ER and the specific health assessment challenges it addresses. The ER method introduces belief rules, offering a robust framework for handling uncertainty and multi-attribute decision analysis. While it builds upon the core ideas of the Dempster–Shafer (D-S) evidence theory, ER extends these concepts to suit more complex and diverse decision-making scenarios. In our study, the ER method will be used to develop a multidimensional health status assessment model, overcoming the limitations of traditional assessment methods and providing a more accurate reflection of the comprehensive health condition of high-speed train running gear. The following subsections will elaborate on the fundamental knowledge of ER and its application in health status assessments.

2.1. Basic Knowledge

ER is built on the D-S evidence theory and enhances its reasoning mechanism, making it suitable for multi-criteria decision analysis (MCDA) problems. It employs a unified belief rule framework that provides an effective solution for addressing uncertainty issues in multi-attribute decision-making processes. By introducing the concept of belief rules, ER transforms subjective uncertainty in the decision-making process into quantifiable and computable information, allowing decision makers to more accurately assess the importance of various attributes. Compared to the traditional analytic hierarchy process (AHP), this method demonstrates higher efficiency and significantly improved rationality in handling multi-attribute decisions.

ER is a reasoning method developed from the D-S evidence theory, inheriting the core ideas of the D-S theory while incorporating additional concepts on handling uncertainty. This method represents an important branch within the MCDA field, specifically designed to address complex decision problems involving multiple attributes and various possible outcomes. By skillfully utilizing belief rules, ER effectively quantifies and integrates these uncertainties, providing more accurate and reliable decision support. For example, AHP’s limitations in effectively and rationally handling uncertain information are addressed by the ER method.

The ER approach is extensively utilized in decision making, risk analysis, and managing the system’s health status. This paper applies the ER approach to evaluate the health condition of high-speed train running gear.

2.2. Description of the Health Status Assessment Problem

Problem 1: In assessing the health condition of high-speed train running gear, indicators such as temperature, pressure, vibration, and ultrasound [24,25] can be utilized to reflect the condition of the running gear. Ideally, more health status indicators should be integrated to establish a more comprehensive and accurate health status assessment model. Traditional assessment models build an indicator system through the object’s ontology to reflect the object’s health level. However, in practical scenarios, other indicators, such as fault indicators, may more objectively reflect health levels than the object. This approach addresses the challenge of obtaining comprehensive health indicators, allowing for a more convenient and objective reflection of the running gear’s health status.

Problem 2: Fault indicators need to be obtained through fault diagnosis. However, existing fault diagnosis models yield one-dimensional results, which are insufficient for observing the health status of the running gear from multiple dimensions. Therefore, this paper establishes a multidimensional fault conclusion health status indicator system, which includes fault types and fault severity, and uses the obtained fault results as the input of the model to achieve a comprehensive assessment of the running gear of high-speed trains. Figure 1 shows the health status assessment indicator system of the running gear constructed in this paper.

3. An Assessment Method of Evidence Reasoning Based on Fault Conclusion

This study begins by presenting the ER assessment model grounded in fault conclusions. Subsequently, it details the model’s objective function and outlines the optimization procedure using the CMA-ES algorithm.

3.1. Assessment Model of ER Based on Fault Conclusion

Before conducting ER and evidence fusion, quantitative data must be transformed into belief distributions relative to reference levels. Assume the p-th reference value of evidence (also attribute) $z_j$ is represented as $z_j$ and $g_{p+1,j}\ge g_{p,j}$, with $g_{M,j}$ being the maximum and $g_{1,j}$ the minimum reference value. Following the acquisition of the initial reference value, the quantitative data can be translated into a belief distribution in the following manner:

$$T\left(z_j\right)=\{(g_{p,j},{\gamma }_{p,j}),p=1,2,\dots ,M\},$$

(1)

where ${\gamma }_{p,j}={\displaystyle \frac{g_{p+1,j}-z_j}{g_{p+1,j}-g_{p,j}}}$, ${\gamma }_{p+1,j}=1-{\gamma }_{p,j}$, $g_{p,j}\le z_j\le g_{p+1,j}$, ${\gamma }_{q,j}=0$, $q=1,2,\dots ,M$, $q\ne p,p+1$.

Quantitative attributes are represented using the following distribution:

$$T\left(z_j\right)=\{(G_{p,j},{\gamma }_{p,j}),p=1,2,\dots ,M\},$$

(2)

where ${\gamma }_{p,j}\ge 0$, ${\displaystyle \sum _{p=1}^M}{\gamma }_{p,j}\le 1$, and ${\gamma }_{p,j}$ represent the belief that the $z_j$ is assessed as the reference level $G_{p,j}$. $T\left(z_j\right)$ denotes the belief distribution of the $z_j$. For quantitative attribute $z_j$, let $n_{p,j}$ represent the basic belief value that the individual is assessed at $G_p$. $n_{G,j}$ denotes the basic belief value that is not allocated to $G_p$, that is,

$$n_{p,j}={\lambda }_j{\gamma }_{p,j},p=1,2,\dots ,M$$

(3)

where ${\lambda }_j$ is the weight of the $z_j$, as well as $0\le \lambda \le 1$, ${\displaystyle \sum _{j=1}^K}{\lambda }_j=1$.

$$n_{G,j}=1-\sum _{p=1}^M{n}_{p,j}=1-{\lambda }_j\sum _{p=1}^M{\gamma }_{p,j}$$

(4)

In the ER method, $n_{G,j}$ can be divided into two components: ${\tilde{n}}_{G,j}$ and ${\overline{n}}_{G,j}$, as follows:

$${\tilde{n}}_{G,j}={\lambda }_j\left(1-\sum _{p=1}^M{\gamma }_{p,j}\right)$$

(5)

$${\overline{n}}_{G,j}=1-{\lambda }_j$$

(6)

where ${\tilde{n}}_{G,j}$ represents the degree of incompleteness in assessing attribute $z_j$. The belief distribution is deemed complete when there is no unassigned belief, indicated by a value of 0, ${\tilde{n}}_{G,j}=0$. ${\overline{n}}_{G,j}$ signifies the remaining basic belief assignment.

Using the ER method to aggregate the first j attributes and the $(j+1)$-th attribute, the belief assignment value at assessment level $G_p$ is

$$\left\{G_p\right\}:n_{p,J\left(j+1\right)}=Q_{J\left(j+1\right)}\left[n_{p,J\left(j\right)}{n}_{p,j+1}+n_{G,J\left(j\right)}{n}_{p,j+1}+n_{G,j+1}{n}_{p,J\left(j\right)}\right]$$

(7)

$$n_{G,J\left(j\right)}={\overline{n}}_{G,J\left(j\right)}+{\tilde{n}}_{G,J\left(j\right)}$$

(8)

$$\left\{G\right\}:{\tilde{n}}_{G,J\left(j+1\right)}=Q_{J\left(j+1\right)}\left[{\tilde{n}}_{G,J\left(j\right)}{\tilde{n}}_{G,j+1}+{\tilde{n}}_{G,J\left(j\right)}{\overline{n}}_{G,j+1}+{\overline{n}}_{G,j+1}{\tilde{n}}_{G,J\left(j\right)}\right]$$

(9)

$$\left\{G\right\}:{\overline{n}}_{G,J\left(j+1\right)}=Q_{J\left(j+1\right)}\left[{\overline{n}}_{G,J\left(j\right)}{\overline{n}}_{G,j+1}\right]$$

(10)

$$Q_{J\left(j+1\right)}={\left[1-\sum _{s=1}^M\sum _{\begin{array}{c}r=1\\ r\ne s\end{array}}^M{n}_{r,j+1}{n}_{s,J\left(j\right)}\right]}^{-1},j=1,\dots ,K-1$$

(11)

When integrating all attributes through the iterative algorithm, the belief is distributed as follows:

$$\left\{G\right\}:{\widehat{\gamma }}_p={\displaystyle \frac{n_{p,J\left(K\right)}}{1-{\overline{n}}_{G,J\left(K\right)}}},(p=1,\dots ,M)$$

(12)

$$\left\{G\right\}:{\widehat{\gamma }}_G={\displaystyle \frac{{\tilde{n}}_{G,J\left(K\right)}}{1-{\overline{n}}_{G,J\left(K\right)}}}$$

(13)

where ${\widehat{\gamma }}_p$ indicates the belief level at which an attribute is assessed at the reference level $G_p$. $G_p$ and ${\widehat{\gamma }}_G$ denote the belief allocated to the global level, with $[{\gamma }_p,({\gamma }_p+{\gamma }_G)]$ representing the belief range.

The overall assessment z can be illustrated through the following belief distribution:

$$T\left(z\right)=\{(G_p,{\widehat{\gamma }}_p),p=1,\dots ,M\}$$

(14)

Next, the health status of the high-speed train running gear is calculated using the following formula:

$$X\left(z\right)=\sum _{p=1}^M{G}_p\times {\widehat{\gamma }}_p$$

(15)

where M represents the number of levels in the assessment result, and $X\left(z\right)$ represents the utility value of individual z after the iterative fusion of K attributes.

3.2. Objective Function of the ER Model

The core idea behind the model design is to use monotonicity and trend indicators as evaluation criteria for quantitatively analyzing performance degradation from a statistical perspective. These indicators were chosen because monotonicity captures the irreversible changes in component degradation, while trend indicators effectively reflect the correlation between degradation and time. This approach is crucial for accurately assessing and predicting the health status of high-speed train running gear under varying operating conditions, providing a solid scientific basis for subsequent maintenance and repair strategies. In this way, it is possible to better understand and predict the degradation trends of mechanisms under different conditions, thus providing a scientific basis for subsequent maintenance and repair.

(1) Monotonicity Indicator

In actual engineering, the degradation of running gear components is irreversible, characterized by a continuous increasing or decreasing trend in degradation features. This characteristic is known as monotonicity. The monotonicity indicator is defined by the following formula:

$$\mathrm{Mon}\left(G_j\right)=\left|{\displaystyle \frac{\mathrm{Num}\mathrm{of}\mathrm{d}{G}_j>0}{R_j-1}}-{\displaystyle \frac{\mathrm{Num}\mathrm{of}\mathrm{d}{G}_j<0}{R_j-1}}\right|$$

(16)

where $d{G}_j$ denotes the difference in the feature sequence values, and $\mathrm{Num}\mathrm{of}\mathrm{d}{G}_j>0$ and $\mathrm{Num}\mathrm{of}\mathrm{d}{G}_j<0$ represent the number of positive and negative values in the feature sequence, respectively, and $R_j$ is the total number. $Mo{n}_j$ ranges from 0 to 1, with larger values indicating better monotonicity.

(2) Trend Indicator

As the service life increases, the running gear gradually deteriorates, a characteristic known as “trend”. The trend indicator assesses the correlation between the progression of degradation and time. The formula for calculating the trend indicator is as follows:

$$\mathrm{Corr}\left(z_q\right)={\displaystyle \frac{\left|{\displaystyle \sum _{q=1}^{R_q}}\left(z_q-\overline{z}\right)\left(m_q-\overline{m}\right)\right|}{\sqrt{{\displaystyle \sum _{q=1}^{R_q}}{\left(z_q-\overline{z}\right)}^2{\displaystyle \sum _{q=1}^{R_q}}{\left(m_q-\overline{m}\right)}^2}}}$$

(17)

where $z_q$ represents the health feature value in the q-th cycle, $\overline{z}$ denotes the average health feature value, $m_q$ denotes the index of the q-th sampling point, $\overline{m}$ represents the mean value of the sampling point indices, and $R_q$ indicates the total number of cycles. Using Formulas (16) and (17), the objective function for optimizing the parameters of the ER assessment model is constructed.

$$\begin{array}{c}max\left\{\mathrm{Mon}(\Phi )\right\}\\ max\left\{\mathrm{Corr}(\Phi )\right\}\\ \mathrm{s}.\mathrm{t}.0\le {\tilde{v}}_j\le 1,j=1,2,\cdots ,R\\ \Phi =\left\{{\tilde{v}}_1,{\tilde{v}}_2,\cdots ,{\tilde{v}}_R\right\}\end{array}$$

(18)

3.3. Introduction to the CMA-ES Optimization Algorithm

This paper utilizes an adaptive ER rule model optimized by the CMA-ES. CMA-ES demonstrates outstanding performance in handling non-convex optimization problems and high-dimensional parameter spaces [26]. It excels in exploring complex solution spaces more comprehensively, particularly when the objective function has multiple local optima, thereby facilitating the identification of global optima. Additionally, CMA-ES efficiently navigates and converges in high-dimensional spaces through adaptive adjustments of the covariance matrix, making it particularly effective for optimizing complex parameters in the ER model. The main purpose of the CMA-ES algorithm is to fine-tune the optimization parameters, especially the step size and covariance matrix, to enhance performance. These adjustments to the optimization parameters play a crucial role in the swift convergence of evolutionary algorithms.

The core principle of the CMA-ES algorithm in parameter optimization is to adjust the covariance matrix, which indirectly guides the evolutionary path across the search space, thus increasing the probability of finding optimal solutions. The objective function in the adaptive ER algorithm model has two components: the monotonicity coefficient (${\mathrm{Mon}}_j$) indicator and the correlation coefficient (${\mathrm{Corr}}_j$) indicator. Initially, all equality conditions are transformed into objective functions. For solution $\Phi$, multiple independent objective functions can be formulated, enabling the CMA-ES algorithm to address each one separately. When using the CMA-ES algorithm to address the equality constraint conditions independently, the following conditions should also be met:

$$\begin{array}{cc}\mathrm{if}\hfill & {\tilde{v}}_j\ge 1\hfill \\ \hfill & {\tilde{v}}_j=1\hfill \\ \mathrm{if}\hfill & {\tilde{v}}_j\le 0\hfill \\ \hfill & {\tilde{v}}_j=0\hfill \end{array}$$

(19)

These conditions ensure that the parameters ${\tilde{v}}_j$ in solution $\Phi$ meet the inequality constraints in the ER model.

This optimization algorithm draws on the concept of multi-objective optimization algorithms, converting each constraint into a specific unconstrained objective function. Each iteration independently seeks the optimal value, ensuring the results fully comply with the established constraints, thereby efficiently solving the model optimization problem. In particular, the execution of this algorithm can be broken down into the following essential steps.

Step 1: Sampling Operation

The sampling operation focuses on selecting a solution within the solution space (comprising all ER parameter vectors $\Phi$) as the central point and creating an initial population based on a normal distribution. This population is denser near the central point and becomes sparser with increasing distance. In the CMA-ES algorithm, this central point represents the expectation. Here, the initial parameter vector ${\Phi }^0$ of the ER model is chosen as the expected value. The population generation method is as follows:

$$\begin{array}{c}{\Phi }_k^{g+1}\sim mea{n}^g+{\eta }^g\mathbb{N}\left(0,{\mathbf{C}}^g\right),k=1,2,\cdots ,\lambda \\ mea{n}^0={\Phi }^0\end{array}$$

(20)

where ${\Phi }_k^{g+1}$ denotes the k-th solution in the population, g represents the evolutionary generation, $mea{n}^g$ denotes the center expectation of the g-th generation population, $\eta$ represents the step size for distribution update, $\mathbb{N}\left(0,{\mathbf{C}}^g\right)$ represents the multivariate normal distribution, and ${\mathbf{C}}^g$ denotes the covariance matrix (positive definite matrix) in the g-th generation’s search distribution. Geometrically, this corresponds to the elliptical distribution of the population in the solution space.

Step 2: Selection Operation

The purpose of this operation is to select solutions with higher fitness values from a pool of potential optimal candidates. Selection is determined by the fitness function value; a higher value indicates a better solution. Additionally, to ensure the diversity and evolutionary efficiency of the population, a weighted averaging method is used to reorder the newly selected solutions. This method calculates the fitness values of each solution and assigns different weights to achieve balance and diversity within the population.

Step 3: Recombination Operation

This operation focuses on updating the population’s mean, guiding it towards the optimal solution, and evolving closer to the optimum as a new population is generated. The population is updated in the following way:

$$mea{n}^{g+1}=\sum _{i=1}^{\epsilon }{\gamma }_i{\Phi }_{i:\lambda }^{g+1}$$

(21)

where ${\gamma }_i$ is the weight of the i-th solution, and their sum equals 1. $\lambda$ is the total number of solutions in the population, and ${\Phi }_{1:\lambda }^{g+1}$ is the i-th solution in the $\lambda$ solutions of the $(g+1)$-th generation.

Step 4: Updating Operation for $\mathbf{C}$

The matrix $\mathbf{C}$ denotes the equiprobable density ellipsoid surface of the population distribution within the solution space. The initial covariance matrix ${\mathbf{C}}^0$ is the identity matrix $\mathit{I}$, corresponding to an equiprobable density surface of the population as a unit sphere. During the evolutionary process, the long axis of the covariance matrix always points towards the optimal solution. The change in direction controls the evolutionary trend of the population, while the length of the long axis controls the search range of the population. The $\mathbf{C}$ is updated using the following formula:

$$C^{g+1}=\left(1-a_1-a_{\epsilon }\right)C^g+a_1{q}^{g+1}{\left(q^{g+1}\right)}^{\mathrm{T}}+a_{\epsilon }\sum _{i=1}^{\epsilon }{\gamma }_i\left({\displaystyle \frac{\left({\Phi }_{1:\lambda }^{g+1}-mea{n}^g\right)}{{\eta }^{\mathrm{g}}+1}}\right){\left({\displaystyle \frac{\left({\Phi }_{1:\lambda }^{g+1}-mea{n}^g\right)}{{\eta }^{\mathrm{g}}}}\right)}^{\mathrm{T}}$$

(22)

where $a_1$ and $a_{\epsilon }$ represent the learning factors, and q represents the evolution path with an initial value of 0. The update rule is as follows:

$$q^{g+1}=\left(1-a_q\right)q^g+\sqrt{a_q\left(2-a_q\right){\left({\sum _{i=1}^{\epsilon }{\gamma }_i}^2\right)}^{-1}}{\displaystyle \frac{mea{n}^{g+1}-mea{n}^g}{{\eta }^g}}$$

(23)

Here, $a_q\le 1$ represents the evolution path parameter. The step size $\eta$ is updated with the following formula:

$${\eta }^{g+1}={\eta }^g\exp \left({\displaystyle \frac{a_{\eta }}{d_{\eta }}}\left({\displaystyle \frac{∥q_{\eta }^{g+1}∥}{E∥\mathbb{N}\left(0,\mathit{I}\right)∥}}-1\right)\right)$$

(24)

Here, $d_{\eta }$ is the damping factor, $E∥\mathbb{N}\left(0,\mathit{I}\right)∥$ is the expected value of the Euclidean norm $∥\mathbb{N}\left(0,\mathit{I}\right)∥$, $\mathit{I}$ stands for the identity matrix, $q_{\eta }$ represents the conjugate evolution path, and $a_{\eta }$ is the parameter for the conjugate evolution path. $q_{\eta }$ is updated using the following formula:

$$q_{\eta }^{g+1}=\left(1-a_{\eta }\right)q_{\eta }^g+\sqrt{a_{\eta }\left(2-a_{\eta }\right){\left({\sum _{i=1}^{\epsilon }{\gamma }_i}^2\right)}^{-1}}{\mathbf{C}}^{\left(g\right)-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{mea{n}^{g+1}-mea{n}^g}{{\eta }^g}}$$

(25)

Repeat the above steps until the accuracy requirement is met, then output the optimal parameters ${\Phi }_{\mathrm{best}}$.

3.4. Health Status Assessment Process for Running Gear

The assessment of the health status of the high-speed train running gear system is carried out through the following seven steps:

Step 1: Define the research problem. Obtain the multi-source data $(z_p,z_q,p=1,2,3$, $q=1,2,3)$ for the assessment model based on fault diagnosis. $z_p$ represents the types of motor faults occurring in the running gear components, and $z_q$ represents the severity of the motor faults in the running gear.

Step 2: Based on the previous mechanistic analysis, select the data of the motor components that significantly impact the running gear as input for the assessment model.

Step 3: Determine the optimization algorithm and objective function. Select the $Mo{n}_j$ and the $Cor{r}_j$ of the health assessment curve as the objective functions. Utilize the CMA-ES optimization algorithm to determine the optimal weight parameters that correspond to the global maxima of the monotonicity and correlation coefficients.

Step 4: Use Formulas (1) to (15) to integrate different attributes and calculate the utility assessment values of the attributes, i.e., the assessment values of the running gear’s health status.

Step 5: Apply the ER model to calculate the utility values across the time series, which allows for the derivation of health status assessment values for the running gear. The health status assessment curve can then be plotted by connecting these points over time. Figure 2 illustrates the flowchart of the health assessment model.

4. Simulation Experiment and Analysis

This paper validates the proposed method on the problem of lack of multi-source feature indicators and difficulty in obtaining feature indicators in a health status assessment by using a case study on a high-speed train’s running gear. Specifically, it examines the health status of the CRH5 model’s running gear. Figure 3 shows the actual CRH5 train’s running gear.

This study utilizes data from a fault log provided by a company on 27 July 2017, focusing on the temperature fault data of small motors in the running gear. By conducting a comparative analysis of the frequency, causes, and impact range of motor temperature faults during this period, the obtained fault diagnosis results, including data on fault types and fault severity, are used for the health status assessment of the running gear. This analysis further validates the accuracy and effectiveness of the proposed evidence reasoning assessment model based on multidimensional fault conclusions in evaluating the health status of the running gear.

The health status classification is grounded in the expertise and experience of technicians and experts who have established a normalization process that maps real-time data, such as temperature, into a health index value between 0 and 1. This normalization process uses predefined thresholds based on historical data and expert knowledge. According to technicians and experts, the health status is categorized into four levels: a value between 0 and 0.3 indicates a healthy status, 0.3 to 0.6 signifies a degraded status, 0.6 to 0.8 represents an abnormal status, and 0.8 to 1 denotes a failed status. The descriptions corresponding to each status of the running gear are shown in

Table 1. Health grade of running gear.

Level	Health Level	Interval Value	Status Description
1	Healthy	[0, 0.3)	The system is in optimal condition.
2	Degraded	[0.3, 0.6)	Some signs of degradation are present, indicating potential need for maintenance.
3	Abnormal	[0.6, 0.8)	The health status has deteriorated, signaling urgent maintenance needs.
4	Failed	[0.8, 1]	The system has failed and requires immediate action.

.

The running gear’s health status is evaluated using the established indicator system and collected fault data. The fault data are fed into the ER model, and the model parameters are optimized using the CMA-ES algorithm. The final outcomes are illustrated in Figure 4.

Figure 4 illustrates that as the number of samples increases over time, the health status of the running gear progressively declines from an initially healthy status to a failure status. This trend aligns with real-world engineering observations, suggesting that the model accurately assesses the running gear’s health status. When the health condition assessment value degrades to a certain extent, maintenance personnel can develop appropriate maintenance plans to ensure running gear’s safe and reliable operation.

To verify the proposed model’s effectiveness, it is compared with several other models, including back propagation neural networks (BPNNs), convolutional neural networks (CNNs), and fuzzy C-means (FCMs). The comparative results are shown in Figure 5, where all methods utilize the same data samples. The initial weight parameters for the optimized ER method proposed in this paper are set to $\omega 1=\omega 2=\omega 3=1$.

The BPNN parameters are set with 100 iterations. The CNN parameters are set with four convolutional layers, where the number of neurons is 8, 16, 32, and 32, respectively, with two average pooling layers and one fully connected layer. In the FCM method, the fuzziness index is set to 2, the maximum number of iterations to 100, and the minimum improvement value to 1 × 10⁻⁶.

From Figure 5, it is evident that the health status assessment value for the high-speed train running gear gradually increases from 0 to 1, indicating a deterioration in the health condition, which aligns well with the fault diagnosis results at the corresponding time points. Additionally, the experimental results obtained from our model are more consistent with the fault diagnosis results. The proposed method’s curve shows a steady upward trend, while the FCM model exhibits significant fluctuations despite being closer in some aspects, making it less reliable. The BPNN and CNN models show greater deviations and minor fluctuations.

This paper uses the monotonicity and correlation coefficients as performance indicators for the assessment model. The detailed results are presented in

Table 2. Experimental results of different methods.

Number	Methods	Correlation Coefficient	Monotonicity Coefficient
1	The method	0.955	0.620
2	BP	0.928	0.373
3	CNN	0.842	0.434
4	FCM	0.868	0.419

. Note that a higher monotonicity coefficient indicates a more significant rising or falling trend in the degradation curve. In contrast, a higher correlation factor signifies a stronger correspondence between the degradation curve and the actual data. Therefore, larger values for these indicators reflect a more effective model.

The proposed method demonstrates significant advantages when analyzing the specific values of the monotonicity and correlation coefficients in Table 2. Among the four methods compared, the proposed method achieves the highest correlation coefficient, indicating a strong alignment between the predicted health status and the actual degradation trend. Additionally, the monotonicity coefficient of the proposed method is notably higher than those of the other methods, reflecting its ability to produce a stable and consistent health assessment curve. In contrast, the other three methods, while achieving relatively high correlation coefficient, suffer from significantly lower monotonicity coefficients. Specifically, the BPNN method has a monotonicity coefficient of 0.373, the CNN method has 0.434, and the FCM method has 0.419. These lower monotonicity values suggest that the assessment curves generated by these methods exhibit intermittent rises and falls, which do not accurately reflect the continuous degradation typically observed in electromechanical equipment. As a result, these methods may not provide reliable assessments of the health status, as they fail to consistently represent the degradation process.

As shown in

Table 3. MSE of different methods.

Number	Methods	MSE
1	The method	0.0219
2	BP	0.0586
3	CNN	0.0478
4	FCM	0.263

, the proposed method demonstrates superior accuracy in the health assessment of high-speed train running gear, with a mean squared error (MSE) of 0.0219, significantly outperforming the other three methods: BPNN, CNN, and FCM. This indicates that the proposed method provides more accurate and reliable predictions of the running gear’s health status. In contrast, the FCM method exhibits the highest MSE, indicating poor performance and insufficient predictive accuracy in this application. Although the BPNN and CNN methods perform relatively well, their accuracy is still inferior to that of the proposed method. Therefore, the proposed method offers a clear advantage in this field, making it a more precise and effective tool for the health assessment of running gear systems.

5. Conclusions

This paper has proposed an ER-based model for assessing the degradation trend and health status of running gear, successfully assessing its condition. The main content is as follows: First, based on the failure mechanism of the running gear, fault type and fault severity indicators were selected as the model’s inputs. Then, the input data of the model were normalized. Next, the ER assessment model was constructed and optimized using the CMA-ES algorithm. By selecting the monotonicity and correlation coefficients as objective functions, the algorithm identified the best weight parameters related to the global maxima of these coefficients, resulting in the model’s optimal parameters. Finally, a case analysis of a specific running gear was conducted. The experimental results revealed that the health status of the running gear deteriorates over time, aligning with real-world conditions. A comparison between the proposed model and other existing models reveals that the proposed model’s correlation coefficient and monotonicity coefficient are closer to 1, indicating superior performance. The lower MSE further supports the effectiveness of the proposed method. This has demonstrated that the ER assessment method based on multidimensional fault conclusions suggested in this study has effectively tackled the issue of assessing the overall health status of running gear, which could not be resolved by solely relying on data-driven or knowledge-driven methods.