Failure Propagation Prediction of Complex Electromechanical Systems Based on Interdependence

Abstract

Interdependence is an inherent feature of the cyber-physical system. Small damage to one component in the system may affect several other components, leading to a series of failures, thus collapsing the entire system. Therefore, the system failure is often caused by the failure of one or more components. In order to solve this problem, this paper focuses on a failure propagation probability prediction method for complex electromechanical systems, considering component states and dependencies between components. Firstly, the key component set in the system is determined based on the reliability measure. Considering the three coupling mechanisms of mechanical, electrical, and information, a topology network model of the system is constructed. Secondly, based on the topology network model and fault data, the calculation method of influence degree between components is proposed. Three state parameters are used to express the risk point state of each component in the system through mathematical representation, and the correlation coefficient between the risk point state parameters is calculated and measured based on the uncertainty evaluation. Then, the influence matrix between the system risk points is constructed, and the fault sequence is predicted by using the prediction function of an Artificial Neural Network (ANN) to obtain the fault propagation probability. Finally, the method is applied to the rail train braking system, which verifies that the proposed method is feasible and effective.

1. Introduction

The modern collaborative electromechanical system is a complex system in which mechanical, electrical, hydraulic, and optical processes are integrated on the same carrier. Since many modern large-scale facilities and equipment belong to complex electromechanical systems, it is very important to predict failure propagation and calculate its probability. Lin et al. [1] proposed a novel framework for assessing the system safety of complex electromechanical systems (CEMSs). From the perspective of system topology, the fault pervasion probability (FPP) is first proposed to analyze fault propagation mechanisms in combination with historical failure data and this approach can easily identify all failure propagation paths. Zhao Yuntian et al. [2] proposed a multi-state health model to predict bearing failures before they occur. The model employs a regression-based method to detect health state transition points and applies an exponential random coefficient model with a Bayesian updating process to estimate time-to-failure distributions. A novel approach that combines technology computer-aided design (TCAD) simulation and machine learning (ML) techniques, is demonstrated to assist the analysis of the performance degradation of GaN HEMTs under hot-electron stress [3].

The tight coupling of the physical and cyber networks is through links and various means of interactions, which makes them intertwined systems with intense interdependencies. Interdependence is a general term that accounts for a relationship among components of a system, where a state of each component influences or is correlated with the state of other components. Interdependence can be categorized into different types and may exist at different levels, i.e., between systems, subsystems, or components [4]. Under the dependent relationship, the system has a polar failure [5,6,7,8,9,10,11].

Lin et al. [5] abstracted the system into a weighted directed network and the topological relationship between nodes are introduced to quantify the initial load and capacity of nodes, and the fault propagation model is constructed to simulate the fault propagation process caused by the failure of a single component. Li et al. [6] introduced the collaborative relationship-aware community in complex networks and establishes a manufacturing service collaboration network model. Then, a variety of fault types are defined, and corresponding detection methods are proposed for the defined faults, revealing the specific cascading propagation characteristics of different faults. Kurzawski et al. [7] proposed a thermal breakdown model in order to design a system to prevent large-scale cascading failures. Liu et al. [8] achieved the goal of a probability-based cascading failure method to quantify the impact of hazards on human errors and determine the relevant propagation paths. Based on a discrete-time model, a failure pattern is proposed wherein the units in a system continuously fail due to cascading events or malevolent attacks [9]. Artime et al. [10] studied the impact of different multiplex topological features on the robustness of the system when subjected to non-local cascade propagation. Li et al. [11] proposed a model that combines two dynamic mechanisms, i.e., the spreading of failure in layer-dependent networks, where each node in a layer depends on one in another layer.

In order to accurately determine the cascading failure, the interdependence and degree between nodes in the system have become a new research point. Aiming at this problem, some of the literature has studied it [4,9,11,12,13]. Koosha Marashi et al. [4] used correlation measures and heuristic causal analysis to identify the interdependence between components of network physical systems. The laboratory tests under long-term hydro-mechanical coupling were carried out based on the cement mortar specimen containing adjacent pre-existing cracks, and the microcrack evolution in the specimen was analyzed by using low-frequency nuclear magnetic resonance technology, the corresponding numerical simulation was carried out to study the propagation and interaction behavior of adjacent cracks and the time-dependent mechanism of creep failure of rocks [12]. Zhang et al. [13] explored the possibility of all failures and quantifies the initiation and extension of failures accordingly.

A large amount of the literature has studied the construction of failure modes [14,15,16] and failure chains [8,17,18,19,20,21,22]. Zhu et al. [14] established a full-stage crack growth rate model of the TBM cutter head through data and main failure modes. Xu et al. [15] presented the fault propagation characteristics of water networks in EIPs. Wang et al. [16] proposed a method to improve the traditional FMEA method by considering the positive and negative effects of failure modes and the attenuation of this effect in the system.

A method of identification of a vulnerable branch considering the propagation characteristics of cascading failures is proposed [17]. Wang et al. [18] proposed an evaluation model for the assessment of N-k contingencies as cascading overloading failures. Bao et al. [19] discussed an integrated simulation approach to simulate the cascading failure propagation process of integrated electricity and natural gas systems (IEGSs). Su et al. [20] proposed a fast real-time probability calculation method for domino fire accidents in storage tank areas. A fault tree model of a blocked transverse drainage pipe in the Chongqing subway tunnel was constructed [21]. Wang Pengjun et al. [22] studied the application of the equipment-status-assessment method based on deep learning in PHM scenarios. After understanding the process of failure propagation, it is also very important to calculate the failure propagation probability to ensure the safe operation of the system [1,8,13,18,20,21].

Based on the above research, we propose a method for predicting the failure propagation probability of complex electromechanical systems based on the state change of the internal parts of the system and the degree of interdependence and influence. In the actual failure propagation process, there is uncertainty in the state of the component itself and uncertainty in the propagation process. Therefore, this paper introduces uncertainty measurement to evaluate the correlation between component states. Finally, the feasibility of the method is verified by taking the rail train braking system as an example. The main contributions of this paper are as follows: (i) a topological network model of rail train braking system based on the coupling mechanism between components is constructed, and the key risk points are expressed in the state; (ii) a multi-dependence measurement operator between risk points based on uncertainty is proposed. On this basis, a prediction method of failure propagation probability of rail train braking system considering dependence is proposed.

The remainder of this paper is structured as follows: Section 2 evaluates the reliability of components. Based on the coupling mechanism and correlation between components, a calculation method of system fault propagation probability is constructed. Section 3 introduces the case study of rail train braking system. Section 4 is the conclusion of this paper.

2. Modeling Method

The model method framework of this paper is shown in Figure 1.

2.1. Determination of Key Component Set Based on Reliability Measurement

This paper uses the Weibull distribution to measure the reliability of system internals. Recently, the Weibull distribution became a hot topic in the study of reliability. It is obtained by the weakest link model or series model. In this distribution, the influence of material defects and stress concentration sources on the fatigue life of materials can be fully reflected. Weibull distribution has different fault distribution and change rates in each stage. It can be used to describe the failure rate bathtub curve of the whole life cycle. It has good fitting performance and can describe the whole life cycle fault of the equipment. The transformation of the Weibull distribution can be used to obtain exponential distribution, normal distribution, etc., and the Weibull distribution can also be used to evaluate the exponential distribution of equipment. Weibull is also widely used in reliability research in mechanical, electronic, aerospace, and other fields.

The Weibull distribution is a commonly used nonlinear reliability evaluation method, such as the reliability evaluation of rolling bearings. In general, the two-parameter model of the Weibull distribution is used to evaluate the reliability of equipment in complex electromechanical systems. There are many modes of the Weibull distribution. This paper introduces the Weibull distribution of the model described by the scale parameter $\alpha$ and the shape parameter $\beta$.

2.1.1. System State Degradation Performance Index

Table 1. List of acronyms and notations in Section 2.

Abbreviations and Symbols	Meaning	Calculation Formula
$R\left(t\right)$	Reliability function	$R\left(t\right)=P\left(T\ge t\right)$
$F\left(t\right)$	Unreliability function	$F\left(t\right)=P(T<t)$
$f\left(t\right)$	Hazard function	$f\left(t\right)=\frac{dF\left(t\right)}{dt}=-\frac{dR\left(t\right)}{dt}$
$\lambda \left(t\right)$	Failure rate	$\lambda \left(t\right)=\frac{f\left(t\right)}{R\left(t\right)}$
MTTF	Mean Time To Failure	$MTTF={{\displaystyle \int }}_0^{\infty }R\left(t\right)dt$
MTBF	Mean Time Between Failures
$T$	Normal working time
$P$	Probability of occurrence
$t$	Working time
$∆t$	The amount of change in time
$\alpha$	Scale parameter
$\beta$	Shape parameter

shows the meaning and calculation formula of abbreviations and symbols in Section 2.

1.

Reliability

(1)

Reliability function $R\left(t\right)$

$R\left(t\right)$ is defined as [23]

$$R\left(t\right)=P\left(T\ge t\right)$$

(1)

where $P$ represents the probability of occurrence.

• (2)

  Unreliability function $F\left(t\right)$

$F\left(t\right)$ is defined as

$$F\left(t\right)=P(T<t)$$

(2)

where $P$ is the same as it is in Equation (1).

At the same time, the implementation and failure of a component or system are opposing events. Therefore, the reliability function $R\left(t\right)$ and the unreliability function $F\left(t\right)$ have the following relationship.

$$R\left(t\right)+F\left(t\right)=1$$

(3)

From the perspective of probability, whether the reliability function or the unreliability function is a probability cumulative distribution function.

• (3)

  Hazard function $f\left(t\right)$

$f\left(t\right)$ is defined as

$$f\left(t\right)=\frac{dF\left(t\right)}{dt}=-\frac{dR\left(t\right)}{dt}$$

(4)

Then there is the following relationship between the unreliability function $F\left(t\right)$, the reliability function $R\left(t\right)$ and the fault density function $f\left(t\right)$.

$$F\left(t\right)={{\displaystyle \int }}_0^{t}f\left(x\right)dx$$

(5)

$$R\left(t\right)={{\displaystyle \int }}_t^{\infty }f\left(x\right)dx$$

(6)

2.

Failure rate $\lambda \left(t\right)$

$\lambda \left(t\right)$ is defined as

$$\begin{array}{l}\lambda \left(t\right)=\underset{∆t\to 0}{\lim }\frac{P\left(t<T\le t+∆t|T>t\right)}{∆t}\\ =\frac{1}{R\left(t\right)}\left[\frac{dF\left(t\right)}{dt}\right]\\ =\frac{1}{R\left(t\right)}\left[-\frac{dR\left(t\right)}{dt}\right]\\ =\frac{f\left(t\right)}{R\left(t\right)}\end{array}$$

(7)

where P is the same as it is in Equation (1), $∆t$ represents the amount of change in time.

The failure rate function $\lambda \left(t\right)$ is based on the number of failures per hour, so the failure rate is generally used to describe the failure law of the component. Different from the failure probability density, the failure rate is generally used to describe the relative failure rate of the component or system, that is, the system or part without failure fails in the subsequent time.

3.

Mean time to failure

Mean time to failure and mean time between failures are important indicators for measuring components or systems, that is, the average time interval between two successive failures. For non-repairable systems, it is called Mean Time To Failure (MTTF); for repairable systems, it is called Mean Time Between Failures (MTBF). In practical engineering, the concept of average fault-free time is more widely used.

Assuming that the average life of the product is $\theta$, the number of products for the life test is $N$, and the life of the $i$th product is $t_i$, the average life of the product is $\theta =\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{t}_i$.

The expectation of life distribution is the mean time to failure, which can be obtained by the following formula.

$$\theta =MTTF=E\left(T\right)={{\displaystyle \int }}_0^{\infty }tf\left(t\right)dt$$

(8)

where $E\left(T\right)$ represents the mathematical expectation of time. It can be seen from the above content:

$$\begin{gathered} \theta =MTTF=-{{\displaystyle \int }}_0^{\infty }t\frac{dR\left(t\right)}{dt}dt \\ =-{{\displaystyle \int }}_0^{\infty }tdR\left(t\right) \\ =-tR\left(t\right){|}_0^{\infty }+{{\displaystyle \int }}_0^{\infty }R\left(t\right)dt \end{gathered}$$

(9)

Because $R(\infty )=0$,$R\left(0\right)=1$, the first half of the formula is equal to 0, then MTTF is

$$\theta =MTTF={{\displaystyle \int }}_0^{\infty }R\left(t\right)dt$$

(10)

2.1.2. Two-Parameter Weibull Function Expression

The fault density function $f\left(t\right)$ of the Weibull function is known as

$$f\left(t\right)=\frac{\beta }{\alpha }{\left(\frac{t}{\alpha }\right)}^{\beta -1}\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\beta }\right]$$

(11)

The fault distribution function $F\left(t\right)$ can be obtained by integrating the fault density function $f\left(t\right)$ from Equations (5) and (6).

$$F\left(t\right)=1-\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\beta }\right]$$

(12)

The reliability function $R\left(t\right)$ is obtained by Equation (3).

$$R\left(t\right)=exp\left[-{\left(\frac{t}{\alpha }\right)}^{\beta }\right]$$

(13)

The failure rate function $\lambda \left(t\right)$ is obtained by Equation (7).

$$\lambda \left(t\right)=\frac{\beta }{\alpha }{\left(\frac{t}{\alpha }\right)}^{\beta -1}=\frac{\beta }{{\alpha }^{\beta }}{t}^{\beta -1}$$

(14)

where $\alpha$ is the scale parameter and $\beta$ is the shape parameter.

The advantages and characteristics of the two-parameter Weibull distribution are that the model parameters are easy to understand and the function analytical form is easy to express.

2.1.3. Parameter Estimation of Weibull Function

Next, the least squares method is used to estimate the parameters of the two-parameter Weibull distribution.

Firstly, the linear transformation of the Weibull function is carried out:

$$\ln \left(\ln \left(\frac{1}{1-F\left(t\right)}\right)\right)=\beta \ln \left(t\right)-\beta \ln \left(\alpha \right)$$

(15)

Let

$$\{\begin{array}{c}y=\ln \left(\ln \left(\frac{1}{1-F\left(t\right)}\right)\right)\\ x=\ln \left(t\right)\\ B=\beta \\ A=-\beta \ln \left(\alpha \right)\end{array}$$

(16)

It can be rewritten as:

$$y=Bx+A$$

(17)

Let

$$\{\begin{array}{c}\overline{x}=\frac{1}{k}{\displaystyle {\displaystyle \sum }_{i=1}^k}{x}_i\\ \overline{y}=\frac{1}{k}{\displaystyle {\displaystyle \sum }_{i=1}^k}{y}_i\end{array}$$

(18)

Next, the estimated values of parameters $A$ and $B$ are calculated by the least square method:

$$\{\begin{array}{c}\widehat{B}=\frac{{{\displaystyle \sum }}_{i=1}^k{x}_i{y}_i-k\overline{xy}}{{{\displaystyle \sum }}_{i=1}^k{x}_i{}^2-k{\overline{x}}^2}\\ \widehat{A}=\overline{y}-\widehat{B}\overline{x}\end{array}$$

(19)

The linear regression equation is:

$$\widehat{y}=\widehat{B}x+\widehat{A}$$

(20)

Then the parameter estimates are:

$$\{\begin{array}{c}\beta =\widehat{B}\\ \alpha =\exp \left(-\left(\frac{\widehat{A}}{\widehat{B}}\right)\right)\end{array}$$

(21)

2.2. A Calculation Method Considering the Dependence of the Topology Model and Fault Data

The dependency relationship is the connection between the two entities. Through it, the state of one entity is associated with the state of the other entity. In this case, this relationship is usually one-way. In this paper, mutual dependence is used to describe two-way dependencies. That is, the impact of the state of one entity or associated with the state of another entity and vice versa. In detail, entity 1 depends on entity 2 through some links, and entity 2 also depends on entity 1, which may pass different links. The interdependence relationship between components can be a causal or simple correlation. In cause and effect, the state of one component should be responsible for the status of another component. On the other hand, when the two components play their own functions, one of them is not the actual reason for another failure, and the state of the component is related.

2.2.1. Construction of Topological Network Model

Couplings between components fall into three categories: mechanical connections, electrical connections, and information connections.

• Mechanical connection relationship $e_{ij}^m$: the mechanical connection relationship between two physical components such as cementation, welding, riveting, and screw connection.
• Electrical connection relationship $e_{ij}^e$: there is a relationship between transmission, conversion, distribution, and utilization of electrical energy between two physical components.
• Information connection relationship $e_{ij}^i$: there is information flow transmission or exchange between two physical components through wired or wireless means.

Consider autonomous entities that assume specific functions, are independent, cannot be subdivided, and interact with other units as components. A system topology coupling network model $G$ with components as nodes and mechanical connection, electrical connection, and information connection between components as edges are constructed, which is expressed as a set:

$$\begin{gathered} \left\{V,E,···\right\}\in G \\ \left\{v_i\right\}\in V \\ \left\{e_{ij}|e_{ij}^m,e_{ij}^e,e_{ij}^i\right\}\in E \end{gathered}$$

The schematic diagram of constructing the system topology coupling network is as follows:

In Figure 2, the lines of three colors represent three coupling mechanisms respectively.

2.2.2. Risk Point Status Parameters Based on Mathematical Characterization Methods

In order to express the state of the risk point, we introduce status parameters and compare the pressure and current of each risk point to compare with the design indicators to determine whether the risk point is healthy or faulty.

Corresponding to the state parameter and coupling mechanism, divided into the following three categories:

• Mechanical status parameters: refers to the basic design indicators of the component, such as volume, density, temperature, pressure, etc. The pressure value can be directly monitored by the pressure sensor.
• Electric status parameters: refers to the flow value of the circuit system, such as current, voltage, etc., where current and voltage are measured.
• Information status parameters: refers to the pass instructions, data packages, etc., and it can judge the state status by passing the time standards of passing instructions.

The information status parameters in the above three state parameters are generally defined as the state of the part with the mechanical status parameter or electrical state parameters. Use the status of the component of the above status parameters and compare the health status of the component to the health state of the component and the standard parameter of the fault, to calculate the dependence between components.

2.3. System Failure Chain Construction Method Based on Component Risk State Representation

2.3.1. Non-Linear Dependencies Based on Uncertainty

The correlation between the components can be divided into linear and nonlinear. This paper only considers the nonlinear correlation. The reason why the linear correlation is not considered in this paper is that the influence between components is divided into direct influence and indirect influence. The correlation of linear correlation is stronger and contains a specific mathematical relationship, which should be attributed to the direct influence relationship. However, in the direct impact, the demarcation point of causality and linear correlation is difficult to calculate accurately according to the determined formula, and causality is more easily reflected in fault cases and sequences, so only causality and nonlinear correlation are considered.

The correlation between two random variables reflects a statistical relationship, that is, the correlation between their values. This dependence can be quantified using various existing metrics. By determining the correlation between these random variables, we introduce the calculation of the correlation coefficient and then use the uncertainty to measure the correlation coefficient.

Uncertainty refers to the degree of uncertainty of the measured value due to the existence of measurement errors, and in turn, it also indicates the degree of reliability of the result. It is an indicator of the quality of the measurement results. The smaller the uncertainty is, the higher the quality, the level, and the higher its value; the greater the uncertainty is, the lower the quality of the measurement results, and the lower the level, the lower its value. MATLAB software can play a very active role in measuring the uncertainty. This paper uses the uncertainty to calculate the correlation coefficient and uncertainty between the two risk point states. The specific calculation process will be carried out in the case study section.

According to the correlation coefficient and fault data calculated in the previous section, the direct impact matrix $D={\left[d_{ij}\right]}^{n\times n}$ of the braking system is obtained, where the element $d_{ij}$ in the matrix represents the degree of influence of component $i$ on component $j$, and n represents the number of components in the system. Among them, the elements of the diagonal in the $D$ matrix are all 0, because the fault of the component itself will not be transmitted to itself, that is why, the fault propagation process does not need to go through the time step.

2.3.2. Component Interdependence Model under Causality

In a system consisting of $n$ components, $m$ failure cases are observed, for a given fault condition $k$, we can construct a fault sequence frequency matrix $W={\left[w_{ij}\right]}^{n\times n}$, where $w_{ij}$ represents the number of time steps for component $j$ to fail after component $i$ fails under all fault conditions, and thus obtain the distribution of $w_{ij}$ in the fault sequence set.

Then we define $a$ to represent the threshold controlling the detection of causality, and the choice of a depends on the distribution of $w_{ij}$ in the collection of fault sequences.

2.3.3. Quantified Inter-Dependencies

Through the above two sections, we divided the relationship between components into correlation and causation. In this section, we quantify this interdependence. First, define the total influence matrix $T$, then give the calculation Formulas (22)–(24) as follows:

$$T=\left[t_{ij}\right]=R·{\displaystyle {\displaystyle \sum }_{k=1}^{\infty }}{\left(\frac{1}{n}D\right)}^k$$

(22)

$$R=\left[r_{ij}\right]$$

(23)

$$r_{ij}=\{\begin{array}{c}\frac{n+1}{n},i\ne j\\ \frac{n+1}{n-1},i=j\end{array},i\ge 1,j\le n$$

(24)

where, $t_{ij}$ represents the degree to which component $j$ is affected by the fault of component $i$ in any number of time steps, revealing the indirect effect. $n$ represents the number of risk points in the system. The matrix $R$ is used to scale $t_{ij}$ to the range [0, 1]. The symbol “$·$ “ represents the dot product operation of the matrix.

In addition, we also define $u_i$ and $v_j$, which respectively represent the weighted out-degree of node $i$ and the weighted in-degree of node $j$, to evaluate the influence of each component on other components, as shown in the following Formulas (25) and (26):

$$u_i=\frac{1}{n}{\displaystyle {\displaystyle \sum }_{j=1}^n}{t}_{ij}$$

(25)

$$v_j=\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n}{t}_{ij}$$

(26)

2.3.4. Affected Component Sets Based on Machine Learning

After calculating the degree of interdependence between risk points, prediction tools can be used to predict possible future failures, to construct the effective propagation chain of the system. Therefore, we introduce ANN artificial neural network.

Artificial Neural Network (ANN) is a complex network structure formed by a large number of connected processing units (neurons). It is an abstraction, simplification, and simulation of the structure and operation mechanism of human brain tissue. It simulates neuronal activity with a mathematical model and is an information processing system based on imitating the structure and function of the brain neural network.

The artificial neural network is divided into multi-layers and single-layer. Each layer contains several neurons, and each neuron is connected by a directed arc with variable weights. The network achieves the purpose of processing information and simulating the relationship between input and output by repeatedly learning and training the known information and gradually adjusting the method of changing the connection weights of neurons. It does not need to know the exact relationship between input and output and does not need a large number of parameters. It only needs to know the non-constant factors that cause output changes, that is the non-constant parameters. Therefore, compared with the traditional data processing method, neural network technology has obvious advantages in dealing with fuzzy data, random data, and nonlinear data. It is especially suitable for systems with large scale, complex structures, and unclear information. Research shows that ANN is a good method to predict failure sequences [6].

For a system with $n$ components, let $X\left(t\right)=\left(X_1\left(t\right),X_2\left(t\right),\dots ,X_n\left(t\right)\right)$ denote the input array to the ANN, where $X_i\left(t\right)$ is the state variable of component $i$ at the time instance $t$. $X\left(t\right)$ is fed to a multilayer fully connected ANN with the architecture shown in Figure 3. The output layer provides $Y=\left(Y_1,Y_2,\dots ,Y_n\right)$, where $Y_i$ represents the probability that component $i$ fails as a result of the disruption specified by the given state variable in the input.

The ANN is trained using a data set generated by simulating some failure cases or data from historical information of previous disruptions. In this paper, the data of the failure case is converted into several input/output data sets, and the failure sequence is converted into an entry for the data set used to train the ANN. In Section 3.5, we demonstrate the application of the ANN and provide its performance in predicting faults in the example of a rail train braking system.

3. Case Study on the Braking System of Rail Train

3.1. Construction of Braking System Topology Network Model Based on Reliability Evaluation

According to the introduction of Section 2.1, this section will use the Weibull distribution to evaluate the reliability of components in the rail transit braking system and determine the key components in the braking system. Limited to data, this section only takes the emergency solenoid valve, EP valve, and relay valve as examples for reliability calculation.

3.1.1. Data Sources

According to the data in the relevant literature [24,25], we can obtain the service life test data of the emergency solenoid valve, EP valve, and relay valve as follows in

Table 2. Test data of emergency solenoid valve and EP valve.

Serial Number	Emergency Solenoid Valve			EP Valve
	${\mathit{t}}_{\mathit{i}}$	${\mathit{n}}_{\mathit{i}}$	${\mathit{s}}_{\mathit{i}}$	${\mathit{t}}_{\mathit{i}}$	${\mathit{n}}_{\mathit{i}}$	${\mathit{s}}_{\mathit{i}}$
1	600	2	30	800	2	18
2	700	2	28	1000	2	16
3	800	2	26	1200	2	14
4	950	3	24	1500	2	12
5	1100	3	21	1750	3	9
6	1200	3	18	2000	3	6
7	1350	3	15	2150	3	3
8	1550	4	12	2300	3	0
9	1800	4	8
10	2000	4	4

and

Table 3. Test data of relay valve.

Test Count	Lifetime t (Ten Thousand Hours)	Type of Data	Test Count	Lifetime t (Ten Thousand Hours)	Type of Data
1	0.67	Fault data	11	2.00	Censored data
2	1.79	Fault data	12	2.00	Censored data
3	1.88	Fault data	13	2.00	Censored data
4	1.89	Fault data	14	2.00	Censored data
5	1.92	Fault data	15	2.00	Censored data
6	2.00	Censored data	16	2.00	Censored data
7	2.00	Censored data	17	2.00	Censored data
8	2.00	Censored data	18	2.00	Censored data
9	2.00	Censored data	19	2.00	Censored data
10	2.00	Censored data	20	2.00	Censored data

.

3.1.2. Component Reliability Analysis

(1)

Reliability analysis of emergency solenoid valve.

According to the experimental data of the emergency solenoid valve, the estimated value of each parameter is $\alpha =1.902\times {10}^{-5}$, $\beta =1.031$. Then, the reliability and failure rate functions of the Weibull function are as follows:

$$\begin{gathered} R\left(t\right)=\exp \left\{-1.902\times {10}^{-5}{t}^{1.031}·\exp \left(1.277\times {10}^{-5}t\right)\right\} \\ \lambda \left(t\right)=1.902\times {10}^{-5}·\left(1.031+1.277\times {10}^{-5}t\right)·t^{1.031}·\exp \left(1.277·{10}^{-5}\right) \end{gathered}$$

The average fault-free time of the emergency solenoid valve is calculated to be 23,563 h. The fault density function, fault distribution function, reliability function, and failure rate function of the emergency solenoid valve from 0 h to 20,000 h can be generated, as shown in Figure 4.

(2)

Reliability analysis of EP valve.

According to the experimental data of the EP valve, the estimated value of each parameter is $\alpha =24,252.05$, $\beta =1.005$. Then the reliability and failure rate functions of the Weibull function are as follows:

$$\begin{gathered} R\left(t\right)=\exp \left\{-{\left(\frac{t}{24252.05}\right)}^{1.005}\right\} \\ \lambda \left(t\right)=\frac{1.005}{{24252.05}^{1.005}}{t}^{0.005} \end{gathered}$$

The fault density function, fault distribution function, reliability function, and failure rate function of the EP valve can be generated as shown in Figure 5.

(3)

Reliability analysis of relay valve.

According to the experimental data of the relay valve, the estimated value of each parameter is $\alpha =2.8$, $\beta =3.84$. Then the reliability and failure rate functions of the Weibull function are $R\left(t\right)=75.33\%$ and $\lambda \left(t\right)=0.31$.

By estimating the obtained parameters, the average fault-free time of the relay valve is 25,600 h. The fault density function, fault distribution function, reliability function, and failure rate function of the relay valve from 0 h to 30,000 h can be generated as shown in Figure 6.

According to the calculation and evaluation of the reliability of the braking system components of the rail train, we obtain the key components of the braking system, the emergency solenoid valve, EP valve, relay valve, empty-load vehicle valve, air source system, brake cylinder; the weak risk point is the anti-skid system.

3.1.3. Coupling Topology Network Model of Rail Train Braking System

According to the schematic diagram of the system coupling topology network constructed in Section 2.2, combined with the connection and structure of the components of the rail train braking system, the coupling topology network model of the braking system is obtained as shown in Figure 7.

In Figure 7, the red line represents a mechanical connection, the blue line represents an electrical connection, and the yellow line represents an information connection. The name of the risk points listed in this article is also bold the name of the risk point on it: the risk point of the wind source system is constituted by the electric bow supply unit, the brake air cylinder, and the air spring cylinder; the brake control device includes four risk points: emergency solenoid valve, EP valve, relay valve, and air heavy truck valve; the brake cylinder is a risk point alone, and the braking speed sensor represents the risk point of the anti-slip system.

3.2. State Expression of Key Risk Point State of the Braking System

In order to construct the state expression in the risk point of the braking system and judge the working state of the risk point, this paper consults the relevant literature on the risk point mentioned in this paper. some characteristics and indexes of related components are introduced in the braking system [26,27,28,29,30,31,32,33,34]. Based on these indicators, this paper constructs the risk point state expression.

3.2.1. Emergency Solenoid Valve

The failure judgment and braking instruction time of the emergency solenoid valve (expressed by $T$, the unit is $S$) and the pressure value of the volume room of the relay valve (kPa is represented by $P$). The braking instructions can detect whether the train has applied emergency brakes, and the pressure value of the relay valve volume room to record the actual working status of the emergency solenoid valve, and there is a conditional relationship between the two. By comparing the pressure value of the circulation volume (CV), the working state of the emergency solenoid valve is obtained. This article only considers emergency construction and non-emergency operating conditions, and the two braking methods of commonly used and fast braking are applied. Therefore, the flow diagram of the diagnosis of emergency solenoid valve fault diagnosis is shown in Figure 8 below:

Combined with the flowchart, the state of the emergency solenoid valve is represented by the mathematical expression of the segmented function (represented by $P1$, $P1=1$ represents health, $P1=0$ represents fault):

In emergency braking conditions,

$$P1=\{\begin{array}{c}1, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t>1, p\ge 150 \mathrm{kPa}\\ 0, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t>1, p<150 \mathrm{kPa}\end{array}$$

(27)

Under non-emergency braking conditions and applying common, rapid braking:

$$P1=\{\begin{array}{c}1, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t>1, p\le 50 \mathrm{kPa}\\ 0, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t>1, p>50 \mathrm{kPa}\end{array}$$

(28)

In the formula, $p$: the pressure of the volume chamber of the relay valve.

3.2.2. EP Valve

EP valve faults are mainly air leakage faults. The specific judgment method is to calculate the difference between the pre-control pressure and the target pressure. If the pressure difference is within 5 kPa, the EP valve is healthy, and if it exceeds 5 kPa, the EP valve is faulty. The state of the EP valve is represented by the mathematical expression of the piecewise function (expressed by $P2$, $P2=1$ represents health, $P2=0$ means failure):

$$P2=\{\begin{array}{c}1, \left|Ps-PCV\right|\le 5 \mathrm{kPa}\\ 0, \left|Ps-PCV\right|>5 \mathrm{kPa}\end{array}$$

(29)

In the formula, $PCV$: pre-control pressure;

$Ps$: target pressure.

3.2.3. Relay Valve

Monitoring the health status of the relay valve lead us to introduce the Kalman filter algorithm. It uses the linear system state equation to optimally estimate the system state through the input and output observation data of the system. During the actual operation of the train, the pressure value output by the relay valve and its corresponding time can be monitored by the system equipment, and these output parameters can be used to estimate the state parameters of the relay valve through the Kalman filter’s function. This process is implemented based on MATLAB.

3.2.4. Empty Truck Valve

Under normal and fast braking conditions, the input pressure of the empty-loaded vehicle valve is low, and the input pressure and output pressure are always connected; under emergency braking conditions, the input pressure of the empty-loaded vehicle valve is high, which limits the output pressure. In this case, the output pressure expression of the empty truck valve is:

$$p_{CV2}=\{\begin{array}{c}{p}_{CV1} p_{CV1}\le p_{CV2\left(MAX\right)}\\ p_{CV2\left(MAX\right)} p_{CV1}>p_{CV2\left(MAX\right)}\end{array}$$

(30)

In the formula, $p_{CV1}$: the input pressure of the empty and heavy truck valve;

$p_{CV2\left(MAX\right)}$: the maximum pressure that can be output.

The design accuracy of the empty-load truck valve is ±15 kPa, and the state of the empty-load truck valve can be judged by detecting the output pressure deviation of the empty-load truck valve under different pre-control pressures. Use the mathematical expression of the piecewise function to express the state of the empty and heavy truck valve (expressed by $P4$, $P4=1$ represents health, $P4=0$ represents failure):

$$P4=\{\begin{array}{c}1, \left|Ps-p_{CV2}\right|\le 15 \mathrm{kPa}\\ 0, \left|Ps-p_{CV2}\right|>15 \mathrm{kPa}\end{array}$$

(31)

In the formula, $p_{CV2}$: output pressure;

$Ps$: target pressure.

3.2.5. Brake Cylinder

In common braking conditions, the braking command is greater than a certain value (currently, the target value of the brake cylinder pressure is usually set to 150 kPa or higher). If the pressure of the brake cylinder does not reach the predetermined pressure value within a certain period of time (generally, the pressure of the brake cylinder cannot be increased to 60 kPa within 3.5 s), the brake system will report a failure of insufficient braking force.

Considering the state of the brake cylinder to be $P5$, the pressure value of the brake cylinder is $p$ (unit is kPa), the duration is $t$ (unit is $s$), $P5=1$ represents health, $P5=0$ represents failure, the mathematical expression for judging the state of the brake cylinder is obtained as follows:

Under normal braking conditions:

$$P5=\{\begin{array}{c}1, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t\le 3.5 \mathrm{s}, p\ge 60 \mathrm{kPa}\\ 0, \mathrm{under} \mathrm{the} \mathrm{condition} \mathrm{of} t>3.5 \mathrm{s}, p<60 \mathrm{kPa}\end{array}$$

(32)

In the formula, $p$: brake cylinder pressure.

3.2.6. Wind Source System

The failures of the air source system mainly include abnormal control and power supply of the air compressor on one side, and the failure of the air compressor and the dryer on another side. Regardless of which component fails, it means that the air source system is faulty. This section uses the hardware structure of the existing wind source system and the mature electronic control principle to describe the operating state of the wind source system. The specific fault diagnosis logic diagram is shown in Figure 9 below.

3.2.7. The Anti-Skid System

The anti-skid valve includes a pressure-maintaining valve and an exhaust valve, which are the core components of the anti-skid system, so the state of the anti-skid valve is very important to the effect of the anti-skid system on the anti-skid protection of the train. Anti-skid failure is generally caused by the failure of a part inside the anti-skid system. As the core part of the anti-skid system, the state of the anti-skid valve can affect the function of the anti-skid system. The failure of the anti-skid valve is mostly caused by its mechanical failure. Based on this, certain parameters such as pre-control ($Cv$) pressure, axle speed, etc. can be found, which can be used as the basis for judging whether the anti-skid valve has a failure. The anti-skid valve fault diagnosis process is shown in Figure 10.

3.3. The Brake System Failure Data Statistics

During the operation of the urban rail train, the monitoring part will collect real-time data and transmit the data to the information system, so people will acquire the fault phenomenon and guess the possible faulty parts from the maintenance point of view. The faults of components are divided into two types: self-faults and faults affected by other component faults. By analyzing fault cases and corresponding faults to corresponding components through fault descriptions, the data statistics are as follows in

Table 4. Fault data statistics table.

Faulty Part	Fault Type Description	Fault Statistics
		Non-Correlated Failure	Correlation Fault
Emergency solenoid valve	BCU failure, emergency braking	3	3
EP valve	BCU failure, booster cylinder failure	2	4
Relay valve	BCU failure, pressure not relieved	2	4
Empty truck valve	BCU failure	1	3
Brake cylinder	Insufficient braking force, braking does not ease	2	6
Air source system	An air compressor failure, dryer failure	3	16
Anti-skid system	Anti-skid valve failure	1	3

, the data used in this paper comes from the fault data of a train faucet manufacturing enterprise.

Through the data in the above table, combined with the coupling mechanism between components and the structure and principle of the air brake, the right-directed influence relationship between each risk point can be obtained, as shown in Figure 11 below.

In Figure 11, $v1$ represents the emergency solenoid valve, $v2$ represents the EP valve, $v3$ represents the relay valve, $v4$ represents the empty-load vehicle valve, $v5$ represents the brake cylinder, $v6$ represents the air source system, and $v7$ represents the anti-skid system. The direction of the arrow in the figure represents the direct impact relationship, and the number on the arrow (indicated by $a$) represents the degree of influence. Taking $v1\to v2$ and $a12$ as examples, it represents that risk point 1 (emergency solenoid valve) directly affects risk point 2 (EP valve), and the degree of influence is $a12$. The degree of influence ($a$) needs to be determined by calculating the correlation coefficients between the component state parameters based on the failure data.

Because the status parameters in the risk point in this chapter are supported by data, compared with the judgment standard should be measured data (such as pressure value, current value, etc.) in the actual operation of the urban rail trains. This data cannot be obtained. Therefore, in the compilation of the MATLAB program code, the measurement data of each risk point is randomly generated within a certain range, and then compared with the judgment standard data, and finally judge the state of the risk points.

3.4. The Components under Correlation Depending on the Model

3.4.1. Non-Linear Correlation Coefficient Calculation Based on Uncertainty

This section only takes the emergency solenoid valve and EP valve as an example to calculate the correlation coefficient and uncertainty of the two risk point states (the correlation coefficient uncertainty between other risk point state parameters is the same). The specific calculation process is as follows:

Input the state parameter $P1$ of the emergency solenoid valve into Matlab, and express it with an $X1$ matrix, and the same for the EP valve, and express it with an $X2$ matrix:

$$\begin{gathered} X1=\left[1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0\right] \\ X2=\left[1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0\right] \\ \left[r,p\right]=corrcoef\left(X1,X2\right) \end{gathered}$$

Obtaining

$$\begin{gathered} r=1.0000\hspace{1em}0.5238\hspace{1em}0.5238\hspace{1em}1.0000 \\ p=1.0000\hspace{1em}0.0178\hspace{1em}0.0178\hspace{1em}1.0000 \end{gathered}$$

The main diagonal elements of the $P$ matrix are all 1,

$$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}=\frac{0.5238\times \sqrt{20-2}}{\sqrt{1-{0.5238}^2}}=2.6088$$

(33)

Using MATLAB to calculate $\left(1-tcdf\left(2.6088,18\right)\right)\ast 2$, the result is 0.0178.

At the significance level of 0.05, 0.0178 is less than 0.05, so the emergency solenoid valve and EP valve are correlated. Define $rlo$ and $rup$, $rlo$ is the lower limit of the correlation coefficient, $rup$ is the upper limit of the correlation coefficient, and the confidence level is 95%.

Use MATLAB to calculate the confidence interval:

$$\begin{gathered} \left[r,p,rlo,rup\right]=corrcoef\left(X1,X2\right) \\ r=1.0000\hspace{1em}0.5238\hspace{1em}0.5238\hspace{1em}1.0000 \\ p=1.0000\hspace{1em}0.0178\hspace{1em}0.0178\hspace{1em}1.0000 \\ rlo=1.0000\hspace{1em}0.1058\hspace{1em}0.1058\hspace{1em}1.0000 \\ rup=1.0000\hspace{1em}0.7845\hspace{1em}0.7845\hspace{1em}1.0000 \end{gathered}$$

Therefore, the 95% confidence interval of the correlation coefficient between the state parameters of the emergency solenoid valve and the state parameters of the EP valve is $\left[0.1058,0.7845\right]$.

3.4.2. Construction of Direct Influence Matrix of the Braking System

According to the introduction of Section 2.3.1 and fault data, the direct influence matrix $D$ of the key risk points of the rail train braking system is constructed, where $n=7$. The weighted directed graph of the braking system dependency is obtained as shown in Figure 12 below, which is the visual expression of the matrix $D$.

Then, we can calculate the total influence matrix between the key risk points of the braking system by Equations (22)–(24), that is, the degree of interdependence between the risk points.

3.4.3. Causal Analysis of the Braking System

According to the introduction of Section 2.3.2, the fault sequence frequency matrix $W$ of the key risk points of the braking system is constructed, and the distribution is shown in Figure 13.

In the figure above we can observe that the elements in the set of failure sequences have a significant separation at $w_{ij}=0.4$, which makes it easy for us to distinguish correlation from causation. Therefore, we assume $a$ is equal to 0.4.

3.5. Construction of the Brake System Failure Propagation Chain

In order to train a neural network, we need to transform the fault data into a data set consisting of multiple input and output elements. Figure 14 below shows an example of a hypothetical failure sequence, where a component status of 1 indicates that the component is normal, and a component status of 0 indicates that the component is faulty. In this fault case, the system initially fails at component 6, and as the fault condition propagates, it affects component 1, which is then affected by the fault in the order of component 2, component 4, component 3, component 7, and finally propagates to component 5.

We convert this failure case into an entry of the data set used to train the neural network and express it in the form of a matrix, which is denoted by $X$, that is,

$$X=\left[\begin{array}{c}1 1 1 1 1 1 1\\ 1 1 1 1 1 0 1\\ 0 1 1 1 1 0 1\\ 0 0 1 1 1 0 1\\ 0 0 1 0 1 0 1\\ 0 0 0 0 1 0 1\\ 0 0 0 0 1 0 0\\ 0 0 0 0 0 0 0\end{array}\right]$$

Based on this fault sequence and the total impact matrix $T$, transform it into input and output data to train the ANN. The trained ANN predicts the probability of fault propagation, and the propagation rate of the failure chain of the braking system is obtained as follows in Figure 15 and Figure 16:

Denote the predicted failure probabilities in the fault propagation chain by matrix $Y$:

$$Y=\left[\begin{array}{c} 0.0265 \\ 0.1185 \\ 0.0823\\ 0.0440\\ 0.1390\\ 0.0406\end{array}\right]$$

The step length in Figure 15 and Figure 16 refers to the process of variable state change propagation, that is, the process from the state change of variable 1 to the state change of variable 2 is a step length, and the step length is 2 when propagating twice, and so on. This propagation chain is a complete failure propagation chain involving all the risk points in this paper, which is connected according to “air source system→ emergency solenoid valve→ EP valve→ empty truck valve→ relay valve→ anti-skid system→ brake cylinder” influence relationship building. The failure probability refers to the probability that the failure of risk point 1 affects the failure of risk point 2, that is, the data in matrix $Y$. The failure probability is 1 when the step length of the risk point propagation path is 0, because the wind source system is the initial fault risk point in the failure propagation chain, and the wind source system must be faulty to make the fault propagate. When the step size is 7, the failure probability is 0 because at this time all risk points have failed and will not continue to propagate. Failure probability data with step sizes 0 and 7 are not in matrix $Y$.

4. Conclusions

This paper proposes a fault propagation probability prediction method for complex electromechanical systems considering component states and dependencies between components. Firstly, the reliability of the internal components of the system is calculated and evaluated by using the two-parameter Weibull distribution, and the set of key components in the system is obtained. Secondly, three coupling mechanisms of mechanical, electrical, and information are proposed. The topological network model of the system is constructed with components as nodes and coupling mechanisms as edges. Based on the three coupling mechanisms, the state parameters of the components are defined, and the state values of the components are calculated by mathematical expressions. In order to quantify the degree of interdependence between components, this paper divides the interaction between components into correlation and causality. Under the correlation, the correlation coefficient between components is calculated and measured by uncertainty, and the direct influence matrix between components is constructed by fault data. Under the causal relationship, the boundary value that distinguishes the two influence relations is found by constructing the fault sequence frequency matrix of the system. Finally, a mathematical expression for calculating the total influence degree between components is proposed, and the quantitative interdependence degree between components is obtained. Finally, the ANN artificial neural network is introduced to predict the propagation probability of the fault based on the known fault sequence and the calculated dependence. In the example of a rail train braking system, the prediction accuracy of ANN is proved, and the probability of failure chain propagation of the braking system is predicted, which proves that the method proposed in this paper is feasible.

In this paper, the research on the uncertainty of the fault propagation of the internal parts of the system is not deep enough. Only the uncertainty is used to measure the correlation between the components. There are other uncertainties in the fault propagation, which can be combined with this problem in the future. For the prediction of failure propagation probability of fault sequence, the method used is not advanced enough, and the prediction method can be updated in the future.