The approaches to the system reliability evaluation with respect to the cases, where the components are independent or the components have interactive relationships within the system, were proposed in this paper. Starting from the higher requirements on system operational safety and economy, the reliability focused optimal models of multiobjective maintenance strategies were built. For safetycritical systems, the pessimistic maintenance strategies are usually taken, and, in these cases, the system reliability evaluation has also to be tackled pessimistically. For safetyuncritical systems, the optimistic maintenance strategies were usually taken, and, in these circumstances, the system reliability evaluation had also to be tackled optimistically, respectively. Besides, the reasonable maintenance strategies and their corresponding reliability evaluation can be obtained through the convex combination of the above two cases. With a highspeed train system as the example background, the proposed method is verified by combining the actual failure data with the maintenance data. Results demonstrate that the proposed study can provide a new system reliability calculation method and solution to select and optimize the multiobjective operational strategies with the considerations of system safety and economical requirements. The theoretical basis is also provided for scientifically estimating the reliability of a highspeed train system and formulating reasonable maintenance strategies.
With the rapid development of Chinese highspeed trains, traditional maintenance strategies may not fit the train organization pattern of “high density, high frequency, high security” due to its disadvantages of short overhaul period, high maintenance costs, and long parking time. The train maintenance work will turn to the trend of “reliabilitycentered maintenance.”
Numerous researches focused on this topic. Jin et al. [
The results from abovementioned studies contribute significantly to the development and investigation of the relationship between maintenance strategy and reliability. But they were difficult to be applied in the area of highspeed train systems. On the one hand, these studies mainly focused on the relatively simple systems, whereas the highspeed train system is a complex system. On the other hand, traditional reliability evaluation methods, such as Fault Tree, Bayesian Network, Markov, and Petri Nets, always assumed that the components are independent. For example, a comprehensive software safety analysis involving a combination of Failure Modes and Effects Analysis (FMEA) and Fault Tree Analysis (FTA) is conducted on the software functions of the critical system to identify potentially hazardous software faults [
The system reliability evaluation approaches for the cases, where the components are independent or the components have interactive relationships within the system, are proposed in this paper. Starting from the higher requirements on system operational safety and costeffectiveness, the reliability focused optimal models of multiobjective maintenance strategies are built. System reliability evaluation is accomplished under different maintenance strategies. Finally, with a highspeed train system as an example background, the proposed method is verified by combining the actual failure data with the maintenance data.
The following hypotheses are made when building reliability models on the system with independent components.
Components (nodes) are independent.
There are only two states: normal and fault.
Edges are independent, which means the failure of one single edge does not affect the others.
A twotuple group
Model of system reliability consisted of independent components.
It is supposed that system
Reliability network modeling on the system with dependent components.
Assuming
It is supposed that system
The lifetime of
The following hypotheses are assumed for building reliability models of the system with dependent components.
The system topology structure is fixed.
There are only two states in the components: normal and fault.
If the functional relationship of the parts of the system is normal, the state of the subsystem is normal, or it is in the failure state.
In the system reliability models, the failure of nodes is equivalent to the failure of the functional relationship among the nodes.
There are two ways to express the functional relationship among the parts, as shown in Table
The way of connection between the components and its formalized expression.
Ways of connection  Graphical representation  Description 

Unidirectional connection 

Oneway functional relationship between two components 


Bidirectional 

The interaction between two components 
In the study of subsystem reliability, the system components were set as nodes, while in the study of system reliability, the subsystems were set as nodes, and the functional relationship between nodes was edge. Directed twolayer network models were built with characteristics of network topology structure.
In the study of subsystem reliability, each subsystem consists of a series of indivisible components. The study sets the components as nodes, the operational reliability of components as the nodes properties, and the functional relationship between components as edges to build a directed network model with characteristics of network topology structure,
The study of system reliability sets the system as nodes and the functional relationship between subsystems as edges, building the directed network models with characteristics of network topology structure,
In order to calculate the reliability of systems which consisted of dependent components, Schweizer and Sklar [
It is assumed that system
It is assumed that system
Considering the fact that this study calculates the system reliability based on the connectivity of the system inside components, the system reliability may be affected by the connecting type of system components.
The calculating methods of system reliability are discussed in different connecting types in this paper. General system connecting types mainly include three ways: single input and single output, single input and multiple outputs, and multiple inputs and multiple outputs (multiple inputs and single output connecting type is the same as single input and multiple outputs connecting type).
As shown in Figure
Model of single input and single output.
All the shortest paths should be figured out among the subsystems to analyze the reliability of subsystem based on the minimal path set. It is supposed that
In conclusion, no matter how many minimal paths are there in the subsystem, the highest reliability of
(1) System reliability is calculated with independent components.
The reliability of every minimal path based on (
The reliability of subsystem is
(2) System reliability is calculated with dependent components.
The reliability of every minimal path based on (
As shown in Figure
Single input and multiple outputs.
For subsystems, the reliability of
This calculation method can also calculate the reliability of subsystem with multiple input nodes and single output node; the details will not be described here.
As shown in Figure
Multiple inputs and multiple outputs.
For subsystems, the reliability of
As can be seen from the abovementioned calculation based on three connecting types, no matter which connecting type is, the reliability of subsystems,
According to comprehensive comparison and analysis on the system reliability estimation methods above, supposing that the system reliability of dependent components is greater than that of independent components and trying to verify it by using the method of Mathematical Induction, the proofs are as follows:
When components are dependent, the system reliability is
When components are independent, the system reliability is
Because
When
The system reliability with dependent components is
The hypotheses are supported,
According to Section
Safetycritical systems should take the relatively conservative method to estimate the reliability of independent components, which requires using the lower limit
From the relationship balance of operational safety and economic efficiency, this paper introduces the accommodation coefficient of system operational strategy,
The two extreme states of
In general, system operational strategy is a combination of the abovementioned two extreme strategies, and the optimal
This paper takes the bogie subsystem which belongs to a highspeed train system as a research example. As shown in Figure
Network reliability model of the case.
Reliability network model of bogie subsystem.
The components’ names of bogie subsystem and their serial numbers are shown in Table
Components’ names and serial numbers of bogie subsystem.
Node number  Component  Node number  Component 


Coupling 

Air spring 

Gearbox 

Draft gear 

Axle 

Beam 

Wheel 

Primary suspension 

Brake disc 

Axle box body 

Brake lining 

Side beam 

Brake clamp 

Intensifier pump 
This paper analyzes the lifetime distribution of five components (
The maximum likelihood estimation of main components in bogie subsystem.
Component  Number  The parameters estimation value of Weibull distribution  Distribution function  


 

1  119246.3925  1.5436 


2  108742.4035  1.2458 


3  476891.2461  1.3325 


4  289653.1456  1.2457 


5  364851.2564  1.4688 

According to [
Substituting
The subsystem reliability is
Figure
Curve graph comparing the reliability distribution of bogie subsystem based on dependent and independent components.
The generalized expression of bogie subsystem reliability estimation is
Combining with maintenance resource of the bogie system, the generalized curve of the bogie system reliability with different
Curve graph about the bogie system reliability with different
For general system, the range of
It can be seen that
The traditional reliability theory only studies the system reliability when components are completely independent, which is too conservative and unfavorable for achieving the system potential. Considering the interaction relationship between components not only can guarantee that the system works well but also accurately reflects the reliability of whole system. Therefore, the safety property of system to accomplish tasks can be enhanced to the maximum extent.
Based on the system reliability calculation of dependent components by applied Copula Function, the convex combination considering operation and maintenance strategy was introduced. For safetycritical systems, it is suitable to estimate the system reliability based on the relatively cautious operational strategy. For nonsafetycritical systems, it is suitable to estimate the system reliability based on the relatively optimistic operational strategy. For most of general system, it is better to estimate the system reliability based on the combination of the above two extreme strategies. Compared with the traditional system reliability estimate value without taking the operational strategy into consideration, this estimate value considers the corresponding relationship between the system reliability and maintenance, and it can meet the need for balance between the higher safety and economy of enterprises and also can offer a theory for scientifically estimating the system reliability of highspeed trains and making rational maintenance strategy.
Besides, with the given operational resource and guaranteed optimum reliability, the system reliability optimal estimate value can be obtained by optimizing
This paper presents a new method to estimate the system reliability based on convex combination considering operation and maintenance strategy. The main conclusions include the following:
A new thought was put forward to calculate the network system reliability of complex dependent components based on convex combination considering operation and maintenance strategy, offering a scientific and practical method to solve the issue of system reliability modeling and estimating on complex electromechanical integration.
Taking the bogie system of highspeed train as the engineering background, the method proposed in this paper was applied and verified. It showed that this method can not only take the safety and economy into consideration but also obtain the rational operational strategy and realize its reliability estimation.
The convex combination equation of system reliability considering operation and maintenance strategy was established in this paper. The subsequent research can focus on the optimization of
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support of the projects of National Science Support Plan of China (2011BAG01802), State Key Laboratory of Rail Traffic Control and Safety (RCS2014ZT23), and CRH3 highspeed train fault data research (I11L00060).