In view of the negative impact of component importance measures based on system reliability theory and centrality measures based on complex networks theory, there is an attempt to provide improved centrality measures (ICMs) construction method with fuzzy integral for measuring the importance of components in electromechanical systems in this paper. ICMs are the meaningful extension of centrality measures and component importance measures, which consider influences on function and topology between components to increase importance measures usefulness. Our work makes two important contributions. First, we propose a novel integration method of component importance measures to define ICMs based on Choquet integral. Second, a meaningful fuzzy integral is first brought into the construction comprehensive measure by fusion multi-ICMs and then identification of important components which could give consideration to the function of components and topological structure of the whole system. In addition, the construction method of ICMs and comprehensive measure by integration multi-CIMs based on fuzzy integral are illustrated with a holistic topological network of bogie system that consists of 35 components.
Recent decades have witnessed not only the rapid development on the highly integrated system of electromechanical systems, but also the significant progress on the system function [
In system reliability theory, importance measures are used as effective tools to evaluate the relative importance of components and identify system weaknesses [
Recent advances indicate that electromechanical systems can be virtually represented as networks, where the components of technological products are easily depicted by the nodes of complex networks and the connections between linkage components are naturally depicted by the links of complex networks [
However, all these researches focused on only one measure, such as component importance measure and one centrality measure, and every measure has its own disadvantage and limitation. In recent years, researchers study a multiattribute ranking problem to evaluate the component importance comprehensively from more than one perspective, which would be a special case of multicriteria decision-making (MCDM). MCDM refers to making decision for alternatives in the presence of multiple and conflicting criteria [
Although the above component importance measures or centrality measures have been widely applied in identifying influential components, there are some limitations and disadvantages. CIMs are built on the assumption of the independence of components and none of them has taken the impact of topological structure between components into account. CMs mentioned above focus only on the components propagation behavior of complex network and are limited to the point of the reliability analysis [
If only one measure is adopted, then the rankings of identifying influential components may be different by using a different measure. In some cases, using different centrality measures may provide different results, even conflicting results [
In this paper, we try to introduce fuzzy integral theory to explore how to identify influential components. Our work makes two important contributions. First, we integrate component importance measures and centrality measures with Choquet integral to define a new kind of improved centrality measures. Second, a novel index, comprehensive measure, of a meaningful fuzzy integral-based is brought into identification of important components for which it could give consideration to function of components and topological structure of the whole system.
The rest of the paper is structured as follows. Section
Currently, complex networks are being studied in many fields of science, such as social sciences, computer sciences, physics, biology, and economics. The majority of systems in reality can be undoubtedly described by models of complex networks. For example, Internet is a complex network composed of web sites [
As mentioned above, electromechanical systems are characterized by large scale, complex structure, nonlinear behavior, various working states, high coupling components, random operation environment, and so forth, which are not easy to be modeled directly to analyze global behaviors. Recently, lots of attempts have been made to model network in engineering, more specifically, electromechanical systems. It can be a complex network, in which the components are depicted by nodes and the physical connections between linkage components are depicted by links between the corresponding nodes. We refer to this representation as a topological structure and a formal model is presented by the following definitions.
We define
The holistic topological network.
For different systems, the properties of nodes or edges may be also different. For instant, assuming that Figure
In Figure
About thirty years ago the concept of fuzzy integral was proposed in Japan by Sugeno [
Let
Let
Let
In this section, the novel importance measures, that is, improved centrality measures (ICMs), are first proposed. It is a series of special importance measures to find the influential components that are really crucial for the normal operation of electromechanical systems. When some components lose or weaken their functions due to a certain mode of failures, there will be a degradation of the holistic system performance. Based on centrality measures, the originality and novelty of proposed ICMs is that they evaluate the importance of a component by taking into account the functional and topological properties in the holistic system. In essence, they are comprehensive indicators in the system, which are proposed for the assessment of the most important component.
In order to measure the importance of components, influential factors that are capable of representing the desired function of the individual components, as well as their structure, are required. Some quantitative information is introduced into the system to improve the accuracy of the important measure of the components. For example, functional properties reflect the ability of the system to perform its intended function, and topological properties describe the stability of the inherent structure of the system.
Through the project (
Influencing factors of the individual components.
A large number of centrality measures have been proposed to identify influential nodes within a complex network. Examples are DC, CC, BC, and EC. However, it cannot be ignored that while most of these centrality measures have been widely used in
Improved centrality measures are a series of importance measures which are extended from centrality measures. In a sense, ICMs can serve as better importance measures of components, as it synthesizes the components functional properties, such as usage reliability, failure rate, and connection strength, meanwhile, by taking systems structure into account, such as CC, BC, DC, and EC.
The following is an efficient and universal construction method to calculate ICMs.
Let us consider
Based on the holistic topological network, functional properties of nodes and edges are integrated into the definition of centrality measures, and then ICMs are constructed, defined as follows:
According to (
Zhang et al. [
Some simple examples are given to explain how ICMs perform.
In a binary network,
The improved closeness centrality of node
Given a directed graph, the length of a path is the number of edges forming it. We define the shortest path as the smallest length among all the paths connecting the source vertex to the target vertex. However, for electromechanical systems, functional properties of nodes and edges also can influence the length of the shortest path. Given that
The improved betweenness centrality of node
The ICMs, such as IDC, ICC, and IBC, reflect the function and structure of system from one aspect and cannot comprehensively reflect the functional and topological characteristics. In some cases, the results of IDC and ICC may be different, even conflicting results. To address this issue, in this paper, comprehensive measure is introduced firstly to explore how to fuse multi-ICMs based on fuzzy integral and then identify influential components. As a well-known fuzzy integral theory, Sugeno integral and Choquet integral have received much interest from researchers and practitioners.
Let us consider a decision matrix
If we choose Choquet integral, the comprehensive measure of node
The specific steps of the method are illustrated as follows.
We can construct a network based on Section
According to Figure
In this step, we apply (
The alternatives with higher
The flow chart of the proposed methods is shown in Figure
The flow chart of the proposed method.
China Railway CRHX Size (CRHX) is designed for a speed of 350 km/h and each car is suspended by two bogies. The bogie system of the 350 km/h EMU train is one of the key parts of CRHX which plays an important role in sustaining the static load from the body weight of a car, carrying the suspensions, brakes, wheels, and axles and controlling wheel sets on curved and straight tracks, in accordance with Figure
Components in bogie system.
Number | Name |
---|---|
|
Bogie frame |
|
Brake Caliper |
|
Brake lining |
|
Brake discs |
|
Booster cylinder |
|
Spring |
|
Axle box body |
|
Vertical shock absorber |
|
Bearing |
|
Wheel |
|
Axle |
|
Secondary vertical shock absorber |
|
Railway coupling |
|
Gearbox |
|
Grounding device |
|
Traction motor |
|
Height adjusting device |
|
Antihunting damper |
|
Air spring |
|
Center pin bush |
|
Traction rod |
|
Transverse shock absorber |
|
Transverse backstop |
|
Anti-side-rolling torsion bar |
|
Control valve |
|
Speed Sensor 1 |
|
Speed Sensor 2 |
|
LKJ2000 |
|
Device for cleaning the tread band of vehicle wheels |
|
Acceleration sensor |
|
Junction box |
|
Temperature sensor bearing |
|
Axle temperature sensor |
|
AG37 |
|
AG43 |
Structural features of bogie system.
In this section, a case concerning the holistic topological network model of bogie system (as shown in Figure
The holistic topological network of bogie system: every node
The attributes of edges and nodes in (
The preprocessed failure data of gear box.
Number | Mileage/105 km | Failure mode |
---|---|---|
( |
6.39853 | Oil leakage |
( |
7.87662 | Oil leakage |
( |
7.90238 | Gear shift |
( |
10.02856 | Oil leakage |
( |
11.26585 | Crackle |
( |
11.29788 | Oil leakage |
( |
12.39568 | Oil leakage |
( |
14.02572 | Crackle |
( |
15.64292 | Crackle |
( |
16.39853 | Oil leakage |
( |
16.64292 | Crackle |
( |
16. 64292 | Oil leakage |
( |
17.66824 | Oil leakage |
( |
18.16762 | Gear shift |
( |
18.2579 | Oil leakage |
( |
19.32587 | Gear shift |
( |
19.90225 | Oil leakage |
( |
21.0191 | Crackle |
( |
22.17948 | Crackle |
( |
22.38788 | Crackle |
( |
23.0191 | Oil leakage |
( |
27.73312 | Oil leakage |
( |
28.62366 | Crackle |
( |
28.87598 | Oil leakage |
( |
29.62366 | Oil leakage |
( |
31.79775 | Crackle |
( |
32.4842 | Crackle |
(28) | 33.43831 | Crackle |
(29) | 34.43831 | Gear shift |
(30) | 34.50838 | Crackle |
(31) | 37.66297 | Oil leakage |
(32) | 38.66297 | Gear shift |
(33) | 40.0016 | Crackle |
Using the preprocessed failure data of CRHX and (
The nodes functional properties.
Node | Life time/year | Failure rate | MTBF |
---|---|---|---|
|
20 | 0.0134 | 2.34 |
|
20 | 0.00798 | 1.25 |
|
15 | 0.0089 | 1.54 |
|
20 | 0.0045 | 2.21 |
|
20 | 0.0079 | 1.72 |
|
20 | 0.0059 | 1.92 |
|
20 | 0.0086 | 1.41 |
|
20 | 0.0081 | 1.69 |
|
20 | 0.0144 | 1.21 |
|
20 | 0.0126 | 1.34 |
|
20 | 0.0176 | 1.47 |
|
20 | 0.0079 | 1.65 |
|
20 | 0.0082 | 1.41 |
|
20 | 0.0103 | 1.52 |
|
20 | 0.0103 | 1.81 |
|
20 | 0.0078 | 1.45 |
|
20 | 0.0116 | 1.44 |
|
20 | 0.0082 | 1.66 |
|
15 | 0.0061 | 2.03 |
|
20 | 0.0052 | 1.55 |
|
20 | 0.0051 | 1.97 |
|
20 | 0.0062 | 1.86 |
|
20 | 0.0049 | 1.93 |
|
20 | 0.0051 | 1.77 |
|
20 | 0.0072 | 1.78 |
|
20 | 0.0077 | 1.68 |
|
20 | 0.0177 | 1.32 |
|
20 | 0.0187 | 1.43 |
|
20 | 0.0107 | 1.50 |
|
20 | 0.0152 | 1.41 |
|
20 | 0.0049 | 1.53 |
|
20 | 0.0165 | 1.32 |
|
15 | 0.0189 | 1.38 |
|
20 | 0.0191 | 1.49 |
|
20 | 0.0169 | 1.53 |
Edges have a striking effect on critical nodes in the network. In essence, edges in holistic topological network also describe components, but these components have different properties from nodes. The fault propagation probability, connection strength, and failure rate of edges are computed according to (
Functional properties of several edges.
Edge | Connection strength | Failure rate | Fault propagation probability | Edge | Connection strength | Failure rate | Fault propagation probability |
---|---|---|---|---|---|---|---|
|
0.4 | 0.0051 | 0.0001 |
|
0.6 | 0.0068 | 0.0003 |
|
0.5 | 0.0087 | 0.0005 |
|
0.4 | 0.0053 | 0.0004 |
|
0.5 | 0.0095 | 0.0006 |
|
0.7 | 0.0071 | 0.0001 |
|
0.8 | 0.0088 | 0.0003 |
|
0.5 | 0.0067 | 0.0015 |
|
0.6 | 0.0082 | 0.0003 |
|
0.3 | 0.0104 | 0.0010 |
|
0.5 | 0.0062 | 0.0007 |
|
0.7 | 0.0045 | 0.0013 |
|
0.7 | 0.0079 | 0.0004 |
|
0.4 | 0.0042 | 0.0011 |
⋯ | ⋯ | ⋯ | … | ⋯ | ⋯ | ⋯ | … |
CMs of nodes in the holistic topological network are necessary to construct improved centrality measures to assess influential components. According to (
CMs of nodes in the holistic topological network.
In the method proposed in this paper, the assessment of influential components requires, firstly, exhaustive and systematic definition and calculation improved centrality measures for any given number of components according to Section
Improved centrality measures.
In order to explain the advantages of ICMs, we compare with the identification results of ICMs and CMs. The most critical node in bogie system is all
The ranking of nodes.
Another interesting fact observed is that, as presented in Figure
In order to overcome the uncertainty and randomness which applies for single measure identifying critical component, we fuse ICMs to explore how to identify influential components based on fuzzy integral. According to Section
Influential components of bogie system.
Meanwhile, how to calculate weight
To better assess the importance of a component in the holistic topological network, the method of integration ICMs is proposed in this paper, which takes functional and topological properties into account. It is much different from integration topological CMs indicators such as DC, BC, CC, and EC mentioned in Section
The results of integration CMs and ICMs.
Table
The evaluation results of different methods.
Number | AHP | TOPSIS | DC | CC | BC | EC | Practical recognition | Sugeno integral | Choquet integral | ||
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With the reduction of the importance of nodes, the difference of all methods in Table
The ranking of partly nodes.
The methods of integration multimeasures are able to overcome the randomness and uncertainty by using a single measure. Different methods of comprehensive multimeasures are selected, and the accuracy of the identification results is different. Figure
The accuracy rate of these methods.
Through the exploratory discussion above, it is shown that the results acquired from comprehensive measure are, to a certain extent, more reasonable and powerful than those traditional importance measures. Just as mentioned, the system function represents the interactions between components; system topology represents the structure relationship. Therefore, the comprehensive gives consideration to both topological features and physical characteristics of a holistic topological network from multiple perspectives.
This paper integrates the literature of mechatronic architecture and complex networks to define holistic topological network. And, based on the notion of complex networks, meaningful improved centrality measures (ICMs) are first brought and then comprehensive measure, with first-time, is constructed to identify important components by integrated multi-ICMs. Indeed, construction ICMs with the consideration of functional and topological properties and their relationship is the originality and novelty of proposed measures. Next, integration multi-ICMs based on fuzzy integral, that is, the combination of multiple influencing factors, is also the novelty of comprehensive measure. This paper has also shown the application of the proposed approach in reliability assessment of bogie system of CRHX EMUs. By applying the comprehensive measure, the components importance of bogie system can be evaluated at reasonable human factors. Results indicate that the ranking of critical components can be not the same in selecting different fuzzy integrals. According to the applicable environment, reasonable choices can be determined. In addition, three conclusions are drawn through an exploratory discussion: The function and topology are of the same importance in electromechanical system. If identifying critical components, these two aspects should be taken into account. The method of integration ICMs with Choquet integral by using The result of comprehensive evaluation is better than that of single measure identification.
Of course, due to the diversification and complexity of the real electromechanical system, the model presented here is just a simplification of what happens in actual systems. Several influential factors of critical components in the model need to be further developed if some additional information can be acquired. As previously mentioned, the robustness of comprehensive measure with respect to the fault propagation damping parameter is still under discussion and valuable for further research.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The work is partially supported by the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China (Grant no. RCS2016ZZ002), and the National Key Research and Development Program of China (Grant no. 2016YFB1200402).