- Research
- Open Access
Reliability analysis of subway vehicles based on the data of operational failures
- Huaixian Yin^{1, 2}Email authorView ORCID ID profile,
- Kai Wang^{3},
- Yong Qin^{1},
- Qingsong Hua^{2} and
- Qibin Jiang^{3}
https://doi.org/10.1186/s13638-017-0996-y
© The Author(s). 2017
- Received: 4 August 2017
- Accepted: 23 November 2017
- Published: 15 December 2017
Abstract
A large quantity of failure data for subway vehicles was collected from long-term field investigations and technical exchanged. These failure data has a guiding significance for preserving subway system. By preprocessing (screening, refining, and classification) the original data and statistical analysis, we establish some selected model, then we use A-D test to verify the degree of fitting in selected model so that we can determine the optimal failure distribution model, and then the reliability characteristic quantities could be calculated by the optimal failure distribution model. These reliability characteristic quantities can predict failure rate, failure number, etc. It can be used to assist proper maintenance scheduling to reduce the occurrence of accidents and significant to important practical guiding.
Keywords
- Reliability analysis
- Survival analysis
- Parameter estimate
- Degree of fitting
1 Introduction
Since the first subway line was put into operation in October 1969, there are more than 20 cities owned their subway systems in China, with a total operating mileage over 2400 km. Chinese subway companies have accumulated large amounts of failure data up to now, these data truly reflect field operating conditions. However, there are certain shortcomings in the data, much of it does not comply with uniform standards or is derived from complex data resources, and it may be missing important information [1, 2]. A large-scale subway system in China requires the successful prevention of major accidents and sudden incident; otherwise, catastrophic results might occur. Therefore, how to analyze and deal with such complex large-scale operation failure data, to ensure the safety of urban rail transit has become a major research topic in the field of subway reliability research.
Wang et al. presented the service life estimation method based on the three-parameter Weibull maximum likelihood estimation, respecting to the component wearing of high speed multiple units [3]. A new product data management method was created in [4] to process the component maintenance and historical failure data of electric multiple units, which resulted in a 30% increase in the reliability. Others performed the reliability analysis in [5] for the Bogie system of Sweden’s railways based on data collected by a wireless sensor network. The problem with this method was that the uncertainty in the time domain was not considered. In [6], they adopted the random process and reliability theory to investigate the failure distribution rules and reliability of rail vehicle components [6]. Yu et al. deduced the safety domain curve of high-speed trains through deducting the extreme sensitivity of system reliability [7]. Some articles used the nonparametric method to estimate the reliability function of the mechanism under extreme impact [8, 9], and Jiang analyzed the application of the proportional risk function in a repairable system [10].
The existing failure-data-based reliability analyses were mainly focused on railway passenger and freight vehicles and high-speed train. However, little analyses attention has been paid to the subway system. In essence, the subway is different from the railway in various aspects, such as the departure intervals, operating cycle, line conditions, the failure position, frequency, and maintenance data.
Parameter estimation method of reliability can only be used in known lifetime distribution. Unknown distribution usually uses probabilistic paper graph method and similar WPP graphic estimation method to study the distribution; these methods need to draw the curve of reliability and failure time. By further studying the shape of the curve, the reliability model of the failure data is determined. If the distribution model of a group failure data is not known, survival analysis theory can assume the group failure data conforms to all models, then each distribution model is fitted and the best fitting distribution model is selected, Finally, the parameter estimation and hypothesis testing are carried out. Survival analysis method can effectively solve uncertain failure time interval problems under the mechanism of censored data on subway vehicles, in order to get more reasonable results of reliability analysis. Therefore, we used survival analysis technology to perform the reliability analysis of the subway vehicles for the purpose of accurately grasping the working status of key subway systems, including identifying failures, performing maintenance, and securing the subway’s operation. The survival analysis method has particular advantages in the processing and analysis of censored data during the application of non-parametric, parametric, and semi-parametric survival analysis.
2 Fault distribution model and methods analysis
2.1 Survival analysis
Survival analysis is a technology of statistical analysis about survival time. Based on data collected via experiment or survey, it statistically analyzes the survival time of living creatures, human, or other things with a survival cycle and represents the results in the form of a survival function, probability density function, danger scale function, and average life [11, 12].
2.1.1 Survival function
On the equation, N is the product sample and n(t) are the numbers of failure at samples time.
2.1.2 Probability density function
2.1.3 Danger scale function
2.1.4 Average life
2.2 Model building and methods
2.2.1 Fault data collection and pretreatment
In terms of fault data collection and pretreatment, we use statistics method, eliminate or merge the fault entry, and eventually determine the effective subway vehicles failure data.
2.2.2 Candidate distributions
A large number of articles were reviewed to determine the candidate distributions, including the exponential distribution, logarithmic normal distribution, two-parameter Weibull distribution, and three-parameter Weibull distribution [16].
2.2.3 The maximum likelihood estimation
The maximum likelihood estimation method was used in this study for the parameter estimation of the optimal distribution. The basic principle of this method is as follows: assuming the known population distribution and an unknown parameter θ, one value \( \hat{\theta} \) is chosen from all possible values, which can result in the maximal probability of the observed results. \( \hat{\theta} \) is then defined as the maximum likelihood estimation value of θ, and the parameter estimation method was named as maximum likelihood estimation method [17].
The \( \widehat{\theta}\left({x}_1,{x}_2,\cdots, {x}_n\right) \) is called the maximum likelihood estimation of θ.
Thus, the problem to determine the maximum likelihood estimation is attributed to seek the maximum in the differential calculus problem.
2.2.4 Degree of fitting
First, the p values of the four candidate distributions were compared. If p > 0.05, it indicated that the corresponding distribution was able to fit the failure data. The distributions with good fitting results were preserved, and then the A-D statistical variable was calculated. The distribution with a minimum A-D value was chosen as the optimal distribution model.
3 Example analysis and results
3.1 Fault data statistics
The structure of subway vehicles includes the running gear, traction system, brake system, control and diagnostic system, and the auxiliary system. All of these subsystems play a significant role in the vehicle’s reliability and safe operation. There are frequent subway failures and accidents due to the rapid development of the urban subway transportation system. Therefore, we investigated the reliability of the key subsystems in subway vehicles in this study.
Because most of the subway vehicles system life distribution data is censored data, we use the survival analysis in the system time between failures to process censored data. In the fault data statistics, censored data mainly includes two categories. One kind is interval-censored data, if the maintenance work is reliable and the failure occurs between the overhaul and the last overhaul, so fault time is an interval, uncertain value, and fault specific time unknown. One kind is the right censored data, statistical period of the beginning and the end will have censored data, and fault time is greater than a certain value of tracked. We use common failure distribution function on censored data for maximum likelihood method of parameter estimation to calculate A-D statistics to select fitting of better distribution function.
3.2 Result and discussion
Fault distribution fit test table of the key subsystems
Subsystems | Candidate distribution model | P | A-D | Optimal distribution | Parameter |
---|---|---|---|---|---|
Running gear | Two-parameter Weibull | 0.249 | 0.704 | Two-parameter Weibull | Shape 0.9124 Scale 13.5450 |
Three-parameter Weibull | 0.007 | 1.325 | |||
Exponential | 0.253 | 0.921 | |||
Logarithmic normal | 0.097 | 0.876 | |||
Traction system | Two-parameter Weibull | 0.086 | 1.380 | Two-parameter Weibull | Shape 0.9940 Scale 10.6495 |
Three-parameter Weibull | < 0.005 | 5.463 | |||
Exponential | 0.021 | 1.705 | |||
Logarithmic normal | 0.052 | 1.700 | |||
Brake system | Two-parameter Weibull | 0.014 | 1.120 | Logarithmic normal | Location 2.3693 Scale 1.3003 |
Three-parameter Weibull | 0.224 | 0.657 | |||
Exponential | < 0.003 | 3.376 | |||
Logarithmic normal | 0.236 | 0.644 | |||
Control and diagnostic system | Two-parameter Weibull | 0.242 | 0.672 | Logarithmic normal | Location 2.5573 Scale 1.3581 |
Three-parameter Weibull | 0.123 | 0.796 | |||
Exponential | 0.005 | 2.277 | |||
Logarithmic normal | 0.353 | 0.595 | |||
Auxiliary system | Two-parameter Weibull | 0.096 | 1.974 | Logarithmic normal distribution | Location 1.4915 Scale 0.9349 |
Three-parameter Weibull | <0.005 | 4.060 | |||
Exponential | 0.010 | 2.080 | |||
Logarithmic normal | 0.080 | 1.016 |
The reliability characteristic functions of each subsystem
Subsystem | Failure density function | Cumulative distribution function |
---|---|---|
Running gear | \( f(t)=\frac{0.9124}{13.5450}{\left(t/13.5450\right)}^{\hbox{-} 0.0876}\exp \left[-{\left(t/13.5450\right)}^{0.9124}\right] \) | F(t) = 1 − exp[−(t/13.5450)^{0.9124}] |
Traction system | \( f(t)=\frac{0.9940}{10.6495}{\left(t/10.6495\right)}^{-0.006}\exp \left[-{\left(t/10.6495\right)}^{0.9940}\right] \) | F(t) = 1 − exp[−(t/10.6495)^{0.9940}] |
Brake system | \( f(t)=\left\{\begin{array}{l}\frac{1}{1.3003\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.3693\right)}^2}{2\times {1.3003}^2}},t>0\\ {}0,t=0\end{array}\right. \) | \( F(t)=\Phi \left(\frac{\ln t-2.3693}{1.3003}\right) \) |
Control and diagnostic system | \( f(t)=\left\{\begin{array}{l}\frac{1}{1.3581\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.5573\right)}^2}{2\times {1.3581}^2}},t>0\\ {}0,t=0\end{array}\right. \) | \( F(t)=\Phi \left(\frac{\ln t-2.5573}{1.3581}\right) \) |
Auxiliary system | \( f(t)=\left\{\begin{array}{l}\frac{1}{0.9349\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-1.4915\right)}^2}{2\times {0.9349}^2}},t>0\\ {}0,t=0\end{array}\right. \) | \( F(t)=\Phi \left(\frac{\ln t-1.4915}{0.9349}\right) \) |
The reliability characteristic functions of each subsystem
Subsystem | Reliability function | Failure rate function | Average life (days) |
---|---|---|---|
Running gear | R(t) = exp[−(t/13.5450)^{0.9124}] | \( \lambda (t)=\frac{0.9124}{13.5450}{\left(t/13.5450\right)}^{\hbox{-} 0.0876} \) | 14 |
Traction system | R(t) = exp[−(t/10.6495)^{0.9940}] | \( \lambda (t)=\frac{0.9940}{10.6495}{\left(t/10.6495\right)}^{-0.006} \) | 11 |
Brake system | \( R(t)=1\hbox{-} \Phi \left(\frac{\ln t-2.3693}{1.3003}\right) \) | \( \lambda (t)=\frac{\frac{1}{1.3003\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.3693\right)}^2}{2\times {1.3003}^2}}}{1-\Phi \left(\frac{\ln t-2.3693}{1.3003}\right)} \) | 25 |
Control and diagnostic system | \( R(t)=1\hbox{-} \Phi \left(\frac{\ln t-2.5573}{1.3581}\right) \) | \( \lambda (t)=\frac{\frac{1}{1.3581\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.5573\right)}^2}{2\times {1.3581}^2}}}{1-\Phi \left(\frac{\ln t-2.5573}{1.3581}\right)} \) | 32 |
Auxiliary system | \( R(t)=1\hbox{-} \Phi \left(\frac{\ln t-1.4915}{0.9349}\right) \) | \( \lambda (t)=\frac{\frac{1}{0.9349\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-1.4915\right)}^2}{2\times {0.9349}^2}}}{1-\Phi \left(\frac{\ln t-1.4915}{0.9349}\right)} \) | 7 |
Tables 2 and 3 shows that the mean operating time between failures for the running gear, traction system, brake system, control and diagnostic system, and auxiliary systems was 14, 11, 25, 32, and 7 days. The mean operating time between failures, namely, the failure rate, increased in the following order: auxiliary systems, traction system, running gear, brake system, and control and diagnostic systems. These results are consistent with the number of failures collected from the field data.
It can be concluded that maintenance should be scheduled every other day in order to meet the reliability requirements of the running gear. Similarly, the maintenance plan for the other subsystem can be formulated.
4 Conclusions
Based on the operational failure data of subway vehicles, a reliability analysis method of subway subsystems was developed based on the survival analysis theory. By filtering, classification, and the preprocessing of the failure data, the numbers of failure and mean operating time between failures were obtained for each subsystem. The results showed that the failure rate increased in the following order: auxiliary systems, traction system, running gear, brake system, and control and diagnostic systems. The optimal failure distribution model of every subsystem was determined by the use of Minitab. We can formulate the vehicle maintenance schedule to direct our daily maintenance work, which could observably reduce the failure of subsystem.
The reliability characteristic functions can be used to obtain a scientific estimation of the reliability characteristic variables. As the rapid construction and increasingly complex of domestic subway system, reliability characteristic function for future subway has guiding significance to the construction and systemic maintenance. In the future, reliability analysis of the subway will get widespread attention and long-term development.
Due to the limitation of time and ability, this article only focuses on the subject of each subsystem. We will analyze the reliability of the specific components to find fault specific reason and provide guidance for train maintenance to reduce the incidence of failure.
Declarations
Acknowledgements
We thank the reviewers for their detailed reviews and constructive comments which have helped to improve the quality of this article.
Funding
This work has been supported by National Key Technology Research and Development Program (2015BAG12B01).
Authors’ contributions
HY gave the original ideas and wrote the manuscript. KW and QJ participated in the establishment of simulation model. YQ and QH provided failure data and analyses. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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