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Research ArticleCentralized Maintenance Time Prediction Algorithm for FreightTrain Wheels Based on Remaining Useful Life Prediction

Hongmei Shi ,1,2 Jinsong Yang ,1,2 and Jin Si 1,2

1School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China2Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology (Beijing Jiaotong University),Ministry of Education, Beijing 100044, China

Correspondence should be addressed to Hongmei Shi; [email protected]

Received 25 June 2019; Revised 6 February 2020; Accepted 13 February 2020; Published 11 March 2020

Academic Editor: Ricardo Branco

Copyright © 2020 Hongmei Shi et al. *is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Many freight trains for special lines have in common the characteristics of a fixed group. Centralized Condition-BasedMaintenance (CCBM) of key components, on the same freight train, can reduce maintenance costs and enhance transportationefficiency. To this end, an optimization algorithm based on the nonlinear Wiener process is proposed, for the prediction of thetrain wheels Remaining Useful Life (RUL) and the centralized maintenance timing. First, Hodrick–Prescott (HP) filtering al-gorithm is employed to process the raw monitoring data of wheel tread wear, extracting its trend components. *en, a nonlinearWiener process model is constructed. Model parameters are calculated with a maximum likelihood estimation and the generaldeterioration parameters of wheel tread wear are obtained. *en, the updating algorithm for the drift coefficient is deduced usingBayesian formula. *e online updating of the model is realized, based on individual wheel monitoring data, while a probabilitydensity function of individual wheel RUL is obtained. A predictionmethod of RUL for centralized maintenance is proposed, basedon two set thresholds: “maintenance limit” and “the ratio of limit-arriving.” Meanwhile, a CCBM timing prediction algorithm isproposed, based on the expectation distribution of individual wheel RUL. Finally, the model is validated using a 500-day onlinemonitoring data on a fixed group, consisting of 54 freight train cars. *e validation result shows that the model can predict thewheels RUL of the train for CCBM. *e proposed method can be used to predict the maintenance timing when there is a largenumber of components under the same working conditions and following the same path of degradation.

1. Introduction

*ere are three main maintenance policies for railwayfreight trains, including corrective maintenance, scheduledmaintenance, and condition-based maintenance. *e con-dition-based maintenance is actually getting increasing at-tention. One of the key points in condition-basedmaintenance of railway freight trains is Remaining UsefulLife (RUL) prediction of vehicle parts, based on reliabilitytheory and online monitoring data. Moreover, determininga reasonable maintenance time is essential for improvingfreight train operation efficiency and maintenance cost re-duction. In a fixed group of a freight train consisting of 54railway freight cars, there is a large number of parts of thesame type and under the same working conditions, such as

wheels and bogies. If the condition of each vehicle part isconsidered independently, then the train maintenance timewill be significantly long, leading to high maintenance costsand low operation efficiency. Hence, Centralized Condition-Based Maintenance approach is proposed in this paper,which means to find a reasonable maintenance time pointwhere all vehicle parts of the same type are repaired si-multaneously, thus improving the maintenance efficiency.

Understandably, the wheel status will directly affect thetrain operation quality and safety, as an important part ofany railway freight car.*e wheel tread wear is one of the keyparameters reflecting the wheel state in relation to the servicetime. *e RUL of wheels can be predicted using a degra-dation model based on offline history tread wear data andonline monitoring data, serving as an important basis for

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 9256312, 12 pageshttps://doi.org/10.1155/2020/9256312

mailto:[email protected]://orcid.org/0000-0001-6004-8784https://orcid.org/0000-0002-9477-7931https://orcid.org/0000-0002-0213-5617https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/9256312

vehicle CCBM. *e dispersion of tread wear data for dif-ferent wheels will gradually increase over operation time,producing different degradation curves for each individualwheel. A set reasonable maintenance time can avoid ex-ceeding the operation limit of the wheel with a fast wearspeed, thus avoiding compromising safety, while alsoavoiding over repair.

At present, there are many researches on the degradationprocess of railway vehicle wheels. Most of them are based onthe mechanism to model the wheel wear, rolling contactfatigue and other degradation processes [1, 2]. Hossein-Niaet al. [3] developed a model to estimate the evolution ofsurface-initiated Rolling Contact Fatigue (RCF). In thismodel for RCF calculations, a shakedown-based theory isapplied locally, and the FaStrip algorithm is used to estimatethe tangential stresses. While considering the mechanism,the prediction of wheel wear needs to consider variousfactors such as lines, rails, vehicles, etc., and it will be difficultto update the model based on real-time degradation, whichis easy to cause error accumulation. *is study uses a data-driven approach that is flexible and easy to achieve real-timeonline prediction.

A lot of research has been carried out on condition-basedmaintenance, applying performance degradation data [4–6].Wiener process is a commonly used method for remaininglife modeling, as the first hitting time is inversely Gauss-distributed and it can, furthermore, reflect nonmonotonicrandom changes [7–9]. *e performance degradation modelderived from standard Wiener process has been widelydiscussed in recent years and can be used to describe theperformance degradation of many typical products. Si et al.[10] proposed a remaining life prediction model based onnonlinear Wiener process. *e unknown parameters in theestablished model are estimated using the maximum like-lihood estimation approach, leading to a probability densityfunction for the RUL. *e model is based on offline mon-itoring data, which means that no real-time updating ofmode parameters is possible. Zhang et al. [11] incorporatethe inspection influence into degradationmodeling based onWiener process. *e impact of the inspection of the systemin the process of obtaining degraded data on the degradationprocess is considered. *e proposed approach was dem-onstrated by a case study using the mechanical gyroscopes.Peng et al. [12] proposed a switchable state-space degra-dation model to characterize degradation paths with non-deterministic switching manner dynamically. *e proposedmethod was applied to a real bearing degradation processwith phase transitions.

*e purpose of modeling the degradation process is toguide the formulation and optimization of maintenancestrategies. *e Wiener process approach is often combinedwith the optimization problem of maintenance decision[13, 14]. Sun et al. [15] studied multicomponent systems,where the degradation of each component was assumed tocomply with the Wiener process. In their study, the optimalmaintenance decision-making problem of multiunit systemswas studied, under the premise of periodic inspection. Weiet al. [16] studied a binary degradation system affected byshock and used Wiener process to simulate degradation.

And according to the system state, the optimal action in-cludes no action, imperfect repair, preventive replacement,or corrective replacement. Zhang et al. [17] published anoverview of the current research status of the Wienerprocess. *e paper reviews recent developments in theWiener process-based modeling methods for degradationdata analysis and RUL estimation, as well as their applica-tions in the field of prognostics and health management(PHM). Specifically, the modifications in theWiener processare introduced considering nonlinearity, multisource vari-ability, covariates, and other multivariable factors involvedin the degradation processes.

In this paper, a prediction algorithm of individual RUL,based on nonlinear Wiener process, is proposed for theprediction of centralized maintenance time of railway freighttrain wheels. *e structure of the article is as follows: Section2 introduces the HP filtering algorithm, which is used toextract the degradation trend of the monitoring data, inorder to model the degradation process. In Section 3, themodeling process is described for the overall degradationmodel of the wheel tread circumferential wear, based on thenonlinear Wiener process. *e model parameters are esti-mated by the maximum likelihood method, while the pa-rameters updating algorithm is derived based on theBayesian formula. *en, the real-time prediction of RUL forthe individual wheel is provided, using the updated pa-rameters. In Section 4, a method for determining themaintenance time of the whole train is proposed, by setting amaintenance limit and a ratio called limit-arriving. Weibulldistribution is employed to fit the RUL distribution ofdifferent wheels on the same train, followed by a predictionfor the centralized maintenance time. Finally, the algorithmwas verified using 500-day data from a train.

2. Degradation Trend Extraction fromMonitoring Data

In this paper, the monitoring data of wheel tread wear areobtained from an online train wheel monitoring system,called TrainWheel Detection System (TWDS), andmountedon the railway line. *e system is used to detect wheelparameters such as tread wear, rim width, wheel diameter,etc. Particularly, as the operation mileage increases, thediameter of a wheel at different positions will vary. In ad-dition, due to wheel vibration and sensor measurementerrors, there is fluctuation in the monitoring data itself. *emeasurements show that even though the actual value ofdegradation is low, the relative fluctuation is quite large.*erefore, it is necessary to apply filtering on the degra-dation data, in order to extract the trend component,achieving better results for the RUL prediction.

HP filtering, proposed by Hodrick and Prescott in 1981,is widely used in economic analysis, but can be generallyapplied on data containing fluctuations, to extract trendcomponents [18, 19]. Ouahilal et al. [20] used HP filter in thestock price forecasting field and proposed an approachcombining Support Vector Regression with HP filter. *eexperimental results confirm that the proposed model ismore powerful in terms of predicting stock prices. Poloni

2 Mathematical Problems in Engineering

and Sbrana [21] extend HP filtering to multidimensionalconditions, offering an interesting option for industrialproduction analysis in several European countries. Dai et al.[22] first used HP filtering for degradation data processing in2018 to extract the trend characteristics of solder joint failuredata. Still, there are only a few studies on HP filtering fordegradation data trend extraction.

In this paper, HP filter is used to process the monitoringdata of wheel tread wear, while the trend component ofwheel wear degradation is extracted for model parametersestimation and individual RUL prediction. *e process ofHP filtering algorithm is described as follows:

Suppose there is a time series Y � Y(1), Y(2),{Y(3), . . . , Y(m)}, which consists of a trend component X(t)and a fluctuation component C(t), as shown in

Y(t) � X(t) + C(t). (1)

Trend component X(t) can be obtained by calculatingthe following equation, as described in [23]:

min m

t�1(Y(t) − X(t))

2+ β

m− 1

t�2[(X(t + 1) − X(t))

− (X(t) − X(t − 1))]2.

(2)

Equation (2) can be divided into two parts, the first partbeing

m

t�1(Y(t) − X(t))

2. (3)

*is part reflects the reductive degree of the originalsequence.*e lower the value of this part, the better trackingperformance of X(t) to the original sequence is.

*e remaining part of (2) is defined as Part Two; i.e.,

m− 1

t�2[(X(t + 1) − X(t)) − (X(t) − X(t − 1))]2. (4)

Part Two measures the smoothness of the new sequence;that is, the lower it is, the higher the smoothness of X(t) is. βis a penalty factor to control the smoothness degree. Its valueis a result of compromise between the fidelity to the raw datatracking and the smoothness degree of the raw data se-quence. According to [24] and the monitoring frequency oftread wear monitoring data, β value is empirically set to1000, as more suitable to the purpose of this study.

Partial derivatives of X(1), X(1), . . . , X(1) are derivedfrom (2) and set to 0, in order to solve for X(t), as shown inthe following equation:

C(1) � β(X(1) − 2X(2) + X(3)),

C(2) � β(− 2X(1) + 5X(2) − 4X(3) + X(4)),

· · ·

C(t) � β(X(t − 2) − 4X(t − 1) + 6X(t) − 4X(t + 1) + X(t + 2)),

· · ·

C(m) � β(X(m − 2) − 2X(m − 1) + X(m)).

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

According to (5), the coefficient matrix F of dimensionsN × N can be obtained as follows:

F �

1 − 2 1 0 · · · 0

− 2 5 − 4 1 0 · · · 0

1 − 4 6 − 4 1 0 · · · 0

0 1 − 4 6 − 4 1 0 · · · 0

· · ·

0 · · · 0 1 − 4 6 4 1 0

0 · · · 0 1 − 4 6 4 1

0 · · · 0 1 − 5 5 − 2

0 · · · 0 1 − 2 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (6)

Equation (5) can be expressed as a matrix, as shown in

C � βFX, (7)

where,

X � (X(1), X(2), . . . , X(m))′,

C � (C(1), C(2), . . . , C(m))′,(8)

where (·)′ denotes the vector transposition. Combining (1)and (7), letting Y � (Y(1), Y(2), . . . , Y(m))′, the followingrelation is derived:

X � (βF + I)− 1Y, (9)

where I is the identity matrix.*e trend seriesX of time seriesY can be obtained by (9).*e degradation trend data of wheel tread wear, after HP

filtering, is used to establish the Wiener process model, asdescribed in the next section.

3. Degradation Modeling and RUL PredictionBased on Wiener Process

*e RUL prediction of wheels includes three steps: degra-dation model establishment, model parameters estimation,and calculation of the RUL. Because the wear rate of wheelsincreases as the service time progresses, a nonlinear Wiener

Mathematical Problems in Engineering 3

process model is selected as basis for the wheel tread wearmodel. *e estimation of model parameters is carried out intwo steps: (1) estimating the overall model parameters, usingthe degradation data of all wheels on the same train, to reflectthe overall characteristics of the degradation process; (2)updating the model parameters, using Bayesian formula, basedon themonitoring data of each individual wheel to better fit thedegradation process of each individual unit. As new moni-toring data are acquired, the individual degradation parametersare updated in the course of online RUL prediction. *en, theprobability density distribution function (PDF) of the RUL isobtained, according to the concept of the first hitting time.

3.1. Nonlinear Wiener Process Modeling. A reasonable per-formance degradation model is the key to accurately predictthe RUL of components. Due to the nonlinearity of wheeltread wear data, a nonlinear Wiener process model isestablished, as shown in the following equation:

X(s) � X(0) + λsb + σBB(s), (10)

where X(s) is the degradation when the mileage is s. X(0) isthe initial degradation at the beginning of monitoring. λ isthe drift coefficient, reflecting the degradation rate. It is oftenset as a random variable, subject to λ ∼ N(μλ, σ2λ), reflectingthe individual difference. b is a nonlinear coefficient, σB is adiffusion coefficient, and B(s) is a standard Brownian mo-tion, subject to B(s) ∼ N(0, s).

Most current studies often assume X(0)� 0 in order tofacilitate the calculations. However, different wheels mayhave different initial degradation due to different servicemileage; thus, X(0) cannot be assumed to be the same on allwheels, in this situation. Equation (10) is rewritten as (11), inorder to simplify the model and take X(0) into account.

X(s) � λsb + σBB(s), (11)

whereX(s) � X(s) − X(0). (12)

In this way, the various values of initial degradation ofdifferent individual wheels are all converted to 0.

3.2. Model Parameters Estimation. *ere are four unknownparameters θ � [μλ, σλ, b, σB] in the wheel degradation model.First, the degradation data of all the wheels on the same train istaken as a sample set, where the overall model parameter θ isestimated, according to the maximum likelihood estimationmethod. Following, the updating algorithm for the mean andvariance of the drift coefficient in the model is deduced,according to the Bayesian formula, and then the parametersupdate of the individual model is realized.

3.2.1. Overall Model Parameter Estimation. *e analyticalsolution of the parameters in the Wiener process model canbe obtained using the maximum likelihood estimation. It isassumed that N wheels are monitored m times up to thecurrent mileage sm. *e degradation data of the N wheels is

denoted as X, and the degradation of the n th wheel at themonitoring mileage s1, s2, . . . , sm is recorded as

Xn � Xn s1( , Xn s2( , . . . , Xn sm( ( ′. (13)

Let

s � sb1, sb2, . . . , s

bm ′. (14)

According to (12), Xn can be described as follows:Xn � Xn s1( − Xn(0), Xn s2( − Xn(0), . . . , Xn sm( − Xn(0)( ′.

(15)

According to (11) and the independent incrementalcharacteristics of the Wiener process, Xn follows the mul-tivariate normal distribution, Xn ∼ N(μ, σ). *e mean andvariance are, respectively,

μ � μλs,

σ � M + σ2λss′,(16)

whereM � σ2BT,

T �

s1 s1 · · · s1s1 s2 · · · s2⋮ ⋮ ⋮ ⋮s1 s2 · · · sm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

(17)

Assuming that the degradation of different wheels isindependent of each other, the logarithmic likelihoodfunction of θ � [μλ, σλ, b, σB] can be obtained [25] as

L(θ | X) � −Nm ln(2π)

2−12

N ln |M| 1 + σ2λs′M− 1s

−12

N

n�1

Xn − μλs( ′σ− 1 Xn − μλs( ,

(18)

where|σ| � |M| 1 + σ2λs′M

− 1s ,

σ− 1 � M− 1 −σ2λ

1 + σ2λs′M− 1s

M− 1ss′M− 1.(19)

By calculating partial derivatives of logarithmic likeli-hood functions for μλ and σλ, we can derive the following:

zl(θ | X)zμλ

�

Nn�1 s′M

− 1 Xn − Nμλs′M− 1s1 + σ2λs′M

− 1s, (20)

zl(θ | X)zσλ

� −Nσλs′M− 1s1 + σ2λs′M

− 1s

+σλ

Nn�1

Xn − μλs( ′M− 1ss′M− 1 Xn − μλs(

1 + σ2λs′M− 1s

2 .

(21)

For a specific set of b and σB, given that the two partialderivatives of equations (20) and (21) equal zero, the


maximum likelihood estimates for μλ and σλ can be cal-culated as

μλ �

Nn�1 s′M

− 1 XnNs′M− 1 Xn

, (22)

σλ � 1

Ns′M− 1 Xn

N

n�1

Xn − μλs( ′M− 1ss′M− 1 Xn − μλs(

−1

s′M− 1s

1/2

.

(23)

Equation (18) can be reformulated by introducingequations (22) and (23) into the maximum likelihoodfunction:

L θ | Xn, μλ, σλ( � −Nm ln(2π)

2−

N

2−12

N ln|M|

−12

N

n�1

X′nM− 1 Xn −

Nn�1 s′M

− 1 Xn( 2

s′M− 1 Xn

−12ln

Nn�1 s′M

− 1 Xn( 2

Ns′M− 1 Xn−

Nn�1 s′M

− 1 Xn 2

N2s′M− 1 Xn

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭.

(24)

*emaximum likelihood estimated value of σB and b canbe derived by calculating the maximum value of equation(24), applying two-dimensional search method. *e esti-mated values of μλ and σλ can be obtained, using equations(22) and (23) with the estimated values of σB and b, aspreviously calculated. *e overall model can only describethe general trend of the degradation process. *e case of thedegradation process of each individual wheel is quite dif-ferent, so the model parameters need to be updated ac-cordingly, in order to accurately characterize thedegradation process of any individual wheel. In this paper,the Bayesian formula is used to solve the posterior distri-bution of drift coefficients λ.

3.2.2. Online Updating of Model Parameters. Assuming thatat the mileage point sm, the model parameters of the nthsample are updated with the m elements of the monitoringdata time series. Degradation increments are represented asfollows:

xi � Xn si( − Xn si− 1( � λ sbi − s

bi− 1 + σB B si( − B si− 1( ( .

(25)

*e updated model parameter λ can be calculated basedon given degradation data and using the conditional dis-tribution P(λ | x1, . . . , xm), as this is expressed according tothe Bayesian theory:

P λ | x1, . . . , xm( ∝f x1, . . . , xm | λ( π(λ). (26)

*e following equation can be acquired based on theindependent incremental properties of the Wiener process:

f x1, x2, . . . , xm|λ(

�1

mi�1

��2πσ2B si − si− 1(

× exp − m

i�1

xi − λ sbi − sbi− 1( (

2

2σ2B si − si− 1( .

(27)

According to equation (27), we can get

P λ | x1, x2, . . . , xm( ∝f x1, x2, . . . , xm | λ( π(λ)

∝ exp − m

i�1

xi − λ sbi − sbi− 1( (

2

2σ2B si − si− 1( ⎛⎝ ⎞⎠exp

− 12σ2λ

λ − μ2λ 2

∝ exp −1

2σ2Bσ2λλ2 sbmσ

2λ + σ

2B − 2λ μλσ

2B + X sm( σ

2λ

∝ exp −sbmσ2λ + σ

2B(

2σ2Bσ2λλ2 − 2λ

μλσ2B + X sm( σ2λsbmσ2λ + σ

2B

.

(28)

Since the parameter λ is assumed to be normallydistributed, the posterior distribution is also normal.Considering the monitoring data, the following relation isderived:

P λ | x1, . . . , xm( �1

��2πσ2λ,sm

exp −1

σ2λ,smλ − μλ,sm

2⎛⎝ ⎞⎠.

(29)

*e posterior expression of the parameter is obtained bycomparing equations (28) and (29).

μλ,sm �μλσ2B + X sm( σ

2λ

sbmσ2λ + σ2B

, (30)

σ2λ,sm �σ2Bσ2λ

sbmσ2λ + σ2B

. (31)

3.3. Individual RUL Prediction. Based on the concept of thefirst hitting time, the remaining life Lm of the component, atthe mileage sm, is defined as

Lm � inf lm: x sm + lm( ≥w , (32)

where w is the preset degradation threshold, which refers tothe “maintenance limit,” as it is defined in this paper. As thecurrent degradation X(sm) is known, the PDF of theremaining life Lm can be obtained [10].


fLm lm( �1

��

2πl2m σ2λη lm( 2

+ σ2Blm

× w − X sm( ( − η lm( − blm lm + sm( b− 1

×w − X sm( ( σ2λη lm( + μλσ

2Blm

σ2λη lm( 2

+ σ2Blm⎛⎝ ⎞⎠⎞⎠

× exp −w − X sm( ( − μλη lm( (

2

2 σ2λη lm( 2

+ σ2Blm ⎡⎢⎢⎣ ⎤⎥⎥⎦,

(33)

where

η lm( � lm + sm( b

− sbm. (34)

*e updated model parameters μλ,sm, σλ,sm are broughtinto equation (33) in order to obtain the individual PDF ofthe RUL, after the parameters update.

4. Centralized Maintenance Timing Prediction

*e calculations, as described in the previous section,provide the PDF of the RUL of each individual wheel atthe current moment. Since the differences between thevarious wheels are significant, they should be taken intoaccount, in order to determine a reasonable time forcentralized maintenance. *is paper proposes that thetime of centralized maintenance of train wheels is de-termined by the maintenance limit w and the ratio oflimit-arriving p. *e ratio of limit-arriving p is the pro-portion of individuals who exceed the maintenance limitw in all wheels. It is suggested that the CCBM be carriedout, when the proportion of individual wheels, withdegradation greater than w, reaches p. Among others, thesetting method for the w and p values should take intoaccount the relevant technical specifications, equipmentmaintenance capability, equipment operation safety,maintenance economy, etc., matters that are not discussedin this paper. In order to ensure the safety, w should beselected less than the use limit, as specified in the re-spective technical specifications.

In the case where w and p are known, a maintenancetiming prediction algorithm, based on the distribution ofremaining life expectation, is proposed. Suppose, that atsome point, N1 out of N wheels, have reached w, while N2wheels have not reached this limit. *e remaining life of N2individual wheels was predicted, under the condition thatthe threshold is w. *e expectation of the remaining lifeprobability density is taken as RUL prediction value.

Ln � +∞

0lm fLm,n lm( dlm, (35)

where fLm,n(lm) represents the PDF of the RUL of the nthindividual wheel, after parameter updating.

*e set of RUL prediction value of the N2 individualwheels will be recorded as

L � L1, L2, L3, . . . , LN2( . (36)

*e distribution function of L is required, in order toaccurately find the remaining life, when the proportion ofindividual wheels whose deterioration reaches w is p.Weibull distribution, having high applicability in productfailure and life analysis, is used to fit the set L. *e distri-bution function and PDF of the set of individual remaininglife prediction values are obtained as

F(l) � 1 − e− (ml)a

,

f(l) �a

m

l

m

a− 1

e− (l/m)a

.

(37)

*e process of solving parameters a and m in Weibulldistribution, by maximum likelihood method, is relativelysimple and thus it is not discussed here.

*e quantile lR(p(l≤ lR) � α) of the probability distri-bution function must be calculated, in order to find aremaining life value, when the percentage of degradation,exceeding the threshold w in all individuals at that mileage,reaches exactly p.

N1 + α · N2N

� p,

α �N · p − N1

N2,

(38)

where lR corresponds to the remaining life of the train, forcentralized maintenance.

Let α � F(lR) provide the remaining life of centralizedmaintenance as follows:

lR �(− ln(1 − α))1/a

m. (39)

5. Model Verification

Tread wear is one of the most important characteristics ofwheel degradation. As shown in Figure 1, the measuringpoint of tread wear is at the tread 70mm from the end face ofwheel flange. *e distance between the current tread wearmeasurement point and the original profile measurementpoint is defined as tread wear.

*e wheel wear monitoring data comes from themonitoring system called TWDS (Train Wheel DetectionSystem), which is mounted on the railway line, as shown inFigure 2. TWDS system uses structural optical sensors tomeasure the profile size of wheels as the trains pass by.TWDS system is mainly composed of a laser source and adigital camera with filter plate in front of the lens. When thelowest point of the wheel set reaches the laser plane, thesystem issues a shooting command to shoot the image.*rough the image processing algorithm, the contour size ofthe wheel is measured. *e measurement error of TWDSsystem is ±0.3mm.

Taking the 500-day monitoring data of tread wear, in a54 fixed group of railway freight cars, as an example, based


on the prediction of individual RUL, we further forecast thetiming of wheel centralized maintenance.*e 54 fixed grouprailway freight cars are special trains for coal transportationof the same type, with rated load of 80 tons. All the freightcars are used under the same conditions and run on fixedrailway lines with an average daily mileage of 550 km.

According to the HP filtering algorithm, introduced inSection 2, the online monitoring data are processed to ex-tract the trend components of the degradation data. *efiltering effect of three wheels wear data is shown in Figure 3.

It can be seen from the graph that the new sequence, afterHP filtering, provides a better sense of the trend in theoriginal sequence, while the fluctuation obviously decreases.*e trend components obtained after filtering can bettermodel the degradation process.

*e obtained trend component data are translatedaccording to (12) and substituted into the parameters esti-mation algorithm of Section 3.2.1. *e estimated values ofthe parameters of the overall degradation process are ob-tained as shown in Table 1.

From the overall model parameters, the overall degra-dation process can be simulated, as demonstrated in Fig-ure 4, along with some individual degradation paths.

It can be seen from Figure 4 that the degradation model,as derived from the overall parameters, can reflect thedegradation trend, but there is still a certain gap between theindividual degradation paths and the overall degradationpath. *erefore, the model parameters should be updated

according to the data of the individual wheels.*e individualdegradation model parameters and the PDF of the RUL,after parameter updating, can be obtained by the algorithmspresented in Sections 3.2 and 3.3.

Each time a group of real monitoring data is obtained,the model parameters are updated according to equations(36) and (37). Tables 2 and 3 show the updates of modelparameters of four wheels at different mileage. For differentwheels, the mean values of drift parameters have their ownupdate paths, and the standard deviation of drift parametersis related to monitoring mileage.

*e degradation paths of these four wheels simulatedby real-time updating model parameters are shown inFigure 5, where the red line represents the overall deg-radation path, the blue line is the true degradation path ofvarious individual wheels, and the green lines representdegradation paths simulated by real-time updating modelparameters of individual wheels. Each time a real mon-itoring data is acquired, the parameters are updated. It canbe seen that the parameter updating method can bettermodify the model parameters, to make them consistentwith the real state.

In order to verify the prediction effect of the proposedmethod on the wheel tread wear degradation path, it wascompared with the modeling methods in other references.*emeasurement data of the wheels with 160,000 kilometersis used for model parameter estimation and parametersupdate. *e model parameters at 160,000 kilometers areused to predict the wear at different mileages thereafter, andthemodel prediction accuracy is measured by calculating the

Initial tread profile

Flange

Profile of worn tread

70mm

Tread wear

Figure 1: *e definition of tread wear.

The structural optical sensors

Figure 2: *e TWDS system mounted on the railway line.

0 0.5 1 1.5 2 2.5×105

0

1

2

3

4

5

6

7

8

Monitoring mileage (km)

Trea

d w

ear (

mm

)

The trend componentsOriginal monitoring data

Figure 3: HP filtering effect on 3 wheels wear data.

Table 1: Model parameters of overall degradation process.

Parameters μλ σλ b σBEstimated values 1.54×10− 6 1.26×10− 6 1.19 3.69×10− 4


Table 2: *e mean value of drift parameters at different mileage after parameter update of four wheels.

Monitoringmileage (km) 5.81× 10

4 1.10×105 1.51× 105 1.91× 105 2.27×105 2.70×105

μλ,tm

Wheel (a) 1.09×10− 6 1.03×10− 6 9.05×10− 7 8.88×10− 7 8.18×10− 7 7.76×10− 7

Wheel (b) 1.55×10− 6 1.22×10− 6 9.97×10− 7 8.76×10− 7 8.14×10− 7 7.97×10− 7

Wheel (c) 9.74×10− 7 9.29×10− 7 8.34×10− 7 7.56×10− 7 6.96×10− 7 6.70×10− 7

Wheel (d) 6.62×10− 7 8.43×10− 7 8.38×10− 7 8.32×10− 7 8.06×10− 7 7.96×10− 7

Table 3: *e standard deviation of drift parameters at different mileage.

Monitoring mileage (km) 106 200 276 348 415 492σλ,tm 2.20×10

− 7 1.07×10− 7 7.00×10− 8 5.32×10− 8 4.20×10− 8 3.47×10− 8

0 0.5 1 1.5 2 2.5×105

–1

0

1

2

3

4

5

6

7

8

9


Trea

d w

ear (

mm

)

Individual degradation pathOverall degradation trends

Figure 4: Overall and individual degradation processes.

0 0.5 1 1.5 2 2.5×105

0

0.5

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1.5

2

2.5

3

3.5

4


Trea

d w

ear (

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)

Predicted value after parameter updatingReal degradation processOverall degradation process

(a)

0 0.5 1 1.5 2 2.5×105

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0.5

1

1.5

2

2.5

3

3.5

4


Trea

d w

ear (

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)

Predicted value a�er parameter updatingReal degradation processOverall degradation process

(b)

Figure 5: Continued.


RMSE (root-mean-square error) at different mileages. RMSEis defined as

RMSEk �

��

ni�1 xi,k − x

p

i,k 2

n

. (40)

RMSEk represents root-mean-square error of predictedvalue of wheel wear at mileage sk. xi,k is the trend componentvalue of the ith wheel tread wear monitoring data after HPfiltering when the mileage is sk, and x

p

i,k is the predicted valueat this time.

As shown in Figure 6, the red line is the prediction errorusing the method in [10]. *is method used the overallmodel parameters to predict the degradation path of dif-ferent individuals, and the model parameters are notupdated. *e blue line is the prediction error obtained withthe method in [8]. Reference [8] adopted a linear Wienerprocess model and updated the model parameters usingmonitoring data based on the overall model parameters. *egreen line is the prediction error obtained by our proposedmethod. It can be seen that all the prediction errors of threemethods gradually increase with the increase of the pre-dicted mileage, but the method proposed in this paper hassmaller prediction error than the other two methods.

After the model parameters are updated, the PDF of theremaining life of the wheel individuals can be obtainedaccording to (39). For example, the RUL probability densityvalue of one wheel at different mileage is shown in Figure 7.If the expectation value of remaining life is taken as thepredicted value, then the predicted remaining life underdifferent using mileage can be obtained. *e predictions oftwo individual wheels are shown in Figure 8.

Figure 8 shows that the variance of the probabilitydensity of remaining life decreases as the updated data

increase. At the initial stage of just a small volume ofmonitoring data, there is a certain deviation between thepredicted value and the real value of remaining life. Asmonitoring data are accumulated, the model parametersare constantly updated to improve the predictionaccuracy.

In order to evaluate the effect of this algorithm on theprediction of centralized maintenance time, the whole trainis predicted to be under maintenance, after monitoring for191000 km. *e parameters of each individual wheel areupdated, using the monitoring data of the first 191000 km,and the probability density of the RUL of each wheel, underthe limit w, is obtained, as shown in Figure 9.

Using Section 4 method, a set of expected values ofremaining life and the distribution functions of remaininglife of each wheel are calculated and demonstrated inFigure 10.

In order to verify the accuracy of the proposed algo-rithm, the prediction of centralized maintenance mileageunder different maintenance thresholds was carried out afterbeing monitored for 191000 km. When the predictedmaintenance mileage was reached, the accuracy of the al-gorithm was judged by comparing the real arrival limit ratioto the set value. For example, w is set to 4mm; the predictedcentralized maintenance mileage is set to 25700 km and82700 km, while p is set to 40% and 50%, respectively. Whenthe recommended maintenance mileage is reached, the truearrival limit ratio is 38.3% and 48.9%, respectively. Similarly,w can be set to 3.5mm. *e experimental prediction resultsare as shown in Table 4.

From the results, we can see that when the scheduledmaintenance mileage is reached, there is a gap between thetrue arrival limit ratio and the expected value. And theaverage error of the four test results is 1.28%. *e error

0 0.5 1 1.5 2 2.5×105

0

0.5

1

1.5

2

2.5

3

3.5

4


Trea

d w

ear (

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)


(c)

0 0.5 1 1.5 2 2.5×105

0

0.5

1

1.5

2

2.5

3

3.5

4


Trea

d w

ear (

mm

)


(d)

Figure 5: Model prediction effect after parameters update. (a–d) are predictive effects of four different wheel degradation processes afterparameter updating.


1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7×105

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


RMSE

(mm

)

Using nonlinear Wiener processUsing linear Wiener process and parameter updateUsing nonlinear Wiener process and parameter update

Figure 6: Prediction effect comparison.

0 0.5 1 1.5 2 2.5 3×105

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0.08

Remaining useful life (km)

Prob

abili

ty d

ensit

y

Used 110,000km

Used 151,000km

Used 191,000km

Used 227,000km

Used 247,000km

Figure 7: *e RUL probability density value of one wheel at different mileage.

11.5

22.5

3

02

46

×105

×105

0.080.070.060.050.040.030.020.01

0

Monitoring mileage (km)Remaining useful life (km)

Prob

abili

ty d

ensit

y

Predicted valueTrue value

(a)

×105×105

Predicted valueTrue value

11.5

22.5

3

02

46

00.020.040.060.08

0.10.120.14

Monitoring mileage (km)Remaining useful life (km)

Prob

abili

ty d

ensit

y

(b)

Figure 8: RUL prediction results. (a) and (b) are the remaining life prediction effects of the two wheels, respectively.


comes from the randomness of wheel wear degradationpath. *e degradation model obtained from the moni-toring data of the first 191000 km may be different fromthe real degradation path behind. *erefore, there aresome differences between the predicted and actual valuesof remaining life. Such prediction accuracy can meet therequirements for the organization and arrangement ofrailway freight car maintenance. According to the ex-perimental results, it is evident that by setting thresholdsw and p, the model can better predict the remaining life ofCCBM of train wheels.

6. Conclusions

Predicting the time of centralized maintenance of wheelsbased on the degradation is of great significance for realizingcondition-based maintenance of railway freight cars. *ispaper presents a prediction algorithm for CCBM timing offreight train wheels, based on the nonlinear Wiener processand, more specifically, on the basis of solving the problem ofRUL prediction for individual wheels.

In order to solve the problem of fluctuations in the directlycollected monitoring data, this paper uses the HP filteringalgorithm to extract the trend components in the monitoringdata. Using trend components for modeling provides a betterfit to the real degradation path. According to the nonlinearity ofwheel degradation, a nonlinear Wiener process model isconstructed to describe the wheel tread wear, with modelparameters that are solved by maximum likelihood estimationmethod. In order to make the model fit the degradation pathsof different individuals better, based on the Bayesian formula,the updating algorithm of model parameters is deduced, andthe real-time updating of individual model parameters is re-alized, providing the PDF of individual RUL. A predictionmethod of centralized maintenance timing is proposed basedon two set thresholds: “maintenance limit” and “the ratio oflimit-arriving.” *e remaining life of each wheel can be ob-tained by setting “maintenance limit,” and the predictedmileage for CCBM can be obtained by setting “the ratio oflimit-arriving.” *e accuracy of the algorithm is verified byusing 500-daymonitoring data from a 54 fixed group of railwayfreight cars. According to the test results, it is considered thefact that the prediction accuracy of this algorithm can meet therequirements of application.

*e method can be extended to other equipment withmultiple parts of the same type and under the same workingconditions, to determine the time for centralized mainte-nance of these parts, reducing thus downtime, increasingoperational lifetime, and improving operation efficiency.

Data Availability

*e data used to support the findings of this study areavailable from the corresponding author upon reasonablerequest.

Conflicts of Interest

*e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

*is research was supported by the China Energy Projectnumber SHGF-17-56-9.

References

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0 1 2 3 4 5 6×105

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0.01

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0.04

0.05

0.06

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abili

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ensit

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p � 40% p � 50%

w � 4 Predicted remaining life (km) 25700 82700Arrival limit ratio 38.3% 48.9%

w � 3.5 Predicted remaining life (km) 21900 49800Arrival limit ratio 40.5% 48.2%


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