Preventive condition-based maintenance model of EMU components based on stochastic differential equation

HUANGFU Lanlan， ZHAO Yifan， SU Hongsheng

(School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract： To improve the operation and maintenance management level of large repairable components, such as electrical equipment, large nuclear power facilities, and high-speed electric multiple unit (EMU), and increase economic benefits, preventive maintenance has been widely used in industrial enterprises in recent years. Focusing on the problems of high maintenance costs and considerable failure hazards of EMU components in operation, we establish a state preventive maintenance model based on a stochastic differential equation. Firstly, a state degradation model of the repairable components is established in consideration of the degradation of the components and external random interference. Secondly, based on topology and martingale theory, the state degradation model is analyzed, and its simplex, stopping time, and martingale properties are proven. Finally, the monitoring data of the EMU components are taken as an example, analyzed and simulated to verify the effectiveness of the model.

Key words： preventive maintenance； stochastic differential equation； simplex； martingale； electric multiple unit components

0 Introduction

With the progress of technology and development of industries, energy resources have become increasingly tense. High-speed electric multiple unit (EMU) are driven by electricity, which has advantages to energy-saving and environmental protection； thus, high-speed EMU are increasingly favored around the world. However, while facilitating travel, increase in the speed and transport competency of EMU also raises certain security issues and increases maintenance costs. Therefore, with the increasing input and operation of high-speed EMU, designing effective preventive maintenance strategies to ensure safe, reliable, efficient, and economic EMU operations is important. Moreover, examining and analyzing the reliability and condition-based maintenance (CBM) of EMU components have considerable theoretical significance and engineering application value.

In the current research, preventive maintenance strategies are divided mainly into time-based maintenance (TBM) and CBM. Goble[1]argued that TBM is a preplanned maintenance activity that can sufficiently prepare corresponding maintenance resources at the planned maintenance time point but does not consider the actual operation status of components, thereby leading to “over maintenance” or “under maintenance.” The CBM strategy makes up for the shortcomings of TBM, as it implements targeted maintenance according to the real operation state of components, which can reduce the possibility of post maintenance, improve the dynamic behavior of a system, and lower preventive maintenance costs[2-3]. Scarf[4] proposed a CBM model with multiple parameters based on the determination of component condition characteristic parameters. Bruns[5]examined an optimal maintenance strategy satisfying the Markov property in the process of equipment deterioration under the condition of partial maintenance and infinite cost. Huang et al.[6]established a CBM decision model for a deteriorating system to determine the optimal maintenance times and preventive maintenance interval of components. Wang et al.[7]designed a dynamic CBM decision model for a system based on the historical data of monitored component condition characteristic parameters. Xiao et al.[8]proposed a CBM strategy of train multi-components based on reliability. However, most of the CBM models in the aforementioned studies are based on deterministic models, which do not comprehensively consider the influence of internal degradation and external random interference on the component state. A stochastic differential equation (SDE) can overcome deterministic models’ inability to describe uncertain factors in the process of component maintenance and describe random interference factors in the maintenance process[9].

Presently, SDEs and martingale theory are widely used in filtering, finance, optimal control, and steady-state analyses. Burgess[10]established the Vasicek short rate model by the SDE. Faried et al.[11]analyzed the transient stability of a power system based on the SDE. Xu et al.[12]proposed the stochastic small signal stability model based on the SDE. But the SDE is used rarely to the maintenance engineering[13]. Doksum et al.[14]regarded the cumulative decay process as a Wiener process with drift and proposed an inverse Gaussian life model. Dual et al.[15]proposed a CBM model for a wind turbine based on a stochastic degradation model. The availability of equipment is analyzed via the update process, and the model is relatively simple. In reference to the problem of gearbox failure, Su et al.[16]established a novel gearbox stochastic differential model that considers the internal degradation and random disturbance of components but is biased toward the verification of the actual data state, which has yet to be proven theoretically. Moreover, the application of an SDE in research on the preventive maintenance of high-speed EMU components is limited. Therefore, this study applies an SDE to the preventive maintenance activities of EMU components. Huangfu et al.[17]proved that the dynamic maintenance process of repairable component belongs to Brownian motion,and the SDE could describe the dynamic processes of the random maintenance caused by the routine, upkeep, lubrication, patrol and the external changing environment. Therefore, in this study, SDE is used to describe the state transfortion model of the EMU component. Firstly, an SDE model of the EMU component state transition is established. The model is composed of two parts, that is, the fault function of the component state and operating mileage. The component state fluctuation is caused by external interference. It considers the internal degradation process of the components and adds the interference of external random factors to its state. Secondly, based on topology theory, the simplex property of the model is proven. Through martingale theory and the stopping time theorem, the martingale and stopping time properties of the model are likewise proven. Finally, the effectiveness of the model is verified through an analysis of an example.

1 Description of SDE theory

SDEs are widely used to describe uncertain dynamic behaviors in physics, economics, finance, and so on, and their form is represented as[9].

dX(t)=b(X(t),t)dt+σ(X(t),t)dB(t),

X(t0)=X0,

(1)

whereX(t)=[X1(t),X2(t), …,Xn(t)] is then-dimensional vector as random state variable on the complete probability space (Ω,F,P) fort∈[t0,T] indicating the state of the system at timet,Ωis a finite sample space, F is theσ-algebra ofΩ;b(X(t),t) is ann-dimensional vector function called the drift vector,σ(X(t),t) is ann×dmatrix function,b(X(t),t) andσ(X(t),t) are Borel measures; andB(t) is a 1-dimensional Brownian motion. Brownian motion is also called the Wiener process, which is a random process caused by random factors in practical applications that can generally be described by Brownian motion. For ∀t≥0, the process obeys a normal distribution.

Lemma1(Existence and uniqueness of strong solutions)： If the coefficient vectorb(X(t),t) and matrix functionσ(X(t),t) satisfy the local Lipschitz condition and linear growth condition, that is, for all |x|, there exists |y|∈Rn, wherexandyare two variable vectors ofX(t).

And when 0≤t≤T, ∃K=const, s.t.

|b(x,t)-b(y,t)|+|σ(x,t)-σ(y,t)|

and ∃KT=const, s.t.

|b(x,t)|+|σ(x,t)|

Then, when the initial valueX(0) is independent of Brownian motionB(t),E[|X(0) |2]<∞, and

where sup|·| is the upper definite bound, which is the maximum value of |x(t)|2in the interval [0,T];Cis a constant only with respect toKandT; and Eq.(1) has a unique strong solution and a continuous path.

2 CBM model of EMU component

2.1 Modeling

The degradation of EMU components generally results from the combined effects of the state degradation of the components and external random interference. The state degradation of the components can be expressed by failure rate, and the external interference is random. Before the state model of the EMU components is established, the following hypotheses are proposed：

Hypothesis1(H1)： The condition detection of the components is considered to be perfect, and no undetectable situation exists.

Hypothesis2(H2)： The external random disturbance is considered as a random fluctuation near a mean point, that is, the disturbance is consistently independent and stable, and its mathematical expectation is zero.

Hypothesis3(H3)： The failure of the components is with respect to their state and operating mileage.

Hypothesis4(H4)： The faulty components are repaired as new components.

Based onH1-H4, an SDE-based state transformation model of the EMU components is established as

dx(t)=b(x(t),t)dt+σ(x(t),t)dB(t).

(2)

In accordance with the definition of “timet” in reliability theory, the concept of “timet” is typically generalized, and the time evaluation scale of a high-speed train is generally the operating mileage. Therefore,tin Eq.(2) represents the operating mileage of the high-speed train in units of 104km. In Eq.(2),x(t) represents the state of the EMU components att. Whenx(t)=1, the components are brand new att. Whenx(t)=0, the components are completely damaged att. In addition,b(x(t),t) is called the drift coefficient, which is the comprehensive failure rate function of the components；σ(x(t),t) is the random disturbance coefficient, which is the volatility function of the components； andB(t) is a 1-dimensional Brownian motion.

WhenLemma1is satisfied, Eq.(2) has a unique strong solution.

The failure of the EMU components is caused by two factors, namely, time and state. Therefore, the comprehensive failure rate is composed of two parts, that is, the basic failure rate and state-related failure rate. The basic failure distribution of the equipment is also called life distribution. Common failure distributions include exponential distribution, normal distribution, gamma distribution, and the Weibull distribution. The Weibull distribution is widely used in reliability analysis, especially in the distribution form of fatigue strength, corrosion life, and accumulated wear failure in machinery[18]. Therefore, the Weibull distribution is selected as the basic failure rate of the EMU components, and it is expressed as

(3)

whereβis the shape parameter,ηis the scale parameter or characteristic life, andtis the operating mileage of the high-speed train.

Let the state-related failure rate beg(x(t)), which is a bounded measurable real-valued continuous function in the interval [0,1]. It is the point where the EMU component statex(t) on theRnspace is mapped on theR1space [0,1]. Whenx(t)=0, that is, when the EMU components are completely damaged att, the failure rateg(x(t))=1. In accordance with Ref. [16],g(x(t)) is obtained as

(4)

The comprehensive failure rate model of the EMU components can be obtained as

(5)

The random interference of the EMU components is stochastic and thus not with respect to the operating mileaget of the train. However, this random interference will affect the state of the components, and the intensity of the impact is recorded as σ, which is the intensity of the state fluctuation. Then, the expression of the volatility function is

σ(x(t),t)=σx(t).

(6)

The state transformation model of the EMU components can be obtained by substituting Eqs.(5) and (6) into Eq.(2), and then we can get

(7)

The integral form of Eq.(7) is

(8)

wherex(0)=1 is the initial value ofx(t), indicating that the components are brand new at initialt=0.

Lety(t) be the state degradation function of the components, which constitutes the stochastic degradation process of the component state {Y(t),t≥0}, then

y(t)=1-x(t)=

(9)

wherey(0)=0 indicates that the components are brand new at initialt=0, andy(t)=1 indicates that the components are completely damaged att.

2.2 Model analysis

Theorem1： Let setA={τ0,τ1,…,τi,…|a=y(τi),i∈Z}, whereτiis the replacement time of the preventive maintenance of the components. When the components satisfyH4, in the Euclidean space, setAis a 0-dimensional simplex.

Proof： Fig.1 presents the schematic diagram of the change curve of the component degradation function. In Fig.1,ypis the preventive replacement threshold, andycis the corrective replacement threshold.θis the detection interval andτiis the preventive maintenance time. The curve 1 and curve 2 respectively represent the state degradation curves in different preventive maintenance cycles. It can be seen that the degradation curves of components show a trend of fluctuation due to the interference of random fluctuations.

Fig.1 Schematic diagram of change curve of component degradation function

Whent=τ0=0,y(0)=1-x(0)=0.

In accordance withH4,y(τi)=0 can be obtained.

Then, setA={τ0,τ1,…,τi,…|a=y(τi),i∈Z}.

In the Euclidean space, in accordance with the definition of simplex[19], setAis a 0-dimensional simplex.

Based onTheorem1, in an ideal case, the components can be in a reliable operating state after reasonable preventive maintenance.

Letkbe the number of component degradation detection,θbe the detection interval, andd(ykθ)=|yp-ykθ| be the difference between the preventive replacement thresholdypand actual degradationykθ.

Definition1： Given the real numberε>0,τε=min{kθ≥0,d(ykθ)<ε} is defined as the best replacement time of the components in the first preventive maintenance.

Theorem2：τεis the stopping time of the stochastic degradation process {Y(t),t≥ 0} of the components under the condition-based preventive maintenance model.

Proof： For ∀t≥0, in order to facilitate the design of the controller, the variables are defined as

{τε=t}={d(ykθ)≥ε,ktc

whereσ(Ykθ,0≤kθ≤t) is theσ-algebra generated by {Y(t), 0≤kθ≤t}.

In accordance with the strict definition of stopping time[20],τεis the stopping time of the stochastic degradation process {Y(t),t≥0} of the components under preventive maintenance.

Based onTheorem2, all the preventive maintenance timesτ1,τ2,…,τi,… in Fig.1 are the stopping times of the stochastic degradation process of the components.

At stopping timeτ, three component maintenance cases may exist. In the first case, the components are not faulty at τ, only minor maintenance is implemented, and no shutdown maintenance is implemented, which will not affect the overall development trend of the state degradation function of the components. In the second case, when the components fail atτ, that is, when the stopping time is hit, maintenance will be implemented. In the third case, beforeτ, the components failed and must be repaired after the failure. The first two cases belong under preventive maintenance, whereas the third case belongs under post maintenance. Therefore, only the first two cases are analyzed.

Theorem3： The first case： Minor maintenance, that is, the components are not faulty,τis arriving, and the stochastic degradation process {Y(t),t≥0} of the components is a submartingale in probability.

Proof： Lett∈[0,τ],s>0, and Ftbe theσ-algebra generated by the progress from 0 tot, that is, Ft=σ({Y(u),0≤u≤t}), then

E[Y(t+s)|Ft]=E[Y(t)+Y(t+s)-Y(t)|Ft]=

E[Y(t)|Ft]+E[Y(t+s)-Y(t)|Ft]=

Y(t)+E[ΔY|Ft].

(10)

In accordance with Fig.1, the state degradation function of the components demonstrates an overall rising trend, and Eq.(10) can be written as

E[Y(t+s)|Ft]=Y(t)+E[ΔY|Ft]≥Y(t),

whereE[ΔY/Ft]≥0, andE[Y(t+s)]≤yn<∞.

Based on the definition of a submartingale[9], {Y(t),t≥0} is a submartingale in probability.

Theorem4： The second case： The stochastic degradation process {Y(t),t≥0} is a local martingale in probability when the components hit the stopping timeτand stoppage maintenance is implemented.

Proof： Let the initial time of the component operation bet0. The stochastic degradation process {Y(t),t≥0} will operate within the maintenance interval [0,τ]. In accordance withH4, there is

E[Y(t0+τ)|Ft0]=

E[Y(t0)+Y(t0+τ)-Y(t0)|Ft0]=

whereE[Y(t0+τ)-Y(t0)|(Ft0)=0.

In accordance with the definition of a local martingale, {Y(t),t≥0} is a local martingale.

Based onTheorem4, the state degradation curve in one stopping time range can be inferred to have the same changing trend as the curve in the other stopping time range. As shown in Fig.1, curve 1 isy(t) in the interval [0,τ1], and curve 2 isf(y(t)) in the interval [τ1,τ2]. Moreover,f(y(t)) is the mapping ofy(t) in the interval [τ1,τ2]. Therefore, only one of the stopping time ranges needs to be considered in the research on the preventive maintenance strategy for the components.

3 Example analysis

The monitoring data of the wheelset wear condition of the EMU components are selected, as shown in Table 1. The data in Table 1 reflect that the wear of train wheelsets increases with the increase of mileage. At the same time, according to the actual data statistics on site, when the running distance of high-speed train reaches 2×105km, the wear of the wheel is about 2 mm, and the cumulative fatigue damage density of the wheel is close to 1, indicating that the fatigue damage of the train wheel may occur, and the normal operation of the train cannot be guaranteed at this time.

Table Wear condition monitoring data of EMU wheelset

In accordance with the maximum likelihood estimation and Newton-Raphson algorithm[21], the parameters in Eq.(7) are obtained asβ=4.44,η=323.49, andα=0.139 3. Volatilityσ=0.003 is selected. In accordance with reliability theory, the reliability function of the EMU components can be obtained as

(11)

In accordance with Eq.(9), the degradation model of the EMU wheelset components can be obtained as

y(t)=1-x(t)=

(12)

The reliability function Eq.(11) is simulated using Matlab, as shown in Fig.2.

Fig.2 Reliability function curve of train components

In Fig.2, as the train operating mileage increases, the reliability function value gradually decreases, which indicates that component wear increases during the train operation, thereby further increasing component fatigue damage and wheelset failure. In addition, at pointAin Fig.2, the reliability value is 0.9, and the safe operation mileage of the high-speed train is approximately 1.9×105km. However, at pointB, the reliability value is 0.8, and the safe operation mileage of the high-speed train is approximately 2.25×105km, which is consistent with the onsite actual reprofiling cycle of 2×105km of the high-speed train wheelset, thereby indicating the effectiveness of the model.

The state degradation function Eq.(12) of the train components is simulated using Matlab, as shown in Fig.3, in which the curve indicates that the degradation process of the EMU components state rises with the increase in operating mileage. However, the curve is not smooth but fluctuates up and down, thereby reflecting the influence of the external random disturbance on the degradation process of the component state, and the influence is with respect to the fluctuation intensityσ.

(a) σ=0.003

(b) σ=0.01Fig.3 Degradation function curve diagram of train component state

In Fig.3(a), the fluctuation intensityσ=0.003, and the external random disturbance has little effect on the degradation function. When it is increased 10 times, as shown in Fig.3(b), the influence of the external random disturbance on the degenerate function intensifies. In addition, Fig.3 indicates that when the operating mileagetis close to 2×105km,y(t) rises sharply, thereby showing that the components are deteriorating rapidly and thus preventive maintenance is necessary, which is consistent with the reprofiling cycle of 2×105km of the train wheelset.

4 Conclusions

Focusing on the failure of repairable components in operation, a condition-based preventive maintenance model based on SDEs is established. The model can predict the actual degradation and preventive maintenance time and considers the degradation of the components and external random interference, which can effectively reflect the actual state change process. Furthermore, the preventive maintenance time is proven to be the stopping time of the stochastic degradation process with the simplex property and stopping time theory. Based on martingale theory, the stochastic degradation process of the repairable components under different preventive maintenance opportunities is proven to belong under submartingale and local martingale. Finally, the effectiveness of the model is verified through the analysis of the example. It provides a scientific basis for the formulation of preventive maintenance strategies for repairable components.