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Answer:

1. Figure 11(a) compares the relative errors between the

proposed method and Gebraeel's method which only

considers a single performance characteristic. Figure 11(b)

compares the relative errors between the proposed method

and the traditional method which consider two

independent performance characteristics. The y-axis is the

relative error defined by the equation under Figure 9. The x-

axis refers to the cycle number.

2. Delete Ref.10.

3. Delete Ref.22.

4. We want to make Tianyu Liu to be the corresponding

author.

Original Article

Proc IMechE Part O:

J Risk and Reliability

1–12

ÓIMechE 2016

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DOI: 10.1177/1748006X16683317

pio.sagepub.com

Residual useful life estimation for

products with two performance

characteristics based on a bivariate

Wiener process

Tianyu Liu, Zhengqiang Pan, Quan Sun, Jing Feng and Yanzhen Tang

Abstract

Residual useful life estimation plays an important role in the field of prognostics and health management, and condition-

based maintenance. This article concerns the issue of residual useful life estimation for degraded components with two

performance characteristics. A bivariate Wiener process with random effects is used to model the evolution of two per-

formance characteristics, which are dependent on each other. A bootstrap method is used to estimate the initial para-

meters with history of degradation data. Once the new degradation information for an individual component is available,

the hyper-parameters of the random effects in the model are first updated by the Bayesian theorem. And then, we use a

Monte Carlo simulation method to estimate the posterior distribution of residual useful life approximately. Via a simula-

tion study and a case study on Lithium-ion batteries, the effectiveness and validity of the proposed approach are

demonstrated.

Keywords

Residual useful life, performance characteristics, bivariate Winner process, Bayesian theorem, degradation

Date received: 4 January 2016; accepted: 14 November 2016

Introduction

In recent years, many high-reliability components are

emerging due to the development of manufacture tech-

niques and material science, especially in the field of

space and electronic industry. For these long lifetime

components, people usually pay special attention to the

issue of prognostic and health management (PHM) in

their entire life span. Prognostic is regarded as the cen-

tral step in PHM. A primary mission of prognostic is

the early estimation of residual useful life (RUL) using

the condition-based information of components being

monitored.

1

The RUL of a component is defined as the

length of time from the present time to the end of use-

ful life. Accurate estimation of RUL furnishes useful

advices for making maintenance strategies, which can

help reduce maintenance cost, improve average system

availability, and even avoid catastrophic disasters.

To make the most effective use of condition infor-

mation, it is helpful to identify one or more quantitative

indicators that can reflect the health status of an oper-

ating component. These indicators are often closely

related to the function of the component and degrade

gradually during either operation or storage, for

example, discharge capacity of the secondary batteries,

crack lengths of the metal, bearing temperature in a

momentum wheel etc. Reliability engineers call these

indictors performance characteristics. The degradation

data of performance characteristics furnish sufficient

information about the current state of the component

and how that condition is likely to evolve in the future,

which construct the basis of RUL estimation.

Centered on the degradation data, there have been a

lot of excellent works on RUL estimation. Gebraeel

et al.

2

proposed a Bayesian approach to update the

stochastic parameters of the exponential random-

coefficient models, and the approximated RUL poster-

ior distribution can be obtained with sensor-based

monitoring signals. Si et al.

3

followed Gebraeel’s work

and obtained an exact and closed-form RUL distribution

College of Information System and Management, National University of

Defense Technology, Changsha, P.R. China

Corresponding author:

Quan Sun, College of Information System and Management, National

University of Defense Technology, Changsha 410073, P.R. China.

Email: sunquan_nudt@163.com

considering Brownian motion error. Wang et al.

4

devel-

oped an approach of RUL estimation based on a gener-

alized Wiener process model, and the strong tracking

filter method was used to update the model parameters

once new degradation data were monitored. Other excel-

lent works about RUL estimation based on linear/non-

linear degradation processes and Bayesian updating can

be found in Gebraeel

5

and Si et al.

6

Sometimes, due to

the extreme complexity of degradation models, updating

the RUL distribution through conventional Bayesian

methods is too difficult to perform. In this case, the par-

ticle filter-based RUL estimation methods are usually

adopted, and the corresponding applications can be

found in the previous studies.

7–9

However, most of the previous researches on RUL

estimation have so far dealt only with a single perfor-

mance characteristic or component failure mechanism.

In engineering practice, some products may involve a

second performance characteristic, which is correlated

with the primary one. These performance characteris-

tics are usually correlated with two underlying physical

degradation processes, respectively. For example, the

degradation of a Lithium-ion battery can be described

by the capacity loss or the energy loss in a fully charge/

discharge cycle. In this case, product fails when at least

one of these performance characteristics exceeds its fail-

ure threshold. Huang and Askin

11

studied the reliabil-

ity assessment of such bivariate degradation process by

assuming that the individual degradation processes of

these performance characteristics are independent.

Sari

12

and Pan et al.

13

considered the dependency of

the two performance characteristics and used a copula

function to model the bivariate degradation process.

Pan and Balakrishnan

14

studied the reliability modeling

of degraded products with two dependent performance

characteristics and used a bivariate Birnbaum–

Saunders distribution and its marginal distributions to

approximate the reliability function.

From the preceding summary, we see that research-

ers have recognized the necessity of degradation model-

ing for degraded components with bivariate

degradation process. Nonetheless, the issue of RUL

estimation for components with two performance char-

acteristics has not been studied thoroughly, especially

when the dependency cannot be ignored. The main

problem is that the joint distribution of bivariate degra-

dation process is very complex, which makes the RUL

updating process quite difficult and complicated.

Despite this, some related researches can be found in

Wang et al.

15,16

The authors assumed that the degrada-

tion of each performance characteristic over time was

governed by a univariate Wiener process and the

dependency between them was characterized by Frank

copula function. Afterward, the RUL estimation was

conducted based on the copula-based degradation

model. However, this method is somewhat restrictive in

engineering practice. First, the copula function is a

pure mathematic method to couple two random vari-

ables, which has a complex formula and often lack

clear physical significance in degradation description.

Second, a key component of RUL estimation is to

update the posterior distribution of parameters by the

Bayesian rule once the new degradation data are avail-

able. In Wang’s work, the analytical expression of

parameters’ posterior distribution is difficult to derive

due to the existence of copula function in the likelihood

function. To solve this, the author used the strong

tracking filter and the Markov chain Monte Carlo

(MCMC) algorithms to update the parameters approx-

imately. These algorithms often consume a lot of time,

and thus, it is difficult to be applied on the online pre-

diction systems. For example, only some ripe packages

such as Winbugs, Openbugs, and MATLAB can solve

the problem of MCMC conveniently. But in practice,

the controller of many online health management sys-

tems is just a single-chip microcomputer, which cannot

execute such a complex procedure.

The bivariate Wiener process, by contrast, has some

advantages in degradation modeling and RUL predic-

tion for bivariate degraded components. In the bivari-

ate Wiener model, the correlation between different

performance characteristics can be described conveni-

ently by the covariance matrix. Moreover, in the updat-

ing procedure, it is easy to make the prior distribution

of random parameters belong to the conjugate family

of the sampling distributions, which can help lead to a

tractable posterior distribution. Barker and Newby

17

took advantage of a degradation model based on the

multivariate Wiener process to describe the state dete-

rioration of a complex multi-component system and

provided an optimal non-periodic inspection policy for

it. Whitmore

18

studied the failure inference problem for

a bivariate Wiener degradation process, where a pro-

cess is the observable marker and the second is latent.

However, most of the literature above only focus on

the issue of reliability estimation based on the bivariate

Wiener process for products tested in the laboratory

and not consider the problem of RUL estimation for

products in field conditions. In this article, we assume

that the degradation process follows a bivariate Wiener

process with random effects and investigate solutions

for the updating of random parameters as well as RUL

estimation. Our approach has two primary compo-

nents: see the schematic diagram displayed in Figure 1.

The first component is an offline work, in which the

initial model parameters reflecting the population char-

acteristics are estimated using the history of degrada-

tion data. The second component is an online updating

process. Once the new degradation data for an individual

product are available, we use the Bayesian theorem to

update the random parameters first, and then estimate

the RUL through the Monte Carlo simulation technique.

The primary contributions of this article are as follows:

(1) we first extend the bivariate Wiener process from off-

line reliability assessment to online RUL prediction for

degraded components with two dependent performance

characteristics and (2) under the Bayesian theorem, the

posterior distribution of random parameters is derived in

2Proc IMechE Part O: J Risk and Reliability

an analytical form. With the advantage of time efficiency

in model updating, the proposed RUL prediction algo-

rithms can be easily embedded into various online

applications.

The remainder of this article is organized as follows.

Section ‘‘Bivariate degradation model’’ constructs a

degradation model based on the bivariate Wiener pro-

cess. In section ‘‘RUL estimation,’’ the update mechan-

isms for random parameters are developed, and a

simulation-based method is presented to estimate the

RUL. Section ‘‘Experimental studies’’ provides a

numerical simulation and a real application to illustrate

the application and validation of the developed

approach. Finally, some conclusions and future

research directions are given in section ‘‘Conclusion.’’

Bivariate degradation model

Bivariate Wiener process

Degradation process of a component is stochastic in

nature due to inherent inconsistency and time-varying

operating conditions. Therefore, it is natural to model a

degradation process as a stochastic process. Among

numerous stochastic processes, the Wiener process and

Gamma process have gained most attentions in degra-

dation modeling for products with only a single perfor-

mance characteristic. A Gamma process is a monotonic

stochastic process that can only deal with observed

degradation with strictly increasing or decreasing incre-

ments. In certain physical situations, it is often the case

that the observed degradation phenomena are not

strictly monotone due to some random factors such as

environment changes and measurement errors. With

the property of non-monotonicity, the Wiener process

is an easy way to model observed non-monotonic

degradation processes.

19

In the two-dimensional case, it

is much easier to extend the Wiener process to a bivari-

ate degradation model than the Gamma process. It is

recommended, therefore, to model degradation of two

performance characteristics in terms of a bivariate

Wiener process.

Consider a two-dimensional process fX(t), Y(t)g0,

for t50. The process is a bivariate Wiener process, pro-

vided it satisfies the following three properties:

20

1. fX(0), Y(0)g0=f0, 0g0and fX(t), Y(t)g0is right-

continuous at t=0:

2. The increment vector fDX(t), DY(t)g0between two

arbitrary times tand t+Dtfollows a bivariate

normal distribution BVN(Dtm,DtS), where mand

Sare the mean vector and covariance matrix,

respectively

m=(a,c)0ð1Þ

S=b2rbd

rbd d2

ð2Þ

3. The increment vectors in any two non-overlapping

intervals ½t1,t2and ½t3,t4, for t1\t24t3\t4, are

independent.

We refer to aand cas the drift coefficients of the two

degradation processes X(t) and Y(t), and band das the

diffusion coefficients. The correlation coefficient rcan

describe the dependency between X(t) and Y(t).

Lifetime distribution

The concept of first passage time (FPT) is often used to

define the lifetime of a Wiener process. For a univariate

Wiener process, the lifetime distribution is inverse-

Gaussian, which has a graceful analytical expression.

Unlike the one-dimensional situation, the analytical

form of lifetime distribution for a bivariate Wiener pro-

cess is too complex to deduce. Suppose the failure

thresholds for performance characteristics X(t) and Y(t)

are w1and w2, respectively. Then, the lifetime Tcan be

defined as the FPT when at least one of the two perfor-

mance characteristics exceeds its threshold, namely

T= inf tX tðÞ

ji

w1orYtðÞ.w2

fg

ð3Þ

The corresponding reliability function is defined as

RtðÞ=Pt:XsðÞ\w1,YsðÞ\w2,04s4t

fg

ð4Þ

Domine and Pieper

21

derived the analytical form of

R(t) by constructing the Kolmogorov forward function.

If both the initial degradation values for X(t) and Y(t)

are equal to zero, the reliability function of a bivariate

Wiener process can be expressed as

RtðÞ=ða

0ð‘

0X

‘

n=1

2r

ad21r2

ðÞtsin np

au0

3

exp da bcr

bd21r2

ðÞ

rcosuc

d21r2

ðÞ

rsinu

3

Figure 1. Schematic diagram of the residual useful life

estimation approach in this article.

Liu et al. 3

exp b2c22acbdr+d2a2

21r2

ðÞb2d2tr2+r2

0

21r2

ðÞd2t

3

sin np

a

Inp

a

rr0

1r2

ðÞdt

drdu

where a= arctan( r1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1r2

p)+p, and r0and u0

jointly satisfy the following pair of equations

r0cosu0=w1b

brw2

r0sinu0=w2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1r2

p

(

and Iv(z) is the modified Bessel function

IvzðÞ=X

‘

n=0

z=2ðÞ

2k+v

k!Gk+v+1ðÞ

Then, the probability density function (PDF) of life-

time can be calculated as

ftðÞ=dRtðÞ

dtð5Þ

It is clear that the PDF of Tis too difficult to repro-

duce analytically. In engineering practice, the Monte

Carlo simulation method is often used to calculate the

approximate distribution of lifetime for a bivariate

Wiener process.

RUL estimation

Updating random parameters

Generally, the degradation process of an individual

product shows some heterogeneity within the popula-

tion as a result of the randomness of materials and

operating environments. RUL estimation focuses on

the evolution of performance characteristics for an

individual component in operation. To describe the

unit-to-unit variability, researchers often divide para-

meters of the degradation model into random effects

and fixed effects. The random parameters are assumed

to follow a probability distribution to capture the var-

iation of the degradation process of an individual com-

ponent, while the fixed parameters represent a constant

physical phenomenon common to all units from a given

population.

23

Particularly, the drift coefficient of the

univariate Wiener process can be treated as the random

realizations following a normal distribution, and the

diffusion coefficient is a constant to present the fixed

effects.

3,4,6,24

Here, inspired by the works above, we assume that

the drift coefficients of the bivariate Wiener process are

s-independent and normally distributed, namely,

a;N(ma,s2

a) and c;N(mc,s2

c), while the diffusion coef-

ficients and correlation coefficients are constant to all

items. It is worth noting that any other forms of distri-

butions can be treated as random effects, but some dif-

ficulty in posterior distribution updating may be

involved.

23

Now, the parameters of the bivariate Wiener process

can be classified into two groups: the random para-

meter vector u=(a,c)0and the fixed parameter vector

ϕ=(b,d,r)0. As for the random parameter vector u,it

can be specified as a prior distribution p(u), which con-

tains hyper-parameters g=(ma,sa,mc,sc)0. Then, the

initial unknown parameters for a given population are

Θ=½g,ϕ.

For an individual component, define X1:k

=fx1,x2,...,xkgand Y1:k=fy1,y2,...,ykgas the

observed degradation of two performance characteris-

tics at time t1,t2,...,tk. Here, for simplicity, we sup-

pose that Xand Yare measured at the same time, and

with the same interval between measurements, namely,

thth1=Dt, where h=1,2, ...,k. Set Dtto be 1 to

represent the unit time. Now, we focus on how to esti-

mate the posterior distribution of uconditional on the

new degradation fX1:k,Y1:kg.

The complete likelihood function conditional on

fX1:k,Y1:kgis

pX1:k,Y1:kjuðÞ=Y

k

h=1

1

2pbd ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1r2

p

exp 1

21r2

ðÞ

DxhaðÞ

2

b22rDxhaðÞDyhcðÞ

bd

"(

+Dyhc

ðÞ

2

d2#) ð6Þ

where Dxh=xhxh1and Dyh=yhyh1with

fx0,y0g=f0, 0g

Under the Bayesian theorem, the posterior distribu-

tion p(ujX1:k,Y1:k) can be expressed as

pujX1:k,Y1:k

ðÞ}pX1:k,Y1:kjuðÞpuðÞ ð7Þ

where p(u) denotes the prior distributions of u.

It is easy to prove that the posterior distribution

p(ujX1:k,Y1:k) is a bivariate normal distribution with

mean (mak,mck), variance (s2

ak,s2

ck), and correlation

coefficient rk. The proof is as follows

pujX1:k,Y1:k

ðÞ}pX1:k,Y1:kja,cðÞpuðÞ

}exp 1

21r2

ðÞ

X

k

h=1

DxhaðÞ

2

b2

"(

2rDxhaðÞDyhcðÞ

bd +DyhcðÞ

2

d2#)

exp ama

ðÞ

2

2s2

a

()

exp cmc

ðÞ

2

2s2

c

()

}exp 1

21r2

ðÞs2

as2

cb2d2s2

cd2s2

ak+1r2

b2

a2+s2

ab2s2

ck+1r2

d2

c2ð8Þ

For simplicity, let

4Proc IMechE Part O: J Risk and Reliability

A=s2

cd2s2

ak+1r2

b2

1r2

ðÞs2

as2

cb2d2

B=s2

ab2s2

ck+1r2

d2

1r2

ðÞs2

as2

cb2d2

C=s2

as2

cd2Pk

h=1 Dxhs2

as2

crbd Pk

h=1 Dyh+1r2

b2d2s2

cma

1r2

ðÞs2

as2

cb2d2

D=s2

as2

cb2Pk

h=1 Dyhs2

as2

crbd Pk

h=1 Dxh+1r2

b2d2s2

amc

1r2

ðÞs2

as2

cb2d2

E=ks2

as2

crbd

1r2

ðÞs2

as2

cb2d2

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

Then, we have

pujX1:k,Y1:k

ðÞ}exp a2A+c2B2aC 2cD 2acE

2

}1

2psaksck ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1r2

k

q

exp 1

21r2

k

amak

ðÞ

2

s2

ak 2rk

amak

ðÞcmck

ðÞ

saksck

+cmck

ðÞ

2

s2

ck

3

7

7

59

>

>

=

>

>

;

2

6

6

4

8

>

>

<

>

>

:ð9Þ

where

mak =BC +DE

AB E2,mck =AD +CE

AB E2,s2

ak =B

AB E2,

s2

ck =A

AB E2, and rk=E

ﬃﬃﬃﬃﬃﬃﬃ

AB

p

The last equation in equation (9) is the PDF of the

bivariate normal distribution with the mean vector

(mak,mck ) and covariance matrix

s2

ak rksaksck

rksaksck s2

ck

Note that although we assume that the random

parameters aand care independent in prior, there is a

correlation coefficient in their posterior distribution.

This may be due to the fact that aand creflect the

mean degradation rates of two dependent degradation

processes. In the posterior distribution, our beliefs

about aand care updated from the correlated dataset

fX1:k,Y1:kg, which contain the degradation rate infor-

mation of the two processes. Thus, it is natural that the

new beliefs about aand cwould be correlated.

Simulation-based RUL estimation

After obtaining the posterior distribution of u, our goal

is to evaluate the posterior distribution of the RUL. In

other words, we focus on determining the distribution

of the length of time from the present time tkto the end

of life. As mentioned above, we assume that failure

occurs when either of the two performance characteris-

tics exceeds the corresponding threshold, which are

denoted as w1and w2, respectively. In practice, the

threshold can be either deterministic or probabilistic.

25

In the remainder of the article, we only concentrate on

the deterministic threshold.

At the current time tk, the degradation characteris-

tics of the component being monitored can be

described by the updated random parameter vector u

and the fixed parameter vector ϕ. As discussed above,

the posterior distribution p(ujX1:k,Y1:k) is a bivariate

normal distribution, namely, ujX1:k,Y1:k;BVN(mak,

s2

ak,mck ,s2

ck,rk). Then, the RUL Lkat time tkcan be

defined as

Lk= inf lkXt

k+lk

ðÞ

ji

w1or Yt

k+lk

ðÞ.w2

fg

ð10Þ

Previous discussion shows that the analytical for-

mula of the FPT distribution for a bivariate Wiener

process with fixed drift coefficients is quite difficult to

derive. With the random effects in drift coefficients, the

analytical formula of the distribution for Lkin equation

(10) is even more intractable. For this reason, we use a

Monte Carlo simulation method to estimate the RUL

distribution approximately. The basic idea is to simu-

late the continued evolution of the two degradation

process fX(t), Y(t)g0in the future with the knowledge at

time tk. For a simulated sample, the s-step prediction of

the future state, that is, fXk+1:k+s,Yk+1:k+sg0,is

given by the following procedure

For j=1,2, ...,s

Step 1. Generate a sample (aj,cj)0from BVN(mak,

s2

ak,mck ,s2

ck,rk);

Step 2. Generate a sample (Dxj,Dyj)0from BVN(aj,

b2,cj,d2,r);

Step 3. Let fxk+j,yk+jg0=fxk+j1,yk+j1g0+

(Dxj,Dyj)0,j=1,2, ...,s.

Liu et al. 5

Conducting the above procedure Ntimes (say

N= 1000), we can obtain Nsimulated samples to pre-

dict the states of the two performance characteristics

after time tk.To obtain the distribution of the RUL, we

must determine when these samples fail, respectively.

According to equation (10), the RUL for a sample is

the least simulated steps when either Xk+1:k+jexceeds

w1or Yk+1:k+jexceeds w2. Then, we can acquire

Nsimulated RULs, which are denoted as fL1

k,

L2

k,...,LN

kg.

Some important indices of RUL like the mean, med-

ian, and confidence intervals enable the construction of

replacement decisions or maintenance strategies under

the framework of PHM. These indices can also be esti-

mated approximately using the simulated RULs. The

corresponding algorithm can be described as follows:

1. Sort the simulated RULs fL1

k,L2

k,...,LN

kgin

ascending order, namely

L1ðÞ

k\L2ðÞ

k\\LNðÞ

k

2. The mean RUL at time tkis estimated approxi-

mately by

1

NX

N

i=1

LiðÞ

k

3. The median RUL at time tkis estimated approxi-

mately by

L

N+1

2

ðÞ

k,Nis odd

L

N

21

ðÞ

k+L

N

2+1

ðÞ

k

=2, Nis even

8

>

<

>

:

4. The approximate 100 (1 a)% confidence inter-

vals for RUL at time tkare estimated approxi-

mately by

LBLðÞ

k,LBUðÞ

k

hi

where BL =a

2N

and BU =1a

2

N

,

bcmeans

round to the nearest integer.

Initial parameters estimation

Now, we move back to the issue of estimating the initial

parameters in Θ=½g,ϕ. Typically, some historical

degradation data of the same type component are

required to determine the fixed parameters ϕ=(b,d,r)0

and the hyper-parameters g=(ma,sa,mc,sc)0. Without

loss of generality, we suppose that nsamples are tested

and mmeasurements for all samples are observed at the

same time (case of balance measurement) before the

component is launched in the market. The intervals

between two measurements are the same and defined to

be unit time, namely, Dtij =tij ti,j1= 1, for

i=1,2, ...,nand j=1,2, ...,m. The degradation

measurements are denoted as Wi,j=(xi,j,yi,j). Then,

the historical bivariate degradation data can be pre-

sented in the form

W2n3m=X

Y

=

x1, 1 x1, m

.

.

...

..

.

.

xn,1 xn,m

y1, 1 y1, m

.

.

...

..

.

.

yn,1 yn,m

0

B

B

B

B

B

B

B

B

@

1

C

C

C

C

C

C

C

C

A

According to the multivariate normal theory, we can

estimate the mean vector and covariance matrix using

the unbiased estimates

^

m=1

mn X

n

i=1X

m

j=1

Dxi,j,Dyi,j

0ð11Þ

^

S=1

mn 1X

n

i=1X

m

j=1

Dxi,j,Dyi,j

0^

m

hi

Dxi,j,Dyi,j

0^

m

hi

0ð12Þ

where Dxi,j=xi,jxi,j1,Dyi,j=yi,jyi,j1.

Comparing equation (11) with equation (1), we can

readily obtain the estimates of aand c. Similarly, con-

trasting equation (12) with equation (2), we can obtain

the estimates of b,d, and r. Due to the existence of ran-

dom effect parameters, it is difficult to directly estimate

the hyper-parameters in gby the traditional maximum

likelihood estimate (MLE) method.

26

In this article,

motivated by the work in Tang et al.,

27

a two-stage

method is adopted to obtain the estimators of all para-

meters in ϕand g. The corresponding algorithm is as

follows:

Stage 1

1. Calculate the MLE of the fixed parameters

ϕ=(b,d,r)0by equation (12).

Stage 2

2. Based on degradation data W2n3m, generate B

bootstrap samples W1

2n3m,...,WB

2n3m.

3. Using equation (11) and each bootstrap sample

Ws

2n3m, obtain a bootstrap estimate, ^

asof a, and

^

csof c,s=1,2, ...,B.

4. Estimate the hyper-parameters g=(ma,sa,mc,sc)0

of the prior distribution for aand cusing the

method of maximum likelihood.

Experimental studies

In this section, we first provide a numerical simulation

to test the performance of the presented approach,

including the procedures of initial parameters estima-

tion, the updating of random parameters, and RUL

estimation. Then, a practical case study on Lithium-ion

batteries is used to illustrate the implementation of

RUL estimation.

6Proc IMechE Part O: J Risk and Reliability

A numerical example

In the experiment, we first simulate the degradation

paths of 20 components based on a bivariate Wiener

process. The procedure of simulation is omitted here

because it is quite similar with the method in section

‘‘Simulation-based RUL estimation.’’ In the simulation,

we let a=0:1, b=0:07, c=0:2, d=0:15, and r=0:9.

The degradation paths are shown in Figure 2 with 50

sampling points. First, the estimate of fixed parameter

vector ϕ=(b,d,r)0is conducted by equation (12), and

the results are as follows: ^

b=0:0690, ^

d=0:1505, and

^r=0:9009. Then, the bootstrap method is used to

estimate the hyper-parameters in g. In the bootstrap

simulation, we find that the estimates of the hyper-

parameters are very stable when the size of bootstrap

sample exceeds 2000. The corresponding estimates are:

^ma=0:0988, ^sa=0:0084, ^mc=0:1917, ^sc=0:0173.

One particular simulated degradation path is empha-

sized in Figure 2, which is used to conduct the RUL

estimation. Suppose that the thresholds for perfor-

mance characteristics X(t) and Y(t) are w1= 4 and

w2= 8. As shown in Figure 2, the time that X(t)

exceeds w1is approximately to be 41, while that Y(t)

exceeds w2is approximately to be 44. According to def-

inition of FPT, the actual lifetime for this unit is

T= 41.

Since the presented approach allows real-time updat-

ing, the RUL distribution as new observations are

available, such updating mechanism should be less sen-

sitive to the selection of parameters in the prior distri-

butions. To test the robustness of the approach, we

assume that the hyper-parameters in the prior distribu-

tion are m

a=1, s

a=0:2, m

c= 2, and s

c=0:4, which

are quite far away from their actual values in the simu-

lation procedure. And the fixed parameters are set to be

the estimates in ^

ϕ. The actual degradation path and the

one step predictions for each performance characteristic

are illustrated in Figure 3. Results show that our model

has a quick and good predictive ability and the degra-

dation path of both actual and predicted value almost

overlap. Due to the inappropriate selection of the

hyper-parameters in the prior distribution, the early

predictions are not accurate enough. Notwithstanding,

the Bayesian updating procedure makes the predictive

ability of our model improve quickly as the new data

are accumulated. The similar phenomenon can be

observed in the updates of the posterior distribution

parameters mak and mck, which are shown in Figure 4.

All these results demonstrate that our approach is quite

robust over the hyper-parameters in the prior distribu-

tion. This is an important character since it can make

the engineering implementation rather reliable. Another

observation is that sak and sck decrease while rk

increases as new data are accumulated. A reasonable

explanation lies in the fact that with more and more

degradation data being collected, the uncertainties

about random parameters aand cdecreases, while the

correlation between them strengthens

Figure 5 compares the mean, median, and confi-

dence intervals of estimated RUL with the actual

RUL. It is evident that inappropriate selection of

Figure 2. Simulated degradation paths of 20 components based on a bivariate Wiener process.

Figure 3. Degradation paths and predicted paths for a

simulated unit.

Liu et al. 7

hyper-parameters leads to significant prediction errors

in the first six sampling points. But the prediction error

decreases quickly as more degradation data are used.

The actual RUL falls with the range of 80% confidence

intervals except for the first six sampling points, and

the estimated mean and median RUL almost overlap

each other.

As mentioned in section ‘‘RUL estimation,’’ the

fixed parameter vector ϕdescribes the population-

based degradation characteristics and will not be

updated with new observed data. Thus, the selection of

ϕ, which is estimated with the population-based histori-

cal data, will also have a direct impact on the accuracy

of RUL estimation. To investigate the sensitivity of the

RUL estimation with respect to the fixed parameters,

we vary each parameter in ^

ϕby multiplying a positive

factor whose value is chosen from (0.9, 0.95, 1.05, 1.1)

sequentially, but keep the other parameters unchanged.

Meanwhile, we set the hyper-parameters to be the esti-

mated values in ^

g. Then, we define the mean relative

error as

MRE =1

lX

l

k=1 d

RLk+tkT

T

where lis the total number of sampling points before

the simulated unit fails, d

RLkis the estimated mean

RUL at time tk, and Tis the actual lifetime. The results

of mean relative errors are shown in Table 1. For com-

parison, we set the fixed parameters to be the estimated

values in ^

ϕand calculate it again. The mean relative

error with appropriate fixed parameters is 0.02839.

Results show that the relative error is quite robust for a

moderate departure from the estimated values in ^

ϕ.

To further demonstrate the superiority of our

approach, we compare it with Gebraeel’s approach with

respect to their accuracy of RUL prediction. Under the

Bayesian theorem, a classic RUL prediction method is

proposed by Gebraeel based on the univariate Wiener

process for products with a single performance charac-

teristic, and the detailed implementation of this

approach can be acquired in Gebraeel et al.

2

In Figure

2, there are 19 samples in which failure occurs at the

end of simulation. According to the failure criterion, 14

of them fail due to Xdegradation, while the other 5 fail

due to Ydegradation. We conduct the RUL prediction

for the 19 samples by our approach, Gebraeel’s

approach only considering Xand Gebraeel’s approach

only considering Y, respectively. In the prediction, the

initial parameters are set to be the estimators by the

simulation data in Figure 2. Figure 6 shows the MRE

of each sample for the three approaches, respectively.

Results show that for most samples, our approach has

a higher prediction accuracy over the traditional ones.

Denote MRE to be the average value of MRE for all

the 19 samples. The MRE of our approach is 0.0522,

while the MRE of the other methods are 0.0621 (only

consider X) and 0.0822 (only consider Y). The results

also demonstrate that our approach has a higher pre-

diction accuracy from the statistics perspective.

A real application: lithium-ion batteries

Lithium-ion batteries have been widely applied to many

portable consumer electronics, such as cell phones, lap-

tops, and digital cameras. Compared with other second-

ary batteries, lithium-ion batteries have a large number

of advantages, for example, no memorability, high

nominal voltage, long cycle life, low self-discharge rate,

high energy density, and low pollution, all of which

make it one of the most ideal power sources in the 21st

century. In many cases, lithium-ion battery is a crucial

Figure 5. Results of RUL estimation for a simulated unit.

Table 1. Sensitivity analysis with respect to the fixed

parameters.

Positive factor 0.9 0.95 1.05 1.1

b0.02866 0.02823 0.02831 0.02801

d0.02911 0.02856 0.02815 0.02768

r0.02818 0.02808 0.02924 0.03314

Figure 4. Update of hyper-parameters mak,sak ,mck,sck ,andrk.

8Proc IMechE Part O: J Risk and Reliability

component of the system and its failure can lead to con-

sequences ranging from operation impairment to even

catastrophic failures.

28

There are many indicators related with lithium-ion

batteries’ aging and degradation, among which discharge

capacity and discharge energy are frequently adopted in

practice. Ideally, in addition to the reaction of the

lithium-ion shuttling between two electrodes, there are

no side reactions inside the battery. Thus, the discharge

capacity and energy of a fully charged battery will not

decline. However, during charge/discharge cycles, cell

capacity and energy fade gradually due to some unex-

pected side reactions, such as oxidation of anode materi-

als, lithium corrosion on cathode, electrolyte

decomposition, and solid electrolyte interface (SEI) for-

mation. Consequently, these unexpected reactions will

consume the active lithium-ion and enhance the battery’s

resistance, which lead to capacity and energy loss over

time.

To study the degradation performance of lithium-

ion battery, several 2400 mAh 18605-type lithium bat-

teries were tested through different testing steps, includ-

ing charging, discharging, and standing by. All

charging processes were carried out with a constant

current 1C until the battery voltage reached 4.2 V, and

then continued the test with a constant voltage until

the current fell to 0.01C. All discharging processes were

carried out with a constant current 2C until the battery

voltage dropped to 2.75V. The standby time between

charging and discharging was 30min. For a 2400-mAh

battery, the 1C corresponds to a current of 2400 mA.

As discussed above, there are some common reasons

causing lithium-ion battery capacity degradation and

energy degradation. So, the degradation paths of the

two performance characteristics may be significantly

correlated. As shown in Figure 7, the capacity and

energy loss of a 18650-type lithium-ion battery are nor-

malized by the initial values, respectively. The Pearson

correlation test result shows that the correlation coeffi-

cient between capacity and energy is 0.94. Generally,

the lifetime of a lithium-ion battery is defined as the

charge/discharge cycles before either the capacity loss

or the energy loss reaches 20%. Under this definition,

the battery fails at the 283rd cycle.

As for this specific application, using the proposed

method, the estimated RUL can be obtained at each

monitoring point. Figure 8 presents the update of the

hyper-parameters over cycle number. In the RUL esti-

mation, 1000 samples are generated to determine the

approximate RUL distribution at each sampling point.

Figure 9 illustrates the mean, median, and confidence

intervals of estimated RUL every 10 sampling points. It

can be observed that our approach can quickly adjust

and update the distribution of the random parameters,

and thus, the estimated RUL matches quite well with

the actual RUL after a few updates. In Figure 10, the

approximate RUL distribution at four sampling points

(tk= 0, 50, 150, 200) are plotted, respectively. Clearly,

the actual RUL falls with the range of the estimated

RUL distribution. Moreover, with the degradation data

accumulating, the variance of RUL is seen to gradually

decrease.

To demonstrate the superiority of the proposed

approach, we first compare the prediction accuracy

Figure 6. Mean relative error comparison of the simulated

samples.

Figure 7. Capacity and energy degradation of a lithium-ion

battery.

Figure 8. Update of hyper-parameters for a lithium-ion battery.

Liu et al. 9

with Gebraeel’s

2

method, in which only a single perfor-

mance characteristic is considered. Define the relative

error at time tkas

REk=d

RLk+tkT

T3100%

where d

RLkis the estimated mean RUL at tk, and Tis

the battery’s actual lifetime. The comparison results are

presented in Figure 11(a). For most of the sampling

points, the prediction accuracy of the proposed approach

is higher than Gebraeel’s method. Then, we further

compare the proposed approach with the traditional

method ignoring the dependency of two performance

characteristics. Under the independence assumption, the

RUL prediction only considering each single perfor-

mance characteristic is conducted respectively, and the

smaller prediction at each sampling point is used to

denote the battery’s RUL. Figure 11(b) shows the rela-

tive errors considering both dependency and indepen-

dency. We can also see that our approach outperforms

the traditional one significantly at most sampling points.

Therefore, to obtain accurate RUL estimations, the

dependency should be taken into account if necessary.

Conclusion

This article proposes an adaptive method for estimat-

ing RUL of partially degraded components with two

performance characteristics. A bivariate Wiener process

with random effects is deployed to capture the nature

of the two performance characteristics. A Bayesian

method is used to update the distribution of the ran-

dom parameters. The effectiveness of this method is

valid through a numerical example and a real applica-

tion on lithium-ion battery. From the examples, we

find that the robustness of our model is quite well. And

the accuracy of our approach is higher than the tradi-

tional methods, which only consider a single perfor-

mance characteristic. Moreover, with more monitoring

data, the RUL estimation is more accurate. Thus, the

RUL of a component should be re-estimated once new

information is obtained. Recently, the function of some

components is much more complex than that before.

As a result, more than one indictor can be used to mea-

sure a component’s degradation and aging. In this case,

Figure 10. Approximate residual useful life distribution at tk= 0, 50, 150, 200 sampling point.

Figure 9. RUL estimation for a lithium-ion battery.

10 Proc IMechE Part O: J Risk and Reliability

AQ2AQ2

our method is more flexible and has a wider application

than the traditional ones.

In this work, we only try to update the distribution of

the random parameters. Actually, the fixed parameters

for a specific component also changes in its operation.

To improve the prediction accuracy, it is necessary to

study how to update the fixed parameters in the future.

This article only considers products with two perfor-

mance characteristics, and it seems to be difficult to

extend our method to the multiple cases (three or more

performance characteristics) due to the complexity in

Bayesian updating of stochastic parameters. Perhaps, the

MCMC approach is a good choice to solve this issue.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest

with respect to the research, authorship, and/or publi-

cation of this article.

Funding

This work was supported by the Chinese National

Science Foundation under grant nos 71271212 and

61304221.

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