Journal of Renewable Energy and Sustainable Development (RESD)      Volume 2, Issue 1, June 2016 - ISSN 2356-8569 
 

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37 

 

Lithium-ion Battery Degradation Assessment and 

Remaining Useful Life Prediction in Hybrid Electric Vehicle 
Nabil LAAYOUJ *, Hicham JAMOULI * 

* LGII Laboratory, National School of Applied Sciences, Ibn Zohr University, 

 Agadir, Morocco 

E-mail: nabil.laayouj@gmail.com 
 
 

Abstract - Prognostic activity deals with prediction of 

the remaining useful life (RUL) of physical systems, 

based on their actual health state and their usage 

conditions. RUL prediction gives operators a potent 

tool in decision making by quantifying how much time 

is left until functionality is lost. In addition, it can be 

used to improve the characterization of the material 

properties, that govern damage propagation for the 

monitored structure. RUL can be predicted by using 

three main approaches, namely model-based, data-

driven and hybrid approaches. Prognostic methods 

used later in this paper are hybrid and data-driven 

approaches, which employ the Particle Filter in the 

first one and the autoregressive integrated moving 

average in the second. The performance of the 

suggested approaches is evaluated in a comparative 

study on data collected from lithium-ion battery of 

hybrid electric vehicle. 

 

Keywords - Remaining useful life; Prognosis; Particle 

Filter; ARIMA. 

  
I. INTRODUCTION 

 

Prognosis and health management (PHM) [1] will 

have significant impact on increasing safety as well 

as reducing operating and maintenance costs. This 

can be achieved by providing an accurate 

quantification of degradation and damage at an early 

stage to reduce or eliminate malfunctions.  

Furthermore, PHM consists of three main routines: 

fault detection, diagnosis and prognosis. A prognosis 

has recently attracted a lot of research interest due to 

the need of models for accurate RUL prediction [2]. 

 

Numerous methods and tools can be employed to 

evaluate the size of damage by predicting the RUL 

value. Prognosis techniques can be categorized 

under three approaches based on the usage of 

information: model-based, data-driven and hybrid 

approaches. The model based approach [3], 

assumes that a model of system behavior is available 

and uses this model to predict the future of the 

system behavior. Some recent developments in the 

model-based approach have been reported in the 

literature. Such as lumped parameter model [4], 

functional models [5] and first principal models [6]. 

The data-driven approach [7] aims at transforming 

the data provided by sensors into relevant models. In 

the literature, there are the following works: relevance 

vector machine [8] and neural network [9]. The hybrid 

approach [10] combines the two approaches cited 

earlier and includes Bayesian techniques [11][12]. 

 

In this work, the researchers will study two main 

approaches to predict RUL. The first approach is a 

hybrid prognosis using a Particle Filter method, which 

employs both, state dynamic model and a 

measurement data. The second approach is a data-

driven prognosis based on routinely collected data, 

using autoregressive integrated moving average 

(ARIMA) model to estimate the system degradation. 

 

The application of Particle Filters to prognosis has 

been reported in the literature. For example, 

prediction of lithium-ion battery capacity depletion 

[13], degradation prediction of a thermal processing 

unit in semiconductor manufacturing [14], and 

remaining useful life prediction of a mechanical 

component subject to fatigue crack growth [15]. The 

reported application results have shown that Particle 

Filters represent a potentially powerful prognosis tool 

due to its capability in handling non-linear dynamic 

systems and non-Gaussian noises, using efficient 

sequential importance sampling [16][17] to 

approximate the future state probability distributions. 

 

Many works around ARIMA model have been 

developed. Among these are mechanical 

deterioration prognosis [18] [19] and economic 

forecasts [20][21]. In the present study, the 

researchers will use the ARIMA model to predict 

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38 
 

degradation by computing RUL. 

 

This paper is organized as follows: section 2 contains 

the descriptions of the approaches at the basis of 

RUL prediction; in section 3, the results of the 

application of the methods are presented, and an 

evaluation of their performance is given; and finally, 

some conclusions on the advantages and limitations 

of the approaches are given in section 4. 

 

II. SUGGESTED APPROACHES BASED ON 

THE RUL PREDICTION 

 

A. Prognosis of degradation and remaining useful 

life 

The term prognosis is originally used in medicine for 

the prediction of a course of an illness. But, later on it 

has been introduced to industry to predict the future 

operating status of equipments, and to set an efficient 

treatment.  

 

The practitioner uses the results of the forecast 

models to determine the most appropriate treatment. 

These forecast models based on simple mathematical 

tools (e.g. decision tree, conditional probability) [22], 

or on more sophisticated (e.g. Markov processes, 

neural networks, and genetic algorithms) [23]. 

 

As mentioned earlier, the main objective of prognosis 

is to determine the time before failure. Note that the 

data prognosis is important information that may be 

used in the decision process. For example, this data 

prognosis can be used to delay the maintenance 

interventions, or stop a machine before its future 

maintenance due to earlier default. 

 

B. Methods of prognosis 

 

1. Particle filter for prognosis 

The Particle Filter method [24] is a Monte Carlo 

technique for the solution of the state estimation 

problem.  The key idea is to represent the required 

posterior density function by a set of random samples 

(particles) with associated weights, and to compute 

the estimations based on these samples and weights. 

As the number of samples becomes very large, this 

Monte Carlo characterization becomes an equivalent 

representation of the posterior probability function, 

and the solution approaches the optimal Bayesian 

estimation. 

2. Particle filter model 

Consider the dynamic system described by the 

following discrete time model [15]: 

 

),(
1 kkkk

vxfx


            (1) 

 

),(
kkkk

xhz             (2) 

 

Where:  

 kf  : is the state transition function (damage model) 

 kv  : is state noise vector of known distribution 

k
h  : is the measurement function 

k
  : is the measurement noise vector 

The goal of tracking is to recursively estimate k
x

by 

using the set of all available measurements

),...,(
1:1 kk

zzz  up to time k, and to create a 

conditional state PDF (probability density function). 

Like any Bayesian estimation, two steps are 

employed: prediction and update. 

 

In the prediction step, the researchers consider that 

the PDF  1:11  kk zxp  previous state estimate at 

time 1k  and the process model (1), both are used 

to obtain the prior PDF of the state at time k  as 
shown in Chapman-Kolmogorov equation. 

11:1111:1   kkkkkkk dxzxpxxpzxp      (3) 
 

In the update step, at time , a measurement   

becomes available (from the likelihood function 

defined by the measurement model (2)), and this may 

be used to update the prior distribution to generate the 

posterior state PDF via Bayes rule (4). 

 

1:1

1:1

:1






kk

kkkk

kk
zzp

zxpxzp
zxp        (4) 

 

In order to obtain exact state estimation solutions for 

Equations (3) and (4), the actual distributions are 

approximated by a set of samples and their 

normalized weights. Consider   SN
i

i

k

i

k
wx

1:0
,


a random 

measure that characterizes the posterior PDF



Journal of Renewable Energy and Sustainable Development (RESD)      Volume 2, Issue 1, June 2016 - ISSN 2356-8569 
 

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39 
 

kk
zxp

:1 , where 
i

k
x

:0
 and 

i

k
w are, respectively a 

set of support points and associated weights. The 

weights are normalized such that 1i
i

k
w . Then, 

the posterior density at time k is approximated [24] as 

 





SN

i

i

kk

i

kkk
xxwzxp

1

:0:0:1
)(          (5) 

 

The weight process based on importance sampling 

[24], such that the weight update equation is given by 

 

k

i

k

i

k

i

k

i

k

i

kki

k

i

k
zxx

xxpxzp
ww

:11:0

1

1









      (6) 

 

When 
S

N the importance density function 

kkk
zxx

:11:0 
  can be approximated by the prior 

PDF 1kk xxp , and weight becomes [16] 

i

kk

i

k

i

k
xzpww

1
           (7) 

 

Another problem arises is the Degeneracy 

phenomenon, where after a few iterations, all but one 

particle will have negligible weight. This degeneracy 

explains that a large number of updating particles is 

around zero. To overcome this problem, considering 

resampling procedure [25] at each step, the 

researchers assign 
S

k
N

w
1

1



 for all the particle weights 

so they have 

i

kk

i

k
xzpw              (8) 

 

To implement SIR (sequential importance resampling) 

filter, as in (7) and (8), the researchers need to know 

process model, measurement model and likelihood 

function kk xzp  

3. RUL Prediction using Particle Filter 

Once the estimated parameter is obtained, the future 

damage state and RUL can be predicted by 

progressing the damage state until it reaches the 

threshold (Fig. 1). The PDF curve represents the 

progress of damage state until it reaches the 

threshold. The distribution of RUL can be obtained by 

subtracting this PDF from the threshold 

 

 
Fig .1. Illustration of RUL calculation 

 

4. ARIMA model for prognosis 

One of the important and widely used time series 

model is the autoregressive integrated moving 

average (ARIMA) model, which is a generalization of 

ARMA model. It requires only the historical time series 

data. These models are fitted to time series data to 

predict future points in the series (forecasting). 

 

5. ARIMA forecasting method 

ARIMA is a forecasting technique, noted as ARIMA 

(p,d,q), the general model was introduced by Box and 

Jenkins [26]. It is a method which allows both 

autoregressive (AR) and moving average (MA). It 

explicitly includes differencing in the formulation of the 

model. Where, p  and q  are, respectively, 

autoregressive parameter and moving average 

parameter, while d  is the number of non-seasonal 

differences. The autoregressive part of the model of 

order p  is written:  

t

p

i

itit
xcx   





1

         (9) 

 

Where 
t

x  is a stationary series, 
it

x


represents lag i

of
t

x , the
i
 , pi ,...,1 are the parameters of the 

model, c is a constant and 
t

  is a white noise. The 

moving average part of the model of order q  is 

written: 

t

q

i

itit
x   





1

       (10) 

 

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Where the 
i

 , qi ,...,1  are the parameters of the 

model,   is the expectation of 
t

x , which is often 

assumed to equal zero, and the 
qttt 

 ,...,,
1

 are 

white noise error terms. After an initial differencing 

step (corresponding to the integrated part of the 

model), the researchers can present the ARIMA 

(p,d,q) as ARMA (p,q) process: 











q

i

iti

p

i

ititt
xcx

11

    (11) 

 

The estimation of the ARIMA model corresponding to 

some learning data is done through Box and Jenkins 

methods [26]. A procedure of the forecasting can be 

summarized as: 

 

 Check stationary: If the data are not stationary, 

they need to be transformed into stationary data 

using the differencing technique. 

 

 Identification: to specify the appropriate number 

of autoregressive term p, and moving average 

term q from the autocorrelation function (acf) and 

partial autocorrelation function (pacf) 

correlograms. 

 

 Forecasting: Based on the forecasting model, 

multi-step-ahead prediction is then conducted to 

forecast the final failure time. 

 

 Verification: If the predictions result in an 

unexpected trend, repeat steps 2 and 3 until the 

model fits the historical data appropriately. 

 

6. RUL prediction using ARIMA 

Prediction of lifetime using ARIMA can be expressed 

in two parts: construction of model and prediction of 

state. In the first step, the researchers construct the 

corresponding coefficients (differencing, 

autoregressive and moving average terms) and they 

obtain an ARIMA model. Next step, when the model is 

built, a sample is selected to be estimated. Remaining 

Useful Life (RUL) is defined as the number of 

predictions from current state until the failure states 

reached the threshold. The following Fig. 2 shows the 

forecasting method for RUL prediction. 

 

7. Hybrid electric vehicle (HEV) 

An electric car is powered by an electric motor instead 

of a gasoline or diesel engine. The electric car (also 

known as electric vehicle EV) uses energy stored in 

its battery (or series of batteries), which are recharged 

by different sources. There is a variety of electric 

vehicles available in the world, among these HEV 

(hybrid electric vehicle), the PHEV (plug-in hybrid 

electric vehicle) and BEV (battery electric vehicle). 

 

The researchers have oriented their work towards the 

HEV (Fig. 3), where a small battery is placed on 

board, and when the vehicle brakes, the energy is 

stored in the battery and it can later be used to power 

the electric motor which assists the gasoline engine. 

HEV typically provides better fuel economy than 

similar conventional vehicles. This is why it is 

necessary to develop a prognosis of battery 

degradation, in order to ensure a proper functionality. 

 

The cell (or battery) studied in this work [27] is a 

lithium-ion with 1Ah capacity and 3.75 V. The cell is 

part of a battery pack (Battery pack= 15 cells) which is 

used to collect and distribute electric power (direct 

current power), mainly to the electrical drive. The 

researchers use the 25°C as the baseline for 

measurement. A simple form of the empirical 

degradation model is expressed by an exponential 

growth model as follows [28] 

 

Degradation Data differencing

Lags selection

AR and MA 

coefficients 

ARIMA Model

Forecasting Accurate

Remaing useful 

life

No

Yes

 
Fig .2. ARIMA prediction Algorithm 

 

)exp( bta         (12) 

 

Where a, b are model parameters, t is the time, and  

  is internal cell performance. The equivalent 

electrical model is introduced as shown in Fig. 4. 

where, DL
C

is the double layer capacitance, 
CT

R is 



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41 
 

the charge transfer resistance, 
W

R is the Warburg 

impedance and ER is the electrolyte resistance. The 

internal cell performance is observed instead of the 

capacity. Additionally, there is a relationship between 

 and 1/C capacity;  is inversely 

proportional to the  1/C [28]. Also, the observed data 

are assumed to be given as a  capacity.  and 

DL
C showed negligible change over the ageing 

process and are excluded from further analysis. The 

threshold for fault declaration has arbitrarily been 

chosen. 

 

 
 

Fig .3. Plan of hybrid electric vehicle 

 

 
 

Fig .4. Equivalent circuit diagram of a Li-ion Cell 

 
Table 1. Data of Degradation 

 

Time (T) Intial,1 2 3 4 5 

C/1 (Ah) 1,000 0.981 0.859 0.811 0.788 

Time(T) 6 7 8 9 10 

C/1 (Ah) 0.714 0.680 0.612 0.56 0.568 

 

 

8. RUL prediction and results 

The Particle Filter uses the exponential growth model 

(12) to obtain the prior PDF (1). The measurement 

and process noise variance 
k

 and 
k

v respectively 

were modeled as Gaussian densities. In the Particle 

Filter, a  and b are incorporated as internal cell 

parameters 
E

R and
CT

R . The values of a  and b in 

the actual state are used as initial estimations. Then, 

the resampling of particles is applied to each iteration 

to solve the degeneracy problem. The predictions are 

progressed until they reach the threshold to get the 

RUL. The failure threshold is defined when the 

capacity fades by 35%. 

 
 

Fig .5. Autocorrelation and partial autocorrelation plots for ARIMA 

model 

 

 
 

Fig .6. Trajectory of degradation for the cell of battery pack using 

Particle Filter 

 

ARIMA (p,d,q) model uses simply the capacity data 

measurement to predict the future degradation. In our 

case, the data are roughly exponential in nature; d is 

CTE
RR 

CTE
RR 

1/C
W

R

E
R

W
R

CT
R

DL
C

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chosen to be 2 in order to remove the non-stationarity, 

both p and q are chosen to be 1 (Fig. 5). The capacity 

data measurement at every 100 charge/discharge 

cycles are given in Table 1, where 1T=100 

charge/discharge cycles. 

 

Fig. 6 shows the real and the estimated values of the 

cell degradation using Particle Filter. Although the 

predicted results are close to the real values 

throughout the prediction area. In Fig. 7, the estimated 

data of the cell degradation obtained by ARIMA model 

are far off the real values after T=15. 

 

 
 

Fig .7. Trajectory of degradation for the cell of battery pack using 

ARIMA model 

 

The performance of both methods has been 

evaluated by the Mean Absolute Scaled Error (MASE) 

[29] and Root Mean Square Error (RMSE). 

 

Definition of MASE: 

 

A scaled error is defined as 












N

i

ii

kk
k

YY
N

YY
q

2

1
1

1

ˆ
    (13) 

 

Where Y is the real value of testing cell, Ŷ is the 
prediction value and N is the number of prediction 

data set. The MASE is simply 

)(
k

qmeanMASE       (14) 

 

Definition of RMSE: 

 





N

t

YY
N

RMSE
1

2
]ˆ[

1
    (15) 

 

The detail results are shown in Table 2. 

 
Table 2. MASE and RMSE measurement of the cell prognosis 

 

predictive 
approach 

Particle Filter ARIMA model 

MASE 0,7345 2,6163 

RMSE 0,0253 0,0719 

 

From Table 2, one can find that the calculation of 

MASE and RMSE for ARIMA model is superior to the 

Particle Filter. Consequently, the Particle Filter is 

more accurate than the ARIMA model. In this study, 

the researchers have illustrated how the Particle Filter 

method can provide accurate predictions of the RUL, 

as presented in [28][30] over conventional data-driven 

methods without physical model. 

 

III. CONCLUSION  

 

In this study, the researchers interested here in two 

approaches for health prognostics using Particle 

Filter and ARIMA model. The goal in applying these 

methods is to calculate RUL. In addition, the RUL 

gives the best idea about the degradation of each 

system. The results showed that the Particle Filter 

was more faithful to the simulated data. The 

researchers considered different frequencies of 

inspection for the measurement.  

 

This study highlights the value of having a physical 

failure model to improve the accuracy of results. In 

contrast, the ARIMA needs great possible historical 

data in order to give proper results without the need 

for physical model. However, the disadvantage of the 

Particle Filter compared to ARIMA model is the 

degradation model requirement, which is not always 

easy in the case of a large-scale system. Remember 

that both approaches require a knowledge of a 

threshold corresponding to the physical system 

failure to plan the actions of preventive maintenance, 

and expect to benefit from the opportunistic 

maintenance. Finally, the obtained results in this 

paper show that the Particle Filter is more efficient 

than the ARIMA model. 

 

 

 



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