Data-driven Prognostics for Lithium-ion Battery Based on Gaussian Process Regression Datong Liu, Jingyue Pang, Jianbao Zhou, Yu Peng
Automatic Test and Control Institute Harbin Institute of Technology Harbin, China
[email protected]
Abstract-Lithium-ion battery is a promising power source for electric vehicles owing to its high specific energy and power. Through monitoring battery health in effective way such as determining
the
operating
conditions,
planning
replacement
interval could increase the reliability and stability of the whole system. However, due to the reliance on integration, errors in terminal
measurement
caused
by
noise,
resolution,
to show the degradation of battery's performance and prevent possible accidents. There are two typical methods to calculate the SOH of the battery [3]. One method uses the battery impedance to indicate the battery SOH. That can be determined using Eq. (1).
the
SOH
uncertainty when we make prognostics for battery health are cumulative, the prediction result is combined with unsatisfied errors.
As
a
result,
the
prognostic
algorithms
supporting
uncertainty representation and management are emphasized. So in this paper, we present the Gaussian process model to realize the prognostics for battery health. Because of the advantages of flexible, probabilistic, nonparametric model with uncertainty
=!ix 100% Ro
(1)
Where R, is the ith impedance measurement varied with the cycles of charging and discharging and Ro is the initial impedance. On the other hand, the battery capacity C can also be used to determine the battery SOH as given in Eq. (2).
predictions, the Gaussian process model can provide variance
SOH
around its mean predictions to describe associated uncertainty in the
evaluation
and
prediction.
To
evaluate
the
proposed
prediction approach, we have executed experiments with lithium ion battery. Experimental results prove its effectiveness and confirm the algorithm can be effectively applied to the battery monitoring and prognostics.
Furthermore, the comparison of
prediction with different amounts of training data has been achieved, and the dynamic model is introduced to improve the prediction for the battery health.
Keywords-Uncertainty; Lithium-ion Battery; Prognostcis
and
Health Management; State of Health; Gaussian Process Regression; Capacity Prediction; Dynamic Model.
I.
INTRODUCTION
The lithium-ion battery is a very important constituent for lots of machines and is critical to the performance of the specific system. Its failure can lead to the performance degradation, operation of damage, and even cause catastrophic accidents. So we must pay more attention to the health monitoring and prognostics for lithium-ion battery, and adopt efforts to monitor battery condition in an effective way such as controlling the operating conditions, replacement interval for the battery aiming to enhance the system reliability and stability [I].The battery on-line operation is a dynamic process, and its performance is strongly influenced by ambient environmental and load conditions [2].State of Health (SOH) estimation is a very important part in the battery prognostics used as qualitative measure for the battery to store and deliver energy in the system. The prognostics of the SOH can be used 978·14577-1911-0/12/$26.00 ©2012 IEEE
Where and
Co
C;
= .s.xlOO% Co
(2)
is the ith capacitance value degenerated with cycles
is the initial capacity. There are several attempts to
estimate the battery SOH using the battery impedance or the battery capacity. The researchers propose a power Li-ion battery SOH prediction concept based on an appropriate SOH definition [4]. Kim developed a new technique to estimate SOH through a dual-sliding-mode observer [5]. The Gaussian process model is a flexible, probabilistic and nonparametric model with uncertainty expression. Furthermore, The GPR can model the behavior of any system through the combination of appropriate Gaussian process and realize prognostics combined with prior knowledge in a flexible and convenient mode based on Bayesian framework [6]. It has become an important part of the algorithms for Prognostics and Health Management (PHM) in the battery system. The choice of covariance functions is very important for different applications. However, there are no clear standards for the selection. In the paper [7], researchers regressed for battery's internal parameters with time and transferred the predicted values to the capacity domain to express capacity decay with time. The results are acceptable, but in the real situation, there are self-recharge phenomena during the usage of lithium-ion battery, the prediction trend should response to this point. In addition, the prediction would obtain less accurate and precise result while less data is
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There are some unknown parameters in the covariance
available. In this paper, we suggest solutions for these two issues.
The paper realizes prognostics for battery health monitoring
based on Gaussian Process Regression (GPR). We deal with
function. e = [O"j'o"n,l,,/ ,vO 2
of three functions which can be used to show the self-recharge
phenomenon
well.
Moreover, we
apply
dynamic
model
compared with the static model to make prediction upon the
training data.
This paper is organized as follows. Firstly, in section II, the
data prediction method with Gaussian Process Model is
introduced. The prognostics for lithium-ion battery based on
GPR are described in details in section III. The conclusions
number
m(x)
and
of
Given a set of training points
variables
defmed in Eq.
function
Gaussian noise E:
which
obey
data
vo) .
points
[9].
The
following
[8].
k(x,x')
Gp[m(x),k(x,x')] .
k
Which
;
vector in the application of prognostics, and
=
(4)
,
is the test
where
Set initial values
Training data input
� H
Training the hyperparameters
�J.
Here we just briefly analyze and list the covariance
Prediction output
functions which we applied in battery health prognostics. The
diagonal squared exponential covariance function:
Figure I.
)
(5)
The periodic covariance function: kj
=
O"� exp( -
- x'))) � sin2( �(x 1 2Jr
of
(6)
2
(7)
The constant covariance function is used as the noise part
that generally considered being a white Gaussian noise.
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(8)
(9)
tA�\r-
D:,'namic model Added new data
I
!I
Update Gaussian process model
L __________
the prediction framework based on GPR which include the static and dynamic model
The most important composition represented the properties Gaussian
process
is
covariance
function, through
its
function values to define the relationship between the different
input points in the application of prediction. Moreover, the forms of covariance function also combine the unknown
The constant covariance function:
k(x;,x) =vo
, I
-l�
discussed in [6].
2/ 2
JJ
p----------
kf
describe noise of the model. The possible selections for kfare
(X - X')2
x')
�l.
the unknown system model, and k" represents the noise part to
0"y2 exp( -
"
Detennine the amounts of hyper-parameters
represents the functional part which would be used to describe
=
k(x
Jl
be interpreted as the
k(x;,x)=k/x;,x)+kn(XI'X)
k(X'X')
Selection of covariance function
The index set
x'
�;
=
and xj• In the system models, it is usually composed of two
kj
the
E[f' I x,y,x'] k(x,x')[k(x,x)+O"�It y, cov(f') = k(x',x')-[k(x',x)+O"�It k(x,x').
I'
measurement of the distance between the input points Xl
parts
describe
1'l x,y,x'�N (f',cov(f'))
are
is the set of input points, which do not need to be a time
main
equations
Posterior:
(3)
k(x;,x) can
[8]:
(�.J (0.( � :��
vector defmitely. However, this set of points should be a time inputs. Covariance function
with the white
Once a posterior distribution is
predictive distribution for GPR
a joint
k(x,x') = E[(f(x)-m(x))· (f(x')-m(x'))] x
�N
y = I(x) + E:
derived, it can be used to estimate predictive values for the test
-
m(x) = E (f(x))
�
(0,
given the effect of noise, namely:
Where
I(x)
we can
Prior:
(3), (4).
And we describe
{(x,y) Ii = l, ... ,n},
applied in general regression problem, we describe the target y
Gaussian distribution that can be fully described by its mean co-variance
hyper
restriction on prior joint distribution. When Gaussian process
GAUSSIAN PROCESS MODEL
random
the
derive the posterior distribution over functions by imposing a
The Gaussian Process (GP) is defmed as a collection of
finite
as
the log-likelihood function.
and future work are discussed in section IV. II.
described
parameters needing to be optimized with the maximization of
the issue of covariance function selection for the training data.
The final choice for the covariance function is the combination
are
{
system noise. Gaussian process requires some prior knowledge
about the form of covariance function, and which should vary
with the different prediction problems. In order to increase the
flexibility of the Gaussian process model, there are some
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2012 Prognostics & System Health Management Conference (PHM-2012 Beijing)
hyper-parameters introduced to define the properties of the covariance function. Although we can select different covariance function for training through a large number of experiments, and realize optimization of parameters using a conjugate gradient method based on optimizer such as maximizing the marginal likelihood of the observed data, select the most suitable covariance function and set initial parameters properly are still challenges in the application of Gaussian process.
rest or relaxation period. For any lithium-ion battery, reaction products build up around the electrodes and slow down the reaction. By letting the battery rest, the reaction products have a chance to dissipate, thus increasing the available capacity for the next cycle [12]. This phenomenon also emerges in other cycles which should be reflected through the prediction models.
The prediction results based on GPR provide variance around its mean value to describe the uncertainty of the model. This paper makes estimation about the lithium-ion capacity directly to describe the state of health for the battery. According to the properties of Gaussian process, the covariance function as a crucial ingredient in a Gaussian Process predictor can affect the results to a great extent. So we should select the covariance function which can fit the training data better, moreover, the dynamic model compared with the static model should be considered to make prediction upon the training data. The framework of the dynamic prediction model including the static model is shown in figure 1. III. A.
PROGNOSTICS FOR
the CUffent trend duting discharging
O.5 ,--���-��--'--r---=--:--��----,
-0.5
I, -1.5 ., , '------' -25 0
XI
40
60
EO
100
120
140
tED
100
200
Sampling point
(2a) the current of discharging the current trend during ch�rging
LITHIUM-ION BATTERY WITH GPR
Raw data ofLithium-ion battery
The battery data used to make prognostics based on the proposed approach is obtained from data repository of NASA Ames Prognostics Center of Excellence (PCoE). This data set has been sampled from a battery prognostics test bed in NASA which comprises of commercially available Li-ion 1850 sized rechargeable batteries, power supply and programmable DC electronic load, voltmeter, thermocouple sensor, environmental chamber, electro-chemical impedance spectrometry (EIS), PXI chassis based on DAQ and experiment control condition [10] [11]. The Li-ion batteries were run through 3 different operational profiles (charge, discharge and impedance) at room temperature. Charging was carried out in a constant current (CC) mode at I.5A until the battery voltage reached 4.2V and then continued in a constant voltage (CV) mode until the charge current dropped to 20rnA. Discharge was carried out at a constant current (CC) level of 2A until the battery voltage fell to 2.7V, 2.5V, 2.2V and 2.5V respectively for batteries No.5, No.6, NO.7 and No.I8. Impedance measurement was carried out through an electrochemical impedance spectroscopy (EIS) frequency sweep from O.IHz to 5kHz. Repeated charging and discharging cycles result in accelerated aging of the batteries. The experiments were stopped when the batteries reached end-of-Iife (EOL) criteria, which was a 30% fade in rated capacity (from 2Ahr to IAAhr) [II].Here the data of No.5 was used to make experiments including training and testing. The details about the trend of current and voltage during charging and discharging cycles about battery No.5 are shown in Figure 2 and Figure 3. The test of SOH produced from battery No.5 is shown in Figure 4. Figure 4 shows that the SOH increased higher at the cycle 90 than previous cycle due to significant self-recharge during
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., ·3
Sampling point
(2b) the current of charging Figure 2.
the current during discharging and charging the yolt�ge trend during discharging
4.2 ,---__ �----'r-,---_-'-;-'__ ____, -
38
36
3.2
'.8
26 0
20
40
60
00
100
120
140
160
100
200
S�mpling pOint
(3a) the voltage of discharging the voltage trend during charging
4.5,-_�_-_____-_-,
r
3.5
Sampling point
(3b) the voltage of charging Figure 3.
the voltage during discharging and charging
2012 Prognostics & System Health Management Conference (PHM-2012 Beijing)
Estimated SOH of battery no.S
GPR prediction
90 ,----�-��-=-��� ;:::::====:::;c=J95% confidence bounds ---+- Prediction
based on GPR
-Ihe aclural
SOH
80 75 70 65 650
20
40
Figure 4.
60
00
100
Number of cycle
120
140
160
180 100
the SOH of battery for No.5 Figure 7.
B.
Experiments and Analysis
The data contains aging infonnation about battery SOH values are from cycle 0 to cycle 168. From the Figure 4, we can see that the estimated SOH of battery No.5 that degrades exponentially with number of cycles similarly, we select different covariance functions including single and combination forms to train the input data because the battery has significant self-recharge phenomenon. As a result, we adopt the combination of the periodic covariance function and diagonal squared exponential covariance function as our covariance function to describe the unknown system model with a white Gaussian noise to describe noise of the model. The results of battery health prognostics based on the GPR algorithm are showed as Figure 5 and Figure 6. The prognostics results also include 95% confidence bounds.
110
1�
Number of cycle
130
140
1�
1m
battery health prognostics based on dynamic data model
From Fig.5 and Fig.6, it may be observed that the prediction at cycle 80 failed to represent the actual trend compared with the prediction at cycle 100. In Fig.6, there are several points of prediction dropped out from accuracy bound. This is known as over-fitting in learning algorithm which has poor predictive perfonnance [12], as it exa�gerate �in�r fluctuations. So it's more accurate when more mformatlOn IS referred. In this case, we can use the dynamic data model to improve the prediction because in real applications some new data would produce during the battery aging. So we implement the SOH prediction at cycle 80 for twenty steps, then we add the twenty updated data to the training model to continue the prediction. The rest can be done in the same manner, we achieve prognostics of SOH to cycle 160, and the results are shown in Figure 7.
GPR prediction
====:::; :: 85r;:;--�-�-��---;::;=;::: c=] 95 %confidence bounds -+- Prediction
-Ihe actural
based on GPR
SOH
5 £ 7
� 'E
a
70 65
�00-L-0-�110��,���,���-7.14�0-�,��-�,m��17 Number of cycle
Figure 5.
battery health prognostics started from 100 cycle with 95% confidence bounds
OO
GPR prediction
�-r.c=J� �%�%= �fi d='",= '� d '-�--��-O CO b," �" �$
--+- Prediction based on GPR
-the actural SOH
� 75 b 70 " 0;65
The Fig.7 shows that the prediction results are better than the static model, though we find two points dropping out from accuracy bound. The predicted health of battery coincides with the actual SOH. In real situation, we can use the dynamic data model to make prediction. The specific procedures are as follows: We make prediction for a few steps firstly, then adding the new real data points to update the model for making new prediction. The prediction includes 95% confidence bounds which consider the effect of noise. It's significant in real conditions, and the accuracy of prediction is measured by two performance matrices as presented in the Table 1. From the above experiments we can see the prediction for the lithium-ion battery health based on GPR can provide a satisfied result. Moreover, the periodic covariance function can express the self-recharge phenomenon of lithium b�ttery so that the prediction can represent the actual trend precisely. With more information and test data used for learning and training of the model, the prediction result is more accurate than the one with less information and data. The quantitative analysis of the experimental result is showed as follows. TABLE!.
THE COMPARISON OF THE THREE GPR MODELS
m
5080
SU
100
110
120
130
Number of cycle
Figure 6.
140
15 0
160
170
battery health prognostics started from 80 cycle with 95% confidence bounds
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Dynamic prediction (at 80 cycle)
Predicted at 100 cycle
Predicted at 80 cycle
RMRSE
0.02
0.07
0.02
RMSE
1.408
4.6
1.41
Error criteria
55
2012 Prognostics & System Health Management Conference (PHM-2012 Beijing)
When there are less data at hand, we can use the dynamic data model to update the training inputs while the new data are available on-line. IV.
B. Saha, K. Goebel, S. Poll, and 1. Christophersen, "An integrated approach to battery health monitoring using Bayesian regression and state estimation," in Proc. of IEEE AUTOTESTCON, 2007, pp. 646-653.
[3]
D. Le and X. D. Tang, "Lithium-ion Battery State of Health Estimation Using Ah-V Characterization," Annual Conference of the Prognostics and Health Management Society, 2011, pp. 367-373.
[4]
H. F. Oai, X. Z. Wei, and Z. C. Sun , "A new SOH prediction concept for the power lithium-ion battery used on HEVs," Vehicle Power and Propulsion Conference, 2009, pp. 1649-1653.
[5]
T. S. Kim, "A technique for estimating the state of health of lithium batteries through a dual-sliding-mode observer," IEEE Trans. Power Electron., vol. 25, issue. 4, 2010, pp. 1013-1022.
[6]
C. E. Rasmussen and C. K. 1. Williams, "Gaussian Processes for Machine Learning," The MIT Press Cambridge MA, 2006.
[7]
K. Goebel, B. Saha, A. Saxena, J. Celaya, and J. Christophersen, "Prognostics in Battery Health Management," In IEEE Instrumentation and Measurement Magazine, vol. 11, issue. 4, 2008, pp. 33-40.
[S]
C. K. I. Williams and C. E. Rasmussen, "Gaussian Processes for Regression", D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo (eds.), Advances in Neural Information Processing Systems, vol. 8, pp. 514-520.
[9]
S. Saha, B. Saha, A. Saxena, and K. Goebel, "Distributed Prognostic Health Management with Gaussian Process Regression," IEEE Aerospace Conference, 20I0, pp. 1-8.
CONCLUSION
In this paper, we realize the data-driven battery SOH estimation approach based on GPR algorithm. Aiming to track the self-recharge phenomenon in the lithium-ion charging and discharging, we propose a new combination of covariance function for GPR to make prognostics about battery health. The experimental results demonstrate that the SOH estimation and prediction could obtain satisfied precision. In the training process, GPR will not provide better results with less learning data. For this case, we present the dynamic data model to update the training inputs while the new data are available on line to achieve more accurate prediction results. Future work involves implementation of other types of batteries for SOH estimation with GPR. Moreover, we estimate the hyper-parameters by conjugate gradient method which is of local optimal phenomenon indicating that more efforts could be concentrated to improve this case. REFERENCES [II
[2]
J. L. Zhang and 1. Lee, "A review on prognostics and health monitoring of Li-ion battery," Journal of Power Sources, vol. 196, issue. 15, 2011, pp. 6007-6014.
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[10] B. Saha and K. Goebel, "Uncertainty management for diagnostics and prognostics of batteries using Bayesian techniques," In Proceedings of the IEEE aerospace conference, Big Sky, MT, 200S, pp. I-S. [II] B. Saha and K. Goebel, "Battery Data Set," NASA Ames Prognostics [http://ti.arc.nasa.gov/project/prognostic-dataRepository, Data repository], NASA Ames, Moffett Field, CA, 2007. [12] A. Widodo, M. C. Shim, W. Caesarendra, and B. S. Yang, "Intelligent prognostics for battery health monitoring based on sample entropy," vol. 38, issue. 9, 2011, pp. 11763-11769.
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