Hindawi Publishing Corporation
Advances in Mechanical Engineering
Volume2013,Article ID154831,7pages
dx.doi/10.1155/2013/154831
Rearch Article
A Method of Remaining Capacity Estimation for
Lithium-Ion Battery
Junfu Li,Lixin Wang,Chao Lyu,Weilin Luo,Kehua Ma,and Liqiang Zhang
School of Electrical Engineering and Automation,Harbin Institute of Technology,Harbin150001,China
Correspondence should be addresd to Lixin Wang;wlx@hit.edu
Received8September2013;Revid22October2013;Accepted22October2013
Academic Editor:Xiaosong Hu
Copyright©2013Junfu Li et al.This is an open access article distributed under the Creative Commons Attribution Licen,which permits unrestricted u,distribution,and reproduction in any medium,provided the original work is properly cited.
Combining particle filter(PF)with sample entropy feature of discharge voltage,a method of remaining capacity estimation for lithium-ion battery is propod.The sample entropy calculated from discharge voltage curve can rve as an indicator for asssing the condition of battery.Under a certain working condition,a functional relationship between sample entropy and discharge capacity is created and estimations computed from the function are taken as obrvations to propagate particles in PF.The results indicate that the algorithm enhances the accuracy.Due to the establishment of functions at different discharge rates and temperature modification,prognostic accuracy of discharge capacity has been improved under multi-operating working conditions.
银行理财产品排行1.Introduction
With the rapid development of industrial technology,the exploration and utilization of new energy have been in urgent need.Electric vehicle occupies a pivotal position in new energy automobile.Batte
ry management system(BMS)is specially designed to improve efficient utilization,to prevent overcharge or overdischarge,to prolong the rvice life,and to monitor the state of the battery.A more sophisticated prognostic of battery health state is much needed for high requirements of reliability,stability,and curity of batteries. Conquently,the prediction of remaining battery life is considered as one of the promising rearch fields.Numerous papers have reported the studies on state of charge(SOC) and state of health(SOH)which are the focus of battery Prognostic and Health Management(PHM).
Battery discharge capacity reaching its criteria without any omen leads to a disastrous failure in some cas.The accurate prediction of remaining uful life(RUL)of battery is esntial for long-time efficient u.The caus of capacity fading are internal factors such as anodic and cathodic active material changes and SEI membrane incrassation [1,2].Accurate battery SOC estimation is of great signi-ficance to battery electric vehicles and hybrid electric vehi-cles.SOC estimation aims at the management of energy flows of electric vehicles and avoiding battery overcharge or undercharge.Lee et al.[3]propod an Extended Kalman Filter(EKF)method along with a measurement noi model and data rejection of lithium-ion battery SOC estimation. The propod algorithm and model approach were verified through veral experiments.An adaptive unscented Kal
man filtering method to estimate SOC of lithium-ion battery was prented[4].The propod SOC estimation method had a better accuracy compared with previous works.Lee et al.[5] estimated the SOC and the capacity of a lithium-ion battery with a modified OCV-SOC model.The method overcame the variation in conventional OCV-SOC.
Methods of battery capacity estimation are propod bad on the following two ideas.One method is feature-bad.In one n,as variations of voltage,current,and tem-perature characteristic curves could reflect the battery aging process or internal resistance variations,some characters are often extracted from them.Salkind et al.[6]propod a practical method that resistances obtained by electrochemical impedance spectroscopy(EIS)measurement and coulomb counting techniques were employed in predicting SOC and SOH.The advantage of the work was that there was no need to know previous discharge or cycling history.Gomez et al.
[7]made a detailed analysis on EIS and pointed out that aging information could be extracted from the parameters of EIS equivalent circuit model.Pincus[8]firstly introduced the concept of approximate entropy mainly to compute the
complexity of time ries.Widodo et al.[9]took sample entropy features obtained from discharge volta
ge curves as inputs of support vector machine(SVM)and relevance vector machine(RVM)for SOH prediction.The results showed that the method propod was plausible.
The other is model-bad.Generally,fault feature is cloly related to the parameters of the model.Correction and adjustment of model parameters can enhance the pre-diction accuracy.The model-bad techniques contribute to an in-depth understanding of the mechanism and have the advantage of real-time fault prediction.A model of battery system state is established to describe the discharge behavior or battery health state.Abbas et al.[10]introduced an integrated methodology bad on both physics of failure models and Bayesian estimation methods for prognosis of electrical components.An empirical formula was propod to depict discharging behavior of lithium-ion batteries[11–13].Simulation results indicated that PF algorithm was appropriate for the prediction of battery health state.Saha et al.[14]prented veral algorithms including ARIMA, RVM,EKF,and PF.A RVM-PF framework had significant advantages over the conventional methods of RUL estimation like ARIMA and EKF.
Some rearchers have also established electrochemical numerical model and thermal model for the study on battery internal characteristics.Porous electrode model with liquid electrolyte was propod by West et al.[15].That electrolyte depletion was the primary limiting factor of capacity was
demonstrated.Park et al.[16]prented an electrochemical heat conduction phenomenal model.A better understanding of conduction phenomena of lithium-ion batteries was pre-nted.Kim et al.[17]extended one-dimensional modeling approach to three dimensions to capture geometrical features such as shapes and dimensions of cell components,to simulate oven tests and to determine how a local hot spot can propagate through the cell.Though some key behaviors of battery cells can be captured in the models,it is complex to deploy a large number of unknown parameters due to the memory and computation.Lumped battery models are likely to be the preferred choice with a relatively fewer parameters.A systematic comparative study of twelve lumped battery models was conducted[18].The developed cell voltage models could be ud in SOC estimation in BMS.
This work is conducted by the combination of the two ideas mentioned above.In the following ction,we firstly introduce the theory about sample entropy and basic uti-lization of particle filter in terms of prognostics of lithium-ion battery RUL.Then,we prent the detailed prediction procedure.
2.Theory and Intelligent Prognostic Method 2.1.Sample Entropy.Sample entropy is defined as generation rate of new information by Richman and Moorman[19] for the calculation of complexity of time ries.It can be expresd as SampEn(m,r,N),where N is a given total number of data,r is the tolerance for accepting matrices, and m is the dimension of vectors.The specific algorithm of sample
entropy is as follows.For a given ries{x i},we form N−m+1vectors as
X(i)=[x(i),x(i+1),...,x(i+m−1)],
for i=1to N−m+1.
(1)
The distance between vectors X(i)and X(j)is defined as d[X(i),X(j)]=max x(i+k)−x(j+k) ,
for i,j=1to N−m+1,
k=0to m−1.
(2)
For a given r,calculate the number when d[X(i),X(j)]< r,for i=j,and define the function
B m i(r)=
1
N−m num{d[X(i),X(j)]<r}.(3) Then,take the average of B m i(r).The result is expresd as
B m(r)=
1小汽车游戏
N−m+1
N−m+1
∑
i=1
B m i(r).(4)
Similarly,replace m with m+1and repeat the steps from the beginning.Afterwards,we can determine the two values B m(r)and B m+1(r).As the sample length is always limited,the sample entropy is estimated by
SampEn(m,r,N)=−ln[
B m+1(r)
m
].(5)
The value of SampEn(m,r,N)is cloly correlated with m,r,and N.Thus,the proper lected parameters could result in more reasonable statistical properties.
2.2.Particle Filter.PF is a Bayesian learning technique using Monte Carlo simulations.The idea is to describe the system state as a probability density function(PDF)approximated by particles that are generated from a priori distribution and updated from obrvations through a measurement model.Model parameters are included as a part of the state vector to be tracked[11].PF framework can be applied to RUL prediction of battery due to its good state tracking performance.
Actual discharge capacity is associated with many factors. It is obvious that charging directly determines the discharge capacity in one cycle.Besides,reaction products forming up around the electrodes will decompo during rest or relaxation period,which lead to the increa of available capacity in next cycle.Primarily,considering the main influ-ence factors of battery capacity,the following state equations are cast to describe the model as follows:
C k+1=β1C k+β2exp(
β3
k
),(6)βi(k+1)=βi(k)+V i(k),i=1,2,3,(7)
where k is cycle index,C k denotes the charge capacity,ΔT k is the relaxation period between the two adjacent cycles,C k+1 is the discharge capacity,β1,β2,andβ3are parameters of the state equation,and V1,V2,and V3are independent zero-mean Gaussian noi terms.
Saha and Goebel[11]established a measurement model and regarded charging capacity as the obrvation to prop-agate particles.A reasonable obrvation for measuring the weights of particles and lectively propagating them plays an important role in prediction accuracy.In the ca of our application,via the fitting method,a functional relationship of sample entropy and discharge capacity is established to obtain an appropriate obrvation.Particularly,sample entropy is calculated from the discharge voltage curve of the cycle number k.The corresponding output of the function is ud as the obrvation in cycle k+1.It is worth mentioning that there is no need to take other experiments to
obtain such features,for the discharge voltage curves can be easily obtained during the monitoring in each cycle.
2.3.Intelligent Prognostic Method.The procedure compris the following.
(1)Data collection is as follows.
(a)Extract battery discharge voltage curves from
training data and the lected parameters m
and r are2and0.1,respectively.The functional
relationship of discharge capacity and sample
entropy is created under the current operating
condition.
(b)Gain discharge current curves,charging capac-
ity,and relaxation time of adjacent discharge
cycles from validation test data.In addition,
some historical capacity data are also needed.
(2)Particle filter initialization is as follows.
(a)Set the starting prediction point T in proportion
to the number of historical capacity data.
(b)Obtain initial parametersβi(i=1,2,3)via
fitting.
(c)500initial particles are generated with values
obtained in(2)-(b)and the variances of noi
term V i(i=1,2,3)are about10,000times
smaller thanβ.
(3)Prediction is as follows.
(a)Particles{x i k}N i=1are updated by(7)and the
priori discharge capacity values in cycle k+1
are calculated through tho updated particles
{x i k+1}N i=1.
(b)Take sample entropy feature as the input of the
function and compute the weight of each parti-
cle per deviation between the calculated obr-
vation and previous discharge voltage value.
Normalize the very particles using the following
formula:
w k+1(x i k+1)=
w k+1(x i k+1)
∑N i=1w k+1(x i k+1)
.(8)
(c)Through the method of random sampling,each
particle{x i k+1}N i=1is copied or abandoned lec-
tively according to its weight and then new
sample{̃x i k+1}N i=1is obtained.
(d)The average of the sample{̃x i k+1}N i=1reprents
the probability density distribution expectation
of each parameter in(6).Then,the final estima-
tion C k+1can be easily figured up by(6).
(e)Repeat the step from(3)-(a)to(3)-(d)until the西湖大学施一公
capacity reaches its criterion which is a30%
fading of rated capacity.
3.Experiment Data
The full t of aging data collected from commercially available18650-size lithium-ion cells provided by NASA Ames Prognostics Center of Excellence was taken as object of study.Battery anode and cathode materials are mostly LiNi0.8Co0.15Al0.05O2and MAG-10graphite,respectively.The electrolyte is1.2M LiPF6in EC:EMC(3:7wt%)and the parator is25μm thick PE.
All testing batteries were run through different working profiles(charge,discharge,and impedance).Batteries No.6 and No.18were tested by the following steps:(1)charging was carried out in a constant current mode at1.5A until the battery voltage reached4.2V,(2)a constant voltage mode was then in operation until the charge current dropped to20mA,(3)batteries were put aside for a period of time,(4)impedance measurement was implemented with an electrochemical impedance sp
黎苗ectroscopy frequency sweep from0.1Hz to5kHz,(5)at24∘C,discharging was carried out at a constant current level of2A until the battery voltage fell to2.5V,(6)the same step as(3),and(7)the same step as(4).Repeated charging and discharging resulted in an accelerated aging process.The experiments were stopped when the batteries reached the end-of-life criteria which was a30%fading in rated capacity(from2Ahr to1.4Ahr).
4.Results and Discussion
4.1.Single Working Condition.Figure1depicts the discharge voltage curves in different cycles.At a constant current of 2A,the voltage drops from4.2V to2.6V.Obviously,the curves vary from cycle to cycle in the aging process.It can be en from Figure1that the lowest voltage point bounces back instantly at the end of discharge and subquently ris slowly until it comes to a stop.The two arrows point out the process mentioned above.Obrving the definition of sample entropy,we can find that when the maximum distance computed from the adjacent vectors constituted by the quential samples is greater than r,the complexity number of the corresponding vector in(3)will not change
V o l t a g e (V )
4.2
3.83.432.6First cycle Second cycle Third cycle Time
Fourth cycle
Figure 1:Battery voltage curves in different cycles and the two voltage variation process were pointed out by the
arrows.
2.1
1.91.71.51.3Actual discharge capacity
去的反义词
Estimated value with obrvation obtained from sample entropy
C a p a c i t y (A h r )
Cycle (—)
Figure 2:Prediction of battery No.6.
E r r o r (%)
0.08
0.060.040.02020
40
60
80100
120
140
Cycle (—)
Figure 3:Relative
errors.
2.1
1.91.71.51.3
1.1
Actual discharge capacity
Estimated value with obrvation obtained from sample entropy C a p a c i t y (A h r )
Cycle (—)
Estimated value with obrvation obtained from charging capacity
Figure 4:Comparative simulation results through different meth-ods.
in statistical calculations.Otherwi,if the noi signal is added to the samples with larger amplitude,it will be ignored by detection,for the distance between the disturbed vectors is longer than others.In that n,sample entropy could capture the features of voltage variance in a constant current mode.
As battery is aging gradually during the usage period,we find an interesting connection between the sample entropy feature and the discharge capacity.In conquence,sample entropy could rve as an indicator for asssing the condition of battery.With training data of battery No.18,a cubic polynomial fitting is introduced to find out the functional relationship between them.When the parameter m and r are deployed to 2and 0.1,respectively,a better fitting effect is obtained with a reasonable statistical result.
The starting point T and predicting length are 25and 115.Figures 2and 3show the prediction result of battery No.6and its errors.From the actual discharge capacity curve,it is evident that battery No.6has faded to its limit 1.405Ahr when it cycles at cycle number 108.Obrving Figure 3,apart from veral points,most relative errors are within 5%.The early prediction has higher precision and errors of some rebound points are less than 2%.
To illustrate the superiority of this work compared with Saha and Goebel [11],Figure 4shows the comparative pre-diction result.
As is showed in Figure 4,some key points of prediction are pointed out by ven arrows on the graph and the contrastive prediction apparently engenders a greater error.Prediction accuracy is measured by the root-mean-squared (RMS)error and peak error.The statistical figures reveal that RMS errors of both predictions are 8.64%and 4.30%,respectively,and the peak errors are 37.86%and 8.28%.
The discharge capacity is not only directly related to charge capacity and rest time of adjacent cycles but is also affected by actual working conditions.When the forecasting and training conditions,such that ambient temperature and discharge rate are inconsistent,it can be easily expected that the estimation points will deviate from the actual ones in each cycle.
4.2.Multioperating Working Condition.Without knowing of aging mechanism,it is hard to make a specific illustration that how the aging process inside the battery is influenced by environmental factors.But,it is certain that as battery aging process,different operational conditions accounts for the discharge capacity fading behaviors.It is required to update or revi the aforementioned function properly to satisfy the requirement of high accuracy when facing a multioperating working condition.The datats provided by NASA only include veral discharge rates.Thus,the paper builds three functions taking different C -rates under each ambient temperature into account summarized in Table 1,where x is sample entropy and F is the estimation capacity ud as obrvation in algorithm PF in our method.
Suppo that the operating ambient temperature is 24∘C.It is interesting to find that the relative mean deviations between estimation values and discharge capacities at actual
Table 1:Capacity estimation functions under different optional conditions.
Discharge rate Ambient temperature
Capacity estimation function
0.5C 4∘C F =(9.6169x 3−0.4326x 2+0.0035x +0.0001)×1041C 24∘C F =(−1.1240x 3+0.0154x 2−0.0166x +0.0018)×1032C
24∘C F =(−7.2590x 3+0.3225x 2−0.0044x +0.00003)×105
O ff s e t (A h r )
0.1−0.1−0.3−0.5
510
15202530354045
Temperature (deg)
罗曼式0−0.4−0.200.2Figure 5:Offts at different temperature.
Actual Estimated
C a p a c i t y (A h r )
1.41.31.21.110.90.80.7
0102030
4050607080
Cycle (—)
Figure 6:Prediction of battery No.55.
Actual Estimated
C a p a c i t y (A h r )
1.821.781.741.71.66
5
10
15
202530
35
40
Cycle (—)
Figure 7:Prediction of battery No.31.
C a p a c i t y (A h r )
1.61.2
0.400
5
15
2025303540
Actual Estimated
0.8
10
Cycle (—)
Figure 8:Prediction of battery No.39.
Table 2:RMS errors and peak errors of battery No.55.Starting point T RMS error (%)
Peak error (%)
从性遗传10 3.2613.2215 2.647.0720 2.30 6.7525
寿比南山的意思2.24
5.63
Table 3:RMS errors and peak errors of battery No.31.Starting point T RMS error (%)
Peak error (%)
10 2.17 5.2712 1.64 4.2215
1.37
3.12
temperature 4∘C and 43∘C are around −0.38and 0.02.As a matter of fact,higher or lower temperature affects the actual discharge capacity.On account of the higher ambi-ent temperature,the internal substances are more active resulting in a larger discharge capacity.On the contrary,the lower temperatures slow down the physicochemical reactions inside the battery leading to the fact that the actual capacity cannot reach the maximum.In a constant discharge current mode,it is reasonable and esntial to modify the capacity obrvations in PF algorithm.Thus,according to the previous calculations,a functional relationship between ambient temperatures and estimation offts is established through quadratic curve fitting.The fitting result is given in Figure 5.
The lected offt benchmark is zero at 24∘C.Figures 6and 7show the prediction results of battery No.55(4∘C,1C )and No.31(43∘C,2C ).Both two offts are parately −0.38and 0.02.As is expected,the prediction curves are basically consistent with the actual ones.
Tables 2and 3show the RMS errors and peak errors at different prediction starting points.The results indicate that as the number of historical capacity data is increasing,errors have the downward trends.
Battery No.39is tested under a multioperating working condition.The first veral discharge cycles are tested at 24∘C,2C and the others at 44∘C,0.5C .The corresponding capacity estimation function should be lected in accordance with the operating condition.As one of the relevant functions is built at 4∘C,0.5C ,the actual offt at 44∘C should be incread to 0.398rather than 0.018in Figure 5.The prediction result of battery No.39is prented in Figure 8and the RMS error is 5.78%.
Figure 9shows the contrastive prediction result.Without the consideration of C -rate and ambient temperature,the estimation performs much wor with 27.56%RMS error.
