Research Article Prognostics of Lithium-Ion Batteries ...

Research ArticlePrognostics of Lithium-Ion Batteries Based onWavelet Denoising and DE-RVM

Chaolong Zhang12 Yigang He1 Lifeng Yuan1 Sheng Xiang1 and Jinping Wang1

1School of Electrical Engineering and Automation Hefei University of Technology Hefei 230009 China2School of Physics and Electronic Engineering Anqing Normal University Anqing 246011 China

Correspondence should be addressed to Chaolong Zhang zhangchaolong126com

Received 8 December 2014 Accepted 10 May 2015

Academic Editor Rafik Aliyev

Copyright copy 2015 Chaolong Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Lithium-ion batteries are widely used inmany electronic systemsTherefore it is significantly important to estimate the lithium-ionbatteryrsquos remaining useful life (RUL) yet very difficult One important reason is that the measured battery capacity data are oftensubject to the different levels of noise pollution In this paper a novel battery capacity prognostics approach is presented to estimatethe RUL of lithium-ion batteries Wavelet denoising is performed with different thresholds in order to weaken the strong noise andremove the weak noise Relevance vector machine (RVM) improved by differential evolution (DE) algorithm is utilized to estimatethe battery RUL based on the denoised data An experiment including battery 5 capacity prognostics case and battery 18 capacityprognostics case is conducted and validated that the proposed approach can predict the trend of battery capacity trajectory closelyand estimate the battery RUL accurately

1 Introduction

Lithium-ion batteries have been widely used as crucial com-ponents and important backup elements for many sys-tems including electric vehicles consumer electronics andaerospace electronics Comparedwith other kind of batteriesthe lithium-ion battery has advantages of high power densityhigh galvanic potential light weight and long cycle lifeHowever irreversible chemical and physical changes takeplace in the lithium-ion battery with usage and aging Asa result the battery health degrades gradually until it isno longer usable eventually The consequence of the batteryfailure would lead to the capacity degradation operation lossdowntime and even catastrophic failure Hence prognosticsand health management (PHM) of the lithium-ion batteryhas been an active field which has attracted an increasingattention today [1ndash12]

PHM is an enabling discipline composed of technologiesand approaches to estimate the reliability of an applicationsystem in its actual life cycle conditions to provide ampleforewarning before a failure occurs andmitigates system risk

PHMof the lithium-ion battery includes evaluating its state ofhealth (SOH) and predicting its remaining useful life (RUL)Meanwhile the gradual decreased capacity of the battery isa universally used SOH indicator that can track its healthdegradation

Model-based and data-driven approaches are two mainkinds of approaches to the battery capacity prognosticsModel-based approaches employ mathematical representa-tions to character the understanding of the battery failureand underlying the battery capacityrsquos degradation modelExtended Kalman filtering (EKF) [1 2] nonlinear model[3 4] and particle filtering (PF) [5 6] are commonly usedmodel-based methods for the battery capacity estimationHowever an accurately analytical and universally acceptedmodel to track the battery capacity degradation and evaluatethe battery RUL is usually difficult to be derived becauseof the complex electronic system noise data availabilityuncertain environments and application constraints Data-driven approaches utilize statistical and machine learningtechniques to evaluate the battery capacity and predict thebattery RUL The approaches avoid constructing complex

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 918305 8 pageshttpdxdoiorg1011552015918305

2 Computational Intelligence and Neuroscience

physical models and have been applied in many relativeworks [7ndash12] Artificial neural network is a widely useddata-driven approach to the battery capacity prognostics[7 8] However it has disadvantages of poor generalizationdifficult structure confirmation and low convergence rateSupport vector machine (SVM) is a machine learning tool[13] characterized by the usage of kernel functions and it hasbeen utilized to estimate the battery RUL [9 10] Relevancevector machine (RVM) is a Bayesian sparse kernel technique[14] with usage of much fewer kernel functions and higherperformance compared to the SVM Meanwhile RVM hasbeen applied to the research field [11 12]

The measured battery capacity data are often subject tothe different levels of noise pollution because of the impactof disturbances measurement errors stochastic load andother unknown behaviors in batteries Capacity prognosticsbased on the noisy data cannot produce accurate predictresults Therefore it is significantly important to preprocessthe measured capacity data for the purpose of extractingthe original data and removing the noise To address theproblem and estimate the battery RUL accurately a novelbattery capacity prognostics approach is presented in thepaper A wavelet denoising method performed with differentthresholds is employed to process the measured data toreduce the uncertainty and extract the useful informationRVM optimized by differential evolution (DE) algorithmis utilized to estimate the battery RUL An experimentincluding battery 5 capacity prognostics case and battery18 capacity prognostics case is conducted which validatesthat the proposed approach can predict the trend of batterycapacity trajectory closely and estimate the RUL accurately

The material in the paper is organized in the followingorder Section 2 describes the strategy of wavelet denois-ing method Section 3 introduces RVM algorithm and itsparameter optimization by using DE algorithm Section 4illustrates the experiment procedure presents the experimentresults and gives discussions Finally conclusions are drawnin Section 5

2 Wavelet Denoising

Themeasured capacity data of batteries often suffer from thedifferent levels of noise pollution Experiment with noisy datacannot yield the accurate RULAs a result it is very importantto preprocess the capacity data for the purpose of extractingthe original data Wavelet denoising method is adopted toaddress the concern

Assume that the measured capacity data capacity(119888) iscomprised by

capacity (119888) = 119909 (119888) + 120590 (119888) (1)

where 119909(119888) is the original data 120590(119888) is the additive noise and119888 refers to the cycle which is a time index

Assume that 119885 is an integers set 119881119905119905isin119885

is an orthogonalmultiresolution analysis and 119882

119905119905isin119885

is the associatedwaveletspace The capacity(119888) projection on 119881

119905is

119875119881119905

= 119875119881119905+1+119875119882119905+1

= sum119894isin119885

119888119894

119905+1120601119905+1119894 +sum119894isin119885

119889119894

119905+1120595119905+1119894 (2)

where 119875119881119905+1

and 119875119882119905+1

denote the capacity(119888) projections on119881119905+1 and 119882

119905+1 at 2119905+1 resolution respectively 119888119894119905+1 and 119889119894

119905+1refer to the scaling coefficient and wavelet coefficient ofcapacity(119888) at 2119905+1 resolution respectively 120601

119905+1 and 120595119905+1

represent the scaling function and wavelet function ofcapacity(119888) at 2119905+1 resolution respectively Therefore 119888

119905+1and 119889

119905+1 characterize the approximations and details ofcapacity(119888) at 2119905+1 resolution respectively Correspondingly119881119905119905isin119885

can be decomposed as

119881119905= 119882119905+1 oplus119881119905+1 = 119882

119905+1 oplus (119882119905+2 oplus119881119905+2)

= 119882119905+1 oplus119882119905+2 oplus (119882119905+3 oplus119881119905+3)

= 119882119905+1 oplus119882119905+2 oplus119882119905+3 oplus sdot sdot sdot

(3)

By using the multilevel wavelet decomposition discreteapproximation coefficients and detail coefficients are pro-duced The detail coefficients with small absolute values areconsidered to be noise Generally the traditional waveletdenoising method is setting the detail coefficients belowa threshold to zero and reconstructing the denoised databy using the rest coefficients Sqtwolog threshold rigorousthreshold heursure threshold and minimax threshold arecommonly used rules to yield the threshold In the workthe wavelet denoising is performed twice with sqtwologthreshold rule and minimax threshold rule respectively

The sqtwolog threshold rule produces the thresholdwhich can yield good performance multiplied by a smallfactor proportional to log(length(capacity))

Thresholdsqtwolog = radic2 log 119899 (4)

where 119899 is the length of the capacity setThe minimax threshold rule brings about the minimum

of the maximum mean square error generated for the worstfunction with a given set by using theminimax principleThethreshold is defined as

Thresholdminimax = 119886+ 119887 lowastlog (119899)log (2)

(5)

where 119886 and 119887 are factors which are generally set to 03936and 01829 respectively

The minimax threshold is obviously lower than the sqt-wolog threshold in magnitude with a signal Wavelet denois-ing with the sqtwolog threshold can weaken the strong noiseobviously Meanwhile wavelet denoising with the minimaxthreshold can remove the weak noise effectively The waveletdenoising strategy in the work is implementing waveletdenoising with the sqtwolog threshold firstly and then per-forming wavelet denoising with the minimax threshold

3 DE-RVM

31 RVM RVM is firstly presented in [14] and has generateddemonstrative effect in prognostics [15ndash17] The algorithm isa Bayesian treatment which provides probabilistic interpre-tation of the output The relevance vectors and weights areobtained by maximizing a marginal likelihood

Computational Intelligence and Neuroscience 3

Assume that x119894 119905119894119873

119894=1 is the input data The target 119905119894is

obtained by

119905119894= 119910 (x

119894w) + 120576

119894 (6)

where w = (1199080 1199081 119908119873)119879 and 120576

119894is the noise with mean

zero and variance 1205902Assume that 119905

119894is independent and the likelihood of com-

plete dataset can be defined as

119901 (t | w 1205902) = (21205871205902)minus1198732

exp minus 121205902

1003817100381710038171003817tminus120593w10038171003817100381710038172 (7)

where t = (1199051 1199052 119905119873)119879 and 120593 is a 119873 times (119873 + 1) design

matrix with 120593 = [120593(x1) 120593(x2) 120593(x119873)]119879 and 120593(x

119894) =

[1 119870(x119894 x1) 119870(x119894 x2) 119870(x119894 x119873)]

119879Maximum likelihood estimations ofw and 1205902 in (7) often

result in overfitting Hence an explicit zero-mean Gaussianprior probability distribution is defined in order to constrainthe parameters as

119901 (w | 120572) =

119873

prod119894=0119873(119908119894| 0 120572minus1119894) (8)

where 120572 is a119873 + 1 hyperparameters vectorUsing Bayesrsquo rule the posterior probability about all of the

unknown parameters can be obtained by

119901 (w120572 1205902 | t)

=119901 (t | w120572 1205902) 119901 (w 120572 1205902)

int 119901 (t | w120572 1205902) 119901 (w120572 1205902) 119889w 119889120572 1198891205902

(9)

However the normalizing integral int119901(t | w120572 1205902)119901(w120572 120590

2)119889w 119889120572 119889120590

2 cannot be easily executedTherefore119901(w1205721205902 | t) can be instead decomposed as

119901 (w120572 1205902 | t) = 119901 (w | t120572 1205902) 119901 (120572 1205902 | t) (10)

Based on the Bayesrsquo rule the posterior distribution ofweights is obtained through

119901 (w | t120572 1205902) =119901 (t | w 1205902) 119901 (w | 120572)

119901 (t | 120572 1205902)

119901 (w | t120572 1205902) = (2120587)minus(119873+1)2 |Σ|minus12

sdot exp minus12(w minus120583)

119879

Σminus1(w minus120583)

(11)

where the posterior mean and covariance are

120583 = 120590minus2Σ120593119879t

Σ = (120590minus2120593119879120593+119860)

minus1

(12)

where 119860 = diag(1205720 1205721 120572119873)

Because of the uniform hyperpriors 119901(t | 120572 1205902) is de-

scribed by

119901 (t | 120572 1205902) = int119901 (t | w 1205902) 119901 (w | 120572) 119889w

= (2120587)minus1198732 100381610038161003816100381610038161205902I+120593119860minus112059311987910038161003816100381610038161003816

minus12

sdot exp minus12t119879 (1205902I+120593119860minus1120593119879)

minus1t

(13)

The maximum posterior (MP) estimate of the weights isdescribed by the posterior mean which depends on the valueof 120572 and 1205902 The estimates of 120572MP and 1205902MP are acquired bymaximizing the marginal likelihood Tipping [14] presentsthe iterative formulas for 120572MP and 120590

2MP as

120572new119894

=1 minus 120572119894Σ119894119894

1205832119894

(1205902)new

=

1003817100381710038171003817119905 minus 12059312058310038171003817100381710038172

119873 minus sum119894(1 minus 120572

119894Σ119894119894)

(14)

where Σ119894119894is the 119894th diagonal element of the posterior weight

covarianceAssume that x

lowastis a new input and the probability distri-

bution of the output 119905lowastis obtained by

119901 (119905lowast| t120572MP 120590

2MP)

= int119901 (119905lowast| w 1205902MP) 119901 (w | t120572MP 120590

2MP) 119889w

(15)

It can be easily obtained for both integrated terms areGaussian and the result is also a Gaussian form

119901 (119905lowast| t120572MP 120590

2MP) = 119873(119905

lowast| 119910lowast 120590

2lowast) (16)

The mean and the variance are

119910lowast= 120583119879120593 (xlowast)

1205902lowast= 120590

2MP +120593 (xlowast)

119879

Σ120593 (xlowast)

(17)

Gaussian radial basis function is selected as the kernelfunction for its powerful nonlinear processing capability andthe function is defined as

119870(x x119894) = exp[minus

(x minus x119894)2

21205742] (18)

where 120574 is the width factor which needs to be predeterminedfor it is crucially important to the predict performance

32 DE Algorithm DE algorithm is a population-based andstochastic search approach [18] and has shown superiorperformance on nonlinear nonconvex and nondifferentiableoptimization problems [19ndash21] DE algorithm starts with aninitial population vector which is randomly generated in asolution space Assume that 119873 is the population size and119883119903119894119866

(119894 = 1 2 119873) is a solution vector of the generation119866


For the classical DE algorithm mutation and crossover areutilized to generate trial vectors and selection is used to selectthe better vectors

Mutation For each vector 119883119903119894119866

a mutant vector 119881119894119866

is gen-erated by

119881119894119866

= 1198831199031119866 +119865 (1198831199032119866 minus1198831199033119866) 1199031 = 1199032 = 1199033 = 119894 (19)

where 1199031 1199032 and 1199033 are random integer indexes selectedfrom 1 2 119873 119865 is the scale fact which determines theamplification of the difference vector (119883

1199032119866 minus 1198831199033119866) and

119865 isin [0 2]

Crossover The crossover operation refers to yielding the trialvector by using the mutant vector 119881

119894119866and target vector119883

119894119866

119880119895119894119866

=

119881119895119894119866

rand119895le 119862119903or 119895 = 119896

119883119895119894119866

otherwise(20)

where 119895 = 1 2 119863 and 119863 is the problem dimension119862119903isin [0 1] is the predefined crossover constant rand

119895is the

119895th evaluation which is randomly generated between 0 and 1119896 isin 1 2 119863 and it is a random index

Selection Assume that 119891(119909) is a minimization problem Thegreedy selection scheme is defined as

119883119894119866+1

=

119880119894119866 if 119891 (119880

119894119866) lt 119891 (119883

119894119866)

119883119894119866 otherwise

(21)

The above three steps are repeated until reaching theterminal condition Then the best vector with minimumfitness value is exported as the result

33 Steps of Optimization DE-RVM refers to the RVM withwidth factor optimized by DE algorithm Mean square error(MSE) is used as the fitness function

MSE =sum119867

120575=1 [119911lowast(120575) minus 119911 (120575)]

2

119867 (22)

whereMSE represents the deviate degree of the predicted dataand the original data 120575 = 1 2 119867 and 119867 is the lengthof the original data 119911(120575) and 119911lowast(120575) are the original data andpredicted data respectively

Theoptimization target is tominimize theMSEvalue andthe optimizing steps are described as follows

(1) Initialize theDE algorithmparameters which includethe population size scale factor crossover rate andmaximum generation

(2) Produce the mutant vector and trial vector accordingto (19) and (20)

(3) Determine the next generation vector according to(21)

(4) Repeat steps (2) and (3) until the terminated criterionis met

(5) Output the optimized value to the RVM and exit theprogram

0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

Battery 5Battery 18

12

13

14

15

16

17

18

19

2

Figure 1 Measured capacity data of batteries

4 Prognostics Experiment

41 Experiment Data An experiment is conducted to dem-onstrate the proposed capacity prognostics approach andthe data were obtained from data repository of NASAAmes Prognostics Center of Excellence [22] In the datacollected procedure lithium-ion batteries were workingunder three different operational profiles charge dischargeand impedance with a temperature of 25∘C Charging wasperformed at a 15 A constant current until the battery voltagereached 42 V and then maintaining the 42 V constant volt-age until the current dropped to 20mADischargingwas run-ning at a 2A constant current until the battery voltage felledto 27 and 25 V which were corresponding to batteries 5 and18 respectively Impedance measurement was implementedwith an electrochemical impedance spectroscopy frequencysweep ranging from 01Hz to 5 kHz Repeated charge anddischarge cycles led to the accelerated aging of batterieswhile impedance measurements discovered the changes ofthe internal battery parameters with aging progresses Theexperiments were terminated when the capacity of batteriesreached its end-of-life (EOL) threshold whichwas about 70rated capacity In the experiments each nominal capacity oflithium-ion battery is 2 Ah and the EOL threshold is set to138 Ah The lithium-ion batteries 5 and 18 capacity data areshown in Figure 1 It can be observed that the capacity gener-ally degrades with usage for the reason of irreversible physicaland chemical changes and at some cycle increases rapidlyand shortly due to the impact of disturbances measurementerrors stochastic load or other unknown behaviors in thebatteries The length of batteries 5 and 18 capacity data are166 cycles and 132 cycles respectivelyMeanwhile their actualcycle lives are 129 and 114 respectively

42 Experiment Procedure The experiment includes a bat-tery 5 capacity prognostics case and a battery 18 capacity


Perform wavelet denoising and yield the denoised data

Divide the denoised data into training data and testing data

Obtain the width factor by using DE algorithm based on the training data

Establish the predict model and estimate the predicted testing data

Generate the estimated RUL

Figure 2 Flowchart of predict steps

prognostics case The detailed predict steps of each case areshown in Figure 2 and described as follows

(1) Perform wavelet denoising based on the measureddata and obtain the denoised data

(2) Separate the denoised data into training data andtesting data The lengths of the training data in thetwo battery cases are set to 80 and 70 respectivelyTherefore the lengths of the testing data in two casesare 88 and 62 respectively

(3) By using DE algorithm a width factor is generatedbased on the training data

(4) A predict model is constructed by RVM with adopt-ing the optimized width factor and the predictedtesting data are estimated

(5) Generate the estimated RUL

43 Experiment Results and Analysis Wavelet denoisingimplemented twice with different thresholds is employed toprocess the measured capacity data Figure 3 displays thecapacity data wavelet denoised with the sqtwolog thresholdand strong peak pulses are weakened obviously comparedto Figure 1 Then the denoised capacity data are processedby using wavelet denoising with the minimax thresholdto remove the weak noise The denoised capacity data areshown in Figure 4 and the trajectory of the denoised data iscontinuous and smooth

The DE algorithm population size and maximum gener-ation are set to 30 and 100 respectively 119865 is equal to 06119862119903is linearly reduced from 09 to 03 Figure 5 shows the

width factor optimization procedures by using DE algorithmbased on batteries 5 and 18 training data respectively Thecorresponding optimized wider factors are 05009 and 02778in the two battery cases respectively

0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Battery 5Battery 18

Figure 3 The batteriesrsquo capacity data wavelet denoised with thesqtwolog threshold

0 20 40 60 80 100 120 140 160 180Cycles

Battery 5Battery 18

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Figure 4 The batteriesrsquo capacity data wavelet denoised with theminimax threshold further

Adopting the optimized width factor RVM is used toperform battery capacity prognostics In order to quantifythe prognostic performance absolute error (AE) and MSEbetween the original testing data and the predicted testingdata relative accuracy (RA) and 120572 minus 120582 accuracy [23] areemployed as the measure metrics The 120572 minus 120582 accuracy isapplied to verify whether the estimated RUL is within theconfidence interval defined by 120572 The metrics are defined as

AE = RULes minusRUL

RA = 1minus1003816100381610038161003816RULes minus RUL1003816100381610038161003816

RUL

120572 minus 120582 accuracy =

Yes if RULes isin [1198621 1198622]

No if others

(23)


Table 1 Actual RULs estimated RULs AEs RAs and MSEs of two cases

Case Estimated RUL Actual RUL AE RA MSE 120572 minus 120582 accuracyBattery 5 45 49 minus4 918 14178119890 minus 04 YesBattery 18 40 44 minus4 909 39249119890 minus 05 Yes

0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case

Figure 5 The width factor optimization procedures

0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case

Figure 6 Prediction results

where RULes refers to the estimated RUL and RUL denotesthe actual RUL 119862

1and 119862

2are confidence intervals which

equal RUL lowast (1 minus 120572) and RUL lowast (1 + 120572) respectively 120572 isa bound which is set to 01 in the experiment

Actual RULs in the two battery cases are 49 and 44respectively The predictions are displayed in Figure 6 Esti-mated RULs actual RULs AEs RAs MSEs and 120572 minus

120582 accuracy of two cases are shown in Table 1 As can be seenfrom Figure 6 the DE-RVM predicts the trend of capacitydegradation trajectories of the two cases successfully Mean-while this can also be verified by the MSEs in Table 1 whichare pretty low in the two cases and this denotes that thepredicted testing data are close to the original testing data

RAs are all beyond 90 in two cases which implies highprediction accuracies produced by the DE-RVMMeanwhilethe estimated RULs in the two cases are both within theconfidence interval as the last row shows

For the purpose of validating the predict performance ofthe presented prognostics approach the DE-RVM approachis compared with ANN optimized by DE algorithm (DE-ANN) [24] approach and SVM improved by DE algorithm(DE-SVM) [25] approach The denoised data of batteries 5and 18 are used as experiment data The assessment indexadopts RA and MSE In order to avoid accidental accidentin the experiment each approach is run 10 times and meanresults are shown in Table 2 As can be seen from the table


Table 2 RAs and MSEs of the referenced approaches

Case DE-ANN DE-SVMRA MSE RA MSE

Battery 5 857 27684119890 minus 04 878 21507119890 minus 04


the DE-RVM provides smaller MSE than the DE-ANN andDE-SVM which implies that the data predicted by the DE-RVM are more close to the original data Meanwhile theDE-RVM yields higher RA than the DE-ANN and the DE-SVMwhich characterizes that the DE-RVM can output moreaccurate prediction than the other two approaches It can beconcluded that the DE-RVM approach significantly outper-forms the DE-ANN approach and the DE-SVM approach onthe problem of the battery capacity prognostics

5 Conclusions

The gradual decreased capacity of lithium-ion batteries hasbeen used as the SOH indicator in the work For the reasonof themeasured battery capacity data often suffering from thedifferent levels of noise pollution awavelet denoisingmethodwith different thresholds has been presented to generate thedenoised data

The RVM with its width factor optimized by DE algo-rithm has been used for battery capacity prognostics Twobattery case results have validated that the approach can pre-dict the trend of capacity degradation trajectory closely andestimate the battery RUL accurately Meanwhile an extendexperiment has demonstrated that the proposed DE-RVMapproach has higher predict accuracy than the referencedapproaches in the battery capacity prognostics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFunds of China for Distinguished Young Scholar underGrant no 50925727 theNationalDefenseAdvancedResearchProject Grant nos C1120110004 and 9140A27020211DZ5102the Key Grant Project of Chinese Ministry of Educationunder Grant no 313018 Anhui Provincial Science and Tech-nology Foundation of China under Grant no 1301022036the Fundamental Research Funds for the Central Universitiesnos 2012HGCX0003 and 2014HGCH0012 National NaturalScience Foundation of China nos 61401139 51407054 and61403115 and Anhui Provincial Natural Science Foundationno 1508085QE85

References

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on the ARX battery modelrdquo Energies vol 6 no 1 pp 444ndash4702013

[2] W He N Williard M Osterman and M Pecht ldquoPrognosticsof lithium-ion batteries using extended Kalman filteringrdquo inProceedings of the IMAPS Advanced Technology Workshop onHigh Reliability Microelectronics for Military Applications pp17ndash19 Linthicum Heights Md USA May 2011

[3] S Tang C Yu X Wang X Guo and X Si ldquoRemaining usefullife prediction of lithium-ion batteries based on the wienerprocess withmeasurement errorrdquo Energies vol 7 no 2 pp 520ndash547 2014

[4] W Xian B Long M Li and HWang ldquoPrognostics of lithium-ion batteries based on the verhulst model particle swarmoptimization and particle filterrdquo IEEE Transactions on Instru-mentation and Measurement vol 63 no 1 pp 2ndash17 2014

[5] B Saha K Goebel S Poll and J Christophersen ldquoPrognosticsmethods for battery health monitoring using a Bayesian frame-workrdquo IEEE Transactions on Instrumentation andMeasurementvol 58 no 2 pp 291ndash296 2009

[6] M E Orchard P Hevia-Koch B Zhang and L Tang ldquoRiskmeasures for particle-filtering-based state-of-charge prognosisin lithium-ion batteriesrdquo IEEE Transactions on Industrial Elec-tronics vol 60 no 11 pp 5260ndash5269 2013

[7] W X Shen C C Chan E W C Lo and K T Chau ldquoA newbattery available capacity indicator for electric vehicles usingneural networkrdquo Energy Conversion and Management vol 43no 6 pp 817ndash826 2002

[8] A Al-Alawi S M Al-Alawi and S M Islam ldquoPredictive controlof an integrated PV-diesel water and power supply system usingan artificial neural networkrdquo Renewable Energy vol 32 no 8pp 1426ndash1439 2007

[9] A Widodo M-C Shim W Caesarendra and B-S YangldquoIntelligent prognostics for battery health monitoring based onsample entropyrdquo Expert Systems with Applications vol 38 no 9pp 11763ndash11769 2011

[10] T Hansen and C J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005

[11] H Li D Pan andC P Chen ldquoIntelligent prognostics for batteryhealthmonitoring using themean entropy and relevanceVectormachinerdquo IEEE Transactions on Systems Man and Cyberneticsvol 44 pp 851ndash862 2014

[12] D Wang Q Miao and M Pecht ldquoPrognostics of lithium-ion batteries based on relevance vectors and a conditionalthree-parameter capacity degradation modelrdquo Journal of PowerSources vol 239 pp 253ndash264 2013

[13] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995

[14] M E Tipping ldquoSparse Bayesian learning and the relevancevector machinerdquoThe Journal of Machine Learning Research vol1 no 3 pp 211ndash244 2001

[15] W Caesarendra A Widodo and B S Yang ldquoApplication ofrelevance vector machine and logistic regression for machinedegradation assessmentrdquo Mechanical Systems and Signal Pro-cessing vol 24 no 4 pp 1161ndash1171 2010

[16] Z Chen ldquoAn overview of bayesian methods for neural spiketrain analysisrdquoComputational Intelligence andNeuroscience vol2013 Article ID 251905 17 pages 2013

[17] C Zhang Y He L Yuan and F Deng ldquoA novel approachfor analog circuit fault prognostics based on improved RVMrdquoJournal of Electronic Testing vol 30 no 3 pp 343ndash356 2014


[18] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuous spacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[19] P Rocca G Oliveri and A Massa ldquoDifferential evolution asapplied to electromagneticsrdquo IEEE Antennas and PropagationMagazine vol 53 no 1 pp 38ndash49 2011

[20] A A Abou El Ela M A Abido and S R Spea ldquoDifferentialevolution algorithm for optimal reactive power dispatchrdquo Elec-tric Power Systems Research vol 81 no 2 pp 458ndash464 2011

[21] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[22] B Saha and K Goebel ldquoBattery Data Setrdquo NASA Ames Prog-nosticsData RepositoryNASAAmesMoffett Field Calif USA2007 httptiarcnasagovtechdashpcoeprognostic-data-repository

[23] A Saxena J Celaya B Saha S Saha and K Goebel ldquoMetricsfor offline evaluation of prognostic performancerdquo InternationalJournal of Prognostics and Health Management vol 1 no 1 20pages 2010

[24] S H Yang U Natarajan M Sekar and S Palani ldquoPredictionof surface roughness in turning operations by computer visionusing neural network trained by differential evolution algo-rithmrdquo The International Journal of Advanced ManufacturingTechnology vol 51 no 9ndash12 pp 965ndash971 2010

[25] J Wang L Li D Niu and Z Tan ldquoAn annual load forecastingmodel based on support vector regression with differentialevolution algorithmrdquo Applied Energy vol 94 pp 65ndash70 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

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Biomedical Imaging


ArtificialNeural Systems

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Human-ComputerInteraction

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physical models and have been applied in many relativeworks [7ndash12] Artificial neural network is a widely useddata-driven approach to the battery capacity prognostics[7 8] However it has disadvantages of poor generalizationdifficult structure confirmation and low convergence rateSupport vector machine (SVM) is a machine learning tool[13] characterized by the usage of kernel functions and it hasbeen utilized to estimate the battery RUL [9 10] Relevancevector machine (RVM) is a Bayesian sparse kernel technique[14] with usage of much fewer kernel functions and higherperformance compared to the SVM Meanwhile RVM hasbeen applied to the research field [11 12]

The measured battery capacity data are often subject tothe different levels of noise pollution because of the impactof disturbances measurement errors stochastic load andother unknown behaviors in batteries Capacity prognosticsbased on the noisy data cannot produce accurate predictresults Therefore it is significantly important to preprocessthe measured capacity data for the purpose of extractingthe original data and removing the noise To address theproblem and estimate the battery RUL accurately a novelbattery capacity prognostics approach is presented in thepaper A wavelet denoising method performed with differentthresholds is employed to process the measured data toreduce the uncertainty and extract the useful informationRVM optimized by differential evolution (DE) algorithmis utilized to estimate the battery RUL An experimentincluding battery 5 capacity prognostics case and battery18 capacity prognostics case is conducted which validatesthat the proposed approach can predict the trend of batterycapacity trajectory closely and estimate the RUL accurately

The material in the paper is organized in the followingorder Section 2 describes the strategy of wavelet denois-ing method Section 3 introduces RVM algorithm and itsparameter optimization by using DE algorithm Section 4illustrates the experiment procedure presents the experimentresults and gives discussions Finally conclusions are drawnin Section 5

2 Wavelet Denoising

Themeasured capacity data of batteries often suffer from thedifferent levels of noise pollution Experiment with noisy datacannot yield the accurate RULAs a result it is very importantto preprocess the capacity data for the purpose of extractingthe original data Wavelet denoising method is adopted toaddress the concern

Assume that the measured capacity data capacity(119888) iscomprised by

capacity (119888) = 119909 (119888) + 120590 (119888) (1)

where 119909(119888) is the original data 120590(119888) is the additive noise and119888 refers to the cycle which is a time index

Assume that 119885 is an integers set 119881119905119905isin119885

is an orthogonalmultiresolution analysis and 119882

119905119905isin119885

is the associatedwaveletspace The capacity(119888) projection on 119881

119905is

119875119881119905

= 119875119881119905+1+119875119882119905+1

= sum119894isin119885

119888119894

119905+1120601119905+1119894 +sum119894isin119885

119889119894

119905+1120595119905+1119894 (2)

where 119875119881119905+1

and 119875119882119905+1

denote the capacity(119888) projections on119881119905+1 and 119882

119905+1 at 2119905+1 resolution respectively 119888119894119905+1 and 119889119894

119905+1refer to the scaling coefficient and wavelet coefficient ofcapacity(119888) at 2119905+1 resolution respectively 120601

119905+1 and 120595119905+1

represent the scaling function and wavelet function ofcapacity(119888) at 2119905+1 resolution respectively Therefore 119888

119905+1and 119889

119905+1 characterize the approximations and details ofcapacity(119888) at 2119905+1 resolution respectively Correspondingly119881119905119905isin119885

can be decomposed as

119881119905= 119882119905+1 oplus119881119905+1 = 119882

119905+1 oplus (119882119905+2 oplus119881119905+2)

= 119882119905+1 oplus119882119905+2 oplus (119882119905+3 oplus119881119905+3)

= 119882119905+1 oplus119882119905+2 oplus119882119905+3 oplus sdot sdot sdot

(3)

By using the multilevel wavelet decomposition discreteapproximation coefficients and detail coefficients are pro-duced The detail coefficients with small absolute values areconsidered to be noise Generally the traditional waveletdenoising method is setting the detail coefficients belowa threshold to zero and reconstructing the denoised databy using the rest coefficients Sqtwolog threshold rigorousthreshold heursure threshold and minimax threshold arecommonly used rules to yield the threshold In the workthe wavelet denoising is performed twice with sqtwologthreshold rule and minimax threshold rule respectively

The sqtwolog threshold rule produces the thresholdwhich can yield good performance multiplied by a smallfactor proportional to log(length(capacity))

Thresholdsqtwolog = radic2 log 119899 (4)

where 119899 is the length of the capacity setThe minimax threshold rule brings about the minimum

of the maximum mean square error generated for the worstfunction with a given set by using theminimax principleThethreshold is defined as

Thresholdminimax = 119886+ 119887 lowastlog (119899)log (2)

(5)

where 119886 and 119887 are factors which are generally set to 03936and 01829 respectively

The minimax threshold is obviously lower than the sqt-wolog threshold in magnitude with a signal Wavelet denois-ing with the sqtwolog threshold can weaken the strong noiseobviously Meanwhile wavelet denoising with the minimaxthreshold can remove the weak noise effectively The waveletdenoising strategy in the work is implementing waveletdenoising with the sqtwolog threshold firstly and then per-forming wavelet denoising with the minimax threshold

3 DE-RVM

31 RVM RVM is firstly presented in [14] and has generateddemonstrative effect in prognostics [15ndash17] The algorithm isa Bayesian treatment which provides probabilistic interpre-tation of the output The relevance vectors and weights areobtained by maximizing a marginal likelihood


Assume that x119894 119905119894119873


obtained by

119905119894= 119910 (x

119894w) + 120576

119894 (6)

where w = (1199080 1199081 119908119873)119879 and 120576





119901 (t | w 1205902) = (21205871205902)minus1198732

exp minus 121205902

1003817100381710038171003817tminus120593w10038171003817100381710038172 (7)



119894) =

[1 119870(x119894 x1) 119870(x119894 x2) 119870(x119894 x119873)]



119901 (w | 120572) =

119873

prod119894=0119873(119908119894| 0 120572minus1119894) (8)



119901 (w120572 1205902 | t)

=119901 (t | w120572 1205902) 119901 (w 120572 1205902)

int 119901 (t | w120572 1205902) 119901 (w120572 1205902) 119889w 119889120572 1198891205902

(9)


2)119889w 119889120572 119889120590


119901 (w120572 1205902 | t) = 119901 (w | t120572 1205902) 119901 (120572 1205902 | t) (10)


119901 (w | t120572 1205902) =119901 (t | w 1205902) 119901 (w | 120572)

119901 (t | 120572 1205902)

119901 (w | t120572 1205902) = (2120587)minus(119873+1)2 |Σ|minus12


119879


(11)


120583 = 120590minus2Σ120593119879t

Σ = (120590minus2120593119879120593+119860)

minus1

(12)

where 119860 = diag(1205720 1205721 120572119873)


scribed by

119901 (t | 120572 1205902) = int119901 (t | w 1205902) 119901 (w | 120572) 119889w

= (2120587)minus1198732 100381610038161003816100381610038161205902I+120593119860minus112059311987910038161003816100381610038161003816

minus12


minus1t

(13)


2MP as

120572new119894

=1 minus 120572119894Σ119894119894

1205832119894

(1205902)new

=

1003817100381710038171003817119905 minus 12059312058310038171003817100381710038172

119873 minus sum119894(1 minus 120572

119894Σ119894119894)

(14)





119901 (119905lowast| t120572MP 120590

2MP)


2MP) 119889w

(15)


119901 (119905lowast| t120572MP 120590

2MP) = 119873(119905


2lowast) (16)


119910lowast= 120583119879120593 (xlowast)

1205902lowast= 120590


119879

Σ120593 (xlowast)

(17)


119870(x x119894) = exp[minus

(x minus x119894)2

21205742] (18)







a mutant vector 119881119894119866

is gen-erated by

119881119894119866

= 1198831199031119866 +119865 (1198831199032119866 minus1198831199033119866) 1199031 = 1199032 = 1199033 = 119894 (19)


1199032119866 minus 1198831199033119866) and

119865 isin [0 2]



119894119866

119880119895119894119866

=

119881119895119894119866

rand119895le 119862119903or 119895 = 119896

119883119895119894119866

otherwise(20)


119895is the



119883119894119866+1

=

119880119894119866 if 119891 (119880

119894119866) lt 119891 (119883

119894119866)

119883119894119866 otherwise

(21)



MSE =sum119867

120575=1 [119911lowast(120575) minus 119911 (120575)]

2

119867 (22)








0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

Battery 5Battery 18

12

13

14

15

16

17

18

19

2





















0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Battery 5Battery 18


0 20 40 60 80 100 120 140 160 180Cycles

Battery 5Battery 18

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2



AE = RULes minusRUL


RUL



No if others

(23)




0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case


0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case



1and 119862













5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Assume that x119894 119905119894119873


obtained by

119905119894= 119910 (x

119894w) + 120576

119894 (6)

where w = (1199080 1199081 119908119873)119879 and 120576





119901 (t | w 1205902) = (21205871205902)minus1198732

exp minus 121205902

1003817100381710038171003817tminus120593w10038171003817100381710038172 (7)



119894) =

[1 119870(x119894 x1) 119870(x119894 x2) 119870(x119894 x119873)]



119901 (w | 120572) =

119873

prod119894=0119873(119908119894| 0 120572minus1119894) (8)



119901 (w120572 1205902 | t)

=119901 (t | w120572 1205902) 119901 (w 120572 1205902)

int 119901 (t | w120572 1205902) 119901 (w120572 1205902) 119889w 119889120572 1198891205902

(9)


2)119889w 119889120572 119889120590


119901 (w120572 1205902 | t) = 119901 (w | t120572 1205902) 119901 (120572 1205902 | t) (10)


119901 (w | t120572 1205902) =119901 (t | w 1205902) 119901 (w | 120572)

119901 (t | 120572 1205902)

119901 (w | t120572 1205902) = (2120587)minus(119873+1)2 |Σ|minus12


119879


(11)


120583 = 120590minus2Σ120593119879t

Σ = (120590minus2120593119879120593+119860)

minus1

(12)

where 119860 = diag(1205720 1205721 120572119873)


scribed by

119901 (t | 120572 1205902) = int119901 (t | w 1205902) 119901 (w | 120572) 119889w

= (2120587)minus1198732 100381610038161003816100381610038161205902I+120593119860minus112059311987910038161003816100381610038161003816

minus12


minus1t

(13)


2MP as

120572new119894

=1 minus 120572119894Σ119894119894

1205832119894

(1205902)new

=

1003817100381710038171003817119905 minus 12059312058310038171003817100381710038172

119873 minus sum119894(1 minus 120572

119894Σ119894119894)

(14)





119901 (119905lowast| t120572MP 120590

2MP)


2MP) 119889w

(15)


119901 (119905lowast| t120572MP 120590

2MP) = 119873(119905


2lowast) (16)


119910lowast= 120583119879120593 (xlowast)

1205902lowast= 120590


119879

Σ120593 (xlowast)

(17)


119870(x x119894) = exp[minus

(x minus x119894)2

21205742] (18)







a mutant vector 119881119894119866

is gen-erated by

119881119894119866

= 1198831199031119866 +119865 (1198831199032119866 minus1198831199033119866) 1199031 = 1199032 = 1199033 = 119894 (19)


1199032119866 minus 1198831199033119866) and

119865 isin [0 2]



119894119866

119880119895119894119866

=

119881119895119894119866

rand119895le 119862119903or 119895 = 119896

119883119895119894119866

otherwise(20)


119895is the



119883119894119866+1

=

119880119894119866 if 119891 (119880

119894119866) lt 119891 (119883

119894119866)

119883119894119866 otherwise

(21)



MSE =sum119867

120575=1 [119911lowast(120575) minus 119911 (120575)]

2

119867 (22)








0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

Battery 5Battery 18

12

13

14

15

16

17

18

19

2





















0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Battery 5Battery 18


0 20 40 60 80 100 120 140 160 180Cycles

Battery 5Battery 18

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2



AE = RULes minusRUL


RUL



No if others

(23)




0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case


0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case



1and 119862













5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






a mutant vector 119881119894119866

is gen-erated by

119881119894119866

= 1198831199031119866 +119865 (1198831199032119866 minus1198831199033119866) 1199031 = 1199032 = 1199033 = 119894 (19)


1199032119866 minus 1198831199033119866) and

119865 isin [0 2]



119894119866

119880119895119894119866

=

119881119895119894119866

rand119895le 119862119903or 119895 = 119896

119883119895119894119866

otherwise(20)


119895is the



119883119894119866+1

=

119880119894119866 if 119891 (119880

119894119866) lt 119891 (119883

119894119866)

119883119894119866 otherwise

(21)



MSE =sum119867

120575=1 [119911lowast(120575) minus 119911 (120575)]

2

119867 (22)








0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

Battery 5Battery 18

12

13

14

15

16

17

18

19

2





















0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Battery 5Battery 18


0 20 40 60 80 100 120 140 160 180Cycles

Battery 5Battery 18

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2



AE = RULes minusRUL


RUL



No if others

(23)




0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case


0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case



1and 119862













5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















0 20 40 60 80 100 120 140 160 180Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

Battery 5Battery 18


0 20 40 60 80 100 120 140 160 180Cycles

Battery 5Battery 18

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2



AE = RULes minusRUL


RUL



No if others

(23)




0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case


0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case



1and 119862













5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






0 20 40 60 80 100Generations

MSE

91

915

92

925

93

935

94

times10minus5

(a) Battery 5 caseM

SE

0 20 40 60 80 100294

295

296

297

298

299

3

301

Generations

times10minus5

(b) Battery 18 case


0 20 40 60 80 100 120 140 160 180Cycles

OriginalPredicted

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(a) Battery 5 case

OriginalPredicted

0 20 40 60 80 100 120 140Cycles

Capa

city

(Ah)

12

13

14

15

16

17

18

19

2

EOL threshold 138Ah

(b) Battery 18 case



1and 119862













5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









5 Conclusions





Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Prognostics of Lithium-Ion Batteries ...

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Transcript of Research Article Prognostics of Lithium-Ion Batteries ...