Research ArticlePrognostics for State of Health of Lithium-Ion BatteriesBased on Gaussian Process Regression

Di Zhou 1 Hongtao Yin 1 Ping Fu1 Xianhua Song 2

Wenbin Lu3 Lili Yuan2 and Zuoxian Fu2

1School of Electrical Engineering and Automation Harbin Institute of Technology Harbin China2School of Science Harbin University of Science and Technology Harbin 150080 China3Shenzhen Academy of Metrology amp Quality Inspection Shenzhen China

Correspondence should be addressed to Hongtao Yin yinhthiteducn

Received 21 October 2017 Revised 27 January 2018 Accepted 6 February 2018 Published 1 April 2018

Academic Editor Anna Vila

Copyright copy 2018 Di Zhou et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Accurate estimation and prediction of the lithium-ion (Li-ion) batteriesrsquo performance has important theoretical and practicalsignificance to make better use of lithium-ion battery and to avoid unnecessary losses State of health (SOH) estimation is usedas a qualitative measure of the capability of a lithium-ion battery to store and deliver energy in a system To evaluate and predict theSOHof batteries the Gaussian process regressionwith neural network (GPRNN) as its variance function is proposed Experimentalresults confirm that the proposed method can be effectively applied to Li-ion battery monitoring and prognostics by quantitativecomparison with basic GPR combination LGPFR combination QGPFR and the multiscale GPR (SMK-GPR P-MGPR and SE-MGPR)The criteria of RMSE andMAPEof the proposed threemodels are reduced significantly compared to those of other existingmethods

1 Introduction

Lithium-ion battery has been widely applied to portableelectronic devices electric and hybrid vehicles even mili-tary electronics aerospace avionics and other automotivevehicles because of the properties of lightness high energydensity high output voltage long lifetime and low self-discharge [1 2] The safety and reliability of Li-ion batteryin actual application are critical factors of the system perfor-mance whose failure would not only lead to the performancedegradation repair costs increase and inconvenience butalso cause catastrophic accidents due to overheating andshort circuiting [3] Therefore the effective monitoring andprognostics for Li-ion battery have been attractedmuchmoreattention and a reliable and effective BMS (battery manage-ment system) is indispensable [4 5] The precondition ofBMS is monitoring and estimation of states which includesestimations of SOC (state of charge) SOH (state of health)and SOL (state of life) The above three states estimations are

still the core content in the research of BMS [4 6ndash8] SOH isthe ratio of the released capacitywith a certain discharge fromfull charge state to cut-off standard voltage under standardcondition and its corresponding initial capacity which is theresponse of the battery performance degradation and theaccidents possibility In general SOC describes the short-term changes of the current parameters and SOH describeslong-term changes SOH does not require continuous mea-surements and only needs periodical measurements in mostcases and the measured periods depend on the applicationsBased on the above advantages of SOH this paper focuses onresearch of SOH to reflect the performance of Li-ion battery

With the development of Li-ion battery SOH estimationplays an important role as a quantitative description for thebattery performance indicators to store and deliver energyin the system [9 10] Generally there are two mainstreammethods to estimate SOH [11] One is experience-basedapproach such as cycle counting method Ah Low and Ahweighted method and event-oriented cumulative method

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 8358025 11 pageshttpsdoiorg10115520188358025

2 Mathematical Problems in Engineering

of aging The other method is performance-based approachwhich utilizes different forms of performance models andconsiders the aging process and stress factors Accordingto different sources of information used the performance-based approach is divided into three categories which aremechanism-based method characteristics-based methodand data-driven method Due to the complex physical andchemical processes of the battery itself a lot of the rulesare difficult to describe directly through the mechanismThemethod which describes the battery performance from theperspective of the test data is called data-driven approachThe acquired data tends to have a strong uncertainty andincompleteness testing all the possible factors affecting thebattery life in the practical application is unrealistic Althoughthe limitations exist data-driven method is still an effectiveprediction method It is used to excavate the rules of batteryperformance for SOH prediction through using battery per-formance test data Data-driven prediction does not requiremechanical knowledge of object systems It excavates theimplied information to predict SOH through a variety ofdata analysis learning methods Therefore it can avoid thedifficulty of the model acquisition so it is a more practicalprediction method

There are many common data-driven algorithms such asParticle Filtering (PF) Support Vector Machine (SVM) andRelevant Vector Machine (RVM) PF can combine availableinformation from systemmeasurements and analytic modelsfor SOH prediction [10 12] Kalman filter and extendedKalman filter (EKF) are applied to SOH prediction of bat-teries by using estimation of equivalent circuit model [1314] Unscented filtering and Bayesian filtering are classicalmethods for battery SOHor SOCestimationThese stochasticfilteringmethods for prognostics are easier to be obtained andhave wider application scope However themethods are diffi-cult in containing the environmental interference and loadingdynamic characteristics factors and so on and the dynamicaccuracy and adaptability are still limitedMoreover PF algo-rithm itself solves the problem of parameters initializationwhich is time-consuming in the case of more parametersSome methods are mostly improvement and optimization ofPF or KF such as sequential Monte Carlo method referringto PF which has attracted attention to SOH [15] In recentyears SVM has been used in SOH prediction of Li-ionbattery [16] In addition to basic SVM algorithm there aresome improved SVMalgorithms and fusion algorithms SVMmethod still has some deficiencies the kernel function mustsatisfy the Mercer condition sparsity is limited the numberof support vectors is sensitive to error boundary and the lossof function and the penalty factor are difficult to determineSVM also lacks the expression of uncertainty managementTo solve this problem RVM approaches have been proposedRVM algorithm has higher calculation accuracy and lowercomputational complexity compared with SVM and canoutput the uncertainty information of prediction results[17] Kim obtained the relevant parameters of exponentialmodel by using EIS test data based on RVM algorithm andgave the prediction outcomes [1] Oai et al developed anew technique to predict SOH using a dual-sliding-modeobserver The researchers presented a method Li-ion battery

SOH prediction concept based on an appropriate SOHdefinition [18] However since the much sparse and dynamicfluctuation of capacity data will lead to forecasting the resultof poor stability if RVM is directly used for predictiononly there are many others methods such as neural fuzzylogic [19] networks [20] regressions [21] and distributedactive learning [22] that lack specific strategies in dealingwith model uncertainties and would lead to prognosticsperformance deterioration when facing long-term changes inenvironmental and operating conditions

To consider modeling flexibility and uncertainty rep-resentations Gaussian process regression (GPR) has beeninvestigated for Li-ion battery prognostics where the degra-dation trends are learnt from battery datasets with thecombination of Gaussian process functions [23] GPR couldprovide a confidence measure of output and has been usedfor SOH predictions as a kind of new machine learningmethod based on Bayesian theory and statistical learningtheory Meanwhile it is a nonparametric method which issuitable for high dimension nonlinear regression problemsand can give the uncertainty representation of SOH predic-tionAn improvedGPRmethodhas been proposed to captureregeneration phenomena [24] Li and Xu proposed a newprognostics method for state of health estimation of Li-ionbatteries based on a mixture of Gaussian process models andparticle filter proposed [25] However forecasting accuracyand uncertainty of the above two methods do not meet therequirements of practical application

In this paper an approach based on GPR with neuralnetwork kernel function is presented to predict the SOHof Li-ion batteries Neural network is ideally suited to theSOH data since it has smaller misspecification than otherstationary covariance functions and draws saturation atdifferent values in the positive and negative directions ofvariable 119909 [23] Meanwhile the log marginal likelihood ismuch larger than other covariance functions The proposedmethod consists of three models the neural network itselfthe sum of the neural network and the Maternard covariancefunction and the product of the neural network and theperiodic covariance function Finally experiments based onthe NASA battery datasets are provided to demonstratethe performance of the proposed three prognostics modelsThe experimental results show that the predictions are nearperfect

The remainder of this paper is organized as follows InSection 2 the background knowledge about the SOH of Li-ion batteries is described Then the proposed methodologyincluding GPR and its parameters is provided in Section 3Experiments and analysis are given in Section 4 to demon-strate the performance of proposed prognostics algorithmFinally conclusions are drawn in Section 5

2 The SOH of Lithium-Ion Batteries

There are four definitions of SOH according to the batterycharacteristics which are shown as follows [26]

Mathematical Problems in Engineering 3

(1) From the perspective of remaining power of battery todetermine the battery SOH its definition can be givenby

SOH = 119876aged119876newtimes 100 (1)

where 119876aged is the current biggest power of batteryand 119876new is the initial power

(2) From the perspective of starting power to define thebattery SOH the expression is given by

SOH = CCAocmp minus CCAmin

CCAnew minus CCAmintimes 100 (2)

where CCAocmp is the real-time starting power andCCAnew is the predicted starting power released bybattery when the SOC is 100 and CCAmin is theminimum starting power needed

(3) From the perspective of impedance measurement todefine the battery SOH the definition is depicted asfollows

SOH = 1198771198941198770 times 100 (3)

where 119877119894 is 119894th impedance measurement varying withthe cycles of charging and discharging and 1198770 is theinitial impedance

(4) From the perspective of battery capacity power todefine the battery SOH this method can be expressedas

SOH = 1198621198941198620 times 100 (4)

where119862119894 is the 119894th capacitance value degenerated withcycles and 1198620 is the initial capacity

In this paper this definition of SOH using the capacity poweris utilized to study the performance of battery

3 Methodology

31 Gaussian Process Regression Gaussian process (GP) isused to describe random variables which are scalars orvectors GP is a stochastic process which also governs theproperties of functions and gets Bayesian inferences with thefunction-space view GP is a set where any finite randomvariable has joint Gaussian distribution and the charactersof GP are completely determined by the mean function andthe covariance function [28]119898(119909) = 119864 [119891 (119909)] 119896 (119909 1199091015840) = 119864 [(119898 (119909) minus 119891 (119909)) (119898 (1199091015840) minus 119891 (1199091015840))] (5)

Consider the simple regression model

119910 = 119891 (119909) + 120576 119891 (119909) = 119883119879119908 (6)

The training set is (119909119894 119910119894) | 119894 = 1 119899 where 119909119894 119894 =1 119899 are the input values 119891 is a function value and 119910119894119894 = 1 119899 are the output values 120576 represents noise whichfollows an independent identically distributed Gaussiandistribution with zero mean and variance 1205902 that is

120576 sim 119873 (0 1205902) (7)

Then the prior distribution of the observation 119910 is obtained119910 | 119883 119908 sim 119873(119883119879119908 119896 (119909 1199091015840) | 1205902119868) (8)

It can be used to estimate the predictive values The jointdistribution of the observation 119910 and the predictive values 119891lowastis derived as follows

( 119910119891lowast) sim 119873(0 ( 119896 (119883119883) 119896 (119883 119909lowast)119896 (119909lowast 119883) 119896 (119909lowast 119909lowast))) (9)

where 119896(119883 119909lowast) represents the covariance of the training datapoints and predictive values and 119896(119909lowast 119909lowast) is the covarianceof the predictive values

Computed by Bayesrsquo rule the posterior distribution overthe weights can be gotten The posterior distribution toinference in the Bayesian linear model can be shown as

119891lowast | 119883 119891 119909lowast sim 119873(119896 (119909lowast 119883) 119896 (119883119883)minus1 119891 119896 (119909lowast 119909lowast)minus 119896 (119909lowast 119883) 119896 (119883119883)minus1 119896 (119883 119909lowast)) (10)

32 Choice of Parameters GP is usually parameterized interms of their covariance functions There are some mostpopular covariance functions [23 29] the squared exponen-tial covariance function

119896 (119909 1199091015840) = 1205902119891 exp(minus(119909119894 minus 119909119895)221198972 ) (11)

the constant covariance function

119896 (119909 1199091015840) = 12059020 (12)

and the periodic covariance function

119896 (119909 1199091015840) = 1205902119891 (minus 21198972 sin2 ( 1206032120587 (119909119894 minus 119909119895))) (13)

These covariance functions are stationary and are usuallyused in battery health prognostic with good results Herein this paper a particular type of neural network is appliedin battery health prognostic which is called neural networkcovariance function see (7) Neural network is ideally suitedto the SOH data since it allows saturation at differentvalues in the positive and negative directions of 119909 [23] Forcomparison the predictive distributions for three covariancefunctions are shown in Figure 1 The 64 data points are pro-duced by a step function with Gaussian noise with standarddeviation 120590 = 01 which is totally the same as the exampletaken in [23] Figure 1 shows the means and 95 confidence

4 Mathematical Problems in Engineering

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14O

utpu

t y

Input x

(a) Squared exponential

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(b) Sum of two squared exponentials

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(c) Neural network

Figure 1 Misspecification example

intervals for the noisy signal in grey Particularly Figure 1(a)expresses the single squared exponential covariance andFigure 1(b) illustrates the sum of two squared exponentialcovariances Obviously this covariance function is moreflexible than a single squared exponential since it has twomagnitudes and two length-scale parameters The predictivedistribution looks a little bit better but they both are notideal for this scenario Figure 1(c) shows the neural networkcovariance function which is ideally suited to this case sinceit allows saturation at different values in the positive andnegative directions of 119909 As shown in Figure 1 the predictionsare also near perfect

New kernels can be generated with this kernel forexample the sum of the neural network covariance functionand the Maternard covariance function is a new kerneland the product of the neural network covariance func-tion and the periodic covariance function is also a newkernel The Maternard covariance function is stationaryand nondegenerate and the periodic covariance function is

selected to approximate the local regeneration phenomenonof SOH data Therefore these two covariance functions areused in combination with the neural network in this paperAccording to the covariance function chosen in our paperthe simulation results are better than othersrsquo It turns out thatthe neural network covariance function can be viewed as asuitable choice of this type of data The covariance functionsused in the proposed paper are as follows [23]

The neural network covariance function

1198961 (119909 1199091015840)= 1205902119891sinminus1( 119909119879Λminus21199091015840radic(1 + 119909119879Λminus2119909) (1 + 1199091015840119879Λminus21199091015840))

(14)

The Maternard covariance function

1198962 (119909 1199091015840) = 1205902119891119891119889 (119903119889) exp (minus119903119889) (15)

Mathematical Problems in Engineering 5

The periodic covariance function

1198963 (119909 1199091015840) = 1205902119891 exp(minus 21198972 sin212058710038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817119901 ) (16)

In the Maternard covariance function 119891 is a function of119903119889 1198911(119905) = 1 1198913(119905) = 1 + 119905 1198915(119905) = 1198913(119905) + 11990523 119903119889 is thedistance between 119909 and 1199091015840 and 119903119889 = radic119889(119909 minus 1199091015840)Λminus2(119909 minus 1199091015840)

Typically the used covariance function has some freeparameters which are generally called free hyperparametersAs these parameters vary the predictions of data are differentwhich will lead to great fitting error In order to get the suit-able hyperparameters optimizationwith themaximization ofthe log-likelihood function is necessary [28 30]

log119901 (119910 | 119883) = minus12119910119879 (119870 + 1205902119868)minus1 119910minus 12 log 10038161003816100381610038161003816119870 + 120590211986810038161003816100381610038161003816 minus 1198992 log 2120587

(17)

The hyperparameters can be determined from training dataNotice that the length-scale 119897 in the kernel varies and thesignal variance 1205902119891 and the noise variance 1205902 would varyThrough experiments it can be found that when differentinitial values are set different fitting curves are obtained andthe error between them would be great According to theproposed issue an appropriate initial value is chosen

A larger number of training data enable GPRmodel to getbetter prognostic prediction According to the experimentaldata the linear mean function 119898(119909) = 119886119909 + 119887 is selected tofit the track of the data The covariance functions discussedabove that is the neural network shown as (7) the sum ofneural network andMaternard given as 119896(119909 1199091015840) = 1198961(119909 1199091015840) +1198962(119909 1199091015840) and the product of neural network and periodicdisplayed as 119896(119909 1199091015840) = 1198961(119909 1199091015840)1198963(119909 1199091015840) are considered Thethree covariance functions are denoted as Model I Model IIand Model III Then the hyperparameters spaces in the threemodels are Θ1 = [119886 119887 l sf]119879 Θ2 = [119886 119887 l sf1 ell sf2]119879 andΘ3 = [119886 119887 l sf1 ell p sf2]119879 where 119886 119887 are coefficients of themean function Λ is 119897 times the unit matrix and sf sf1 andsf2 control the variance1205902119891 in threemodes respectivelyThesethree models are all effective and are designed to fit SOH dataand make the prediction more accurate

4 Experiments and Analysis

41 Raw Data Description of Li-Ion Batteries To comprehen-sively illustrate the efficiency of the proposed method thedataset about the aging of Li-ion batteries is selected whichis obtained from the data repository of the NASA AmesPrognostics Center of Excellence (PCoE) [31]

The dataset of the NASA Ames PCoE consists of 36Li-ion batteriesrsquo data No 5 No 6 No 7 No 18 andNos 25ndash56 Because the lengths of SOHs extracted fromthe 36 batteries are different Li-ion batteries of No 5No 6 and No 7 are the most commonly used in relatedliteratures [24 25 27] because they are much longer Theywere run through 3 different operational profiles (charge

0 20 40 60 80 100 120 140 16055

60

65

70

75

80

85

90

95

100

Cycle number

SOH

()

No 5No 6No 7

Figure 2 SOHs of Li-ion batteries No 5 No 6 and No 7

discharge and impedance) at room temperature Chargingwas carried out in a constant current (CC) mode at 15 Auntil the battery voltage reached 42 V and then continuedin a constant voltage (CV) mode until the charge currentdropped to 20mA Discharge was carried out at a constantcurrent (CC) level of 2 A until the battery voltage fell to27 V 25 V 22 V and 25 V for batteries Nos 5 6 7 and18 respectively Impedance measurement was carried outthrough an electrochemical impedance spectroscopy (EIS)frequency sweep from 01Hz to 5 kHz Repeated charge anddischarge cycles result in accelerated aging of the batterieswhile impedance measurements provide insight into theinternal battery parameters that change as aging progressesThe experiments were stopped when the batteries reachedend-of-life (EOL) criteria which was a 30 fade in ratedcapacity (from 2Ahr to 14 Ahr) This dataset can be used forthe prediction of SOC SOH and the remaining useful life(RUL) of batteries [23]

In this paper the data from batteries No 5 No 6 andNo 7 is processed using (4) to implement the experimentsincluding training and testing Figure 2 shows the SOHcurves of these three batteries The total chargedischargecycles are all 168 Figure 2 indicates that the SOH showsan obvious global degradation trend and local regenerationphenomenon

42 Training with GPR For No 5 No 6 and No 7 the first100 SOH points are utilized to train the proposed GPRmodeland the remaining 68 points are used for prediction Forthe training data from the above three batteries the meanfunction is chosen as the linear function 119898(119909) = 119886119909 + 119887 andthe covariance function is chosen as three functions discussedabove according to these three models the algorithm for thetraining and prediction based on GPR is produced as follows

6 Mathematical Problems in Engineering

Table 1 Prediction errors comparison of different methods for batteries Nos 5 6 and 7

(a)

Battery number Error criteria Basic GPR LGPFR QGPFR Combination LGPFR Combination QGPFR SMK-GPR

5 MAPE () 1210 230 190 160 210 165RMSE () 1303 171 150 136 180 138

6 MAPE () 270 1030 770 1020 290 1060RMSE () 2251 690 512 686 2044 708

7 MAPE () 1920 190 540 170 260 191RMSE () 2070 159 552 173 269 188

(b)

Battery number Error criteria P-MGPR SE-MGPR Model I Model II Model III

5 MAPE () 155 138 080 094 091RMSE () 136 120 074 083 083

6 MAPE () 296 293 100 099 218RMSE () 212 211 082 081 171

7 MAPE () 109 102 074 073 112RMSE () 114 107 082 081 171

Note Some experimental results are obtained from [24 27] LGPFR Gaussian process functional regression with linear mean function [24] QGPFRGaussian process functional regressionwith quadratic polynomialmean function [24] Combination LGPFR LGPFRwith combination of squared exponentialcovariance function and periodic covariance function [24] Combination QGPFR QGPFR with combination of squared exponential covariance functionand periodic covariance function [24] SMK-GPR the GPR method with spectral mixture kernels [27] SE-MGPR multiscale GPR methods with squaredexponential function [27] P-MGPR multiscale GPR methods with periodic covariance function [27]

(a) Determine the training datasets 119894 119909119894119873119894=1 where 119894is the number of chargedischarge cycles and 119909119894is the corresponding value of SOH at the 119894thchargedischarge cycle For batteries No 5 No 6 andNo 7119873 equals 100

(b) Initialize the hyperparameters included in the meanfunction and covariance function for the differenttraining sets Here the initial hyperparameters forModel I are Θ1 = [119886 119887 l sf]119879 = [01 0 rand 2]119879for Model II are Θ2 = [119886 119887 l sf1 ell sf2]119879 =[01 0 rand 2 09 2]119879 and for Model III are Θ3 =[119886 119887 l sf1 ell p sf2]119879 = [05 1 rand 2 09 2 2]119879wherein signal ldquorandrdquo which is the value of parameterldquo119897rdquo in these threemodels is a randomnumber between0 and 1To compare with other methods the hyperparam-eter in basic GPR is selected as Θ1015840 = [l sf]119879 =[1 1]119879 and in combination LGPFR is Θ10158401015840 =[119886 119887 sf1 l sf2 ell 120596]119879 = [0 0 01 1 02 1 5]119879 whichis the same as the hyperparameters used in [24]

(c) Optimize the hyperparameters with the maximiza-tion of the log-likelihood function

(d) Set the prediction steps119898 for No 5 No 6 and No 7119898 = 68(e) Apply the training data 119894 119909119894119873119894=1 and the testing data119894 119909119894119873+119898119894=119873+1 to Models I II and III with the optimal

hyperparametersThen the prediction outputs will bederived

43 Prediction and Comparison In this part the predictionresults of five GPR models basic GPR combination LGPFR

and Models I II and III for batteries No 5 No 6 and No 7are shown in Figures 3ndash5 respectively As shown in Figure 3the red lines are the actual SOH value the blue scatter plotsare the prediction result and the grey regions represent the95 confidence intervals Taking Figure 3 for example forthe method of basic GPR shown in Figure 3(a) the meanprediction output is further away from the actual SOH for thesubsequent cycles Meanwhile the 95 confidence intervalsincrease significantly The above two aspects demonstratethe poor prediction effect of basic GPR For the methodof combination LGPFR shown in Figure 3(b) although theprediction values are close to the actual SOH data points the95 confidence intervals are still wide The wide confidenceintervals indicate the high uncertainty of the predictionresult Experiments have shown the limitations of Li-ionbatteriesrsquo SOH prediction based on the basic GPRmodel andthe combination LGPFRmodelThe simulationswithModelsI II and III for No 5 are shown in Figures 3(c)ndash3(e) it is clearthat both the point prediction results and the uncertaintyrepresentations are improved greatly

The concrete prediction errors comparisons of the fiveGPRmodels for three batteries are indicated in Table 1 Heretwo criteria of root mean square error (RMSE) and the meanabsolute percentage error (MAPE) are introduced to evaluatethe prediction performance which are shown as follows

RMSE = radicsum119898119894=1 (119910119894 minus 119910119894)2119898 MAPE = 1119898 119898sum

119894=1

10038161003816100381610038161003816100381610038161003816119910119894 minus 11991011989411991011989410038161003816100381610038161003816100381610038161003816

(18)

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

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120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

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Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

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Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

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Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

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95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

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Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

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()

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(b) Combination LGPFR

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()

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(c) Model I

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(d) Model II

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Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

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Mathematical Problems in Engineering 3

(1) From the perspective of remaining power of battery todetermine the battery SOH its definition can be givenby

SOH = 119876aged119876newtimes 100 (1)

where 119876aged is the current biggest power of batteryand 119876new is the initial power

(2) From the perspective of starting power to define thebattery SOH the expression is given by

SOH = CCAocmp minus CCAmin

CCAnew minus CCAmintimes 100 (2)

where CCAocmp is the real-time starting power andCCAnew is the predicted starting power released bybattery when the SOC is 100 and CCAmin is theminimum starting power needed

(3) From the perspective of impedance measurement todefine the battery SOH the definition is depicted asfollows

SOH = 1198771198941198770 times 100 (3)

where 119877119894 is 119894th impedance measurement varying withthe cycles of charging and discharging and 1198770 is theinitial impedance

(4) From the perspective of battery capacity power todefine the battery SOH this method can be expressedas

SOH = 1198621198941198620 times 100 (4)

where119862119894 is the 119894th capacitance value degenerated withcycles and 1198620 is the initial capacity

In this paper this definition of SOH using the capacity poweris utilized to study the performance of battery

3 Methodology

31 Gaussian Process Regression Gaussian process (GP) isused to describe random variables which are scalars orvectors GP is a stochastic process which also governs theproperties of functions and gets Bayesian inferences with thefunction-space view GP is a set where any finite randomvariable has joint Gaussian distribution and the charactersof GP are completely determined by the mean function andthe covariance function [28]119898(119909) = 119864 [119891 (119909)] 119896 (119909 1199091015840) = 119864 [(119898 (119909) minus 119891 (119909)) (119898 (1199091015840) minus 119891 (1199091015840))] (5)

Consider the simple regression model

119910 = 119891 (119909) + 120576 119891 (119909) = 119883119879119908 (6)

The training set is (119909119894 119910119894) | 119894 = 1 119899 where 119909119894 119894 =1 119899 are the input values 119891 is a function value and 119910119894119894 = 1 119899 are the output values 120576 represents noise whichfollows an independent identically distributed Gaussiandistribution with zero mean and variance 1205902 that is

120576 sim 119873 (0 1205902) (7)

Then the prior distribution of the observation 119910 is obtained119910 | 119883 119908 sim 119873(119883119879119908 119896 (119909 1199091015840) | 1205902119868) (8)

It can be used to estimate the predictive values The jointdistribution of the observation 119910 and the predictive values 119891lowastis derived as follows

( 119910119891lowast) sim 119873(0 ( 119896 (119883119883) 119896 (119883 119909lowast)119896 (119909lowast 119883) 119896 (119909lowast 119909lowast))) (9)

where 119896(119883 119909lowast) represents the covariance of the training datapoints and predictive values and 119896(119909lowast 119909lowast) is the covarianceof the predictive values

Computed by Bayesrsquo rule the posterior distribution overthe weights can be gotten The posterior distribution toinference in the Bayesian linear model can be shown as

119891lowast | 119883 119891 119909lowast sim 119873(119896 (119909lowast 119883) 119896 (119883119883)minus1 119891 119896 (119909lowast 119909lowast)minus 119896 (119909lowast 119883) 119896 (119883119883)minus1 119896 (119883 119909lowast)) (10)

32 Choice of Parameters GP is usually parameterized interms of their covariance functions There are some mostpopular covariance functions [23 29] the squared exponen-tial covariance function

119896 (119909 1199091015840) = 1205902119891 exp(minus(119909119894 minus 119909119895)221198972 ) (11)

the constant covariance function

119896 (119909 1199091015840) = 12059020 (12)

and the periodic covariance function

119896 (119909 1199091015840) = 1205902119891 (minus 21198972 sin2 ( 1206032120587 (119909119894 minus 119909119895))) (13)

These covariance functions are stationary and are usuallyused in battery health prognostic with good results Herein this paper a particular type of neural network is appliedin battery health prognostic which is called neural networkcovariance function see (7) Neural network is ideally suitedto the SOH data since it allows saturation at differentvalues in the positive and negative directions of 119909 [23] Forcomparison the predictive distributions for three covariancefunctions are shown in Figure 1 The 64 data points are pro-duced by a step function with Gaussian noise with standarddeviation 120590 = 01 which is totally the same as the exampletaken in [23] Figure 1 shows the means and 95 confidence

4 Mathematical Problems in Engineering

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14O

utpu

t y

Input x

(a) Squared exponential

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(b) Sum of two squared exponentials

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(c) Neural network

Figure 1 Misspecification example

intervals for the noisy signal in grey Particularly Figure 1(a)expresses the single squared exponential covariance andFigure 1(b) illustrates the sum of two squared exponentialcovariances Obviously this covariance function is moreflexible than a single squared exponential since it has twomagnitudes and two length-scale parameters The predictivedistribution looks a little bit better but they both are notideal for this scenario Figure 1(c) shows the neural networkcovariance function which is ideally suited to this case sinceit allows saturation at different values in the positive andnegative directions of 119909 As shown in Figure 1 the predictionsare also near perfect

New kernels can be generated with this kernel forexample the sum of the neural network covariance functionand the Maternard covariance function is a new kerneland the product of the neural network covariance func-tion and the periodic covariance function is also a newkernel The Maternard covariance function is stationaryand nondegenerate and the periodic covariance function is

selected to approximate the local regeneration phenomenonof SOH data Therefore these two covariance functions areused in combination with the neural network in this paperAccording to the covariance function chosen in our paperthe simulation results are better than othersrsquo It turns out thatthe neural network covariance function can be viewed as asuitable choice of this type of data The covariance functionsused in the proposed paper are as follows [23]

The neural network covariance function

1198961 (119909 1199091015840)= 1205902119891sinminus1( 119909119879Λminus21199091015840radic(1 + 119909119879Λminus2119909) (1 + 1199091015840119879Λminus21199091015840))

(14)

The Maternard covariance function

1198962 (119909 1199091015840) = 1205902119891119891119889 (119903119889) exp (minus119903119889) (15)

Mathematical Problems in Engineering 5

The periodic covariance function

1198963 (119909 1199091015840) = 1205902119891 exp(minus 21198972 sin212058710038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817119901 ) (16)

In the Maternard covariance function 119891 is a function of119903119889 1198911(119905) = 1 1198913(119905) = 1 + 119905 1198915(119905) = 1198913(119905) + 11990523 119903119889 is thedistance between 119909 and 1199091015840 and 119903119889 = radic119889(119909 minus 1199091015840)Λminus2(119909 minus 1199091015840)

Typically the used covariance function has some freeparameters which are generally called free hyperparametersAs these parameters vary the predictions of data are differentwhich will lead to great fitting error In order to get the suit-able hyperparameters optimizationwith themaximization ofthe log-likelihood function is necessary [28 30]

log119901 (119910 | 119883) = minus12119910119879 (119870 + 1205902119868)minus1 119910minus 12 log 10038161003816100381610038161003816119870 + 120590211986810038161003816100381610038161003816 minus 1198992 log 2120587

(17)

The hyperparameters can be determined from training dataNotice that the length-scale 119897 in the kernel varies and thesignal variance 1205902119891 and the noise variance 1205902 would varyThrough experiments it can be found that when differentinitial values are set different fitting curves are obtained andthe error between them would be great According to theproposed issue an appropriate initial value is chosen

A larger number of training data enable GPRmodel to getbetter prognostic prediction According to the experimentaldata the linear mean function 119898(119909) = 119886119909 + 119887 is selected tofit the track of the data The covariance functions discussedabove that is the neural network shown as (7) the sum ofneural network andMaternard given as 119896(119909 1199091015840) = 1198961(119909 1199091015840) +1198962(119909 1199091015840) and the product of neural network and periodicdisplayed as 119896(119909 1199091015840) = 1198961(119909 1199091015840)1198963(119909 1199091015840) are considered Thethree covariance functions are denoted as Model I Model IIand Model III Then the hyperparameters spaces in the threemodels are Θ1 = [119886 119887 l sf]119879 Θ2 = [119886 119887 l sf1 ell sf2]119879 andΘ3 = [119886 119887 l sf1 ell p sf2]119879 where 119886 119887 are coefficients of themean function Λ is 119897 times the unit matrix and sf sf1 andsf2 control the variance1205902119891 in threemodes respectivelyThesethree models are all effective and are designed to fit SOH dataand make the prediction more accurate

4 Experiments and Analysis

41 Raw Data Description of Li-Ion Batteries To comprehen-sively illustrate the efficiency of the proposed method thedataset about the aging of Li-ion batteries is selected whichis obtained from the data repository of the NASA AmesPrognostics Center of Excellence (PCoE) [31]

The dataset of the NASA Ames PCoE consists of 36Li-ion batteriesrsquo data No 5 No 6 No 7 No 18 andNos 25ndash56 Because the lengths of SOHs extracted fromthe 36 batteries are different Li-ion batteries of No 5No 6 and No 7 are the most commonly used in relatedliteratures [24 25 27] because they are much longer Theywere run through 3 different operational profiles (charge

0 20 40 60 80 100 120 140 16055

60

65

70

75

80

85

90

95

100

Cycle number

SOH

()

No 5No 6No 7

Figure 2 SOHs of Li-ion batteries No 5 No 6 and No 7

discharge and impedance) at room temperature Chargingwas carried out in a constant current (CC) mode at 15 Auntil the battery voltage reached 42 V and then continuedin a constant voltage (CV) mode until the charge currentdropped to 20mA Discharge was carried out at a constantcurrent (CC) level of 2 A until the battery voltage fell to27 V 25 V 22 V and 25 V for batteries Nos 5 6 7 and18 respectively Impedance measurement was carried outthrough an electrochemical impedance spectroscopy (EIS)frequency sweep from 01Hz to 5 kHz Repeated charge anddischarge cycles result in accelerated aging of the batterieswhile impedance measurements provide insight into theinternal battery parameters that change as aging progressesThe experiments were stopped when the batteries reachedend-of-life (EOL) criteria which was a 30 fade in ratedcapacity (from 2Ahr to 14 Ahr) This dataset can be used forthe prediction of SOC SOH and the remaining useful life(RUL) of batteries [23]

In this paper the data from batteries No 5 No 6 andNo 7 is processed using (4) to implement the experimentsincluding training and testing Figure 2 shows the SOHcurves of these three batteries The total chargedischargecycles are all 168 Figure 2 indicates that the SOH showsan obvious global degradation trend and local regenerationphenomenon

42 Training with GPR For No 5 No 6 and No 7 the first100 SOH points are utilized to train the proposed GPRmodeland the remaining 68 points are used for prediction Forthe training data from the above three batteries the meanfunction is chosen as the linear function 119898(119909) = 119886119909 + 119887 andthe covariance function is chosen as three functions discussedabove according to these three models the algorithm for thetraining and prediction based on GPR is produced as follows

6 Mathematical Problems in Engineering

Table 1 Prediction errors comparison of different methods for batteries Nos 5 6 and 7

(a)

Battery number Error criteria Basic GPR LGPFR QGPFR Combination LGPFR Combination QGPFR SMK-GPR

5 MAPE () 1210 230 190 160 210 165RMSE () 1303 171 150 136 180 138

6 MAPE () 270 1030 770 1020 290 1060RMSE () 2251 690 512 686 2044 708

7 MAPE () 1920 190 540 170 260 191RMSE () 2070 159 552 173 269 188

(b)

Battery number Error criteria P-MGPR SE-MGPR Model I Model II Model III

5 MAPE () 155 138 080 094 091RMSE () 136 120 074 083 083

6 MAPE () 296 293 100 099 218RMSE () 212 211 082 081 171

7 MAPE () 109 102 074 073 112RMSE () 114 107 082 081 171

Note Some experimental results are obtained from [24 27] LGPFR Gaussian process functional regression with linear mean function [24] QGPFRGaussian process functional regressionwith quadratic polynomialmean function [24] Combination LGPFR LGPFRwith combination of squared exponentialcovariance function and periodic covariance function [24] Combination QGPFR QGPFR with combination of squared exponential covariance functionand periodic covariance function [24] SMK-GPR the GPR method with spectral mixture kernels [27] SE-MGPR multiscale GPR methods with squaredexponential function [27] P-MGPR multiscale GPR methods with periodic covariance function [27]

(a) Determine the training datasets 119894 119909119894119873119894=1 where 119894is the number of chargedischarge cycles and 119909119894is the corresponding value of SOH at the 119894thchargedischarge cycle For batteries No 5 No 6 andNo 7119873 equals 100

(b) Initialize the hyperparameters included in the meanfunction and covariance function for the differenttraining sets Here the initial hyperparameters forModel I are Θ1 = [119886 119887 l sf]119879 = [01 0 rand 2]119879for Model II are Θ2 = [119886 119887 l sf1 ell sf2]119879 =[01 0 rand 2 09 2]119879 and for Model III are Θ3 =[119886 119887 l sf1 ell p sf2]119879 = [05 1 rand 2 09 2 2]119879wherein signal ldquorandrdquo which is the value of parameterldquo119897rdquo in these threemodels is a randomnumber between0 and 1To compare with other methods the hyperparam-eter in basic GPR is selected as Θ1015840 = [l sf]119879 =[1 1]119879 and in combination LGPFR is Θ10158401015840 =[119886 119887 sf1 l sf2 ell 120596]119879 = [0 0 01 1 02 1 5]119879 whichis the same as the hyperparameters used in [24]

(c) Optimize the hyperparameters with the maximiza-tion of the log-likelihood function

(d) Set the prediction steps119898 for No 5 No 6 and No 7119898 = 68(e) Apply the training data 119894 119909119894119873119894=1 and the testing data119894 119909119894119873+119898119894=119873+1 to Models I II and III with the optimal

hyperparametersThen the prediction outputs will bederived

43 Prediction and Comparison In this part the predictionresults of five GPR models basic GPR combination LGPFR

and Models I II and III for batteries No 5 No 6 and No 7are shown in Figures 3ndash5 respectively As shown in Figure 3the red lines are the actual SOH value the blue scatter plotsare the prediction result and the grey regions represent the95 confidence intervals Taking Figure 3 for example forthe method of basic GPR shown in Figure 3(a) the meanprediction output is further away from the actual SOH for thesubsequent cycles Meanwhile the 95 confidence intervalsincrease significantly The above two aspects demonstratethe poor prediction effect of basic GPR For the methodof combination LGPFR shown in Figure 3(b) although theprediction values are close to the actual SOH data points the95 confidence intervals are still wide The wide confidenceintervals indicate the high uncertainty of the predictionresult Experiments have shown the limitations of Li-ionbatteriesrsquo SOH prediction based on the basic GPRmodel andthe combination LGPFRmodelThe simulationswithModelsI II and III for No 5 are shown in Figures 3(c)ndash3(e) it is clearthat both the point prediction results and the uncertaintyrepresentations are improved greatly

The concrete prediction errors comparisons of the fiveGPRmodels for three batteries are indicated in Table 1 Heretwo criteria of root mean square error (RMSE) and the meanabsolute percentage error (MAPE) are introduced to evaluatethe prediction performance which are shown as follows

RMSE = radicsum119898119894=1 (119910119894 minus 119910119894)2119898 MAPE = 1119898 119898sum

119894=1

10038161003816100381610038161003816100381610038161003816119910119894 minus 11991011989411991011989410038161003816100381610038161003816100381610038161003816

(18)

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14O

utpu

t y

Input x

(a) Squared exponential

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(b) Sum of two squared exponentials

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

minus04

minus02

0

02

04

06

08

1

12

14

Out

put y

Input x

(c) Neural network

Figure 1 Misspecification example

intervals for the noisy signal in grey Particularly Figure 1(a)expresses the single squared exponential covariance andFigure 1(b) illustrates the sum of two squared exponentialcovariances Obviously this covariance function is moreflexible than a single squared exponential since it has twomagnitudes and two length-scale parameters The predictivedistribution looks a little bit better but they both are notideal for this scenario Figure 1(c) shows the neural networkcovariance function which is ideally suited to this case sinceit allows saturation at different values in the positive andnegative directions of 119909 As shown in Figure 1 the predictionsare also near perfect

New kernels can be generated with this kernel forexample the sum of the neural network covariance functionand the Maternard covariance function is a new kerneland the product of the neural network covariance func-tion and the periodic covariance function is also a newkernel The Maternard covariance function is stationaryand nondegenerate and the periodic covariance function is

selected to approximate the local regeneration phenomenonof SOH data Therefore these two covariance functions areused in combination with the neural network in this paperAccording to the covariance function chosen in our paperthe simulation results are better than othersrsquo It turns out thatthe neural network covariance function can be viewed as asuitable choice of this type of data The covariance functionsused in the proposed paper are as follows [23]

The neural network covariance function

1198961 (119909 1199091015840)= 1205902119891sinminus1( 119909119879Λminus21199091015840radic(1 + 119909119879Λminus2119909) (1 + 1199091015840119879Λminus21199091015840))

(14)

The Maternard covariance function

1198962 (119909 1199091015840) = 1205902119891119891119889 (119903119889) exp (minus119903119889) (15)

Mathematical Problems in Engineering 5

The periodic covariance function

1198963 (119909 1199091015840) = 1205902119891 exp(minus 21198972 sin212058710038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817119901 ) (16)

In the Maternard covariance function 119891 is a function of119903119889 1198911(119905) = 1 1198913(119905) = 1 + 119905 1198915(119905) = 1198913(119905) + 11990523 119903119889 is thedistance between 119909 and 1199091015840 and 119903119889 = radic119889(119909 minus 1199091015840)Λminus2(119909 minus 1199091015840)

Typically the used covariance function has some freeparameters which are generally called free hyperparametersAs these parameters vary the predictions of data are differentwhich will lead to great fitting error In order to get the suit-able hyperparameters optimizationwith themaximization ofthe log-likelihood function is necessary [28 30]

log119901 (119910 | 119883) = minus12119910119879 (119870 + 1205902119868)minus1 119910minus 12 log 10038161003816100381610038161003816119870 + 120590211986810038161003816100381610038161003816 minus 1198992 log 2120587

(17)

The hyperparameters can be determined from training dataNotice that the length-scale 119897 in the kernel varies and thesignal variance 1205902119891 and the noise variance 1205902 would varyThrough experiments it can be found that when differentinitial values are set different fitting curves are obtained andthe error between them would be great According to theproposed issue an appropriate initial value is chosen

A larger number of training data enable GPRmodel to getbetter prognostic prediction According to the experimentaldata the linear mean function 119898(119909) = 119886119909 + 119887 is selected tofit the track of the data The covariance functions discussedabove that is the neural network shown as (7) the sum ofneural network andMaternard given as 119896(119909 1199091015840) = 1198961(119909 1199091015840) +1198962(119909 1199091015840) and the product of neural network and periodicdisplayed as 119896(119909 1199091015840) = 1198961(119909 1199091015840)1198963(119909 1199091015840) are considered Thethree covariance functions are denoted as Model I Model IIand Model III Then the hyperparameters spaces in the threemodels are Θ1 = [119886 119887 l sf]119879 Θ2 = [119886 119887 l sf1 ell sf2]119879 andΘ3 = [119886 119887 l sf1 ell p sf2]119879 where 119886 119887 are coefficients of themean function Λ is 119897 times the unit matrix and sf sf1 andsf2 control the variance1205902119891 in threemodes respectivelyThesethree models are all effective and are designed to fit SOH dataand make the prediction more accurate

4 Experiments and Analysis

41 Raw Data Description of Li-Ion Batteries To comprehen-sively illustrate the efficiency of the proposed method thedataset about the aging of Li-ion batteries is selected whichis obtained from the data repository of the NASA AmesPrognostics Center of Excellence (PCoE) [31]

The dataset of the NASA Ames PCoE consists of 36Li-ion batteriesrsquo data No 5 No 6 No 7 No 18 andNos 25ndash56 Because the lengths of SOHs extracted fromthe 36 batteries are different Li-ion batteries of No 5No 6 and No 7 are the most commonly used in relatedliteratures [24 25 27] because they are much longer Theywere run through 3 different operational profiles (charge

0 20 40 60 80 100 120 140 16055

60

65

70

75

80

85

90

95

100

Cycle number

SOH

()

No 5No 6No 7

Figure 2 SOHs of Li-ion batteries No 5 No 6 and No 7

discharge and impedance) at room temperature Chargingwas carried out in a constant current (CC) mode at 15 Auntil the battery voltage reached 42 V and then continuedin a constant voltage (CV) mode until the charge currentdropped to 20mA Discharge was carried out at a constantcurrent (CC) level of 2 A until the battery voltage fell to27 V 25 V 22 V and 25 V for batteries Nos 5 6 7 and18 respectively Impedance measurement was carried outthrough an electrochemical impedance spectroscopy (EIS)frequency sweep from 01Hz to 5 kHz Repeated charge anddischarge cycles result in accelerated aging of the batterieswhile impedance measurements provide insight into theinternal battery parameters that change as aging progressesThe experiments were stopped when the batteries reachedend-of-life (EOL) criteria which was a 30 fade in ratedcapacity (from 2Ahr to 14 Ahr) This dataset can be used forthe prediction of SOC SOH and the remaining useful life(RUL) of batteries [23]

In this paper the data from batteries No 5 No 6 andNo 7 is processed using (4) to implement the experimentsincluding training and testing Figure 2 shows the SOHcurves of these three batteries The total chargedischargecycles are all 168 Figure 2 indicates that the SOH showsan obvious global degradation trend and local regenerationphenomenon

42 Training with GPR For No 5 No 6 and No 7 the first100 SOH points are utilized to train the proposed GPRmodeland the remaining 68 points are used for prediction Forthe training data from the above three batteries the meanfunction is chosen as the linear function 119898(119909) = 119886119909 + 119887 andthe covariance function is chosen as three functions discussedabove according to these three models the algorithm for thetraining and prediction based on GPR is produced as follows

6 Mathematical Problems in Engineering

Table 1 Prediction errors comparison of different methods for batteries Nos 5 6 and 7

(a)

Battery number Error criteria Basic GPR LGPFR QGPFR Combination LGPFR Combination QGPFR SMK-GPR

5 MAPE () 1210 230 190 160 210 165RMSE () 1303 171 150 136 180 138

6 MAPE () 270 1030 770 1020 290 1060RMSE () 2251 690 512 686 2044 708

7 MAPE () 1920 190 540 170 260 191RMSE () 2070 159 552 173 269 188

(b)

Battery number Error criteria P-MGPR SE-MGPR Model I Model II Model III

5 MAPE () 155 138 080 094 091RMSE () 136 120 074 083 083

6 MAPE () 296 293 100 099 218RMSE () 212 211 082 081 171

7 MAPE () 109 102 074 073 112RMSE () 114 107 082 081 171

Note Some experimental results are obtained from [24 27] LGPFR Gaussian process functional regression with linear mean function [24] QGPFRGaussian process functional regressionwith quadratic polynomialmean function [24] Combination LGPFR LGPFRwith combination of squared exponentialcovariance function and periodic covariance function [24] Combination QGPFR QGPFR with combination of squared exponential covariance functionand periodic covariance function [24] SMK-GPR the GPR method with spectral mixture kernels [27] SE-MGPR multiscale GPR methods with squaredexponential function [27] P-MGPR multiscale GPR methods with periodic covariance function [27]

(a) Determine the training datasets 119894 119909119894119873119894=1 where 119894is the number of chargedischarge cycles and 119909119894is the corresponding value of SOH at the 119894thchargedischarge cycle For batteries No 5 No 6 andNo 7119873 equals 100

(b) Initialize the hyperparameters included in the meanfunction and covariance function for the differenttraining sets Here the initial hyperparameters forModel I are Θ1 = [119886 119887 l sf]119879 = [01 0 rand 2]119879for Model II are Θ2 = [119886 119887 l sf1 ell sf2]119879 =[01 0 rand 2 09 2]119879 and for Model III are Θ3 =[119886 119887 l sf1 ell p sf2]119879 = [05 1 rand 2 09 2 2]119879wherein signal ldquorandrdquo which is the value of parameterldquo119897rdquo in these threemodels is a randomnumber between0 and 1To compare with other methods the hyperparam-eter in basic GPR is selected as Θ1015840 = [l sf]119879 =[1 1]119879 and in combination LGPFR is Θ10158401015840 =[119886 119887 sf1 l sf2 ell 120596]119879 = [0 0 01 1 02 1 5]119879 whichis the same as the hyperparameters used in [24]

(c) Optimize the hyperparameters with the maximiza-tion of the log-likelihood function

(d) Set the prediction steps119898 for No 5 No 6 and No 7119898 = 68(e) Apply the training data 119894 119909119894119873119894=1 and the testing data119894 119909119894119873+119898119894=119873+1 to Models I II and III with the optimal

hyperparametersThen the prediction outputs will bederived

43 Prediction and Comparison In this part the predictionresults of five GPR models basic GPR combination LGPFR

and Models I II and III for batteries No 5 No 6 and No 7are shown in Figures 3ndash5 respectively As shown in Figure 3the red lines are the actual SOH value the blue scatter plotsare the prediction result and the grey regions represent the95 confidence intervals Taking Figure 3 for example forthe method of basic GPR shown in Figure 3(a) the meanprediction output is further away from the actual SOH for thesubsequent cycles Meanwhile the 95 confidence intervalsincrease significantly The above two aspects demonstratethe poor prediction effect of basic GPR For the methodof combination LGPFR shown in Figure 3(b) although theprediction values are close to the actual SOH data points the95 confidence intervals are still wide The wide confidenceintervals indicate the high uncertainty of the predictionresult Experiments have shown the limitations of Li-ionbatteriesrsquo SOH prediction based on the basic GPRmodel andthe combination LGPFRmodelThe simulationswithModelsI II and III for No 5 are shown in Figures 3(c)ndash3(e) it is clearthat both the point prediction results and the uncertaintyrepresentations are improved greatly

The concrete prediction errors comparisons of the fiveGPRmodels for three batteries are indicated in Table 1 Heretwo criteria of root mean square error (RMSE) and the meanabsolute percentage error (MAPE) are introduced to evaluatethe prediction performance which are shown as follows

RMSE = radicsum119898119894=1 (119910119894 minus 119910119894)2119898 MAPE = 1119898 119898sum

119894=1

10038161003816100381610038161003816100381610038161003816119910119894 minus 11991011989411991011989410038161003816100381610038161003816100381610038161003816

(18)

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

The periodic covariance function

1198963 (119909 1199091015840) = 1205902119891 exp(minus 21198972 sin212058710038171003817100381710038171003817119909 minus 119909101584010038171003817100381710038171003817119901 ) (16)

In the Maternard covariance function 119891 is a function of119903119889 1198911(119905) = 1 1198913(119905) = 1 + 119905 1198915(119905) = 1198913(119905) + 11990523 119903119889 is thedistance between 119909 and 1199091015840 and 119903119889 = radic119889(119909 minus 1199091015840)Λminus2(119909 minus 1199091015840)

Typically the used covariance function has some freeparameters which are generally called free hyperparametersAs these parameters vary the predictions of data are differentwhich will lead to great fitting error In order to get the suit-able hyperparameters optimizationwith themaximization ofthe log-likelihood function is necessary [28 30]

log119901 (119910 | 119883) = minus12119910119879 (119870 + 1205902119868)minus1 119910minus 12 log 10038161003816100381610038161003816119870 + 120590211986810038161003816100381610038161003816 minus 1198992 log 2120587

(17)

The hyperparameters can be determined from training dataNotice that the length-scale 119897 in the kernel varies and thesignal variance 1205902119891 and the noise variance 1205902 would varyThrough experiments it can be found that when differentinitial values are set different fitting curves are obtained andthe error between them would be great According to theproposed issue an appropriate initial value is chosen

A larger number of training data enable GPRmodel to getbetter prognostic prediction According to the experimentaldata the linear mean function 119898(119909) = 119886119909 + 119887 is selected tofit the track of the data The covariance functions discussedabove that is the neural network shown as (7) the sum ofneural network andMaternard given as 119896(119909 1199091015840) = 1198961(119909 1199091015840) +1198962(119909 1199091015840) and the product of neural network and periodicdisplayed as 119896(119909 1199091015840) = 1198961(119909 1199091015840)1198963(119909 1199091015840) are considered Thethree covariance functions are denoted as Model I Model IIand Model III Then the hyperparameters spaces in the threemodels are Θ1 = [119886 119887 l sf]119879 Θ2 = [119886 119887 l sf1 ell sf2]119879 andΘ3 = [119886 119887 l sf1 ell p sf2]119879 where 119886 119887 are coefficients of themean function Λ is 119897 times the unit matrix and sf sf1 andsf2 control the variance1205902119891 in threemodes respectivelyThesethree models are all effective and are designed to fit SOH dataand make the prediction more accurate

4 Experiments and Analysis

41 Raw Data Description of Li-Ion Batteries To comprehen-sively illustrate the efficiency of the proposed method thedataset about the aging of Li-ion batteries is selected whichis obtained from the data repository of the NASA AmesPrognostics Center of Excellence (PCoE) [31]

The dataset of the NASA Ames PCoE consists of 36Li-ion batteriesrsquo data No 5 No 6 No 7 No 18 andNos 25ndash56 Because the lengths of SOHs extracted fromthe 36 batteries are different Li-ion batteries of No 5No 6 and No 7 are the most commonly used in relatedliteratures [24 25 27] because they are much longer Theywere run through 3 different operational profiles (charge

0 20 40 60 80 100 120 140 16055

60

65

70

75

80

85

90

95

100

Cycle number

SOH

()

No 5No 6No 7

Figure 2 SOHs of Li-ion batteries No 5 No 6 and No 7

discharge and impedance) at room temperature Chargingwas carried out in a constant current (CC) mode at 15 Auntil the battery voltage reached 42 V and then continuedin a constant voltage (CV) mode until the charge currentdropped to 20mA Discharge was carried out at a constantcurrent (CC) level of 2 A until the battery voltage fell to27 V 25 V 22 V and 25 V for batteries Nos 5 6 7 and18 respectively Impedance measurement was carried outthrough an electrochemical impedance spectroscopy (EIS)frequency sweep from 01Hz to 5 kHz Repeated charge anddischarge cycles result in accelerated aging of the batterieswhile impedance measurements provide insight into theinternal battery parameters that change as aging progressesThe experiments were stopped when the batteries reachedend-of-life (EOL) criteria which was a 30 fade in ratedcapacity (from 2Ahr to 14 Ahr) This dataset can be used forthe prediction of SOC SOH and the remaining useful life(RUL) of batteries [23]

In this paper the data from batteries No 5 No 6 andNo 7 is processed using (4) to implement the experimentsincluding training and testing Figure 2 shows the SOHcurves of these three batteries The total chargedischargecycles are all 168 Figure 2 indicates that the SOH showsan obvious global degradation trend and local regenerationphenomenon

42 Training with GPR For No 5 No 6 and No 7 the first100 SOH points are utilized to train the proposed GPRmodeland the remaining 68 points are used for prediction Forthe training data from the above three batteries the meanfunction is chosen as the linear function 119898(119909) = 119886119909 + 119887 andthe covariance function is chosen as three functions discussedabove according to these three models the algorithm for thetraining and prediction based on GPR is produced as follows

6 Mathematical Problems in Engineering

Table 1 Prediction errors comparison of different methods for batteries Nos 5 6 and 7

(a)

Battery number Error criteria Basic GPR LGPFR QGPFR Combination LGPFR Combination QGPFR SMK-GPR

5 MAPE () 1210 230 190 160 210 165RMSE () 1303 171 150 136 180 138

6 MAPE () 270 1030 770 1020 290 1060RMSE () 2251 690 512 686 2044 708

7 MAPE () 1920 190 540 170 260 191RMSE () 2070 159 552 173 269 188

(b)

Battery number Error criteria P-MGPR SE-MGPR Model I Model II Model III

5 MAPE () 155 138 080 094 091RMSE () 136 120 074 083 083

6 MAPE () 296 293 100 099 218RMSE () 212 211 082 081 171

7 MAPE () 109 102 074 073 112RMSE () 114 107 082 081 171

Note Some experimental results are obtained from [24 27] LGPFR Gaussian process functional regression with linear mean function [24] QGPFRGaussian process functional regressionwith quadratic polynomialmean function [24] Combination LGPFR LGPFRwith combination of squared exponentialcovariance function and periodic covariance function [24] Combination QGPFR QGPFR with combination of squared exponential covariance functionand periodic covariance function [24] SMK-GPR the GPR method with spectral mixture kernels [27] SE-MGPR multiscale GPR methods with squaredexponential function [27] P-MGPR multiscale GPR methods with periodic covariance function [27]

(a) Determine the training datasets 119894 119909119894119873119894=1 where 119894is the number of chargedischarge cycles and 119909119894is the corresponding value of SOH at the 119894thchargedischarge cycle For batteries No 5 No 6 andNo 7119873 equals 100

(b) Initialize the hyperparameters included in the meanfunction and covariance function for the differenttraining sets Here the initial hyperparameters forModel I are Θ1 = [119886 119887 l sf]119879 = [01 0 rand 2]119879for Model II are Θ2 = [119886 119887 l sf1 ell sf2]119879 =[01 0 rand 2 09 2]119879 and for Model III are Θ3 =[119886 119887 l sf1 ell p sf2]119879 = [05 1 rand 2 09 2 2]119879wherein signal ldquorandrdquo which is the value of parameterldquo119897rdquo in these threemodels is a randomnumber between0 and 1To compare with other methods the hyperparam-eter in basic GPR is selected as Θ1015840 = [l sf]119879 =[1 1]119879 and in combination LGPFR is Θ10158401015840 =[119886 119887 sf1 l sf2 ell 120596]119879 = [0 0 01 1 02 1 5]119879 whichis the same as the hyperparameters used in [24]

(c) Optimize the hyperparameters with the maximiza-tion of the log-likelihood function

(d) Set the prediction steps119898 for No 5 No 6 and No 7119898 = 68(e) Apply the training data 119894 119909119894119873119894=1 and the testing data119894 119909119894119873+119898119894=119873+1 to Models I II and III with the optimal

hyperparametersThen the prediction outputs will bederived

43 Prediction and Comparison In this part the predictionresults of five GPR models basic GPR combination LGPFR

and Models I II and III for batteries No 5 No 6 and No 7are shown in Figures 3ndash5 respectively As shown in Figure 3the red lines are the actual SOH value the blue scatter plotsare the prediction result and the grey regions represent the95 confidence intervals Taking Figure 3 for example forthe method of basic GPR shown in Figure 3(a) the meanprediction output is further away from the actual SOH for thesubsequent cycles Meanwhile the 95 confidence intervalsincrease significantly The above two aspects demonstratethe poor prediction effect of basic GPR For the methodof combination LGPFR shown in Figure 3(b) although theprediction values are close to the actual SOH data points the95 confidence intervals are still wide The wide confidenceintervals indicate the high uncertainty of the predictionresult Experiments have shown the limitations of Li-ionbatteriesrsquo SOH prediction based on the basic GPRmodel andthe combination LGPFRmodelThe simulationswithModelsI II and III for No 5 are shown in Figures 3(c)ndash3(e) it is clearthat both the point prediction results and the uncertaintyrepresentations are improved greatly

The concrete prediction errors comparisons of the fiveGPRmodels for three batteries are indicated in Table 1 Heretwo criteria of root mean square error (RMSE) and the meanabsolute percentage error (MAPE) are introduced to evaluatethe prediction performance which are shown as follows

RMSE = radicsum119898119894=1 (119910119894 minus 119910119894)2119898 MAPE = 1119898 119898sum

119894=1

10038161003816100381610038161003816100381610038161003816119910119894 minus 11991011989411991011989410038161003816100381610038161003816100381610038161003816

(18)

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Table 1 Prediction errors comparison of different methods for batteries Nos 5 6 and 7

(a)

Battery number Error criteria Basic GPR LGPFR QGPFR Combination LGPFR Combination QGPFR SMK-GPR

5 MAPE () 1210 230 190 160 210 165RMSE () 1303 171 150 136 180 138

6 MAPE () 270 1030 770 1020 290 1060RMSE () 2251 690 512 686 2044 708

7 MAPE () 1920 190 540 170 260 191RMSE () 2070 159 552 173 269 188

(b)

Battery number Error criteria P-MGPR SE-MGPR Model I Model II Model III

5 MAPE () 155 138 080 094 091RMSE () 136 120 074 083 083

6 MAPE () 296 293 100 099 218RMSE () 212 211 082 081 171

7 MAPE () 109 102 074 073 112RMSE () 114 107 082 081 171

Note Some experimental results are obtained from [24 27] LGPFR Gaussian process functional regression with linear mean function [24] QGPFRGaussian process functional regressionwith quadratic polynomialmean function [24] Combination LGPFR LGPFRwith combination of squared exponentialcovariance function and periodic covariance function [24] Combination QGPFR QGPFR with combination of squared exponential covariance functionand periodic covariance function [24] SMK-GPR the GPR method with spectral mixture kernels [27] SE-MGPR multiscale GPR methods with squaredexponential function [27] P-MGPR multiscale GPR methods with periodic covariance function [27]

(a) Determine the training datasets 119894 119909119894119873119894=1 where 119894is the number of chargedischarge cycles and 119909119894is the corresponding value of SOH at the 119894thchargedischarge cycle For batteries No 5 No 6 andNo 7119873 equals 100

(b) Initialize the hyperparameters included in the meanfunction and covariance function for the differenttraining sets Here the initial hyperparameters forModel I are Θ1 = [119886 119887 l sf]119879 = [01 0 rand 2]119879for Model II are Θ2 = [119886 119887 l sf1 ell sf2]119879 =[01 0 rand 2 09 2]119879 and for Model III are Θ3 =[119886 119887 l sf1 ell p sf2]119879 = [05 1 rand 2 09 2 2]119879wherein signal ldquorandrdquo which is the value of parameterldquo119897rdquo in these threemodels is a randomnumber between0 and 1To compare with other methods the hyperparam-eter in basic GPR is selected as Θ1015840 = [l sf]119879 =[1 1]119879 and in combination LGPFR is Θ10158401015840 =[119886 119887 sf1 l sf2 ell 120596]119879 = [0 0 01 1 02 1 5]119879 whichis the same as the hyperparameters used in [24]

(c) Optimize the hyperparameters with the maximiza-tion of the log-likelihood function

(d) Set the prediction steps119898 for No 5 No 6 and No 7119898 = 68(e) Apply the training data 119894 119909119894119873119894=1 and the testing data119894 119909119894119873+119898119894=119873+1 to Models I II and III with the optimal

hyperparametersThen the prediction outputs will bederived

43 Prediction and Comparison In this part the predictionresults of five GPR models basic GPR combination LGPFR

and Models I II and III for batteries No 5 No 6 and No 7are shown in Figures 3ndash5 respectively As shown in Figure 3the red lines are the actual SOH value the blue scatter plotsare the prediction result and the grey regions represent the95 confidence intervals Taking Figure 3 for example forthe method of basic GPR shown in Figure 3(a) the meanprediction output is further away from the actual SOH for thesubsequent cycles Meanwhile the 95 confidence intervalsincrease significantly The above two aspects demonstratethe poor prediction effect of basic GPR For the methodof combination LGPFR shown in Figure 3(b) although theprediction values are close to the actual SOH data points the95 confidence intervals are still wide The wide confidenceintervals indicate the high uncertainty of the predictionresult Experiments have shown the limitations of Li-ionbatteriesrsquo SOH prediction based on the basic GPRmodel andthe combination LGPFRmodelThe simulationswithModelsI II and III for No 5 are shown in Figures 3(c)ndash3(e) it is clearthat both the point prediction results and the uncertaintyrepresentations are improved greatly

The concrete prediction errors comparisons of the fiveGPRmodels for three batteries are indicated in Table 1 Heretwo criteria of root mean square error (RMSE) and the meanabsolute percentage error (MAPE) are introduced to evaluatethe prediction performance which are shown as follows

RMSE = radicsum119898119894=1 (119910119894 minus 119910119894)2119898 MAPE = 1119898 119898sum

119894=1

10038161003816100381610038161003816100381610038161003816119910119894 minus 11991011989411991011989410038161003816100381610038161003816100381610038161003816

(18)

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

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Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 condidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17055

60

65

70

75

80

85

90

95

Cycle number for No 5

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 3 Prediction results of five GPR models for battery No 5

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

100 110 120 130 140 150 160 170minus80minus60minus40minus20

020406080

100120

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 170404550556065707580859095

Cycle number for No 6

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 4 Prediction results of five GPR models for battery No 6

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

100 110 120 130 140 150 160 170minus40

minus20

0

20

40

60

80

100

120

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on basic GPRActual SOH

(a) Basic GPR

100 110 120 130 140 150 160 17050

60

70

80

90

100

110

120

130

Cycle number for No 7

SOH

()

95 confidence boundsPrediction based on combination LGPFRActual SOH

(b) Combination LGPFR

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IActual SOH

(c) Model I

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIActual SOH

(d) Model II

100 110 120 130 140 150 160 17060

65

70

75

80

85

90

95

100

Cycle number for No 7

SOH

()

95 confidence boundsPrediction using Model IIIActual SOH

(e) Model III

Figure 5 Prediction results of five GPR models for battery No 7

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

where 119898 is the prediction steps 119910119894 represents the actualSOH and 119910119894 represents the predicted SOH Table 1 showsthe prognostic RMSEs () and MAPEs () of differentSOH prediction methods respectively Compared with theother popular published methods in [24 27] both RMSEand MAPE values have been reduced greatly It is foundthat for three batteries the prognostics accuracies of SMK-GPR SE-MGRP and P-MGPR in [27] are improved evidentlycompared with the combination LGPFR and combinationQGPFR in [24] It is also seen that for the proposed ModelsI II and III the maximum prediction errors of batteriesNos 5 6 and 7 are averagely less by 50 than SE-MGRPand P-MGPR models respectively Furthermore the MAPEand RMSE values are less than 1 for three batteries usingModel II In conclusion it is found that the proposedmethodshows a much better prediction performance than the othermethods The small values of MAPE and RMSE indicate thatthe proposed approach can be suitable to perform accurateSOH prediction and satisfy the engineering requirement

The prediction results for No 6 and No 7 are similar toNo 5rsquos The prediction results of five GPR models are shownin Figures 4 and 5

From the above three experiments for different Li-ionbatteries the efficiency of the proposed method is verified

5 Conclusions

In this paper a battery SOH estimation approach based onGPRNN is presented In order to improve the long-termprediction accuracy and uncertainty based on basic GPR andcombination GPFR and multiscale GPR three GPR modelsbased on neural network are used Compared with the othermixturemethods of GPR andmultiscale GPR algorithms theproposed GPRNNmodels are simpler therefore they satisfythe need of real-time prognostics Moreover the experimen-tal simulations show that the proposed models have higherprediction accuracy and lower prediction uncertainty for Li-ion batteries with the complicated phenomenon To furtherverify the efficiency of the proposed algorithm using datasetwith greater number of degradation data samples in realindustrial applications is necessary Finally exploring themethodologywith better prognostic performances will be thefuture work

Abbreviations

SOH State of healthSOC State of chargeGPRNN Gaussian process regression with neural

networkGPR Gaussian process regressionPF Particle FilteringSVM Support Vector MachineRVM Relevant Vector MachineEKF Extended Kalman filterRMSE Root mean square errorMAPE Mean absolute percentage error

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Key Researchand Development Program of China under Grant no2016YFF0201204 the National Natural Science Foundationof China (61501148) and the Natural Science Foundation ofHeilongjiang Province (QC2017075)

References

[1] I-S Kim ldquoA technique for estimating the state of health oflithium batteries through a dual-sliding-mode observerrdquo IEEETransactions on Power Electronics vol 25 no 4 pp 1013ndash10222010

[2] Y Nishi ldquoLithium ion secondary batteries Past 10 years and thefuturerdquo Journal of Power Sources vol 100 no 1-2 pp 101ndash1062001

[3] D Liu J Pang J Zhou and Y Peng ldquoData-driven prognosticsfor lithium-ion battery based on Gaussian process regressionrdquoin Proceedings of the 3rd Annual IEEE Prognostics and SystemHealth Management Conference PHM-2012 Beijing ChinaMay 2012

[4] J Zhang and J Lee ldquoA review on prognostics and healthmonitoring of Li-ion batteryrdquo Journal of Power Sources vol 196no 15 pp 6007ndash6014 2011

[5] B Saha S Poll K Goebel and J Christophersen ldquoAn inte-grated approach to battery health monitoring using bayesianregression and state estimationrdquo in Proceedings of the IEEEAutotestcon pp 646ndash653 September 2007

[6] R Xiong F Sun H He and T D Nguyen ldquoA data-drivenadaptive state of charge and power capability joint estimator oflithium-ion polymer battery used in electric vehiclesrdquo Energyvol 63 pp 295ndash308 2013

[7] X Hu S E Li Z Jia and B Egardt ldquoEnhanced sample entropy-based health management of Li-ion battery for electrifiedvehiclesrdquo Energy vol 64 pp 953ndash960 2014

[8] L Pei C Zhu T Wang R Lu and C C Chan ldquoOnline peakpower prediction based on a parameter and state estimator forlithium-ion batteries in electric vehiclesrdquo Energy vol 66 pp766ndash778 2014

[9] 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

[10] B E Olivares M A Cerda Munoz M E Orchard andJ F Silva ldquoParticle-filtering-based prognosis framework forenergy storage devices with a statistical characterization ofstate-of-health regeneration phenomenardquo IEEE Transactions onInstrumentation and Measurement vol 62 no 2 pp 364ndash3762013

[11] Y Xing E W M Ma K L Tsui and M Pecht ldquoBatterymanagement systems in electric and hybrid vehiclesrdquo Energiesvol 4 no 11 pp 1840ndash1857 2011

[12] M E Orchard L Tang B Saha K Goebel and G Vachtse-vanos ldquoRisk-sensitive particle-filtering-based prognosis frame-work for estimation of remaining useful life in energy storage

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

devicesrdquo Studies in Informatics and Control vol 19 no 3 pp209ndash218 2010

[13] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[14] C Hu B D Youn and J Chung ldquoA multiscale frameworkwith extended Kalman filter for lithium-ion battery SOC andcapacity estimationrdquoApplied Energy vol 92 pp 694ndash704 2012

[15] W He NWilliardM Osterman andM Pecht ldquoPrognostics oflithium-ion batteries based on Dempster-Shafer theory and theBayesian Monte Carlo methodrdquo Journal of Power Sources vol196 no 23 pp 10314ndash10321 2011

[16] D Andre C Appel T Soczka-Guth andDU Sauer ldquoAdvancedmathematical methods of SOC and SOH estimation forlithium-ion batteriesrdquo Journal of Power Sources vol 224 pp 20ndash27 2013

[17] 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

[18] H F Oai X Z Wei and Z C Sun ldquoA new SOH predictionconcept for the ower lithium-ion battery used on HEVsrdquo inProceedings of the Vehicle Power and Propulsion Conference pp1649ndash1653 2009

[19] K Qian C Zhou Y Yuan and M Allan ldquoTemperature effecton electric vehicle battery cycle life in vehicle-to-grid applica-tionsrdquo in Proceedings of the China International Conference onElectricity Distribution pp 1ndash6 IEEE Press 2010

[20] A J Salkind C Fennie P Singh T Atwater and D EReisner ldquoDetermination of state-of-charge and state-of-healthof batteries by fuzzy logic methodologyrdquo Journal of PowerSources vol 80 no 1 pp 293ndash300 1999

[21] B Pattipati K Pattipati J P Christopherson S M NamburuD V Prokhorov and L Qiao ldquoAutomotive batterymanagementsystemsrdquo in Proceedings of the IEEE Autotestcon 2008 pp 581ndash586 Salt Lake City Utah USA September 2008

[22] H Chen and X Li ldquoDistributed active learning with applicationto battery health managementrdquo in Proceedings of the 14th Inter-national Conference on Information Fusion pp 1ndash7 Chicago IllUSA 2011

[23] C E Rasmussen and C K I Williams Gaussian Process forMachine Learning The MIT Press Boston Mass USA 2006

[24] D Liu J Pang J Zhou Y Peng and M Pecht ldquoPrognosticsfor state of health estimation of lithium-ion batteries based oncombination Gaussian process functional regressionrdquo Micro-electronics Reliability vol 53 no 6 pp 832ndash839 2013

[25] F Li and J Xu ldquoA new prognostics method for state of healthestimation of lithium-ion batteries based on a mixture ofGaussian process models and particle filterrdquo MicroelectronicsReliability vol 55 no 7 pp 1035ndash1045 2015

[26] D Le and X D Tang ldquoLithium-ion Battery State of HealthEstimation Using Ah-V Characterizationrdquo in Proceedings of theAnnual Conference of the Prognostics and Health ManagementSociety pp 367ndash373 2011

[27] Y-J He J-N Shen J-F Shen and Z-F Ma ldquoState of healthestimation of lithium-ion batteries A multiscale Gaussianprocess regression modeling approachrdquo AIChE Journal vol 61no 5 pp 1589ndash1600 2015

[28] C K I Williams and C E Rasmussen ldquoGaussian processesfor regressionrdquo in Advances in neural information processingsystems vol 8 MIT Press 1996

[29] D MacKay Introduction to Gaussian Processes CambridgeUniversity Cambridge UK 1997

[30] J Q Shi B Wang R Murray-Smith and D M TitteringtonldquoGaussian process functional regression modeling for batchdatardquo Biometrics vol 63 no 3 pp 714ndash723 2007

[31] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom