Assessing the Performance Degradation of Lithium-Ion

Research ArticleAssessing the Performance Degradation of Lithium-IonBatteries Using an Approach Based on Fusion of MultipleFeature Parameters

AnchenWang1 Ying Zhang 12 and Hongfu Zuo2

1College of Automobile and Traffic Engineering Nanjing Forestry University Nanjing 210037 China2College of Civil Aviation Nanjing University of Aeronautics and Astronautics Nanjing 210037 China

Correspondence should be addressed to Ying Zhang zhangyingrms163com

Anchen Wang and Ying Zhang contributed equally to this work

Received 28 January 2019 Revised 17 April 2019 Accepted 2 May 2019 Published 19 May 2019

Academic Editor Konstantinos Karamanos

Copyright copy 2019 Anchen Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Amethod based on fusion ofmultiple features is proposed to assess and accurately describe the performance degradation of lithium-ion batteries in this paper First the discharge voltage signal of lithium-ion batteries under real-time monitoring is analyzed fromthe perspective of time domain and complexity to obtain the values ofmultiple featuresThen themulti-feature parameters undergoa spectral regression process to reduce the number of dimensions and to eliminate redundancy and on the basis of this regressiona Gaussian mixture model is established to model the health state of batteries Thus the degree of lithium-ion battery performancedegradation can be quantitatively assessed using the Bayesian inference-based distance metric A case calculation experiment iscarried out to verify the effectiveness of the method proposed in this paper The experimental results demonstrate that comparedwith other assessment methods the performance degradation assessment method proposed in this paper can be used to monitorthe degradation process of lithium-ion batteries more effectively and to improve the accuracy of condition monitoring of batteriesthereby providing powerful support for making maintenance decisions

1 Introduction

As a kind of ideal power supply energy which can bereused lithium-ion battery has been widely used in manyfields However the performance of lithium battery willdecline over time which will not only lead to system fail-ure but also lead to serious safety accidents [1] Workingcondition monitoring performance degradation assessmentand remaining useful life (RUL) prediction of lithium-ionbattery are of great significance for improving the safetyand reliability of the battery system and preventing batteryaccidents The RUL prediction of lithium-ion battery canbe divided into two steps battery health status recognitionand performance prediction In terms of prediction modelZou [2] used parameter degradation model and tracelessparticle filter algorithm to predict RUL of battery discoveringthat the method for predicting the RUL of the battery was

more accurate when considering the degradation of theperformance parameters of the battery capacity state Ma[3] proposed an improved trackless Kalman filter based onMarkov chain-Monte Carlo to predict RUL and an empiricalmodel of four-parameter power function reflecting batterycapacitywas established Linear least squaremethodwas usedto estimate capacity parameters and finally life predictionwas realized with high prediction accuracy Yang [4] usedtwo logarithmic empirical capacity models to capture two-phase nonlinear sag degradation problem of lithium-ionbatteries and an improved particle filter algorithm wasused to update the modelrsquos state parameters to predict theRUL The prediction model has achieved good results butmostly used the battery capacity or single parameter as afeature to predict However the battery capacity is difficultto directly measure online and a single parameter cannotdescribe the performance degradation of battery Therefore

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3091071 12 pageshttpsdoiorg10115520193091071

2 Mathematical Problems in Engineering

it is advantageous to accurately reflect the state of batteryperformance and improve the prediction accuracy of residuallife by extracting characteristic parameters from multipleperspectives and fusing them into a health index and thispaper focuses on performance degradation assessment oflithium batteries

To avoid problems such as insufficient maintenance orover-maintenance of lithium-ion batteries it is necessary tocarry out condition-based maintenance for batteries Thequantitative assessment of the degree of performance degra-dation is a prerequisite for condition-based maintenance oflithium-ion batteries It is difficult to measure the residualcapacity of batteries in real time [5] many researcherstry to assess the performance of the battery using theapproach of analyzing variations in the discharge voltageSbarufatti proposed a method that combines particle filtersand radial basis function neural networks to predict thedischarge duration of lithium-ion batteries [6] Xu proposedan effective model based on state space for modeling theperformance degradation of batteries The model adoptsan analysis method based on the sequentially observeddischarging profiles and uses the expectation-maximization(EM) algorithm in conjunction with the extended Kalmanfilter (EKF) algorithm to estimate and update the modelparameters and state [7] Xu also analyzed the distributionof discharge voltage and combined the discharge process andthe degradation process to develop a hierarchical model forcomprehensive prediction of discharge duration in differentcycles and the remaining effective cycles [8] Widodo usedthe sample entropy of the discharge voltage to predict theperformance of the battery [9]

Although some of the methods proposed in the abovestudies can be used tomake effective assessment they still fallshort of the expectation of accurately and comprehensivelyassessing the degradation process of the batteries This isbecause most of these methods only process a single fea-ture of the discharge voltage which is not very sensitiveto variations in the battery state To solve this challengethis paper proposes a method based on the fusion ofmultiple features for assessing the performance degradationof lithium-ion batteries In fact the multi-feature fusionapproach has been widely used in assessing the performanceof bearings but relatively fewer attempts have been madeto adopt this approach for assessing the performance ofbatteries

In researching the performance degradation assessmenteffectively extracting multi-dimensional features and estab-lishing an effective degradation assessment model are twocritical issues A lot of work has been done with regard tothe first critical issue Principal component analysis (PCA)is one of the most widely used data compression algorithmsDong used PCA to reduce the dimensions of the parametersin bearing degradation process and they obtained goodresults [10] Yu proposed a method based on local preservingprojection (LPP) to reduce the dimensions of the multi-dimensional features in the time domain frequency domainand time-frequency domain The method is applied to theperformance degradation assessment of bearings and verifiedthe feasibility of the method through experiments [11] Cai

proposed a spectral regression (SR) method which is anew type of model for supervised unsupervised and semi-supervised regression Using SR to reduce the dimensionsmakes it possible to solve the projection function usingthe regression framework thereby circumventing the thornyissue of decomposing the features of the dense matrix [12]With regard to the second critical issue Yu proposed amethod based on multiple two-layer sparse autoencoderneural networks to fuse the features and used experimentto verify the superiority of this method [13] Yu appliedthe Gaussian mixture model (GMM) model to the healthassessment of bearings and achieved quantitative analysisof performance degradation of the bearings [11] Batistaproposed an algorithm based on improved support vectormachine (SVM) reducing the error rate in diagnosis [14]Ai proposed a fusion method based on n-dimensionalcharacteristic parameters distance [15] Hassani proposed afusion model utilizes zSlices-based representation of generaltype-2 fuzzy logic systems to combine different SVMs [16]Zhang proposed an exergy-based fusion method to fusemultiple transient states with the information of multiplesensors obtained in state monitoring [17] Shao proposeda feature fusion method based on the deep belief network(DBN) taking advantage of the excellent learning ability ofDBN [18] Zhou proposed an NCA-based feature extractionmethod to reduce dimensionality and used a coupled hiddenMarkov model based on multichannel data acquisition forfault diagnosis [19] In addition Wang [20] also summarizedmethods to evaluate the health indicators based on Markovmodel self-organizingmapping wavelet change and fuzzy c-means transformation All of the abovemethods have yieldedgood assessment results

On the basis of the above analysis this paper proposesa performance degradation assessment method based onthe fusion of multiple features First the discharge voltagesignals of lithium-ion batteries under real-time monitoringare analyzed to obtain features in the dimensions of timedomain and complexity Then the multiple features undergoan SR processing to reduce the number of dimensions andeliminate redundancy based on these processes GMM isestablished to model the health status of batteries Thusthe degree of performance degradation of batteries can bequantitatively assessed using the Bayesian inference-baseddistance (BID) metric The experimental results show thatthe assessment method proposed in this paper can be usedto monitor the degradation process of lithium-ion batteriesmore effectively and to improve the accuracy of conditionmonitoring of batteries To the best of our knowledge singlecharacteristic parameters are used in most studies of lithiumbattery status assessment However there is redundancyand difference among parameters so the characterization oflithium battery state is not accurate enough Therefore inthis paper the characteristic parameters are extracted fromthe perspectives of statistical measurement and complexityafter that the extracted features are fused by the GMM-BID model proposed in this paper Compared with a singleparameter the health index can characterize the performancedegradation degree of lithium battery more accurately

Mathematical Problems in Engineering 3

2 Feature Extraction

The capacity of battery is a parameter that can accuratelyevaluate the performance but it is difficult to obtain directlywhich can only be obtained in the laboratory or offline stateTherefore it is necessary to select some parameters thatcan be monitored online to estimate the degradation stateof lithium-ion batteries During the charging process thecharging mode of the battery is generally fixed that is thecharging voltage and current are also fixed The dischargecurrent depends on the system payload which can affect thehealth of the battery but cannot reflect the degradation ofthe health state of the battery At present most studies havefound that the discharge voltage of the battery will tend todegrade with an increase in the number of discharge cycles[6ndash11 21] Therefore this paper focuses on analyzing theeasy-to-measure discharge voltage of lithium-ion batteries tomake a more accurate assessment of the battery performancedegradation

21 Time Domain Features The voltage values of all of thedischarge cycles of a lithium-ion batteries can be regarded asa time series of voltage It is necessary to analyze this timeseries to extract the features that reflect the statistical patternof the time series The most common method for analyzinga time series is time domain analysis Charakopoulos AK[22] estimated mainly nonlinear measures such as mutualinformation combinedwith descriptive statisticsmeasures aswell as some linear and nonlinear dynamic detectors such asHurst exponent detrended fluctuation analysis and Hjorthparameters

With the increase of charging and discharging timesof lithium-ion battery its internal resistance and energyloss increase which shows that the discharge duration willbe shorter and shorter after the battery is fully chargedTherefore the article adopted the method of statistical mea-surement such as the mean and RMS values to describe thechanging trend of battery discharge voltage Also regardingthe trend of the battery voltage at the same time intervalincreasing with the number of charge and discharge cyclesfixed-interval voltage difference can be constructed to char-acterize the degree of battery performance degradation

211 Statistical Measurement There are some conventionaltime domain features based on statistical measurement(see Table 1)

212 Fixed-Interval Voltage Difference When a series of dis-charge voltage values are obtained by sampling the dischargevoltage at fixed intervals the voltage discrepancy betweentwo neighboring samples is called the fixed-interval voltagedifference As the battery is used successive times (that isthe number of discharge cycles of a lithium-ion batteryincreases) the battery ages continuously which is reflectedby the increasingly shorter discharge duration During eachdischarge cycle the discharge voltage of a battery decreasesfrom a higher voltage value to a lower value

Each fixed time interval is denoted as 119905 and the fixed-interval voltage difference in the 119894-th discharge cycle is

Table 1 Statistical measurement

Feature Equation

Mean 119909119898119890119886119899 = sum119873119899=1 119909(119899)119873

Root meansquare 119909119903119898119904 = radicsum

119873119899=1 (119909 (119899))2119873

Standarddeviation 119909119904119905119889 = radicsum

119873119899=1 (119909 (119899) minus 119909119898)2119873 minus 1

Skewness 119909119904119896119890 = sum119873119899=1 (119909 (119899) minus 119909119898)3(119873 minus 1)1199093

119904119905119889

Kurtosis 119909119896119906119903 = sum119873119899=1 (119909 (119899) minus 119909119898)4(119873 minus 1)1199094

119904119905119889119909(119899) is the voltage value

119881119894 = 119881119905119898119886119909 minus 119881119905119898119894119899 119894 = 1 2 119899 (1)

where119881119905119898119886119909 is the voltage value at the end of the time interval119905 and 119881119905119898119894119899 is the voltage value at the beginning of thetime interval 119905 The sequence of fixed-interval voltage valuesextracted from the discharge cycles can then be expressed as

119881 = 1198811 1198812 119881119899 (2)

where 119899 is the number of discharge cycles

22 Complexity Entropy is a way to describe the complexityof a time series or the degree of internal chaos Originallya concept in physics the concept of entropy graduallyexpanded with the development of information theory andhas become an indicator of system instability Entropy is nowwidely used in system state monitoring and signal analysisAt present the various forms of entropy (such as informationentropy sample entropy and approximate entropy) are usedas the characteristic parameters for measuring complexity

Derived from the approximate entropy the sampleentropy provides a better entropy representation in sig-nal analysis Compared with other metrics such as relateddimensions sample entropy is better suited for classifyingcomplex systems including characterization of the certaintyand randomness of the time series of finite data points [9]Sample entropy has been applied in bearing fault diagnosis[23ndash25] Therefore in this paper we chose sample entropyas a useful complement to the characteristic parameters Thealgorithm works in the following way(1) Assume that there is an 119873-dimensional time series119906(1) 119906(2) 119906(119873) obtained via sampling at fixed timeintervals(2) Define the associated parameters 119898 119903 of the sampleentropy where the real number 119898 is the length of thecomparative vector and the real number 119903 is the metric ofsimilarity(3) Reconstruct the 119898-dimensional vector 119883(1) 119883(2) 119883(119873minus119898+1)where119883(119894) = [119906(119894) 119906(119894+1) 119906(119894+119898minus1)]


(4) For 1 le 119894 le 119873 minus 119898 + 1 count the number of vectorsthat satisfy the following condition

119861119898119894 (119903)= 119899119906119898119887119890119903 of119883(119895) 119904119906119888ℎ 119905ℎ119886119905 119889 [119883 (119894) 119883 (119895)] le 119903119873 minus 119898

119894 = 119895(3)

where 119889[119883(119894) 119883(119895)] is defined as

119889 [119883 (119894) 119883 (119895)] = max119886

1003816100381610038161003816119906 (119886) minus 119906lowast (119886)1003816100381610038161003816 119883 (119894) = 119883 (119895) (4)

where 119906(119886) is an element of the vector 119883 119889 is the distancebetween the vectors 119883(119894) and 119883(119895) is determined from themaximum difference of the corresponding elements and thevalue range of 119895 is [1119873 minus 119898 + 1] (119894 = 119895)(5) Calculate the average of 119861119898119894 (119903) for all 119894 values anddenote the average value as 119861119898(119903) which can be expressedas

119861119898 (119903) = (119873 minus 119898 + 1)minus1 119873minus119898+1sum119894=1

119861119898119894 (119903) (5)

(6) Let 119896 = 119898 + 1 and repeat steps (3) and (4) to obtain119860119896 (119903) = (119873 minus 119896 + 1)minus1 119873minus119896+1sum

119894=1

119860119896119894 (119903) (6)

where 119860119896119894 (119903) = (119899119906119898119887119890119903 of119883(119895) 119904119906119888ℎ 119905ℎ119886119905 119889[119883(119894) 119883(119895)] le119903)(119873 minus 119896) 119894 = 119895(7)The sample entropy is defined as

119878119886119898119901119864119899 = lim119899997888rarrinfin

[minus ln 119860119896 (119903)119861119898 (119903)] (7)

In practical applications 119873 will never beinfin Thereforewhen 119873 is a finite value the sample entropy is estimated tobe

119878119886119898119901119864119899 = minus ln 119860119896 (119903)119861119898 (119903) (8)

23 Spectral Regression Given a data set 1199111 1199112 119911119898 isin 119877119899the mapping element of ℎ1 ℎ2 ℎ119898 isin 119877119897(119897 ltlt 119899) can befound in the low dimension space for a given point119898 and wecan find a transformation matrix 119860 that satisfies ℎ119894 = 119860119879119911119894

A graph 119866 was constructed with 119898 samples using theidea of graph embedding a process of mapping the dataof a graph (usually a high dimensional dense matrix) intoa low dimensional dense vector Each vertex of the graphcorresponds to a sample (the 119894-th point is 119911119894) If all of theweights of the connections between sample 119894 and sample 119895are represented by a weight matrix119882 then119882119894119895 is the weightof the connection between the 119894-th point and 119895-th point Theprocess of finding the linear projection function using the

SR method is divided into two steps (1) Solve 119882119910 = 120582119863119910where D is the diagonal matrix 119863119894119895 = sum119895119882119894119895 The featurevector corresponding to the first 119897 feature value is defined as1199100 1199101 119910119897minus1 (2) Obtain the projection vector 119886 by solvingthe following regularized least-square

119886 = argmin119886[ 119898sum119894=1

(a119879119911119894 minus 119910119894)2 + 120582 1198862] (9)

where 120582 is the regularized parameter The original data canthen be dimension-reduced to an 119897-dimensional vector

119911 997888rarr ℎ = 119860119879119911 119860 = (1198860 1198861 119886119897minus1) (10)

3 Performance Degradation AssessmentModel Based on Fusion of Multiple Features

Data fusion can be defined as a combination of multiplesources to obtain improved information in this contextimproved information means less expensive higher qualityor more relevant informationThe available data fusion tech-niques can be classified into three nonexclusive categories (i)data association (ii) state estimation and (iii) decision fusion[26]The data fusionmethod described in this paper is a stateestimation fusion method

31 GMM GMM is a clustering algorithm that uses a Gaus-sian distribution as the parameter model and is trained usingthe EM algorithm As a simple extension of the Gaussianmodel GMM uses a combination of multiple Gaussiandistributions to characterize the distribution of data whichis expressed as

119901 (119909) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (11)

where 119872 is the number of model mixing 120587119898 is a weightcoefficient of the model (sum120587119898 = 1) 119901(119909 | 120579119898) isthe Gaussian distribution function (the 119898-th average valueis 120583119898 and the covariance matrix is 119878119898) denoted as 120601 =1205871 120587119898 1205831 120583119898 1198781 119878119898 The model can be rewrit-ten in the following form

119901 (119909 | 120601) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (12)

After the GMMmodel was established using the data setsfor the normal operation state a quantitative indicator basedon BID [27] was used to perform quantitative assessment ofthe degradation degree of lithium-ion batteries

Suppose that there are119870Gaussian classes The 119896-th com-ponent can be denoted as 119862119896 and its occurrence probabilitycan be denoted as 120572119896 The probability of the test point 119909119905 thenbelongs to the 119896 component 119862119896 (denoted as 119901(119862119896 | 119909119905)) andcan be calculated using the following formula

119901 (119862119896 | 119909119905) = 120572119896119901 (119909119905 | 119862119896)119901 (119909119905) = 120572119896119901 (119909119905 | 119862119896)sum119870119894=1 120572119894119901 (119909119905 | 119862119896) (13)


Train data Feature extraction

Spectral regression

Establish a healthy state GMM model

Calculate the BID value of the test point

Get parameters from GMM

modelTest data


Spectralregression

Establish ahealthy stateGMM model

Calculate theBID value of the test point

Get parametersfrom GMM

modelTest data Feature

extraction

Offline

Online

Figure 1 Flowchart of proposed performance degradation assessment method

where 120572119896 is the a priori probability that can be calculatedusing the model data 119901(119909119905 | 119862119896) can be calculated using thefollowing formula

119901 (119909119905 | 119862119896)= 1(2120587)12 1003816100381610038161003816119878119896100381610038161003816100381612 exp [minus

12 (119909119905 minus 120583119896)119879 119878minus1119896 (119909119905 minus 120583119896)](14)

where 120583119896 is the average value of the 119896-th Gaussiancomponent 119878119896 is the covariance matrix of the 119896-th Gaussiancomponent Next the distance from 119909119905 to each component119862119896is defined as

119863119862119896 (119909119905) = (119909119905 minus 120583119896)119879 119878minus1119896 (119909119905 minus 120583119896) (15)

The BID value can be obtained by performing a weightedsummation over all of the component distance values of a testpoint

119861119868119863 = 119870sum119896=1

119901 (119862119896 | 119909119905)119863119862119896 (119909119905) (16)

32 Performance Degradation Assessment Model Based onFusion of Multiple Features On the basis of the aboveanalysis this paper proposes a method based on the fusion ofmultiple features for assessing the performance of lithium-ionbatteries The flowchart of the method is shown in Figure 1

Figure 1 shows that the performance degradation assess-ment process is divided into an offline phase and onlinephase The specific steps of each phase are as follows

(1) Off-Line Phase (A) Extract initial features from thetraining data anduse the SRmethod to reduce the dimensionsof the original features

(B) Select the health state training data to establish aGMMmodel and obtain the parameters of the model

(C) Calculate the BID value of the test data using theGMM parameters to describe the performance degradationprocess of lithium-ion batteries

(2) Online Phase (A) Extract initial features from the trainingdata and use the SR method to reduce the dimensions of theoriginal features

(B) Calculate the BID value between the test data and thehealth state GMM

4 Lithium-Ion Battery PerformanceDegradation Assessment Experiment andResults Analysis

41 Data Source A case calculation experiment was carriedout to verify the reliability and effectiveness of the proposedmethod in assessing the performance degradation of lithium-ion batteries In the experiment the experimental data setswere provided by NASA [28] A set of four Li-ion batteries( 5 6 7 and 18) were run through 3 different operationalprofiles (charge discharge and impedance) at room tem-perature Charging was carried out in a constant current(CC) mode at 15A until the battery voltage reached 42Vand then continued in a constant voltage (CV) mode untilthe charge current dropped to 20mA Discharge was carriedout at a constant current (CC) level of 2A until the batteryvoltage fell to 27V 25V 22V and 25V for batteries 5 6 7and 18 respectively Impedancemeasurement was carried outthrough an electrochemical impedance spectroscopy (EIS)frequency sweep from 01Hz to 5kHz 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 14Ahr)

In order to save space the batteries No 5 No6 and No18 of the four batteries are randomly selected for analysisFigure 2 depicts the relationship between the dischargevoltage and the discharge time of the three batteries As seenin the figure the different discharge cycles share a commonpattern In the early stage of discharge the discharge voltagedeclines gradually and the discharge time has a gentle slopeThe slope of the decline increases gradually over time Inthe last stage of discharge the voltage drops rapidly Asthe number of discharge cycles increases the slope of thedecline of the discharge voltage becomes increasingly steepThe variation of the discharge voltage in each discharge cyclediffers from the voltage variation in any other discharge cyclehence it is necessary to assess and manage the battery

42 Comparison of Clustering Ability Concerning DegradationDegree In the experiment 132 original data sets of batterynumber 18 are extracted After a normalization process 25sets of normal state data 25 sets of slight degradation dataand 25 sets of severe degradation data are used to verify


Battery Number 5

Cycle30Cycle60Cycle90

Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)

(a) Variation of discharge voltage of battery number 5

Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)

(b) Variation of discharge voltage of battery number 6

Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)

(c) Variation of discharge voltage of battery number 18

Figure 2 Variation of discharge voltage of battery numbers 5 6 and 18

the clustering ability and correctness of the SR dimensionreduction method proposed in this paper Furthermore theSR algorithm is compared with the LPP and PCA algo-rithms Figure 3 depicts the effects of the three algorithmsin dimension reduction (only the first two dimensions areshown for each algorithm) To compare the three featureextraction methods more directly the clustering algorithmK-means is used as the degradation degree classifier tocalculate the cluster recognition rates of the three methodsThe results are shown in Table 2 Figure 3 shows that afterthe dimension reduction the PCA method failed to clusterthe three states (ie the three states overlap with each other)which means that PCA has the worst clustering effect amongthe three algorithms The LPP and SR algorithms have betterclustering effects as indicated by their ability to clearlydistinguish the three states Between these two algorithmsSR is superior to LPP Table 2 also shows that PCA has thelowest cluster recognition rate among the three algorithmsThe cluster recognition rate of SR is 9733 which is 4higher than that of the LPP algorithm In summary theSR dimension reduction algorithm has stronger clusteringability and recognition rateTherefore the SRmethod ismoresuitable for on-line monitoring

Table 2 Clustering recognition rate of the three dimension reduc-tion methods

Dimension reduction algorithm Recognition ratePCA 9067LPP 9333SR 9733

43 Results of Performance Degradation Assessment Sevenrepresentative features (namely root mean square (RMS)standard deviation mean skewness kurtosis fixed-intervalvoltage difference and sample entropy) are used to assessthe performance degradation of battery numbers 5 6 and 18using the existing data sets After several trials the numberof the length of the comparative vector and the metric ofsimilarity of sample entropy are set to 119898 = 1 119903 = 01respectively The variation curves of the six features areshown in Figure 4 The following can be observed fromthe diagrams For the same battery different features exhibitdifferent variation trends In terms of overall trends RMSmean and kurtosis tend to decline with fluctuations whereasthe standard deviation skewness fixed-interval voltage


NormalSlight degradation

Severe degradationCenter of K-Means

minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1

(a) Results of PCA-based clustering

minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1

(b) Results of LPP-based clustering

minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1

(c) Results of SR-based clustering

Figure 3 Clustering results of the three dimension reduction methods

difference and sample entropy tend to rise with fluctuationsTherefore the extracted features could be used to characterizethe performance of Lithium-ion batteries however thesefeatures have some drawbacks Standard deviation skewnessand kurtosis fluctuate violently and are not sensitive to thedegradation of performance making it difficult to determinethe degree of degradation It is observed that the standarddeviation skewness kurtosis and sample entropy of batterynumbers 5 and 6 have a ldquorun-inrdquo period in the earlystage during which the values fluctuate violently without asignificant trend When the battery is diagnosed during thisldquorun-inrdquo period it is hard to see a trend towards degradation

To enable more accurate online monitoring of the batterydegradation this paper proposes a quantitative indicatorobtained by fusing multiple multi-dimensional features toeffectively describe battery degradation

As soon as the battery is used it begins to degrade Inorder to reduce the error of the health state model the first4 of all cycles were selected for training On the basis ofthe above analysis the first 4 sets of health state data ofbattery numbers 5 6 and 18 are chosen to establish theGMM model A GMM with 2 Gaussian components is usedto model the normal state data and the first two dimensionsafter dimension reduction by SR are used as the input dataAfter the GMM model is trained with the normal state

data all of the test data is input into the model in groupsand one BID value is calculated using each data group asshown in Figures 5 6 and 7 Figure 5 shows that the BIDvalue of battery number 5 effectively and clearly reflects theperformance degradation process of the lithium-ion batterySimilarly the trend towards battery degradation is reflectedin Figures 6 and 7The BID curve increases with the increaseof the number of battery discharge cycles

The assessment method based on SR-GMM-BID is supe-rior to the methods that use only a single feature FirstBID value is sensitive to the performance degradation of thebattery and the trend of lithium-ion battery degradation canbe clearly seen Second the fused indicator does not fluctuateviolently in the early stage and improves the reliability ofassessment

44 Comparative Analysis

441 Comparison with the Assessment Method Based onthe Support Vector Data Description (SVDD) Algorithm TheSVDDalgorithm is an algorithm that is well-suited for assess-ing the performance of bearings [29] In the experimentthe first 4 of all data groups were taken as the normalstate data to train the SVDD model the kernel functionwas defined as a Gaussian kernel and the penalty factor


34342344346348

35352354356358

36

RMS

20 40 60 80 100 120 140 160 1800Number of discharge cycle

Battery No5Battery No6Battery No18

(a) Variation of RMS values with an increase in thenumber of discharge cycles


021022023024025026027028

Stan

dard

dev

iatio

n


(b) Variation of standard deviation values with an increasein the number of discharge cycles


34342344346348

35352354356358

Mea

n


(c) Variation of mean values with an increase in thenumber of discharge cycles


3456789

1011

Kurt

osis


(d) Variation of kurtosis values with an increase in thenumber of discharge cycles


Skew

ness


minus18minus16minus14minus12minus1

minus08minus06minus04minus02

(e) Variation of skewness values with an increase in thenumber of discharge cycles


06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)


(f) Variation of fixed-interval voltage difference valueswith an increase in the number of discharge cycles


0001002003004005006007008009

01

Sam

ple e

ntro

py


(g) Variation of sample entropy values with an increase inthe number of discharge cycles

Figure 4 Variation trends of the 6 features exhibited by battery numbers 5 6 and 18


times104

0 20 40 60 80 100 120 140 160 1800

Number of discharge cycle

05

1

15

2

25

3

BID

Figure 5 Assessing battery number 5 using BID values

0 20 40 60 80 100 120 140 160 180Number of discharge cycle

0

05

1

15

2

25

3

35

BID

times105


0 20 40 60 80 100 120 140Number of discharge cycle

0

5

10

15

BID

times1018


was set to 03 The value of the kernel distance betweenthe test data and the trained hypersphere was used asthe quantitative performance assessment parameter and theassessment results based on SR-SVDD are shown in Figures8 9 and 10

442 Comparison with the Assessment Method Based onAnother GMM Based Indicator A negative logarithmic like-lihood probability (NLLP) value was used to evaluate theperformance of the bearing [30] that is after establishing theGMM model in normal state a logarithmic likelihood valueis obtained for each set of test data The larger the value isthe farther the deviation is In general logarithmic likelihoodvalues are less than 0 so negative logarithmic likelihoodvalues are used as assessment parameter and the assessment


0

001

002

003

004

005

006

007

DI

Figure 8 Assessing battery number 5 using the SVDD-basedmethod


0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI


results based on SR-GMM-NLLP are shown in Figures 11 12and 13

443 Comparison with the Assessment Method Based onSelf-Organizing Map(SOM) A data fusion method based onSOM is used to evaluate the bearing state [31] SOMwas usedfor network training and multi-feature fusion model wasconstructed After weight vector was obtained the minimumquantization error (MQE) from each sample to the weightvector was calculated as the indicator and the assessmentresults based on SOM are shown in Figures 14 15 and 16

444 e Results of Comparative Analysis Spearman rankcorrelation coefficient is a typical analytical method todescribe the linear relationship between two data sequences


Table 3 Results of correlation analysis

Battery No 5 Battery No 6 Battery No 18BID -09969 -09949 -09926DI -09926 -09627 -09105NLLP -09941 -09790 -09418MQE -09927 -09483 -09288


minus20

0

20

40

60

80

100

NLL

P

Figure 11 Assessing battery number 5 using the NLLP-basedmethod


minus200

20406080

100120140160180

NLL

P


[32] In order to prove the superiority of themethod proposedin this paper the four fusion parameters and battery capacitywere respectively analyzed by Spearman correlation Theinterval of coefficient r is [-1 1] When r is greater than 0it means positive correlation when r is less than 0 it meansnegative correlation The closer the absolute value of r is to 1the more linearly dependent it is

The correlation analysis results are shown in Table 3 Thefour health indicators are negative correlated with batterycapacity The absolute value of correlation coefficient of BIDis significantly higher than DI NLLP and MQE It indicatesthat the BID index has the best correlation with the batterycapacity and can be used for the input of battery remaininglife prediction (RUL) to improve the prognosis accuracy

5 Conclusions

This paper proposes a model based on fusion of multiplefeatures for assessing performance degradation of lithium-ion batteries The study yielded the following findings


minus40minus20

020406080

100120140

NLL

PFigure 13 Assessing battery number 18 using the NLLP-basedmethod


0

002

004

006

008

01

012

MQ

E

Figure 14 Assessing battery number 5 using the SOM-basedmethod


0002004006008

01012014016018

MQ

E


(1) The method based on fusion of multiple featuresproposed in this paper can be used to conduct effectiveassessment of the performance degradation of lithium-ionbatteries(2) Compared with the PCA and LPP methods thedimension reduction method based on SR can better cluster



0

002

004

006

008

01

012

014

MQ

E


the measurement data into different degradation stages forlithium-ion batteries(3) The SR-GMM-BID-based quantitative performanceassessment method proposed in this paper can better char-acterize the degradation trend of batteries than the SVDDNLLP and SOM based methods

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare no conflicts of interest The funders hadno role in the design of the study in the collection analysesor interpretation of data in the writing of the manuscript orin the decision to publish the results

Authorsrsquo Contributions

Anchen Wang and Ying Zhang contributed equally to thiswork Conceptualization is done by Anchen Wang YingZhang and Hongfu Zuo Data curation is done by AnchenWang and Ying Zhang Formal analysis is done by AnchenWang and Ying Zhang Methodology is done by AnchenWang Ying Zhang and Hongfu Zuo Software is done byAnchenWang and Ying Zhang Validation is done by HongfuZuo Writing ndash original draft is done by Anchen WangWriting ndash review amp editing are done by Ying Zhang andHongfu Zuo

Acknowledgments

This research was supported by National Natural ScienceFoundation of China grant number 51505229

References

[1] 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

[2] Y Zou X Hu H Ma and S E Li ldquoCombined State of Chargeand State of Health estimation over lithium-ion battery cellcycle lifespan for electric vehiclesrdquo Journal of Power Sources vol273 pp 793ndash803 2015

[3] Y Ma Y Chen X Zhou and H Chen ldquoRemaining usefullife prediction of lithium-ion battery based on gauss-hermiteparticle filterrdquo IEEETransactions onControl Systems Technologyvol 3 pp 1ndash8 2018

[4] F Yang D Wang Y Xing and K-L Tsui ldquoPrognosticsof Li(NiMnCo)O2-based lithium-ion batteries using a novelbattery degradation modelrdquoMicroelectronics Reliability vol 70pp 70ndash78 2017

[5] M A Hannan M S H Lipu A Hussain and A MohamedldquoA review of lithium-ion battery state of charge estimationand management system in electric vehicle applications Chal-lenges and recommendationsrdquo Renewable amp Sustainable EnergyReviews vol 78 pp 834ndash854 2017

[6] C Sbarufatti M Corbetta M Giglio and F Cadini ldquoAdaptiveprognosis of lithium-ion batteries based on the combinationof particle filters and radial basis function neural networksrdquoJournal of Power Sources vol 344 pp 128ndash140 2017

[7] X Xu and N Chen ldquoA state-space-based prognostics modelfor lithium-ion battery degradationrdquo Reliability Engineering ampSystem Safety vol 159 pp 47ndash57 2017

[8] X Xu Z Li and N Chen ldquoA hierarchical model for lithium-ion battery degradation predictionrdquo IEEE Transactions onReliability vol 65 no 1 pp 310ndash325 2016

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

[10] S Dong and T Luo ldquoBearing degradation process predictionbased on the PCA and optimized LS-SVM modelrdquo Measure-ment vol 46 no 9 pp 3143ndash3152 2013

[11] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[12] D Cai X He and J Han ldquoSpectral regressiona unifiedapproach for sparse subspace learningrdquo in IEEE InternationalConference on Data Mining pp 73ndash82 IEEE 2007

[13] Z Chen andW Li ldquoMultisensor feature fusion for bearing faultdiagnosis using sparse autoencoder and deep belief networkrdquoIEEE Transactions on Instrumentation and Measurement vol66 no 7 pp 1693ndash1702 2017

[14] L Batista B Badri R Sabourin and M Thomas ldquoA classifierfusion system for bearing fault diagnosisrdquo Expert Systems withApplications vol 40 no 17 pp 6788ndash6797 2013

[15] Y T Ai J Y Guan W C Fei et al ldquoFusion informationentropy method of rolling bearing fault diagnosis based onn -dimensional characteristic parameter distancerdquo MechanicalSystems amp Signal Processing vol 88 pp 123ndash136 2017

[16] H Hassani J Zarei M M Arefi and R Razavi-Far ldquoZSlices-based general type-2 fuzzy fusion of support vector machineswith application to bearing fault detectionrdquo IEEE Transactionson Industrial Electronics vol 64 no 9 pp 7210ndash7217 2017

[17] B Zhang L Zhang J Xu and P Wang ldquoPerformance degrada-tion assessment of rolling element bearings based on an indexcombining SVD and information exergyrdquo Entropy vol 16 no10 pp 5400ndash5415 2014

[18] H Shao H Jiang X Zhang and M Niu ldquoRolling bearingfault diagnosis using an optimization deep belief networkrdquo


Measurement Science and Technology vol 26 no 11 Article ID115002 2015

[19] H T Zhou J Chen G M Dong H C Wang and HD Yuan ldquoBearing fault recognition method based on neigh-bourhood component analysis and coupled hidden Markovmodelrdquo Mechanical Systems and Signal Processing Article IDS0888327015002204 2015

[20] D Wang K-L Tsui and Q Miao ldquoPrognostics and healthmanagement a review of vibration based bearing and gearhealth indicatorsrdquo IEEE Access vol 6 pp 665ndash676 2017

[21] D H Liu H Wang Y Peng W Xie and H T Liao ldquoSatellitelithium-ion battery remaining cycle life prediction with novelindirect health indicator extractionrdquo Energies vol 6 no 8 pp3654ndash3668 2013

[22] A K Charakopoulos T E Karakasidis P N Papanicolaou andA Liakopoulos ldquoNonlinear time series analysis and clusteringfor jet axis identification in vertical turbulent heated jetsrdquoPhysical Review E Statistical Nonlinear and SoMatter Physicsvol 89 no 3 Article ID 032913 2014

[23] M Han and J Pan ldquoA fault diagnosis method combined withLMD sample entropy and energy ratio for roller bearingsrdquoMeasurement vol 76 pp 7ndash19 2015

[24] M Seera M L D Wong and A K Nandi ldquoClassification ofball bearing faults using a hybrid intelligentmodelrdquoApplied SoComputing vol 57 pp 427ndash435 2017

[25] Y Gao M Karimi A A Kudreyko and W Song ldquoSpareoptimistic based on improved ADMM and the minimumentropy de-convolution for the early weak fault diagnosis ofbearings in marine systemsrdquo ISA Transactions Article IDS0019057817306377 2017

[26] C Federico ldquoA review of data fusion techniquesrdquoe ScientificWorld Journal vol 2013 Article ID 704504 19 pages 2013

[27] X Xie and H Shi ldquoGlobal monitoring strategy for multimodechemical processesrdquo Journal of Chemical Industry and Engineer-ing(China) vol 63 no 7 pp 2156ndash2162 2012

[28] B Saha andKGoebelBatteryData Set NASAAmes PrognosticsData Repository NASA Ames Moffett Field Calif USA 2007httpstiarcnasagovtechdashgroupspcoeprognostic-data-repositorybattery

[29] Y N Pan J Chen and L Guo ldquoRobust bearing performancedegradation assessment method based on improved waveletpacket-support vector data descriptionrdquo Mechanical Systemsand Signal Processing vol 23 no 3 pp 669ndash681 2009

[30] J Yu ldquoBearing performance degradation assessment usinglocality preserving projections and Gaussian mixture modelsrdquoMechanical Systems and Signal Processing vol 25 no 7 pp2573ndash2588 2011

[31] R Huang L Xi X Li C Richard Liu H Qiu and J LeeldquoResidual life predictions for ball bearings based on self-organizing map and back propagation neural network meth-odsrdquoMechanical Systems and Signal Processing vol 21 no 1 pp193ndash207 2007

[32] C Spearman ldquoThe proof and measurement of associationbetween two thingsrdquo e American Journal of Psychology vol100 no 3-4 pp 441ndash471 1987

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Submit your manuscripts atwwwhindawicom


it is advantageous to accurately reflect the state of batteryperformance and improve the prediction accuracy of residuallife by extracting characteristic parameters from multipleperspectives and fusing them into a health index and thispaper focuses on performance degradation assessment oflithium batteries

To avoid problems such as insufficient maintenance orover-maintenance of lithium-ion batteries it is necessary tocarry out condition-based maintenance for batteries Thequantitative assessment of the degree of performance degra-dation is a prerequisite for condition-based maintenance oflithium-ion batteries It is difficult to measure the residualcapacity of batteries in real time [5] many researcherstry to assess the performance of the battery using theapproach of analyzing variations in the discharge voltageSbarufatti proposed a method that combines particle filtersand radial basis function neural networks to predict thedischarge duration of lithium-ion batteries [6] Xu proposedan effective model based on state space for modeling theperformance degradation of batteries The model adoptsan analysis method based on the sequentially observeddischarging profiles and uses the expectation-maximization(EM) algorithm in conjunction with the extended Kalmanfilter (EKF) algorithm to estimate and update the modelparameters and state [7] Xu also analyzed the distributionof discharge voltage and combined the discharge process andthe degradation process to develop a hierarchical model forcomprehensive prediction of discharge duration in differentcycles and the remaining effective cycles [8] Widodo usedthe sample entropy of the discharge voltage to predict theperformance of the battery [9]

Although some of the methods proposed in the abovestudies can be used tomake effective assessment they still fallshort of the expectation of accurately and comprehensivelyassessing the degradation process of the batteries This isbecause most of these methods only process a single fea-ture of the discharge voltage which is not very sensitiveto variations in the battery state To solve this challengethis paper proposes a method based on the fusion ofmultiple features for assessing the performance degradationof lithium-ion batteries In fact the multi-feature fusionapproach has been widely used in assessing the performanceof bearings but relatively fewer attempts have been madeto adopt this approach for assessing the performance ofbatteries

In researching the performance degradation assessmenteffectively extracting multi-dimensional features and estab-lishing an effective degradation assessment model are twocritical issues A lot of work has been done with regard tothe first critical issue Principal component analysis (PCA)is one of the most widely used data compression algorithmsDong used PCA to reduce the dimensions of the parametersin bearing degradation process and they obtained goodresults [10] Yu proposed a method based on local preservingprojection (LPP) to reduce the dimensions of the multi-dimensional features in the time domain frequency domainand time-frequency domain The method is applied to theperformance degradation assessment of bearings and verifiedthe feasibility of the method through experiments [11] Cai

proposed a spectral regression (SR) method which is anew type of model for supervised unsupervised and semi-supervised regression Using SR to reduce the dimensionsmakes it possible to solve the projection function usingthe regression framework thereby circumventing the thornyissue of decomposing the features of the dense matrix [12]With regard to the second critical issue Yu proposed amethod based on multiple two-layer sparse autoencoderneural networks to fuse the features and used experimentto verify the superiority of this method [13] Yu appliedthe Gaussian mixture model (GMM) model to the healthassessment of bearings and achieved quantitative analysisof performance degradation of the bearings [11] Batistaproposed an algorithm based on improved support vectormachine (SVM) reducing the error rate in diagnosis [14]Ai proposed a fusion method based on n-dimensionalcharacteristic parameters distance [15] Hassani proposed afusion model utilizes zSlices-based representation of generaltype-2 fuzzy logic systems to combine different SVMs [16]Zhang proposed an exergy-based fusion method to fusemultiple transient states with the information of multiplesensors obtained in state monitoring [17] Shao proposeda feature fusion method based on the deep belief network(DBN) taking advantage of the excellent learning ability ofDBN [18] Zhou proposed an NCA-based feature extractionmethod to reduce dimensionality and used a coupled hiddenMarkov model based on multichannel data acquisition forfault diagnosis [19] In addition Wang [20] also summarizedmethods to evaluate the health indicators based on Markovmodel self-organizingmapping wavelet change and fuzzy c-means transformation All of the abovemethods have yieldedgood assessment results

On the basis of the above analysis this paper proposesa performance degradation assessment method based onthe fusion of multiple features First the discharge voltagesignals of lithium-ion batteries under real-time monitoringare analyzed to obtain features in the dimensions of timedomain and complexity Then the multiple features undergoan SR processing to reduce the number of dimensions andeliminate redundancy based on these processes GMM isestablished to model the health status of batteries Thusthe degree of performance degradation of batteries can bequantitatively assessed using the Bayesian inference-baseddistance (BID) metric The experimental results show thatthe assessment method proposed in this paper can be usedto monitor the degradation process of lithium-ion batteriesmore effectively and to improve the accuracy of conditionmonitoring of batteries To the best of our knowledge singlecharacteristic parameters are used in most studies of lithiumbattery status assessment However there is redundancyand difference among parameters so the characterization oflithium battery state is not accurate enough Therefore inthis paper the characteristic parameters are extracted fromthe perspectives of statistical measurement and complexityafter that the extracted features are fused by the GMM-BID model proposed in this paper Compared with a singleparameter the health index can characterize the performancedegradation degree of lithium battery more accurately










Feature Equation

Mean 119909119898119890119886119899 = sum119873119899=1 119909(119899)119873


119873119899=1 (119909 (119899))2119873


119873119899=1 (119909 (119899) minus 119909119898)2119873 minus 1


119904119905119889



119881119894 = 119881119905119898119886119909 minus 119881119905119898119894119899 119894 = 1 2 119899 (1)


119881 = 1198811 1198812 119881119899 (2)






119861119898119894 (119903)= 119899119906119898119887119890119903 of119883(119895) 119904119906119888ℎ 119905ℎ119886119905 119889 [119883 (119894) 119883 (119895)] le 119903119873 minus 119898

119894 = 119895(3)


119889 [119883 (119894) 119883 (119895)] = max119886

1003816100381610038161003816119906 (119886) minus 119906lowast (119886)1003816100381610038161003816 119883 (119894) = 119883 (119895) (4)



119861119898119894 (119903) (5)


119894=1

119860119896119894 (119903) (6)


119878119886119898119901119864119899 = lim119899997888rarrinfin

[minus ln 119860119896 (119903)119861119898 (119903)] (7)


119878119886119898119901119864119899 = minus ln 119860119896 (119903)119861119898 (119903) (8)




119886 = argmin119886[ 119898sum119894=1

(a119879119911119894 minus 119910119894)2 + 120582 1198862] (9)


119911 997888rarr ℎ = 119860119879119911 119860 = (1198860 1198861 119886119897minus1) (10)




119901 (119909) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (11)


119901 (119909 | 120601) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (12)



119901 (119862119896 | 119909119905) = 120572119896119901 (119909119905 | 119862119896)119901 (119909119905) = 120572119896119901 (119909119905 | 119862119896)sum119870119894=1 120572119894119901 (119909119905 | 119862119896) (13)



Spectral regression




modelTest data


Spectralregression





extraction

Offline

Online



119901 (119909119905 | 119862119896)= 1(2120587)12 1003816100381610038161003816119878119896100381610038161003816100381612 exp [minus





119861119868119863 = 119870sum119896=1

119901 (119862119896 | 119909119905)119863119862119896 (119909119905) (16)













Battery Number 5


Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)










minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018

















Feature Equation

Mean 119909119898119890119886119899 = sum119873119899=1 119909(119899)119873


119873119899=1 (119909 (119899))2119873


119873119899=1 (119909 (119899) minus 119909119898)2119873 minus 1


119904119905119889



119881119894 = 119881119905119898119886119909 minus 119881119905119898119894119899 119894 = 1 2 119899 (1)


119881 = 1198811 1198812 119881119899 (2)






119861119898119894 (119903)= 119899119906119898119887119890119903 of119883(119895) 119904119906119888ℎ 119905ℎ119886119905 119889 [119883 (119894) 119883 (119895)] le 119903119873 minus 119898

119894 = 119895(3)


119889 [119883 (119894) 119883 (119895)] = max119886

1003816100381610038161003816119906 (119886) minus 119906lowast (119886)1003816100381610038161003816 119883 (119894) = 119883 (119895) (4)



119861119898119894 (119903) (5)


119894=1

119860119896119894 (119903) (6)


119878119886119898119901119864119899 = lim119899997888rarrinfin

[minus ln 119860119896 (119903)119861119898 (119903)] (7)


119878119886119898119901119864119899 = minus ln 119860119896 (119903)119861119898 (119903) (8)




119886 = argmin119886[ 119898sum119894=1

(a119879119911119894 minus 119910119894)2 + 120582 1198862] (9)


119911 997888rarr ℎ = 119860119879119911 119860 = (1198860 1198861 119886119897minus1) (10)




119901 (119909) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (11)


119901 (119909 | 120601) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (12)



119901 (119862119896 | 119909119905) = 120572119896119901 (119909119905 | 119862119896)119901 (119909119905) = 120572119896119901 (119909119905 | 119862119896)sum119870119894=1 120572119894119901 (119909119905 | 119862119896) (13)



Spectral regression




modelTest data


Spectralregression





extraction

Offline

Online



119901 (119909119905 | 119862119896)= 1(2120587)12 1003816100381610038161003816119878119896100381610038161003816100381612 exp [minus





119861119868119863 = 119870sum119896=1

119901 (119862119896 | 119909119905)119863119862119896 (119909119905) (16)













Battery Number 5


Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)










minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










119861119898119894 (119903)= 119899119906119898119887119890119903 of119883(119895) 119904119906119888ℎ 119905ℎ119886119905 119889 [119883 (119894) 119883 (119895)] le 119903119873 minus 119898

119894 = 119895(3)


119889 [119883 (119894) 119883 (119895)] = max119886

1003816100381610038161003816119906 (119886) minus 119906lowast (119886)1003816100381610038161003816 119883 (119894) = 119883 (119895) (4)



119861119898119894 (119903) (5)


119894=1

119860119896119894 (119903) (6)


119878119886119898119901119864119899 = lim119899997888rarrinfin

[minus ln 119860119896 (119903)119861119898 (119903)] (7)


119878119886119898119901119864119899 = minus ln 119860119896 (119903)119861119898 (119903) (8)




119886 = argmin119886[ 119898sum119894=1

(a119879119911119894 minus 119910119894)2 + 120582 1198862] (9)


119911 997888rarr ℎ = 119860119879119911 119860 = (1198860 1198861 119886119897minus1) (10)




119901 (119909) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (11)


119901 (119909 | 120601) = 119872sum119898=1

120587119898119901 (119909 | 120579119898) (12)



119901 (119862119896 | 119909119905) = 120572119896119901 (119909119905 | 119862119896)119901 (119909119905) = 120572119896119901 (119909119905 | 119862119896)sum119870119894=1 120572119894119901 (119909119905 | 119862119896) (13)



Spectral regression




modelTest data


Spectralregression





extraction

Offline

Online



119901 (119909119905 | 119862119896)= 1(2120587)12 1003816100381610038161003816119878119896100381610038161003816100381612 exp [minus





119861119868119863 = 119870sum119896=1

119901 (119862119896 | 119909119905)119863119862119896 (119909119905) (16)













Battery Number 5


Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)










minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










Spectral regression




modelTest data


Spectralregression





extraction

Offline

Online



119901 (119909119905 | 119862119896)= 1(2120587)12 1003816100381610038161003816119878119896100381610038161003816100381612 exp [minus





119861119868119863 = 119870sum119896=1

119901 (119862119896 | 119909119905)119863119862119896 (119909119905) (16)













Battery Number 5


Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)










minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









Battery Number 5


Cycle120Cycle150

242628

332343638

44244

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 6


Cycle120Cycle150

2

25

3

35

4

45

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)


Battery Number 18

Cycle30Cycle60

Cycle90Cycle120

22242628

332343638

442

Volta

ge (V

)

500 1000 1500 2000 2500 3000 35000Time (s)










minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018











minus015

minus01

minus005

0

005

01

015

02

PCA

2

0075 008 0085 009 0095 01 0105 011007PCA1


minus007

minus0065

minus006

minus0055

minus005

minus0045

LPP2



minus001 0 001 002 003 004 005 006 007minus002LPP1


minus0145minus014

minus0135minus013

minus0125minus012

minus0115minus011

minus0105minus01

SR2



minus002 0 002 004 006 008 01 012minus004SR1











34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









34342344346348

35352354356358

36

RMS





021022023024025026027028

Stan

dard

dev

iatio

n




34342344346348

35352354356358

Mea

n




3456789

1011

Kurt

osis




Skew

ness






06070809

1111213

Fixe

d-in

terv

al v

olta

ge d

iffer

ence

(V)




0001002003004005006007008009

01

Sam

ple e

ntro

py





times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









times104

0 20 40 60 80 100 120 140 160 1800


05

1

15

2

25

3

BID



0

05

1

15

2

25

3

35

BID

times105



0

5

10

15

BID

times1018





0

001

002

003

004

005

006

007

DI



0

002

004

006

008

01

012

014

DI



0001002003004005006007008009

01

DI









minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018












minus20

0

20

40

60

80

100

NLL

P



minus200

20406080

100120140160180

NLL

P




5 Conclusions



minus40minus20

020406080

100120140

NLL



0

002

004

006

008

01

012

MQ

E



0002004006008

01012014016018

MQ

E





0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










0

002

004

006

008

01

012

014

MQ

E



Data Availability






Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Assessing the Performance Degradation of Lithium-Ion