research article multistage cc-cv charge method...

Research ArticleMultistage CC-CV Charge Method for Li-Ion Battery

Xiaogang Wu1 Chen Hu1 Jiuyu Du2 and Jinlei Sun3

1College of Electrical and Electronic Engineering Harbin University of Science and Technology Xue Fu Road 52 Harbin 150080 China2State Key Laboratory of Automotive Safety and Energy Tsinghua University Beijing China3School of Electrical Engineering and Automation Harbin Institute of Technology Harbin China

Correspondence should be addressed to Xiaogang Wu xgwuhrbusteducn

Received 18 August 2015 Revised 26 September 2015 Accepted 28 September 2015

Academic Editor Xiaosong Hu

Copyright copy 2015 Xiaogang Wu 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

Charging the Li-ion battery with constant current and constant voltage (CC-CV) strategy at minus10∘C can only reach 4847 of thenormal capacity To improve the poor charging characteristic at low temperature the working principle of charging battery at lowtemperature is analyzed using electrochemical model and first-order RC equivalent circuit model moreover the multistage CC-CV strategy is proposed In the proposed multistage CC-CV strategy the charging current is decreased to extend the chargingprocess when terminal voltage reaches the charging cut-off voltageThe charging results of multistage CC-CV strategy are obtainedat 25∘C 0∘C and minus10∘C compared with the results of CC-CV and two-stage CC-CC strategies The comparison results show thatat the target temperatures the charging capacities are increased with multistage CC-CV strategy and it is notable that the chargingcapacity can reach 8532 of the nominal capacity at minus10∘C also the charging time is decreased

1 Introduction

With the advantages of zero pollution high energy efficiencyand pluralistic energy sources electric vehicle (EV) has beenthe new development point of motor industry [1ndash3] Li-ion battery has been widely used in EV for its high energydensity long cycle life and high safety level [4] But thebattery technology still cannot meet the EV demand of longtravel distance fast capacity recovery and low temperatureutilization [5] At low temperature battery chemical activitydecreases resistance increases and capacity is decreasedCharging process is more difficult than the dischargingprocess at low temperature [6 7]

Muchwork has been done on charging strategies in recentyears In [8] a three-step charging method for NiMH batterywas proposed to obtain the rapid charge In [9] an optimumcurrent charging strategy based on the boundary chargingcurrent curves was proposedThe boundary charging currentcurves were obtained by analysis of temperature rise andpolarization voltage in charging processThe charging periodwas decreased and capacity was increased with the strat-egy Reference [10] proposed a duty-varied voltage chargingstrategy that can detect and dynamically track the suitableduty of the charging pulse Compared with conventional

CC-CV strategy the charging speed was increased by 14and charging efficiency was increased by 34 Reference[11] constructed a SOC estimation model and the CC-CV charging process was controlled by battery SOC Thecharging capacity can be monitored to gain a higher levelcharging degree and avoid being overcharged In [12] an AntColony System algorithm was used to select the optimumcharging current among five charging states and the chargingtime was decreased and battery cycle life was extendedby 25 In [13] a Taguchi-based algorithm was used toobtain rapid charge With the charging strategy the batterycapacity could reach to 75 in 40min In [14] a constant-polarization-based fuzzy-control charging method was pro-posed to adapt charging current acceptance with batterySOC stages The charging strategy could shorten chargingtime with no obvious temperature rise Ruan et al andZhao et al [15 16] studied the temperature characteristic ofcharging and discharging processThe temperature increasedmore in discharging process compared to the temperatureincrease in charging process The pulse chargingdischargingprocess was added before charging process so the batterycould be preheated The battery could start charging processat relatively high temperature and charging capacity wasincreased at low temperature

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 294793 10 pageshttpdxdoiorg1011552015294793

2 Mathematical Problems in Engineering

Table 1 Equipment parameters

Battery equipmentMaximum test current 20AMaximum test voltage 5V

Test accuracy 01

Temperature chamberMaximum temperature 150∘CMinimum temperature minus40∘CTemperature tolerance 001∘C

T

PCBattery tester

Battery

Temperature chamber

Battery data

Temperature data

Figure 1 Experimental setup

All the charging strategies increase the battery chargingcharacteristic at different degrees proposed in [8ndash14] But thecharging performance at low temperature is not consideredAlthough the preheating charging strategy at low temperatureproposed in [15 16] can increase the charging capacity theself-preheating process costs toomuch time and cannot workat low SOC condition This paper analyzes the chargingcharacteristic of a Li-ion battery at different temperature useselectrochemical model and first-order 119877119862 equivalent circuitmodel to analyze the bad low temperature characteristic ofLi-ion battery in theory and proposes a multistage CC-CVstrategy The multistage CC-CV strategy is compared withCC-CV and two-stage CC-CV strategies at 25∘C 0∘C andminus10∘C

2 Experimental

21 Battery and Equipment The battery used is 18650 cylin-drical Li-ion battery with normal capacity of 137 Ah anormal voltage of 32 V and a cut-off voltage of 36 V Themaximum charging and discharging rates are 1 C and 2Crespectively The positive electrode material is LiFePO

4 and

negative electrode material is LiC6 The battery tester is LD

battery tester with 8 test channels and the test process can beprogrammed and monitored by computer The battery wastested in a temperature chamber to ensure the temperatureparameter to be constant The detailed parameters of batterytester and temperature chamber are shown in Table 1 Theexperimental setup can be described as in Figure 1

22 Experimental Process The battery charging strategiestested in experiments were CC-CV two-stage CC-CV andmultistage CC-CV The test temperature points were 25∘C0∘C and minus10∘C The charging strategies are explained asfollows

14

13

12

11

10

09

08

07

06

05

04

03

02

01

00

Capa

city

(Ah)

25 0 minus10

(∘C)

Figure 2 CC-CV strategy charging capacities at different tempera-ture

For the CC-CV strategy the constant current processwas charging at 03 C to the cut-off voltage of 36 V and theconstant voltage process was charging at 36 V for 5min

For the two-stage CC-CV strategy the first constantcurrent process was charging at 1 C to the cut-off voltageof 36 V Then in the second constant current process thecharging current was decreased to 05 C Since the chargingcurrent was decreased the terminal voltage was decreasedbelow 36V allowing the constant current process to beextended until the terminal voltage reached the cut-offvoltage once againThe constant voltage process was chargingat 36 V for 5min [17]

For the multistage CC-CV strategy the constant currentprocess was divided into ten stages The maximum and min-imum rates were 1 C and 01 C respectively and the chargingcurrent was decreased by 01 Cwhen terminal voltage reachedthe cut-off voltageThe constant voltage process was chargingat 36 V for 5min

3 Charging Characteristic of Battery atLow Temperature

31 Charging Capacity Characteristic at Different Tempera-ture The selected battery was charged by CC-CV strategyat 25∘C 0∘C and minus10∘C to obtain the charging capacitycharacteristic at low temperature Before every chargingprocess at different temperature the battery was dischargedempty at 25∘C and kept for six hours to ensure the wholebattery temperature to be uniform As shown in Figure 2the charging capacities at 25∘C 0∘C and minus10∘C are 1309Ah1196Ah and 0664Ah respectively The charging capacity isdecreased by 86 at 0∘C and 493 at minus10∘C compared withthat at 25∘C The charging capacity has a great decrease atminus10∘C

32 OCVCharacteristic at Different Temperature Thebatterywas tested by hybrid pulse power characteristic (HPPC)rule that is detailed in ldquoFreedom CAR Battery Test Manualrdquo[18] to obtain OCV ohmic resistance (119877

119903) and polarization

Mathematical Problems in Engineering 3

340

335

330

325

3200 10 20 30 40 50 60 70 80 90 100

OCV

(V)

SOC ()

25∘C0∘Cminus10∘C

Figure 3 OCV curves at different temperature

resistance (119877119901) SOC can be calculated by the following

formula

SOC = SOC0minus

1

AHCint

119905

0

119894 119889120591(1)

where SOC0is the initial SOC of the battery AHC is the

normal capacity of the battery at 25∘C and 119894 is the discharge(positive 119894) or charge (negative 119894) current As the OCV curvesshown in Figure 3 theOCV reflects increasing tendencywithtemperature decreasing and the difference ofOCVat differenttemperature is relatively more obvious at low SOC

33 119877

119903and 119877

119901Characteristic at Different Temperature

As shown in Figures 4 and 5 both 119877

119903and 119877

119901increase

with temperature decreasing 119877119903remains steady with SOC

increasing and the increase is nearly 258 at minus10∘C 119877119901

increases with SOC increasing and temperature decreasingand themaximum increase in119877

119901is nearly 257 atminus10∘Cwith

90 of SOC

4 Electrochemical and First-Order 119877119862Equivalent Circuit Model

41 Electrochemical Li-Ion Battery Model Doyle et al haveproposed the porous electrode theory for the analysis ofelectrochemical process of Li-ion battery [19] The one-dimensional geometry consists of negativepositive currentcollector negativepositive electrodes and separator Thenegative current collector material is copper and positivecurrent collector material is aluminum The positive elec-trode active material is LiFePO

4 and negative electrode

active material is LiC6 The separator is polyolefin porous

membrane The electrolyte is lithium salt dissolved in 1 1 or2 1 liquid mixture of ethylene carbonate (EC) and dimethylcarbonate (DMC) The one-dimensional geometry example

Rr

(Ω)

0050

0045

0040

0035

0030

0025

0020

00150 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 4 119877119903

curves at different temperature

Rp

(Ω)

0040

0035

0030

0025

0020

0015

0010

0 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 5 119877119901


of charging process is shown in Figure 6 [20] and the chargingchemical equation is

Negative Electrode Li119909minus119911

119862

6+ 119911Li+ + 119911119890minus

Charge997888997888997888997888997888rarr Li

119909119862

6

Positive Electrode Li119910FePO

4

Charge997888997888997888997888997888rarr Li

119910minus119911FePO

4+ 119911Li+ + 119911119890minus

(2)

In the charging process the electrons move from thepositive electrode to the negative electrode through theexternal circuit and Li+ moves from the positive electrodeto the negative electrode through the separator in electrolyte


SeparatorPositive electrode Negative electrode

Positivecurrent

collector

Negativecurrent

collector

x

I ILi+

Li+

Li+

Li+

Figure 6 One-dimensional geometry example of charging process

As the charging process is a chemical reaction the reactioncharacteristic is influenced by concentration and Li+ diffu-sion The Li-ion concentration in electrolyte phase changeswith time and can be described by Fickrsquos second law alongthe 119909-coordinate shown in Figure 6 [21]

120597119862

119890

120597119905

=

120597

120597119909

(119863

119890

120597119862

119890

120597119909

) +

1 minus 119905

0

+

119865

119895

Li

(3)

where 119862119890is the concentration of Li-ion in electrolyte phase

119863

119890is Li+ diffusion coefficient in electrolyte phase 1199050

+

is thetransference number of lithium ions with respect to thevelocity of the solvent 119865 is Faraday constant and 119895

Li ischarging transfer current density

The distribution of Li-ion in solid state phase is alsodescribed by Fickrsquos second law of diffusion in polar coordi-nates [21]

120597119862

119904

119889119905

=

119863

119904

119903

2

120597

120597119903

(119903

2119862

119904

120597119903

)

(4)

where 119862119904is the concentration of Li-ion in solid119863

119904is the Li+

diffusion coefficient in solid state phase and 119903 is radius ofspherical particle

TheArrhenius formula shows the Li+ diffusion coefficient119863

119904in solid state phase as shown below [21]

119863

119904(119879) = 119863ref exp [

119864

119886119863

119877

(

1

119879refminus

1

119879

)] (5)

where 119864

119886119863is the activation energy for diffusion 119877 is the

universal gas constant 119863ref is the reference diffusion coeffi-cient at 119879ref 119879ref is the reference temperature and 119879 is thetemperature Formula (5) shows that the diffusion coefficientdecreases with the temperature decreasing Reference [14]indicates that the solid state phase diffusion polarizationdominates the total polarization and the solid state phasepolarization is increased with diffusion coefficient decreas-ingThe increase of polarization results in higher polarizationvoltage compared with that of normal temperature theterminal voltage increasing space during constant currentcharging process is decreased and the charging capacity willbe decreased

The charging transfer current density can be obtainedusing the following Butler-Volmer formula [20]

119895

Li= 119895

0exp [

120572

119886119865

119877119879

120578] minus exp [120572

119888119865

119877119879

120578] (6)

where 1198950is the exchange current density 120572

119886and 120572

119888are the

transfer coefficients of anode and cathode and 120578 is the surfaceover potential which can be obtained using the followingformula [20]

120578 = 120601

119904minus 120601

119890minus 119880ocv (7)

where120601119904is the solid phase potential120601

119890is the electrolyte phase

potential and 119880ocv is the open circuit voltage119895

0can be described as shown below [20]

119895

0= 119865119896

0119862

120572119886

119890

(119862

119904max minus 119862119904surf)120572119886

119862

120572119886

119904surf (8)

where 1198960is the reaction rate coefficient 119862

119904max is the maxi-mum Li-ion concentration in the electrodes and 119862

119904surf is theLi-ion concentration on the active particles surface

119896

0can be obtained using the following formula [20]

119896

0(119879) = 119896

0ref exp [119864

119886119877

119877

(

1

119879refminus

1

119879

)] (9)

where 119864

119886119903is the reaction activation energy and 119896

119900ref isthe reaction rate coefficient at 119879ref With the temperaturedecreasing reaction rate coefficient is decreased As formula(7) shows 119896

0is decreased with temperature decreasing The

charging reaction is impeded for the reaction rate coefficientdecreasing As the parameter is time-invariant the chargingobstruction can be considered as a resistive process Theincrease of impedance also results in the terminal voltageincrease and the decrease of charging capacity

The electrochemistry model analysis of the chargingprocess at low temperature shows that the main obstruc-tion consists of polarization and impedance increase Thisincrease can be analyzed by the equivalent circuit model thepolarization can be modeled by capacitance and resistance inparallel and the impedance can be modeled by resistance Afirst-order 119877119862 equivalent circuit model is used in the nextpart


Up

Cp

Rp

UoOCV

+

minus

Rr

Ur I

Figure 7 First-order 119877119862 equivalent circuit model

42 First-Order 119877119862 Equivalent Circuit Model The first-order119877119862 equivalent circuit model is used to analyze the chargingprocess [21 22] As shown in Figure 7 119877

119903represents the

ohmic resistance 119880119903is the voltage on 119877

119903 119862119901and 119877

119901 respec-

tively represent the polarization capacity and polarizationresistance 119880

119901is the voltage on 119862

119901and 119877

119901 OCV is the open

circuit voltage119880119900is the terminal voltage and 119868 is the charging

current The following formulas can be obtained

119862

119901

119889119906

119901(119905)

119889119905

+

119906

119901(119905)

119877

119901

= 119894 (119905)

119906

0(119905) = 119906ocv (119905) + 119894 (119905) 119877119903 + 119906119901 (119905)

(10)

With assumption of 119894(0) = 119868 and 119906119901(0) = 0 the following

can be obtained

119906

119901(119905) = 119868119877

119901(1 minus 119890

minus119905120591

)

119906

0(119905) = 119906ocv (119905) + 119868119877119903 + 119906119901 (119905)

(11)

where 120591 = 119877

119901119862

119901

It can be seen from formulas (10)-(11) that 119880119900is deter-

mined by OCV 119877119901 119877119903 and 119868 As is mentioned above

OCV changes little with temperature decreasing while 119877119903

and 119877

119901increase significantly with temperature decreasing

The increase of 119877119901can be explained by the slow kinetics

of electrochemical reaction influenced by temperature Theconstant current process of CC-CV strategy is limited bycut-off voltage and the charging capacity mainly dependson the constant current process At low temperature 119877

119903

and 119877

119901increase making 119880

119903and 119880

119901increase and 119880

0is

higher than that at normal temperature The cut-off voltageis reached earlier and the constant current process is stoppedearlier [23] The increasing of 119877

119901and 119877

119903depends on the

battery design parameters and cannot be controlled duringthe charging process The only parameter which can becontrolled is the charging current As proposed in [17] fora two-stage CC-CV strategy the constant current chargingprocess was divided into two stagesThe first stage is chargingbattery with the maximum charging rate until the cut-offvoltage is reached The second stage charging current wasdecreased to half of the maximum charging rate and theterminal voltage can be decreased to extend the constant

0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

15

10

05

00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

Figure 8 Charging curves of CC-CV strategy at 25∘C

current charging process to increase capacity According tothe current decrease process of the two-stageCC-CV strategyamultistage CC-CV strategy withmore detailed current ratesis proposed in this paper Once the cut-off voltage is rapidlyreached at a low temperature the terminal voltage can bedecreased with charging current decreasing and the constantcurrent charging process can be repeatedly extended toincrease charging capacity Meanwhile the charging currentis decreased from the maximum rate and the multistage canautomatically and degressively select the optimal chargingcurrent to use high charging rate as far as possible and shortenthe charging period

5 Result and Discussion

51 Different Charging Strategy Analysis at 25∘C Figures 8ndash10 show the terminal voltage curves with different chargingstrategies at 25∘C The terminal voltage of CC-CV strategyincreases to 325V at the low SOC range of 0ndash10while theterminal voltages of two-stage CC-CV and multistage CC-CV strategies increase to near 34 V The terminal voltage ofCC-CV strategy increases to 34 V with SOC reaching 90and has a huge increase to 36 V at the end of chargingThe terminal voltages of two-stage CC-CV and multistageCC-CV strategies increase to 36 V with SOC of 85 Withcurrent decreasing the terminal voltage of two-stage CC-CVstrategy decreases to 349V and increases to 36 V again withSOC increasing of 7 Unlike two-stage CC-CV strategythe terminal voltage of multistage CC-CV strategy has moredecreasing times to extend the charging SOC to a higher level

Figure 11 shows the SOC curves of different chargingstrategies at 25∘C The charging capacities of CC-CV two-stage CC-CV and multistage CC-CV charging strategies are1309Ah 1299Ah and 1368Ah respectively The capacitiesof two-stage CC-CV and multi-CC-CV strategies are higherthan that of CC-CV strategy for current decreasing processThe multi-CC-CV has the highest charging capacity becausethe current decrease process of multistage CC-CV strategy


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05

Figure 9 Charging curves of two-stage CC-CV strategy at 25∘C

0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05

Figure 10 Charging curves of multistage CC-CV strategy at 25∘C

has more gradients than two-stage CC-CV strategy Thecharging periods of CC-CV two-stage CC-CV and multi-stage CC-CV charging strategies are 223min 674min and947min respectively It is obvious that the CC-CV chargingstrategy has the longest charging period for a low constantcharging rate Although the whole charging period of two-stage CC-CV is shorter than that of the multistage CC-CVmultistage CC-CV charging strategy has a larger chargingcapacity The charging period of multistage CC-CV strategyis shorter than that of two-stage CC-CV strategy at the samecharging SOC point 948

52 Different Charging Strategy Analysis at 0∘C As shown inFigure 12 unlike the terminal voltage at 25∘C the terminalvoltage of CC-CV strategy increases to 335V at low SOCrange of 0ndash10The terminal voltage increase slope duringSOC range of 10ndash80 is enhanced The terminal voltageincrease towards the cut-off voltage and sharp increase

0 5000 10000 15000

Time (s)

CC-CVSOC = 948

Two-stage CC-CVMultistage CC-CV

0

10

20

30

40

50

60

70

80

90

100

SOC

()

Figure 11 SOC curves of different charging strategies at 25∘C

at 25∘C with SOC range beyond 80 are vanished Theincrease in terminal voltage indicates that the normal CC-CVcharging process has been changed by the increase in internalresistance at low temperature

As shown in Figures 13-14 the first charging stage of two-stage CC-CV strategy does not last long before the cut-offvoltage is reached for the increase in internal resistance andthe high charging rate The second charging stage decreasesthe charging current rate by 05 C and the terminal voltagedecreases by 023V and keeps on increasing until the cut-offvoltage is reached Unlike the two-stage CC-CV strategy thecurrent decreases by 01 C of the multistage CC-CV strategyThe terminal voltages of 1 Cndash07 C constant current chargingstages increase rapidly with SOC below 22 Terminal volt-ages of 06 Cndash04 C constant current charging stages increaseslower with SOC between 22 and 734 Terminal voltagesof 03 Cndash01 C constant current charging stages increaserapidly again with SOC beyond 734 The terminal voltagecurve of multistage CC-CV indicates that multistage CC-CVstrategy can automatically select the optimal charging currentby cut-off voltage limiting and current decreasing

The charging result at 0∘C shows that the capacities ofCC-CV two-stage CC-CV and multistage CC-CV chargingstrategies are 1196Ah 0758Ah and 1246Ah respectivelyCompared with the charging result at 25∘C the chargingcapacities of CC-CV two-stage CC-CV and multistage CC-CV charging strategies decrease by 82 395 and 89respectively As the main charging rate of two-stage CC-CVstrategy is 05 C higher than 03 C of CC-CV strategy andthe charging rate does not decrease further the two-stageCC-CV strategy has the largest decrease in charging capacitydecrease at 0∘C As multistage CC-CV strategy has 02 C and01 C charging rate lower than 03 C of CC-CV strategy thecharging capacity of multistage CC-CV strategy is higherthan that of CC-CV strategy

Figure 15 shows the SOC curves of different chargingstrategies at 0∘C The charging periods of CC-CV two-stage CC-CV and multistage CC-CV charging strategies


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)

Voltage (V)Current (A)

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()


are 1834min 687min and 148min The curve tendencyof multistage CC-CV charging strategy shows the obviouscapacity increasing speed although the speed is sloweddown for the current decrease at later period As the dottedline shows the charging period of multistage CC-CV isshorter than that of two-stage CC-CV strategy with the samecharging SOC of 5532 The multistage CC-CV still has themaximum charging capacity and minimum charging periodat 0∘C

53 Different Charging StrategyAnalysis atminus10∘C Figures 16ndash18 show the terminal voltage curves with different chargingstrategies at minus10∘C The terminal voltage of CC-CV strategyreaches cut-off voltage at SOC point of 4867 The termi-nal voltage of the first stage of two-stage CC-CV strategyincreases straightly towards the cut-off voltage and the secondstage only extends the SOC to 3226 All the terminalvoltages of charging current at 1 Cndash06 C of multistage CC-CV strategy increase rapidly to the cut-off voltage with SOCgrowth less than 10 The terminal voltages of chargingcurrent at 05 Cndash01 C increase slower with near 75 of SOCgrowth All the terminal voltage curves indicate that thevoltage increases faster at the lower temperature and highercharging current rate

The charging result at minus10∘C shows that the capacities ofCC-CV two-stage CC-CV and multistage CC-CV chargingstrategies are 0664Ah 0442Ah and 1169Ah respectivelyCompared with the charging result at 25∘C the chargingcapacities of CC-CV two-stage CC-CV and multistage CC-CV charging strategies decrease by 4708 6256 and1453 respectively It can be indicated that the chargingcapacity of CC-CV decreases badly and the first stage oftwo-stage CC-CV strategy oppositely becomes the capacitylimit The multistage CC-CV strategy can keep the chargingcapacity beyond 80 even at minus10∘C

Figure 19 shows SOC curves of different charging strate-gies at minus10∘C and the charging periods of CC-CV two-stage


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01

Figure 16 Charging curves of CC-CV strategy at minus10∘C

0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1

Figure 17 Charging curves of two-stage CC-CV strategy at minus10∘C

0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1

Figure 18 Charging curves of multistage CC-CV strategy at minus10∘C

0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()

Figure 19 SOC curves of different charging strategies at minus10∘C

CC-CV and multistage CC-CV charging strategies are1017min 3938min and 1971min respectively The curve oftwo-stage CC-CV charging strategy shows the obvious dif-ficulty of capacity increasing at such temperature Althoughthe terminal voltage increasing slope of two-stage CC-CVstrategy is close to that of multistage CC-CV strategy thecharging capacity is significantly differentThe comparison ofthe SOC curves shows that the charging period of multistageCC-CV strategy is still the shortest at the same SOC pointThe multistage CC-CV still has maximum charging capacityat minus10∘C

54 Analysis of Multistage CC-CV Strategy Figure 20 showsthe capacity curves of different charging current rates ofmultistage CC-CV strategy at different temperature and thehigh charging capacity corresponding charging current ratedecreaseswith temperature decreasingThe charging capacityof 1 C is 1162Ah beyond 80 of battery capacity and theother charging rates only need to recover the rest of capacityat 25∘CWhile the high charging rate does not work well withtemperature decreasing the charging current rate with themaximum charging capacity of 028Ah is 05 C at 0∘C Thecharging current rate with the maximum charging capacityof 0266Ah is 03 C at minus10∘C The main capacity is chargedwith a range of charging current rates at low temperatureThe multistage CC-CV can automatically select the optimalcharging current rate for two reasons (1)The cut-off voltagelimit can stop the charging stage of the not optimal chargingcurrent rate (2) The multistage has ten charging currentrates from the maximum 1C to the minimum 01 C ensuringthe charging demands at different temperature points Themultistage CC-CV strategy is a wide temperature rangecharging strategy that keeps high charging capacity and lowcharging period

6 Conclusion

It can be seen from the presentation above that the chargingcapacity of the CC-CV strategy can be only 4847 of the


14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01


Figure 20 Capacity curves of different charging current rates ofmultistage CC-CV strategy at different temperature

normal capacity at minus10∘C The charging process is analyzedby electrochemical Li-ion battery model and first-order 119877119862equivalent circuitmodelThe increase in internal resistance isthe main limitation of charging capacity at low temperatureThe proposed multistage CC-CV strategy can extend theconstant current charging process to obtain a larger capacityby decreasing the charging rate when the terminal voltagereaches the cut-off voltage Experimental results indicate thatthe charging capacities with multistage CC-CV strategy at25∘C 0∘C and minus10∘C are 1368Ah 1246Ah and 1169Ahrespectively Compared with CC-CV and two-stage CC-CVstrategies the multistage CC-CV strategy has the largestcharging capacities and the shortest charging periods at thetarget temperatures

Conflict of Interests

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

Acknowledgment

This work is supported by National Natural Science Founda-tion (NNSF) of China (Grant no 51105220)

References

[1] H Zhang X Zhang and J Wang ldquoRobust gain-schedulingenergy-to-peak control of vehicle lateral dynamics stabiliza-tionrdquo Vehicle System Dynamics no 52 pp 309ndash340 2014

[2] H Zhang and J Wang ldquoVehicle lateral dynamics controlthrough AFSDYC and robust gain-scheduling approachrdquo IEEETransactions on Vehicular Technology no 19 2015

[3] FMengH Zhang D Cao andH Chen ldquoSystemmodeling andpressure control of a clutch actuator for heavy-duty automatic

transmission systemsrdquo IEEE Transactions on Vehicular Technol-ogy vol 99 pp 1ndash9 2015

[4] D Ansean M Gonzalez M V Garcia C J Viera C JAnton and C Blanco ldquoEvaluation of LiFePO

4

batteries forelectric vehicle applicationsrdquo IEEE Transactions on IndustryApplications vol 2 pp 1855ndash1863 2015

[5] J Fan and S Tan ldquoStudies on charging lithium-ion cells at lowtemperaturesrdquo Journal of the Electrochemical Society vol 153no 6 pp A1081ndashA1092 2006

[6] H-S Song J-B Jeong B-H Lee et al ldquoExperimental studyon the effects of pre-heating a battery in a low-temperatureenvironmentrdquo in Proceedings of the IEEE Vehicle Power andPropulsion Conference (VPPC rsquo12) pp 1198ndash1201 IEEE SeoulRepublic of Korea October 2012

[7] L Liao P Zuo Y Ma et al ldquoEffects of temperature onchargedischarge behaviors of LiFePO

4

cathode for Li-ionbatteriesrdquo Electrochimica Acta vol 60 pp 269ndash273 2012

[8] T Ikeya N Sawada J-I Murakami et al ldquoMulti-step constant-current charging method for an electric vehicle nickelmetalhydride battery with high-energy efficiency and long cycle liferdquoJournal of Power Sources vol 105 no 1 pp 6ndash12 2002

[9] YGaoC ZhangQ Liu Y JiangWMa andYMu ldquoAnoptimalcharging strategy of lithium-ion batteries based on polarizationand temperature riserdquo in Proceedings of the IEEE Conferenceand Expo Transportation ElectrificationAsia-Pacific (ITECAsia-Pacific rsquo14) pp 1ndash6 IEEE Beijing China September 2014

[10] L-R Chen ldquoDesign of duty-varied voltage pulse charger forimproving Li-ion battery-charging responserdquo IEEE Transac-tions on Industrial Electronics vol 56 no 2 pp 480ndash487 2009

[11] S J Huang B G Huang and F S Pai ldquoFast charge strategybased on the characterization and evaluation of LiFePO

4

bat-teriesrdquo IEEE Transactions on Power Electronics vol 28 no 4pp 1555ndash1562 2013

[12] Y-H Liu J-H Teng andY-C Lin ldquoSearch for an optimal rapidcharging pattern for lithium-ion batteries using Ant ColonySystem algorithmrdquo IEEE Transactions on Industrial Electronicsvol 52 no 5 pp 1328ndash1336 2005

[13] Y-H Liu and Y-F Luo ldquoSearch for an optimal rapid-chargingpattern for Li-ion batteries using the Taguchi approachrdquo IEEETransactions on Industrial Electronics vol 57 no 12 pp 3963ndash3971 2010

[14] J Jiang C Zhang J Wen W Zhang and S M SharkhldquoAn optimal charging method for Li-ion batteries using afuzzy-control approach based on polarization propertiesrdquo IEEETransactions on Vehicular Technology vol 62 no 7 pp 3000ndash3009 2013

[15] H J Ruan J C Jiang B X Sun N N Wu W Shi and Y RZhang ldquoStepwise segmented charging technique for lithium-ion battery to induce thermal management by low-temperatureinternal heatingrdquo in Proceedings of the IEEE TransportationElectrification Conference and Expo (ITEC rsquo14) Beijing ChinaSeptember 2014

[16] X W Zhao G Y Zhang L Yang J X Qiang and Z Q ChenldquoA new charging mode of Li-ion batteries with LiFePO

4

Ccomposites under low temperaturerdquo Journal ofThermal Analysisand Calorimetry vol 104 no 2 pp 561ndash567 2011

[17] DAnseanMGonzalez J CViera VMGarcıa C Blanco andM Valledor ldquoFast charging technique for high power lithiumiron phosphate batteries a cycle life analysisrdquo Journal of PowerSources vol 239 pp 9ndash15 2013


[18] G Hunt and C Motloch Freedom Car Battery Test Manual forPower-Assist Hybrid Electric Vehicles Idaho National Engineer-ing and Environmental Laboratory (INEEL) Idaho Falls IdahoUSA 2003

[19] M Doyle J Newman A S Gozdz C N Schmutz and J-MTarascon ldquoComparison of modeling predictions with experi-mental data from plastic lithium ion cellsrdquo Journal of theElectrochemical Society vol 143 no 6 pp 1890ndash1903 1996

[20] M Xu Z Q Zhang X Wang L Jia and L X Yang ldquoApseudo three-dimensional electrochemical-thermal model of aprismatic LiFePO

4

battery during discharge processrdquo Energyvol 80 pp 303ndash317 2015

[21] B Wu V Yufit M Marinescu G J Offer R F Martinez-Botas and N P Brandon ldquoCoupled thermal-electrochemicalmodelling of uneven heat generation in lithium-ion batterypacksrdquo Journal of Power Sources vol 243 pp 544ndash554 2013

[22] J Kim S Lee and B H Cho ldquoComplementary cooperationalgorithm based on DEKF combined with pattern recognitionfor SOCcapacity estimation and SOH predictionrdquo IEEE Trans-actions on Power Electronics vol 27 no 1 pp 436ndash451 2012

[23] J Kim J Shin C Chun and B H Cho ldquoStable configurationof a Li-ion series battery pack based on a screening process forimproved voltageSOC balancingrdquo IEEE Transactions on PowerElectronics vol 27 no 1 pp 411ndash424 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Equipment parameters

Battery equipmentMaximum test current 20AMaximum test voltage 5V

Test accuracy 01

Temperature chamberMaximum temperature 150∘CMinimum temperature minus40∘CTemperature tolerance 001∘C

T

PCBattery tester

Battery

Temperature chamber

Battery data

Temperature data

Figure 1 Experimental setup

All the charging strategies increase the battery chargingcharacteristic at different degrees proposed in [8ndash14] But thecharging performance at low temperature is not consideredAlthough the preheating charging strategy at low temperatureproposed in [15 16] can increase the charging capacity theself-preheating process costs toomuch time and cannot workat low SOC condition This paper analyzes the chargingcharacteristic of a Li-ion battery at different temperature useselectrochemical model and first-order 119877119862 equivalent circuitmodel to analyze the bad low temperature characteristic ofLi-ion battery in theory and proposes a multistage CC-CVstrategy The multistage CC-CV strategy is compared withCC-CV and two-stage CC-CV strategies at 25∘C 0∘C andminus10∘C

2 Experimental

21 Battery and Equipment The battery used is 18650 cylin-drical Li-ion battery with normal capacity of 137 Ah anormal voltage of 32 V and a cut-off voltage of 36 V Themaximum charging and discharging rates are 1 C and 2Crespectively The positive electrode material is LiFePO

4 and

negative electrode material is LiC6 The battery tester is LD

battery tester with 8 test channels and the test process can beprogrammed and monitored by computer The battery wastested in a temperature chamber to ensure the temperatureparameter to be constant The detailed parameters of batterytester and temperature chamber are shown in Table 1 Theexperimental setup can be described as in Figure 1

22 Experimental Process The battery charging strategiestested in experiments were CC-CV two-stage CC-CV andmultistage CC-CV The test temperature points were 25∘C0∘C and minus10∘C The charging strategies are explained asfollows

14

13

12

11

10

09

08

07

06

05

04

03

02

01

00

Capa

city

(Ah)

25 0 minus10

(∘C)

Figure 2 CC-CV strategy charging capacities at different tempera-ture

For the CC-CV strategy the constant current processwas charging at 03 C to the cut-off voltage of 36 V and theconstant voltage process was charging at 36 V for 5min

For the two-stage CC-CV strategy the first constantcurrent process was charging at 1 C to the cut-off voltageof 36 V Then in the second constant current process thecharging current was decreased to 05 C Since the chargingcurrent was decreased the terminal voltage was decreasedbelow 36V allowing the constant current process to beextended until the terminal voltage reached the cut-offvoltage once againThe constant voltage process was chargingat 36 V for 5min [17]

For the multistage CC-CV strategy the constant currentprocess was divided into ten stages The maximum and min-imum rates were 1 C and 01 C respectively and the chargingcurrent was decreased by 01 Cwhen terminal voltage reachedthe cut-off voltageThe constant voltage process was chargingat 36 V for 5min

3 Charging Characteristic of Battery atLow Temperature

31 Charging Capacity Characteristic at Different Tempera-ture The selected battery was charged by CC-CV strategyat 25∘C 0∘C and minus10∘C to obtain the charging capacitycharacteristic at low temperature Before every chargingprocess at different temperature the battery was dischargedempty at 25∘C and kept for six hours to ensure the wholebattery temperature to be uniform As shown in Figure 2the charging capacities at 25∘C 0∘C and minus10∘C are 1309Ah1196Ah and 0664Ah respectively The charging capacity isdecreased by 86 at 0∘C and 493 at minus10∘C compared withthat at 25∘C The charging capacity has a great decrease atminus10∘C

32 OCVCharacteristic at Different Temperature Thebatterywas tested by hybrid pulse power characteristic (HPPC)rule that is detailed in ldquoFreedom CAR Battery Test Manualrdquo[18] to obtain OCV ohmic resistance (119877

119903) and polarization


340

335

330

325

3200 10 20 30 40 50 60 70 80 90 100

OCV

(V)

SOC ()




formula

SOC = SOC0minus

1

AHCint

119905

0

119894 119889120591(1)



33 119877

119903and 119877



119903and 119877

119901increase





90 of SOC






Rr

(Ω)

0050

0045

0040

0035

0030

0025

0020

00150 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 4 119877119903


Rp

(Ω)

0040

0035

0030

0025

0020

0015

0010

0 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 5 119877119901




119862

6+ 119911Li+ + 119911119890minus

Charge997888997888997888997888997888rarr Li

119909119862

6


4

Charge997888997888997888997888997888rarr Li


4+ 119911Li+ + 119911119890minus

(2)




Positivecurrent

collector

Negativecurrent

collector

x

I ILi+

Li+

Li+

Li+



120597119862

119890

120597119905

=

120597

120597119909

(119863

119890

120597119862

119890

120597119909

) +

1 minus 119905

0

+

119865

119895

Li

(3)


119863


+




120597119862

119904

119889119905

=

119863

119904

119903

2

120597

120597119903

(119903

2119862

119904

120597119903

)

(4)


119904is the Li+




119863

119904(119879) = 119863ref exp [

119864

119886119863

119877

(

1

119879refminus

1

119879

)] (5)

where 119864




119895

Li= 119895

0exp [

120572

119886119865

119877119879

120578] minus exp [120572

119888119865

119877119879

120578] (6)


119886and 120572

119888are the


120578 = 120601

119904minus 120601

119890minus 119880ocv (7)





119895

0= 119865119896

0119862

120572119886

119890

(119862

119904max minus 119862119904surf)120572119886

119862

120572119886

119904surf (8)




119896


119896

0(119879) = 119896

0ref exp [119864

119886119877

119877

(

1

119879refminus

1

119879

)] (9)

where 119864







Up

Cp

Rp

UoOCV

+

minus

Rr

Ur I





119903 119862119901and 119877

119901 respec-



119901and 119877




119862

119901

119889119906

119901(119905)

119889119905

+

119906

119901(119905)

119877

119901

= 119894 (119905)

119906

0(119905) = 119906ocv (119905) + 119894 (119905) 119877119903 + 119906119901 (119905)

(10)


can be obtained

119906

119901(119905) = 119868119877

119901(1 minus 119890

minus119905120591

)

119906

0(119905) = 119906ocv (119905) + 119868119877119903 + 119906119901 (119905)

(11)

where 120591 = 119877

119901119862

119901




and 119877




119903

and 119877


119903and 119880


0is


119901and 119877



0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

15

10

05

00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05




0 5000 10000 15000

Time (s)

CC-CVSOC = 948


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










340

335

330

325

3200 10 20 30 40 50 60 70 80 90 100

OCV

(V)

SOC ()




formula

SOC = SOC0minus

1

AHCint

119905

0

119894 119889120591(1)



33 119877

119903and 119877



119903and 119877

119901increase





90 of SOC






Rr

(Ω)

0050

0045

0040

0035

0030

0025

0020

00150 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 4 119877119903


Rp

(Ω)

0040

0035

0030

0025

0020

0015

0010

0 10 20 30 40 50 60 70 80 90 100

SOC ()


Figure 5 119877119901




119862

6+ 119911Li+ + 119911119890minus

Charge997888997888997888997888997888rarr Li

119909119862

6


4

Charge997888997888997888997888997888rarr Li


4+ 119911Li+ + 119911119890minus

(2)




Positivecurrent

collector

Negativecurrent

collector

x

I ILi+

Li+

Li+

Li+



120597119862

119890

120597119905

=

120597

120597119909

(119863

119890

120597119862

119890

120597119909

) +

1 minus 119905

0

+

119865

119895

Li

(3)


119863


+




120597119862

119904

119889119905

=

119863

119904

119903

2

120597

120597119903

(119903

2119862

119904

120597119903

)

(4)


119904is the Li+




119863

119904(119879) = 119863ref exp [

119864

119886119863

119877

(

1

119879refminus

1

119879

)] (5)

where 119864




119895

Li= 119895

0exp [

120572

119886119865

119877119879

120578] minus exp [120572

119888119865

119877119879

120578] (6)


119886and 120572

119888are the


120578 = 120601

119904minus 120601

119890minus 119880ocv (7)





119895

0= 119865119896

0119862

120572119886

119890

(119862

119904max minus 119862119904surf)120572119886

119862

120572119886

119904surf (8)




119896


119896

0(119879) = 119896

0ref exp [119864

119886119877

119877

(

1

119879refminus

1

119879

)] (9)

where 119864







Up

Cp

Rp

UoOCV

+

minus

Rr

Ur I





119903 119862119901and 119877

119901 respec-



119901and 119877




119862

119901

119889119906

119901(119905)

119889119905

+

119906

119901(119905)

119877

119901

= 119894 (119905)

119906

0(119905) = 119906ocv (119905) + 119894 (119905) 119877119903 + 119906119901 (119905)

(10)


can be obtained

119906

119901(119905) = 119868119877

119901(1 minus 119890

minus119905120591

)

119906

0(119905) = 119906ocv (119905) + 119868119877119903 + 119906119901 (119905)

(11)

where 120591 = 119877

119901119862

119901




and 119877




119903

and 119877


119903and 119880


0is


119901and 119877



0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

15

10

05

00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05




0 5000 10000 15000

Time (s)

CC-CVSOC = 948


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Positivecurrent

collector

Negativecurrent

collector

x

I ILi+

Li+

Li+

Li+



120597119862

119890

120597119905

=

120597

120597119909

(119863

119890

120597119862

119890

120597119909

) +

1 minus 119905

0

+

119865

119895

Li

(3)


119863


+




120597119862

119904

119889119905

=

119863

119904

119903

2

120597

120597119903

(119903

2119862

119904

120597119903

)

(4)


119904is the Li+




119863

119904(119879) = 119863ref exp [

119864

119886119863

119877

(

1

119879refminus

1

119879

)] (5)

where 119864




119895

Li= 119895

0exp [

120572

119886119865

119877119879

120578] minus exp [120572

119888119865

119877119879

120578] (6)


119886and 120572

119888are the


120578 = 120601

119904minus 120601

119890minus 119880ocv (7)





119895

0= 119865119896

0119862

120572119886

119890

(119862

119904max minus 119862119904surf)120572119886

119862

120572119886

119904surf (8)




119896


119896

0(119879) = 119896

0ref exp [119864

119886119877

119877

(

1

119879refminus

1

119879

)] (9)

where 119864







Up

Cp

Rp

UoOCV

+

minus

Rr

Ur I





119903 119862119901and 119877

119901 respec-



119901and 119877




119862

119901

119889119906

119901(119905)

119889119905

+

119906

119901(119905)

119877

119901

= 119894 (119905)

119906

0(119905) = 119906ocv (119905) + 119894 (119905) 119877119903 + 119906119901 (119905)

(10)


can be obtained

119906

119901(119905) = 119868119877

119901(1 minus 119890

minus119905120591

)

119906

0(119905) = 119906ocv (119905) + 119868119877119903 + 119906119901 (119905)

(11)

where 120591 = 119877

119901119862

119901




and 119877




119903

and 119877


119903and 119880


0is


119901and 119877



0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

15

10

05

00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05




0 5000 10000 15000

Time (s)

CC-CVSOC = 948


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Up

Cp

Rp

UoOCV

+

minus

Rr

Ur I





119903 119862119901and 119877

119901 respec-



119901and 119877




119862

119901

119889119906

119901(119905)

119889119905

+

119906

119901(119905)

119877

119901

= 119894 (119905)

119906

0(119905) = 119906ocv (119905) + 119894 (119905) 119877119903 + 119906119901 (119905)

(10)


can be obtained

119906

119901(119905) = 119868119877

119901(1 minus 119890

minus119905120591

)

119906

0(119905) = 119906ocv (119905) + 119868119877119903 + 119906119901 (119905)

(11)

where 120591 = 119877

119901119862

119901




and 119877




119903

and 119877


119903and 119880


0is


119901and 119877



0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

15

10

05

00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05




0 5000 10000 15000

Time (s)

CC-CVSOC = 948


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30

29

28

27 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

50

45

40

35

30

25

20

15

10

05




0 5000 10000 15000

Time (s)

CC-CVSOC = 948


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31

30 00

Volta

ge (V

)

Curr

ent (

A)


10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32

31 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 5532


0

10

20

30

40

50

60

70

80

90

100

SOC

()







0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 00

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

10

09

08

07

06

05

04

03

02

01


0 10 20 30 40 50 60 70 80 90 100

SOC ()

37

36

35

34

33

32 0

Volta

ge (V

)

Curr

ent (

A)

VoltageCurrent

5

4

3

2

1


0 10 20 30 40 50 60 70 80 90 100

SOC ()

36

37

35

34

32

33

0

Volta

ge (V

)

VoltageCurrent

5

4

3

2

1


0 5000 10000 15000

Time (s)

CC-CVSOC = 3226


0

10

20

30

40

50

60

70

80

90

100

SOC

()




6 Conclusion



14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

12

10

08

06

04

02

00

minus02

minus04

Current rate (C)

Capa

city

(Ah)

00101112 09 08 07 06 05 04 03 02 01






Acknowledgment


References






4





4






4







4







4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra













4












Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article multistage cc-cv charge method...

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