Energy Conversion and Management 51 (2010) 2857–2862

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Identification and modelling of Lithium ion battery

K.M. Tsang, L. Sun, W.L. Chan *

Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:Received 6 January 2009Received in revised form 4 October 2009Accepted 7 June 2010Available online 29 June 2010

Keywords:Lithium ion batteriesUniversal battery modelBattery charging

0196-8904/$ - see front matter Crown Copyright � 2doi:10.1016/j.enconman.2010.06.024

* Corresponding author. Tel.: +852 27666186; fax:E-mail addresses: steve.tsang@polyu.edu.hk (K.M

edu.hk (W.L. Chan).

a b s t r a c t

A universal battery model for the charging process has been identified for Lithium ion battery working atconstant temperature. Mathematical models are fitted to different collected charging profiles using theleast squares algorithm. With the removal of the component which is related to the DC resistance ofthe battery, a universal model can be fitted to predict profiles of different charging rates after time scal-ing. Experimental results are included to demonstrate the goodness of fit of the model at different charg-ing rates and for batteries of different capacities. Comparison with standard electrical-circuit model isalso presented. With the proposed model, it is possible to derive more effective way to monitor the statusof Lithium ion batteries, and to develop a universal quick charger for different capacities of batteries toresult with a more effective usage of Lithium ion batteries.

Crown Copyright � 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Lithium ion (Li-ion) batteries are widely used in many portableelectronic devices such as mobile telephones, digital cameras andlaptop computers because of their excellent performance, compact,high energy density and high reliability. To fully utilize and have abetter management system on the Li-ion battery, models whichcorrectly describe its characteristics are necessary. Researchershave developed a wide variety of mathematical models to describethe dynamics of a battery. The battery models can be divided intofour categories [1]: empirical models, electrochemical models,electrical-circuit models and abstract models using artificial intel-ligence (AI). Empirical models are easy to configure, but the com-putational results are the least accurate. So they are moreapplicable to the imprecise capacity evaluation. Electrochemicalmodels are complex and time consuming to produce predictionsalthough they may be more accurate. Moreover, they need a lotof physical and chemical parameters which are not easily availablefor simulations [2]. Consequently, more efficient battery modelsshould be proposed. Electrical-circuit models are based on a com-bination of voltage sources, resistors, and capacitors to approxi-mate the electrochemical processes and dynamics of a battery[3]. Recently, per unit representation is also proposed to facilitatecomparison among batteries of different capacities [4]. AI basedlearning approach includes artificial neural network (ANN) model-ling [5] as well as support vector machine (SVM) [6] could be veryaccurate but they are very much depending on the training data. It

010 Published by Elsevier Ltd. All r

+852 23301544.. Tsang), eewlchan@polyu.

may not be extended to other batteries without additional trainingdata. A new modelling technique which based on simple mathe-matical equations and least squares algorithm is proposed. Mathe-matical models are fitted to charging profiles of different chargingrates at constant temperature using the least squares algorithm.The fitted mathematical model can easily be realized by simpletransfer function blocks. Experimental results are included to dem-onstrate the goodness of fit of the model at different charging ratesand for batteries of different capacities. Comparison with standardelectrical-circuit model [3] is also presented.

2. Re-charging of Li-ion batteries

The most widely used charging strategy for the Li-ion battery isthe constant-current and constant-voltage (CC–CV) charge strat-egy. Hence it is natural to make use of the charging profile duringthe constant-current charging phase for the identification andmodelling of Li-ion batteries.

2.1. Charging tests

A series of charging tests have been performed on a 3.7 V,700mAh, Li-ion battery at 25�C. The initial states of charge for allcharging tests are set to be the same at 3 V. Constant charging cur-rent of magnitudes 1C, 0.5C, 0.2C and 0.1C where C represents thecapacity of the battery are used and the charging processes arestopped when the battery voltage reached the cut-off voltage at4.2 V. The constant charging currents correspond to 700 mA,350 mA, 140 mA and 70 mA. Fig. 1 shows the charging profiles ofthe battery at 1C, 0.5C, 0.2C and 0.1C charging rate. Clearly the timerequires to charge up the battery at 1C is much quicker than the

ights reserved.

0 1 2 3 4 5 6 7 8 9 102.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Vol

tage

(V)

0.1C0.2C0.5C1C

Fig. 1. Charging profiles at 1C, 0.5C, 0.2C and 1C charging rates.

2858 K.M. Tsang et al. / Energy Conversion and Management 51 (2010) 2857–2862

one at 0.5C, 0.2C and 0.1C. It is very difficult to extract useful infor-mation from the charging profiles at different charging ratesshown in Fig. 1. Since the charging time is inversely proportionalto the charging rate to certain extent, different charging profilescan be brought into a similar time span by proper time scaling.

2.2. Time scaling

In order to compare different charging profiles on similar timescale and extract useful information from different charging pro-files, the charging times of different profiles are multiplied by theircorresponding charging rates to give

T ¼ ct ð1Þ

where t is the actual charging time, c is the percentage of the charg-ing rate and T is the normalized charging time. If the charging rate is0.1C, c becomes 0.1. Fig. 2 shows the charging profiles of the batteryat 1C, 0.5C, 0.2C and 0.1C charging rates after time normalization.Apart from some DC offsets, the four charging profiles are very sim-ilar. According to [7], the DC resistance of Li-ion battery varies verylittle during the charging process and it can be regarded as constant.Also, it does not vary with the amplitude of the charging currentand temperature.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

0.1C0.2C0.5C1C

Fig. 2. Charging profiles at 1C, 0.5C, 0.2C and 1C charging rates after timenormalization.

3. Battery charging model

The following sections report the development of the universalbattery model.

3.1. Removal of voltage rise due to DC resistance

The voltage rise due to the DC resistance of the Li-ion batterydepends on the amplitude of the charging current and the DC resis-tance R. A simple approach to determine the DC resistance of Li-ionbattery is based on the Ohm’s law [8].

vR ¼ IR ð2Þ

where vR is the voltage rise due to a constant charging current I. TheDC resistance can be obtained by measurement or it can be ob-tained from the differences between charging profiles. From thefour charging profiles shown in Fig. 2, the amplitudes of the charg-ing current are 700 mA, 350 mA, 140 mA and 70 mA. It can be ar-gued that the differences in the DC offsets are caused by thevoltage rise due to the amplitude of charging current and the DCresistance. Fig. 3 shows the differences between the charging pro-files V(1C) and V(0.1C), V(0.5C) and V(0.1C), and V(0.2C) and(0.1C), where V(1C), V(0.5C), V(0.2C) and V(0.1C) are the normalizedcharging profiles with 1C, 0.5C, 0.2C and 0.1C charging rates respec-tively. From Fig. 3, the variations of voltage rise due to the DC resis-tance are relatively small. The mean value of V(1C)–V(0.1C) is0.1438 V and the difference in the charging current is 630 mA. From(2), an estimate of the DC resistance is 0.2283 X. The mean value ofV(0.5C)–V(0.1C) is 0.0615 V and the difference in the charging cur-rent is 280 mA, an estimate of the DC resistance is 0.2196 X. Fromthe mean value of V(0.2C)–V(0.1C), an estimate of the DC resistanceis 0.2134 X. A mean estimate for the DC resistance for the Li-ionbattery R = 0.2204 X. If the DC resistance is taken as R = 0.2204 Xand the voltage rise due to the DC resistance at charging rates 1C,0.5C, 0.2C and 0.1C are 0.1543 V, 0.0771 V, 0.03086 V and0.01543 V respectively. Fig. 4 shows the normalized charging pro-files with the removal of voltage rise due to the DC resistance.The four charging curves are very similar.

3.2. Modelling of battery dynamics

With the removal of voltage rise due to DC resistance, all nor-malized charging profiles are very similar. Hence a single modelis sufficient to describe the dynamics at different charging rates.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

Time (Hr)

Volta

ge (V

)

V(1C)-V(0.1C)

V(0.5C)-V(0.1C)

V(0.2C)-V(0.1C)

Fig. 3. Voltage rise according to DC resistance.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Vol

tage

(V)

0.1C0.2C0.5C1C

Fig. 4. Normalised charging profiles with the removal of voltage rise due to DCresistance.

Table 1Comparison of the predicted output and the actual charging profile.

1Ccharging

0.5Ccharging

0.2Ccharging

0.1C charging

Maximum error 0.0223 V 0.0405 V 0.0642 V 0.0759 VMean error 0.0085 V 0.0071 V 0.0026 V 1.94 � 10�6 VStandard

deviation0.0191 V 0.0243 V 0.0260 V 0.0281 V

K.M. Tsang et al. / Energy Conversion and Management 51 (2010) 2857–2862 2859

3.2.1. First order model with an integratorConsider the modelling of the normalized charging profile

shown in Fig. 4 with a first order polynomial and an exponentialof the form

vðTÞ ¼ a0 þ a1T þ a2ebT ð3Þ

where a0, a1, a2 and b are some unknown coefficients, T is the nor-malized charging time and v(T) is the voltage across the chargingbattery with the removal of voltage rise due to the DC resistance.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

model0.1C data

(a) 0.1C charging

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

model0.5C data

(c) 0.5C charging

Fig. 5. Predicted output superimpo

When the least squares method [9,10] is applied to the 0.1C charg-ing profile, the fitted equation becomes

vðTÞ ¼ 3:6341þ 0:4760T � 0:5523e�53:93T ð4Þ

Fig. 5 shows the predicted output of (4) superimposed on thedifferent normalized charging profiles. Table 1 shows the erroranalysis between the fitted equation and different charging pro-files. The maximum modelling error from the four charging profilesis 0.075 V. This clearly demonstrates that the fitted equation cap-tures the behaviour of the battery rather well.

Since (4) is equivalent to the output step response of a batterywhen the charging rate is 1C, the derivative of the charging currentwith respect to T is an impulse function and its Laplace transform isequal to 1. Taking derivative of (4) with respect to T gives

_vðTÞ ¼ 0:4760þ 29:79e�53:93T ð5Þ

and the transfer function of the battery with the removal of DCresistance can be obtained by taking Laplace transform of (5) to give

HðsÞ ¼ 0:4760s

þ 29:79sþ 53:93

ð6Þ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

model0.2C data

(b) 0.2C charging

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

model1C data

(d) 1C charging

sed on actual battery output.

CIRI

I

V

2860 K.M. Tsang et al. / Energy Conversion and Management 51 (2010) 2857–2862

which is a first order model with an integrator. (6) can further bedecomposed to a storage element given by

H1s ðsÞ ¼

0:4760s

ð7Þ

and a transient element

H1t ðsÞ ¼

29:79sþ 53:93

ð8Þ

Cb

R

Fig. 6. Circuit model for Li-ion battery.

3.2.1.1. Normalized full battery model. Combining the DC resistanceof the battery with the (6) to form the normalized full battery mod-el at 1C charging.

H1bðsÞ ¼ Rn þ H1

s ðsÞ þ H1t ðsÞ ð9Þ

where Rn = RC, R is the DC resistance of the battery, C is the capacityof the battery and the time unit T is in hour.

3.2.1.2. Realization of the full battery model. The normalized full bat-tery model can be realized using circuit model or block diagramwith simple transfer function blocks. In circuit modelling, the nor-malized DC resistance can easily be realized by simple resistor. Thestorage element H1

s ðsÞ in (9) is an integrator which has an incre-ment of 0.4760 V at 1C charging. The transfer function can be rep-resented by a capacitor. If the capacity of the battery is C and thetime scale is in second, the capacitance Cb for the storage elementis given by

Cb ¼3600C0:476

F ð10Þ

If C is 700 mAh, the capacitance Cb becomes 5294.1 F. If thecapacity of the battery is large, the corresponding capacitance be-comes large as well. The transient element H1

t ðsÞ in (9) is a first or-der model and it can be represented by a simple resistor andcapacitor in parallel. If RI and CI are the resistance and capacitanceof the resistor and capacitor network, the transfer function of thenetwork becomes

HRCðsÞ ¼RI

RICIsþ 1ð11Þ

Based on the charging profiles shown in Fig. 4, the voltage riseaccording to the transient element at the steady state is fixed at0.5523 V and the magnitude of the charging current will not affectthis steady value. To capture this characteristic and to model thenormalized H1

t ðsÞ by (11), the resistance RI has to be current depen-dent. To realize the resistor and capacitor network with a chargingcurrent of I A, the resistance and the capacitance of the networkcan be obtained as

RI ¼0:5523

IX ð12Þ

and

3600CRICII

¼ 53:93) CI ¼ 120:85CF ð13Þ

The equivalent circuit model [3] is shown in Fig. 6 and the initialcharge of the storage capacitor is taken as the terminal voltage ofthe battery at no load condition. The full battery model can alsobe realized using simple transfer functions after denormalized (9)to give

H1Ib ðsÞ ¼ Rþ H1

s ð3600CsÞ þ 1I

H1t

3600CsI

� �ð14Þ

where I is the magnitude of the charging current, initial value ofH1

s ð3600CsÞ is taken as the terminal voltage of the battery at no loadcondition and the time unit is in second.

3.2.2. First order model with double integratorFrom the charging profiles in Fig. 4, the voltage exponentially

increases towards the end of the charging. Hence a higher orderpolynomial is proposed to capture the charging characteristics.Consider the modelling of the normalized charging profile shownin Fig. 4 with a second order polynomial and an exponential ofthe form.

vðTÞ ¼ a0 þ a1T þ a2T2 þ a3ebT ð15Þ

where a0, a1, a2, a3 and b are some unknown coefficients, T is thenormalized charging time and v(T) is the voltage across the chargingbattery with the removal of voltage rise due to the DC resistance.When the least squares method is applied to the 0.1C charging pro-file, the fitted equation becomes

vðTÞ ¼ 3:7439� 0:0423T þ 0:4738T2 � 0:5972e�32T ð16Þ

Fig. 7 shows the predicted output of (16) superimposed on thedifferent normalized charging profiles. Table 2 shows the erroranalysis between the fitted equation and different charging pro-files. The maximum modelling error from the four charging pro-files is 0.0201 V. This clearly demonstrates that (16) capturesthe behaviour of the battery much better than (4). By comparingTables 1 and 2 it is obvious that the proposed model is outper-forming standard electrical-circuit model with smaller modellingerrors.

Since (16) is equivalent to the output step response of a batterywhen the charging rate is 1C, the derivative of the charging currentwith respect to T is an impulse function and its Laplace transform isequal to 1. Taking derivative of (16) with respect to T gives.

_vðTÞ ¼ �0:0423þ 0:9476T þ 19:11e�32T ð17Þ

and the transfer function of the battery with the removal of DCresistance can be obtained by taking Laplace transform of (17) togive

HðsÞ ¼ �0:0423s

þ 0:9476s2 þ 19:11

sþ 32ð18Þ

which is very close to a first order model with a double integrator.(18) can further be decomposed to a storage element given by

H2s ðsÞ ¼ �

0:0423s

þ 0:9476s2 ð19Þ

and a transient element

H2t ðsÞ ¼

19:11sþ 32

ð20Þ

model0.1C data

model0.2C data

model0.5C data

model1C data

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Volta

ge (V

)

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Volta

ge (V

)

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Volta

ge (V

)

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Volta

ge (V

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (Hr)

(a) 0.1C charging

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (Hr)

(b) 0.2C charging

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (Hr)

(c) 0.5C charging

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (Hr)

(d) 1C charging

Fig. 7. Predicted output superimposed on actual battery output.

Table 2Comparison of the predicted output and the actual charging profile.

1Ccharging

0.5Ccharging

0.2Ccharging

0.1C charging

Maximum error 0.0167 V 0.0177 V 0.0186 V 0.0201 VMean 0.0020 V 0.0037 V 0.0020 V 7.45 � 10�5 VStandard

deviation0.0163 V 0.0125 V 0.0104 V 0.0099 V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Vol

tage

(V)

530mAh1700mAh

Fig. 8. Charging profiles for 530 mAh and 1700 mAh batteries at 1C charging rate.

K.M. Tsang et al. / Energy Conversion and Management 51 (2010) 2857–2862 2861

3.2.2.1. Normalized full battery model. Combining the DC resistanceof the battery with the (20) to form the normalized full batterymodel at 1C charging.

H2bðsÞ ¼ Rn þ H2

s ðsÞ þ H2t ðsÞ ð21Þ

where Rn = RC, R is the DC resistance of the battery, C is the capacityof the battery and the time unit T is in hour.

3.2.2.2. Realization of the full battery model. Since the storage ele-ment of (19) has a double integrator, there is no passive circuit ele-ment to model the double integrator action and a circuit modelconsists of passive elements cannot be realized for (21). However,the full battery model can easily be realized using simple transferfunction blocks. The full battery model can also be realized usingsimple transfer functions after denormalized (21) to give

H2Ib ðsÞ ¼ Rþ H2

s ð3600CsÞ þ 1I

H1t

3600CsI

� �ð22Þ

where I is the magnitude of the charging current, initial value ofHs(3600Cs) is taken as the terminal voltage of the battery at no loadcondition and the time unit is in second.

The actual charging profile can easily be obtained from the nor-malized model by adding the initial state of charge into the simu-lation and scale the time axis accordingly.

4. Model validation

To verify the accuracy and uniqueness of the fitted battery mod-el, two 3.7 V Li-ion batteries with capacities of 530 mAh and1700 mAh are selected. Fig. 8 shows the charging profiles at 1Ccharging. The measured DC resistances for the 530 mAh and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Vol

tage

(V)

530mAh1700mAh

Fig. 9. Charging profiles with the removal of voltage rise due to DC resistance.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (Hr)

Volta

ge (V

)

530mAh1700mAhfit

Fig. 10. Predicted output superimposed on actual battery output.

Table 3Comparison of the predicted output and the actual charging profile at 1C charging.

530 (mAh) 1700 (mAh)

Maximum error 0.0179 V 0.0197 VMean error 0.0016 V 0.0028 VStandard deviation 0.0219 V 0.0193 V

2862 K.M. Tsang et al. / Energy Conversion and Management 51 (2010) 2857–2862

1700 mAh batteries are 0.38 X and 0.145 X respectively. The volt-age rises due to the DC resistances are 0.2014 V and 0.2465 V. Fig. 9shows the charging profiles with the removal of voltage rises due

to DC resistances. Again, the two charging profiles are very similar.Fig. 10 shows the predicted output of (18) which describes thedynamical part of the battery superimposed on the two chargingprofiles and Table 3 shows the error analysis for the fitted model.The maximum modelling error is 0.02 V which clearly shows thatthe fitted full battery model of (21 and 22) captures the chargingprocess of Li-ion batteries at different rates and of different capac-ities very well.

5. Conclusions

In this paper, a universal Lithium-ion battery charging model ispresented. Least squares algorithm has been successfully appliedto obtain the unknown coefficients for the normalized batterymodel. It has been found that the charging model can be decoupledto a DC resistance, a storage element and a transient element. Withproper denormalization, the model can be denormalized to capturecharacteristics of batteries at different charging rates and of differ-ent capacities. Experimental results also verify that the fitted mod-el can capture the charging process at different rates and cope withbatteries of with different capacities. With the proposed model, itis possible to derive more effective way to monitor the status ofLithium ion batteries, and to develop a universal quick chargerfor different capacities of batteries to result with a more effectiveusage of Lithium ion batteries.

Acknowledgement

The authors gratefully acknowledge the support of the HongKong Polytechnic University.

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