Hysteresis Modeling for Estimation ofState-of-Charge in NiMH BatteryBased on Improved Takacs Model

Novie Ayub Windarko, Jaeho Choi

School of Electrical and Computer Engineering.,Chungbuk National University, Korea

E-mail: choi@chungbuk.ac.kr

Abstract- This paper describes about the electrical modelingof NiMH battery to estimate the State-of-Charge (SOC). Therelationship between the SOC and the Open Circuit Voltage(OCV) has the hysteresis characteristic and it is modeled basedon the Takacs model in this paper. The modeling includes both ofmajor loop and minor loop. The original Takacs model isimproved with additional polynomial function to approach theexperimental data as similar as possible. To verify theimprovement by using the proposed Takacs model, the results ofthe original Takacs model and the improved one for the majorand minor loops are compared. The modeling error in majorloop based on the improved Takacs model is less than 4°A. SOC.In minor loop modeling, the error of modeling is less than 6%SOC, but there is no significant difference with the improvedTakacs model.

I. INTRODUCTION

In decades, the increasing of oil prices has motivated thepower electronics engineers to focus on energy efficiency.One of the methods to increase the energy efficiency in thearea of power utility system is the application of energystorage systems. Batteries as electrochemical energy storagesystems have become popular for energy efficiency due totheir fast response. So far, many applications have been usedthe battery energy storage systems from hundreds watts toseveral megawatts. Especially in the microgrid system, thebattery energy storage system plays the main role for thepower quality regulation and energy efficiency. Wind powerand solar power are one of the uncontrollable energy sourcesand so the power utility could be adjusted to meet the timeband of demand load and could be kept the efficiency high byapplying the battery energy storage.

The term of State-of-Charge (SOC) is used to determinethe battery capacity. In the energy management system, theSOC monitoring has the following functions that it couldmaintain the battery not to enter the modes of over-charge orover-discharge. These conditions may lead to the damage ofbattery. In some applications, the SOC has to provide the userany information how long he could use the battery. Such as inthe HEV application, the driver should know the residualdriving distance.

So far, Coulomb counting method is the most popularmethod to estimate the SOC. The residual capacity could bemeasured by a simple algorithm of the product of the currentand the duration time of charge or discharge. This method may

be more precise when it is considered the temperature and thecharging or discharging efficiencies [1]. On the other hand, ithas some disadvantages as followings: There is no possibilityto estimate the initial SOC and it depends on the precision rateof sensor. And, by the time, the error is integrated and resultedin high errors.

The measurement method of Open Circuit Voltage (OCV)is found as an accurate one to determine the SOC. The OCVcould be measured under no load condition. This method isvery suitable to estimate the initial SOC.

In the battery measurement, there has been used two termsof voltage generally: Battery Terminal Voltage and OCV. Theterminal voltage is only equals to the OCV when no currentflows and the voltage have relaxed to its equilibrium state [2].Especially for Li-ion battery, the equilibrium state is reachedafter no current flows for more than ten hours. The voltageafter 30 min differs by approximately 15 mV from the voltageafter 600 min [2].

At the present, there are many battery types with theiradvantages and disadvantages. NiMH battery is popular one.NiMH type is reported favorable one by several reasons, suchas high energy density, low prices, non-toxic and safety.However, the hysteresis phenomenon is strongly appeared inOCV ofNiMH battery [3].

So far, hysteresis modeling is often discussed in magneticissue. With similarities of hysteresis characteristic betweenbattery and magnetic, several modeling methods used inmagnetic system could be adopted for the modeling ofbatteries. In paper [4], Preisach model is applied for thehysteresis modeling of NiMH battery. However this model isinvolved the long computation and memory resources. Severalresearchers have proposed the hysteresis modeling based onthe mathematical function. Most of the modeling methodsbased on the mathematical function involved the magneticparameters in its modeling. One of the latest models is Takacsmodel [5]. The Takacs model with the several parameters ofhysteresis curve is much simpler. The Takacs modelparameters have no direct physical meaning in magnetic dataexperiments. With this reason, Takacs model is suitable for thehysteresis modeling ofbattery.

In this paper, a simple NiMH battery model of hysteresischaracteristic has been proposed for SOC estimation based onTakacs model. Takacs model is provided for both of major andminor hysteresis loops and the improved one is provided less

Fig.. 1.Simple battery model.

Fig.2.Thevenin battery model.

+Eo-T__Fig. 3.Resistive Thevenin battery model.

Rj

+EoT _where

SOC/: SOC at time tSOCo: SOC at starting time17: discharging or discharging efficiencyI: battery currentt: time

The capacity could be changeable by the temperature. Theefficiency is influenced by the internal resistance and otheraspects such as temperature. Based on this definition in (1),Coulomb counting method is very convenient although theprecision of modeling is influenced by the several conditionsas described previously.

There have been proposed many battery models. In thissection the advantage and disadvantage of the conventionalbattery models are described .

error of estimation. By applying the improved Takacs model,it is more suitable to estimate the SOC by online due to itssimplicity which reduces the computation difficulties .

II. BATTERY MODELING

The capacity of a battery could be described by the productof the current and the duration of charge or discharge. TheSOC is determined from the previous battery capacity aftersubtracting the discharging capacity or adding the chargingcapacity. The SOC is expressed as:

SOC/ =SOCo- fn -I -tdt (1)

B. Simple Battery Model

This model is the most commonly used. As shown in Fig.1. This model consists of Eo, OCV or Electro Motive Force,and R, series internal resistance . VI is the battery terminalvoltage. Eo is obtained from the measurement of an opencircuit voltage. Although this model has been commonly used,it could not cover the varying characteristic of the internalresistance and OCV with the charging condition of SOC. Thismodel is only applicable in a simulation where the energydrawn from the battery is unlimited or the model element isindependent on SOC [6].

C. Thevenin Battery Model [7lThevenin battery model is shown in Fig. 2. It consists of Eo,

OCV, RiO' internal resistance , and RC parallel network of CT,

transient capacitance and RT, transient resistance . This addedRC parallel circuit is considered for the prediction of a batteryresponse to the transient load changes. In another paper [8],RC parallel circuit is represented for a physical component,which CT is represented as a parallel plates and RT isrepresented as a non-linear resistance contributed from thecontact resistance of the plate of electrolyte. The disadvantageof this model is that the model elements are assumed to beconstant in any changing condition of battery such as SOC,temperature , and etc.

D. Resistive Thevenin Battery model

This model has two internal resistant based on dischargingand charging states as shown in Fig. 3. Ric and Rid are

+Eo

Fig. 4. Modified battety model.

associated with the internal resistance under the chargingmode and discharging mode, respectively. Both of theseresistances are represented by energy losses which include theelectrical and non-electrical losses. The diodes are implied thatduring the charging or discharging mode the only one of theresistances Ric or Rid could be conducted. In any states ofcharging or discharging modes, the conducting diode isforward biased while another diode is reverse biased. There isno physical meaning for the diodes without modeling purposes.This model is not dependent on the SOC.

E. Modified ModelIn paper [9], a modified battery model as shown in Fig. 4

has been proposed by the authors. The configuration of theproposed model is same as the previous resistive Theveninbattery model, but all of the elements are described as thefunction of SOC. The OCV values are different from eachother in the charging and discharging modes. In case of thecharging mode, OCV is higher than that of the dischargingmode.

The model of the charging and discharging modes isdescribed as in (2), respectively.

where z is the variable of SOC, and I, and Id are the chargingand discharging currents, respectively. The elements of model,Eo, V" and R; are obtained from the various currents of L, andi,

1.25

· Averaged OCV

• 1-hr OCY oata aner lC charge

· l- hr OCV data aner tC discharge

I\. - - OCY equatlon

....... _-.... -

~- - ~- --..,<,

1.45

1.5

(2a)

(2b)

Eo(z) =~(z)-IcR;c(z)

Eo(z) =~(z)+ IdR;Az)

where f, is ascending branch and f. is descending branch, a, isx when f+(x) =0, and b l is value obtained from (7):

1.2a 10 20 30 40 50 60 70 80 90 100

Percent State of Discharge

where c. and C2 is constant.The difference between the Langevin function and the

Brillouin one is so small and the Brillouin function approachesthe Langevin one as a limit when C2 tends to zero.

Fig. 5.Average OCVforcharge anddischarge [10].

Based on a quantum mechanical approach, Brillouinmodified the Langevin model [11]. Brillouin model isdescribed by (5).

B(x) =C1 corhrc.x) -c2 cothtc.x) (5)

(6a)

(6b)

f+(x) =tanh(x-ao)+b.

f_(x) = tanh(x+aO)-bl

B. Takacs's Hysteresis Modeling

Takacs model is based on a mathematical function forhysteresis modeling [5]. Takacs model is followed the severalprevious models based on mathematical models such asLangevin or Brillouin. In Fig. 6, it is shown the properties ofhysteresis with parameter in Takacs's model.

By using this model, major loop and minor loop ofhysteresis curve could be modeled. In Takacs model, majorloop could be divided into two branches, ascending branchand descending branch. The equations are shown in (6):

IV. HYSTERESIS MODELING FOR OCV

A. Average OCVfor Charging and Discharging Modes

In paper [10], it is described the application of NiMHbatteries for HEV. The electrical model of battery is based onThevenin battery model with SOC dependent. The SOCmonitoring algorithm is based on average OCV during thecharging or discharging mode. In this modeling, major loopand minor loop is avoided. Between OCV and SOC has asingle relationship. With an algorithm based on a simpleaverage model, it is reported that HEV could operate at theculminate efficiency with 80-mile-per-gallon. The black linein Fig. 5 is the regression line for averaged OCV

B. Preisach Model

The Preisach model has been used widely to estimate themodel of hysteresis in many areas such as smart materials,sensors, actuators, electric-magnetic relays and transformers,and etc., but this model involves a complex calculation. Inpaper [4], the Preisach model is used for the HEV application.The continuous Preisach model is modified into a discretemodel which is more suitable for onboard hysteresisestimation due to the limited computational and memorysources. It is reported that the error of algorithm within10%SOC.

III. OPEN CIRCUIT VOLTAGE MODELING

Comparing with another battery types, NiMH type hasmuch stronger hysteresis characteristic. With this uniquecharacteristic, several authors have been applied severalapproach to develop the model for OCV ofNiMH battery.

(3)

Fig. 6. Hysteresis modeling based onTakacs model anditsproperties.

A. Overview on Hysteresis Modeling

A number of mathematical models of hysteresis have beenproposed by many researchers. Langevin proposed theLangevin function which described the saturation effect forthe hysteretic curve. The equation is shown below:

L(x) = coth(x)-XThe Langevin model leads to the well known Curie law for

paramagnetic susceptibility [11]. Generally, Curie law couldbe described by (4).

C(x) = A(coth ; - :) (4)

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-4 -3 -2 -1 a 2 3 4

where A and B is constant.

where xmis x at maximum value of fux).For minor loop, the curve could be described by the

following equation:

f+(x) =tanh(x-ao)+cu

L (x) = tanh(x +ao) - Cd

(7)

(8a)

(8b)

y

Y1o

The expression for Cn and Cd is given as follows:

tanh(xm - ao)- tanh(x- ao)C =C1

u tanlux; -ao)-tanh(xr-ao)

tanh(-xm +ao)-tanh(x+ao)Cd =C1 tanh(-xm +ao)-tanh(xr +ao)

(9a)

(9b)

(9c)

oFig. 7. Observed value, Yi andcurve model, ttx;).

The sum of squared residuals is given as:

x

(12)

where Co. ci, ...cnare the polynomial coefficients.

where t;.(x) is an error function between the measurement dataand the original Takacs model. As the motivation to develop asimple model, it is proposed t;.(x) as a polynomial functionexpressed as shown (II):

(13)

D. Scaling Technique

Takacs model is based on a tangent hyperbolic function.The typical curve is shown in Fig. 8. As shown in Fig. 8, the xaxis could be set from any values and the maximum value ofthe y axis is unity. In a battery model, the x axis could beOCV and the y axis could be SOC. The OCV value is higherthan zero. For single cell NiMH battery, the OCV isapproximated 1.2-1.4V. The SOC value is in the range of0-100%. The difference between the original Takacs modeland its application for battery model can be solved by scaling.

The scaling technique to solve the different betweenoriginal Takacs model and battery modeling is shown in Fig. 8.

The residual is defmed as the difference between the valueof the dependent variable and the predicted value from theestimated model which expressed as:

(11)

(10a)

(10b)

f+(x) =tanh(x - ao) +b, + f+r (x)

f: (x) = tanh(x +ao) - hI + [.; (x)

where x, is the reversal point of minor loop.However, in naturally, the hysteresis curve obtained from

the experimental data might not fully followed by the Takacsmodel. So it is proposed to add an additional polynomialfunction to cover any possibilities of a hysteresis characteristicwhich is not fully followed by the Takacs model. Theimprovement ofthid model is shown in (10):

V. RESULT

Batte Model

SOCma

----------------_ _-----

Basic TakacsModel

OCVmaxocv.;

soc.; --------:;

Fig. 8. Illustration forscaling technique.

Ymax ---------- ;!;

! Xmax

;: ------- Ymin

A. OCV Measurement

In this experiment, Sebang GMH 100 NiMH Battery ratingof I.2V and 100Ah is used. The experiment is done with the

C. Least Square Error Method

The Least Square Error method is described firstly by CarlFriedrich Gauss around 1794. The method of least squares isused to solve the overdetermined systems approximately, ofwhich system equations have more equations than unknowns.Least squares are often applied in statistical contexts.

Least squares could be interpreted as a method to fit thedata. The best modeling in the least-squares error methodcould be reached when the sum of squared residuals has itsleast value. The residual is the difference between an observedvalue, Yi and the value given by the model, f(Xi)'

By adjusting the parameters of a model function, f(Xi), itwould get the most similar model function to a data set. Asimple data set consists of n points (data pairs) (x., Yi), i = I,..., n, where Xi is an independent variable and Yi is a dependentvariable whose values are found by observation. The modelfunction has the form shown in Eqs. (6) and (7). The leastsquares method defmes that the similarity is the best when thesum of squared residuals, S, is minimum .

80.0070.0060 .0050 .00

.............., , ., ,

40.00

0.80 ..

0.40 '-----__~ ~ ~ ~__------'

30 .00

0 ' 0 ..

1,00 • •••• ••• • • ••• •

V1.' 0 ,----~---~---~---~--______,

1.40

1.20

70.0060 ,0050.00

••• •••••• ••••••• !'•••••••• •• ••••• •:••••• •• •• •••••••, ,, ,, ,, ,, ,, ,

---------------- .----------------,----------------, ,, ,, ,, ,, ,, ,, ,

40.0030.00

1.40

1.30 ......: .

1.20 -- -------------,---------------

V1.' 0 ,------,-----,-----,-----,------,

1,50 --------------- -------------- --------------- ------------

Time (Ks) T ime (Ks)

Fig. 9. OCVmeasurement with10"10 SOCcharging step. Fig. 11. OCVmeasurement with10"10 SOCdischarging step.

v, , , ,. , , ,, , , ,

:: .- --.:-~ ~ : : : : :T : : : - - ..-.. --i ·..·::T::::::-..--'r - --, .-_._-_.....•.__.... _.... ,.... _..... -, ,, ,, , ,... - ....... ..........................•............

, ,, , ,1 ------:------------: ------------: ------------, --- -------

, , . ,, , . ,.... _ _.,.. _ , _ ,.. _ _ , _ _., , , ., , , . ,. , . . ,................................................._., _ , .

1.22

1.20

1.25

1.27

1.30

, , . . ,, , , . ,, , , . ,1.35 •••••••••••• ~ •••••••••••• ~ •••••••••••• ~ •••••••••••• ~ •••••••••••• ~ ••••••.•••••

, " ,, , ,1.32 •••••••• •. •• • ••••••••• ~ •••••••••••• t··········· .~ .

, , ,

1.37 ,----~--~--_--_--_-----,

\.

-----~----' .36~ _

1.38

1.40

47.0046 .0045 .0044 .00Time (Ks)

43 .0042 .00

1.18 '-----__~__~ ----J

41 .0043,0042 .004 1.0040.00

Tim e (l<s)

30 .0038,00

1.34

37 .00

Fig. 10.Zoom picture ofsquare inFig. 9.

charging current of 1C at 30°C. The experimental results areshown in Fig. 9. The battery is charged with lO%SOC step,and then the battery is open for one hour. The starting point ofa charging process step is indicated with square and thestopping point is indicated with a triangular as shown in Fig10.

After one hour in open circuit condition, there is nosignificant changing in the battery voltage. The battery voltagechanging after one hour open circuit is approximated ImV orless. In this condition, it could be assumed that the batteryvoltage reaches at the equilibrium state or be assumed that thebattery voltage as OCV. The OCV is obtained at the positionindicated with a circle in Fig 10. The battery is assumed as fullcapacity or 100%SOC when the battery voltage reaches at1.5V.

In Fig. 11, it is shown the OCV measurement for thedischarging process. The experimental condition is same asthe charging process. The starting point and stopping point for

Fig. 12.Zoom picture ofsquare inFig. 11.

a step discharging process is shown in Fig. 12. The startingpoint is indicated with square and the stopping point isindicated with triangular. As one step of discharging process,the SOC is reduced to ten percent lower. The empty conditionis recognized by the sharp drop voltage near to the end ofdischarge process. Generally, for a single cell NiMH battery,the empty condition or O%SOC is reached when the batteryvoltage reaches at 0.9V in discharge process . In this dischargeprocess , when the battery voltage is reached less than 0.9V,the discharge process is stopped to avoid any damage tobattery.

B. OCV Modeling

Figure 13 shows the OCV vs SOC experiment data and themodeling result based on the Takacs model. The uniqueness ofOCV in NiMH battery is obvious comparing to anotherbattery.The OCV between the charging state and thedischarging one is different. Non-linearity of curve is very

I I I I I I

100 --- T--- '--- ~----r--- r-- - -- ~---I I I I I

I I I I I

I I I I I I

00 ---~---~----1----~ 8 ~---~---I I I I I I

: : ::0 .: :I I I I I I I

00 --- T---'----I---- ---r- 1---~---

()~ I I I I I II I I I I I

~ I I I I I I I

40 - - - -I- - - - --I - - - -I - - - -0- - - - .... - -+ - - - -I - - -1 1 1 1 I I 1

1 1 1 0 1 I -+ I 11 1 1 1 I 1

20 - - - +- - - ~ - - - -: - - :- - - - - - - J - - - -: - - -1 1 1 1 1

1 1 1 -+ 1

1 1 1 I I 1

o ---+--- - - - - ~ - - -~ - - - ~ - - - ~ - - -1 1 1 I I 1

Fig. 13. Modeling ofmajor loop based onTakacs model.

1.51.451.41.21.151.1 1.25 1.3 1.35OGV(V)

Fig. 14. Modeling ofmaior loop based onimproved Takacs model.

1 1 1 1 I 1

100 --- T---'--- ~---- r---r--- -- ~---1 1 1 1 1

1 1 1 1 1

1 1 1 1 I 100 L ~ I ~ ~--- ~---

1 1 1 1 I 1

1 1 1 1 I 1

1 1 1 1 I 1

1 1 1 1 I 100 --- T--- '----I---- 1--- ~---

()~ 1 1 1 I 11 1 1 I 1

o 1 1 1 I I 1

00 ~ ~---~----I--- ~ - - -~ - - - -I - - -1 1 1 I I 1

1 1 1 I I 1

1 1 1 I I 1

20 - - - +- - - ~ - - - -: - - - ~ - - - -: - - -1 1 1 I 1

1 1 1 I 1

1 1 1 1 I I 1

o --- +--- --I----~--- ~---~---~---1 1 1 I I 11.51.451.41.25 1.3 1.35

OGV(V)1.21.151.1

minor loop1 20 ,-----~-~-~--~-~-~--~-,

II

I I I I I I100 --- T--- l-------- ~---r---l--- ~---

I I I I I II I I I I I I

00 --- +--- ~----I---- ~---~-- --- ~---

I I I I II I I I I

~ 00 ---+---~----:----~ 1 ~---b' I I I I

~ 40 - - - ~ - - - ~ - - - - :- - - - -t - - - - :- --I I I I II I I I I

W l J I_ _ __ J ~ _I I' I II I I II I I I I I

o ---T--- --I----r---r--- '---,---I I I I I II I I I I I

-20 '-- --'-- ----'-- -'- - -'---'----'- --'-- -'1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

OCV(V)

Fig. 15.Modelingof minorloopbasedon improved Takacsmodel.

serious in 0-20%SOC and 80-100%SOC. Between 20%SOCand 80%SOC, the curve is lean to be linear.

As shown in Fig. 13, the modeling could be presentedaccurately for SOC. By applying the Takacs model describedas in (6), the maximum error is up to 10% in the full range ofO-lOO%SOC. Around the full or empty capacity, the modelingerror could be close to zero.

To reduce the modeling error, the improved Takacs modeldescribed as in (10) is used. The polynomial function is usedto cover the residual error between (6) and experiment data.The modeling result using (10) is shown in Fig. 14. The errorin this modeling is reduced to less than 4% SOC.

The minor loop of hysteresis is shown in Fig. 15. Itconsists of:- Discharging from 80%SOC to O%SOC- Discharging from 60%SOC to O%SOC- Discharging from 40%SOC to O%SOC- Discharging from 20%SOC to O%SOC

The OCV in minor loop is obtained by a procedure assame as the previous explanation. The modeling error in minorloop is less than 6%SOC but there is no significant differencewith the application of the improved Takacs model.

VI. CONCLUSION

In this paper, the electrical modeling for a hysteresischaracteristic of OCV of NiMH battery is proposed. Themodeling is based on the Takacs model but improved with theadditional polynomial functions. The modeling is provided forboth of major and minor loops. By the proposed model, asimple mathematical model could be given with themeasurement data of open circuit voltage. The modeling errorfor major loop is less than 4%SOC and minor loop is less than6%SOC. This model can be used in any SOC estimationalgorithm of batteries which has the hysteresis characteristic.

ACKNOWLEDGMENT

This research was supported by the MKE (The Ministry ofKnowledge Economy), Korea, under the ITRC (InformationTechnology Research Center) support program supervised bythe lITA(Institute for Information Technology Advancement)"(lITA-2009-C 1090-0904-0007)

REFERENCES

[l] Kong Soon Ng, Chin-Sien Moo, Vi-Ping Chen, and Yao-ching Hsieh,"Enhanced Coulomb Counting Method for Estimating State-Of-Chargeand State-Of-Health of Litihium-Ion Batteries," Journal of AppliedEnergy, vol. 86, no. 9, pp. 1506-1511, Sept. 2009.

[2] V. Pop, H. 1. Bergveld , J. H. G. Op het Veld, P. P. L. Regtien , D.Danilov, and P. H. L. Notten , "Modeling Battery Behavior for AccurateState-of-Charge Indication," Journal of The Electrochemical Society,153 (11), A2013-A2022 , 2006.

[3] Antoni Szumanowski and Yuhua Chang, "Battery Management SystemBased on Battery Nonlinear Dynamics Modeling ," IEEE Trans onVehicular Tech, vol. 57, no. 3, pp. 1425-1432, May 2008.

[4] X. Tang, X. Zhang, B. Koch, and D. Frisch, "Modeling and Estimationof Nickel Metal Hydride Battery Hysteresis for SOC Estimation ," Proc.International Conference on PHM, 2008.

[5] J. Takacs, "A Phenomenologi cal Mathemati cal Model of Hysteresis," Int.J. for Computation and Mathematics in Electrical and ElectronicsEngineering, vol. 20, no.4, pp. 1022-1014,2001.

[6] Salameh et. AI, "A Mathematical Model for Lead-Acid Batteries ," IEEETrans. on Energy Conversion, vol. 7, no.I , pp. 93-97, March 1992.

(7] Min Chen and Gabriel A. Rincon-Mora, "Accurate Electrical BatteryModel Capable of Predicting Runtime and I-V Performance ," IEEETrans. on Energy Conversion, vol. 21, no. 2, pp. 504-511,2006.

[8] D Sutanto and HL Chang, "A New Battery Model for use with BatteryEnergy Storage Systems and Electric Vehicles Power Systems," Conf.Proc. of IEEE Power Engineering Society Winter Meeting, vol. 1, pp.470-475 ,2000.

[9] Novie Ayub Windarko , Jaeho Choi, and Deoksu Hyun, "SOCEstimation based on OCV for Continuous Charging/Discharging Processin NiMH Battery," Proc. of;rdJapan-Korea Workshop Joint TechnicalWorkshop on Semiconductor Power Converter, pp. 1-6,2009.

[101 Mark Verbrugge and E. Tate, "Adaptive State of Charge Algorithm forNickel Metal Hydride Batteries including Hysteresis phenomena,"Journal ofPower Sources, 126, pp. 246-249,2004.

[Ill Amalia Ivanyi, Hysteresis Models in Electromagnetic Comp utation, Int 'ISpecialized Book Service Inc., 1997.

Appendix

Table 1. Parameters for major loop.

Charg ing mode Discharging modeScale :-2.sX~ Scale :-2.4.$X~.2

a,r1.8 , a,rO.3,Or iginal Takacs bF400 .8121387e-003, b,=10 .277758532e-003 ,mode I paraneter xnF2 xnF2.2

«». min 1.20SV, o». min 1.2051/ ,max 1.403V max 1.403V

1.370.$Xs1.4031.315.$Xs1 .403

dt~2 . 55568899316082e+003c t 1=2ZJ

dt1=-7.01542603433868e+OctO=-321

dt0=4.81266304689473e+003

Polynomial for 1.313.$Xs1.3701.296.$Xs1.315

0'Z?-0-51 .823217920938ge+003irrprovement c21=-202 .9383

d21= 136.401466204330e+003caF270.6711

daF-89.7447602244507e+003

1.205.$Xs1 .3131.205.$Xs1.296

~-3. 49373888793905e+003c31=34.8241

d31=8 .61994532792247e+003c3?=-41 .9630

1130=-5.31395291040670e+003

Table 2. Parameters for minor loop.

Discharge20-0% 40-0% 60-0% 80-0%

minor loopScale: Scale: Scale: Scale:

Original-2.35Xs1 .0 -2.3~ 1.2 -2.3~ 1.6 -2.3~.0

Takacs mode Ix,=l .348V x,=l .368V x,=1 .378V x,=1 .391V

Parameterx,r1.205V x,r1.205V x,r1.205V x,r1.205V

a,rO .3 a,rO.3 a,rO.3 a,rO.3b,=12e-003 b,=12e-003 b,=12e-003 b,=12e-003