IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

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Abstract—For a battery powered safety critical system the safe

duration of power for executing a specific task is extremelyimportant. It is necessary to avoid unacceptable consequences dueto unwanted battery power failure. An early stage estimation ofthis duration reduces the overall risk through optimization ofcurrent consumption by switching off noncritical load ahead ofdelivery of power to a critical load. In order to address this issue,an online battery state of charge estimator on chip is conceivedand implemented using Kalman filter. The Kalman filterestimates the true values of measurements by predicting a value,considering the estimated uncertainty of the predicted value, andthen computing a weighted average of the predicted value and themeasured value. The basic idea is more accurate state predictionis possible when the state predicted value is fused with sensorprediction under any uncertain disturbance. The state estimatoris developed in the form of an algorithm and stored into a singlechip microcontroller. It is finally used to generate an early stagewarning signal against battery failure. The paper presents amethodology for creating energy aware system that would avoidsudden system failure due to power outage. The authors used ageneralized state space model of the battery to estimate the effectof unobserved battery parameters for duration estimation. Anexperiment was conducted in this regard through discharging thebattery under constant load. Subsequently the internalparameters of battery were calculated. The model was simulatedthrough MATLAB/simulink R2008a software and efficiency wastested. The program for prediction was finally emulated in amicrocontroller and found satisfactory result.(Abstract)

Index Terms— Kalman filter, Microcontroller, Recovery effectin battery, State of charge (SOC), State estimation, State spacemodel, and Surface resistance

I. INTRODUCTION

n general online estimation of battery health for a batterybacked embedded systems is difficult task. Here battery

operates as an auxiliary or a secondary power source whichautomatically connects to the processor when primary or mainpower source fails. This is essential to avoid losses like dataloss due to unwanted power interruption. In this case durationof effective discharge time of the battery plays a critical roleand hence puts an upper bound on the systems operationalperformance. The investigation shows that main problem ofearly stage battery state estimation arises due to unnoticedbattery health degradation due to aging of internal structure ofthe battery like resistance and capacitance values, Thisdegradation of hidden battery parameters leads to inaccurateprediction of discharge time across a same load. Theperformance is measured by the earliness of prediction. Theproposed methodology significantly improve performances ofthe system through prediction of discharge time more ahead ofactual taking these hidden or otherwise unobserved states intoconsideration through a mathematical model.

According to Battery Association of Japan in 2010, 20% ofrechargeable batteries were NiMH type [1]. NiMH typebatteries have approximately 2-3 times more capacity thantheir NiCd counterparts. NiMH batteries have nearly the samevolumetric energy density as Li-ion batteries. A significantlimitation of NiMH battery is its high rate of self discharge.However after the evolution of low self discharge NiMHbattery in 2005 by Sanyo [2], these batteries have gainedsignificant popularity. The cost of Li-ion battery is higher thanthat of NiMH battery which makes NiMH battery a lucrativechoice in certain applications. In a nutshell, research on NiMHbattery paves to be a productive and challenging area ofresearch both in product development and economicperspective.

Probabilistic Duration of Power Estimationfor Nickel- metal- hydride (NiMH) Battery

under Constant Load using Kalman Filter onChip

1Anirudh Kumar, 2Joydeb Roychaudhury, 3Soumen MandalCSIR- Central Mechanical Engineering Research Institute

Durgapur, West Bengal,India1anirudhkumar@cmeri.res.in,2 jrc@cmeri.res.in, 3somandal88@gmail.com

I

IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

ISBN: 978-81-909042-2-3 ©2012 IEEE

644

II. EARLY WORK

A substantial amount of work for prediction of dischargeprofile has been carried out by Bhangu et al on Lead Acidbatteries [3]. The analysis results for a NiMH battery aredifferent due to non uniform and step-type discharge profilecompared to flat discharge profile for Lead acid battery. UsualNiMH discharge profiles consist of a normal discharge period,followed by over discharge if the normal discharge period isexceeded. These two stages are separated by a point known asknee of discharge as shown in Figure 1. At this point thepolarity of cell reverses itself.

III. GOAL OF THE WORK

The uneven battery discharge profile, due to corner point(knee point) creates major problem for prior to outage ofpower across a load. Hence accurate prediction of state ofcharge for NiMH battery prior to reaching the knee voltage isimportant. However under this condition the code complexityand software execution timing overheads for accurate stateestimation in real time needs to be taken care of. As peravailable literature, implementation of microcontroller basedKalman filter for real time battery state estimation is relativelynew area. This motivates to design this filter on chip using amicrocontroller for prediction of discharge pattern of NiMHbatteries across a load which is applied for early prediction ofknee voltage.

IV. SCOPE OF WORK

The primary scope of work is to estimate probabilisticduration of discharge voltage of a NiMH battery against aconstant load on the basis of battery internal parameters whichchanges due to aging. This is estimated from battery model interms of an equivalent circuit described in this section. Theother part of the work is to develop a suitable methodology todetect the knee voltage at an early stage of occurrence. Thework describes a methodology for early stage estimation ofremaining using useful battery life ahead of 5 sec of actualtime of reaching knee voltage which can be used to executesome critical task. This can be utilized to execute 2.712673*106 single machine cycles instructions to perform a safetycritical task assuming the controller operates at 11.0592Mhzwith 6 clock cycle for executing each machine cycle.

A. Battery Model

A model of a system describes its behavior in absence ofactual system. Hence one important task is to describe amathematical model of the battery to understand the agingproblem under operation with reference to time. In literaturenumber of battery models are available such asElectrochemical model (Doyle et al) [4], Electric circuit model(Hageman) [5], Analytical model (Peukert) [6], KineticBattery Model (Manwell et al) [7]. All these models have theirown advantages and disadvantages. Electrochemical modelrequires numerous battery parameters which are difficult to beknown. The Electric circuit model is based on look up tableswhich increases data complexity. The analytical model(Peukert’s model) is simple but ignores the recovery effect inbattery which makes it less accurate [9]. The Kinetic BatteryModel is not suitable for modern batteries which have slopeddischarge profile.

To maintain simplicity in the battery performance analysisthe model as described by Johnson et al [8] is considered. Thebattery equivalent circuit is shown in Figure 2 which consistsof a simple circuit with resistances and capacitances as electriccomponents with their values tabulated in Table 1. Anexperiment has been performed in the laboratory to confirmthe effectiveness of the circuit to suit the basic objective ofstate estimation and for comparison a model was developed insimulink for the battery equivalent circuit. The experiment wasconducted with periodic discharge of 50 seconds across a loadof 100 ohm resistor followed by a period of 100 seconds withno load and simulink model was also simulated under sameconditions of load. Digital multimeter was used to log thebattery terminal voltage to computer during the experiment. Asshown in figure 3 the battery terminal voltage obtained frommodel (blue-dotted line) during recovery and discharge periodof the battery is close to the extreme values of experimentallyobtained battery terminal voltage (red-continuous line). Goodagreement was observed between the terminal voltage dataobtained from the model and that obtained experimentally,with a maximum error of 1.4%.

Figure 2: Equivalent Battery Circuit in Simulink

Figure1: Discharge profile of NiMH Battery

IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

ISBN: 978-81-909042-2-3 ©2012 IEEE

645

Figure 4: Discharge of battery under periodic load for Ccs calculation

(b) Discharge of Battery

(a) Periodic load

Where, Ccb is bulk capacitor representing battery storagecapacity and changes with time indicating the age of battery,Ccs is the surface capacitor, Re is end resistance, Rs is surfaceresistance and Rt is internal resistance which changes withtime due to aging.

It has been observed that battery discharge pattern doesn’tchange so frequently when charged and discharged in sameconditions, a significant amount of time is required to changebulk capacitance (Ccb) and internal resistance (Rt) of thebattery, so the values of Ccb and Rt can be assumed constantfor a single short time operational discharge. The assumptiongreatly reduces the computational overhead for battery stateestimation.. The aging effect of battery increases the internalresistance of battery (Rt) and decrease the bulk capacitance(Ccb). This in turn reduces the charge holding capacity of thebattery which renders it unable to support a load for claimedduration. At present this work doesn’t takes the aging effectinto consideration and deals with new battery.

Equations 1 and 2 provide the state space model [3] ofbattery equivalent circuit shown in figure 2.

.

V 0cb.

V = 0

1 1 Rs-

Ccb(Re + Rs) Ccb(Re + Rs) Ccb(Re + Rs)Vcb1 1 Re

- Vcs × I

VoCcs(Re + Rs) Ccs(Re + Rs) Ccs

× +cs. X 0 Y

V

(Re

0

+ Rs)Z

(1)

Where,

2Rs Re Rs Rs

X = - + - +2 2 2 2

CcbRe(Re + Rs) Ccs(Re + Rs)Ccb Re + Rs Ccs Re + Rs

Rs 1Y = -

CcbRe(Re + Rs)) Ccs(Re + Rs)

2Re RsRt Rt ReRs

Z = - + +2 2CcbRe(Re + Rs) Ccs(Re + Rs)

Ccs(Rs + Re) Ccs(Rs + Re)

Vcb

Vo = 0 0 1 × Vcs

Vo

(2)

Where Vcb (Voltage across bulk capacitance), Vcs (Voltageacross surface capacitance) and Vo (Terminal voltage ofbattery) are the state variables, I (discharge current throughbattery) is the control input and Vo is the measured output.

From the state equations the state matrix (Ac), the inputmatrix (Bc), the output matrix (Cc) and the feed through matrix(Dc) are calculated and shown as follows (for values of Ccb,Vcs, Rt, Re and Rs refer to Table 1)

1 1-

-0.

0Ccb(Re+Rs) Ccb(Re+Rs)

01

001 0.0010

0.6958 -0.691

A = - 0 = 0c Ccs(Re+Rs)58

0.6949 -0.694Ccs(Re+Rs)

090

X Y

Rs

Ccb(Re + Rs)Re

=

0.0009

B = 0.6263c1.147Ccs(Re R 4+ s)

Z

, C =c 0 0 1 , D = [0]c and

Vcb

X = Vcs

Vo

V. CALCULATION OF BATTERY PARAMETERS

An experiment was conducted using a 9-Volt 170 milli-ampere-hour, NiMH battery manufactured by UnirossCompany, France to calculate the values of Bulk capacitance(Ccb), Surface capacitance (Ccs), Surface resistance (Rs), Endresistance (Re) and internal resistance (Rt) as shown in batterymodel in figure 2. The discharge profile for the battery underperiodic load of 100 ohm with time period 10 seconds andduty cycle of 50% is considered and is as shown in Figure 4.

The parameters Ccb and Ccs were calculated usingequations 3, 4 and 5 and the procedure for evaluating them

Figure 3: Comparison of Terminal Voltage form model with measuredBattery’s Terminal Voltage.

IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

ISBN: 978-81-909042-2-3 ©2012 IEEE

646

were adapted from Bhangu et al [3]. Where, Vx%SOC denotesOCV of battery at X% SOC. τ is the time constant.

Δt denotes the time for which load is on (here 5 seconds).V1, V2, V3, V4 are as shown in Figure 3(b).

Rated(Amp - Sec) × V100%SOC

Ccb =1 2 2

V - V100%SOC 0%SOC2

(3)

V - V4 3τ = -Δtln 1 -

V - V1 3

(4)

τCcs =

R + Re s

(5)

From the experimental data shown in Figure 6 V100%SOC=9.48 V and V0%SOC= 8.48 V, the rated ampere-hour of thebattery used being 0.17 mAh, the value of Ccb= 575.9171Farads. The terminal voltage profile shows V1= 9.08 V, V2=8.97 V, V3= 9.06 V, V4= 9.07 V with Δt = 5 seconds yieldCcs= 0.7984 Farads.

Rt and Rs were calculated from battery resistance measuredseparately by calculating the difference between the OCV ofbattery and voltage drop across the load connected to thebattery and dividing it by the current drawn from battery. Themeasured net resistance of battery was 1.2 ohm. For simplicitythe value of Re was considered equal to Rs, bearing a numericvalue as 75% of measured net resistance. We obtain Re=Rs=0.9 ohm. So, Rt = 1.2 – (0.9/2) = 0.75 ohm.

Table 1 Calculated Battery Parameters

VI. DESIGN OF KALMAN FILTER

Kalman filter has been used to predict the value of batteryterminal voltage before the value is actually reached.

B. Formulation

First step for application of Kalman filter to our system is toconvert the continuous linear system derived from equation (1)and (2) to discrete system. Taking assumption that thedischarge current remains constant for the sampling period δtand solving equation (1) we obtain (6) and (7).

A δtcA = ed (6)

'A Γ 'cδt

B = e dΓ B0d c (7)

And from equation (2) we have,

C = Cd c (8)

In the equation (6) and (7) an approximation for expansionof exponential series given by equation (9) has been used toget the final expression given by equations (10), (11) and (12).

A δt A δt A δtA δt 2 3 4c c cc

e = (I + A δt + ( ) + ( ) + ( ) + ...) (I + A δt)c c

2! 3! 4!

(9)

A = I + A × δtd c (10)

B = B × δtd c (11)

C Cd= c (12)

Where Ad is the discrete state space matrix, Ac is thecontinuous state space matrix, Bd is the discrete input matrix,Bc is the continuous input matrix, Cd is the discrete outputmatrix, Cd is the continuous output matrix, δt is the samplingtime and I is identity matrix.

Calculated values of Ad, Bd for Cd for NiMH battery are

0.9952 0.0048 0

A = 3.4792 -2.4792 0d3.4744 0 -2.4744

0.0043

B = 3.1313d5.7370

C = 0 0 1d

From here onwards symbols A, B, C will be used torepresent Ad, Bd for Cd respectively.

Equations (13) to (17) show the formulation of Kalmanfilter. This needs initial state matrix X, Error covariancematrix Po, Q and R. Q and R are the process noise andmeasurement noise which are assumed as Gaussian, havingmean zero and variance Sw and Sz respectively. Kalman filteris used as predictor/corrector algorithm. Here the statepredictor and sensor predictor fused to estimate the predictedbattery state. In the predictor stage priori estimated values ofXK and PK are calculated using the known values at (K-1) th

instant.-

x = A x +BuK K-1 K (13)

Using the value of control input (here discharge current ofbattery) at Kth time and posteriori estimate of X at (K-1)thtime, X at Kth instant of time is predicted which is the prioriestimate of X at Kth instant.

- TP = AP A + QK K-1 (14)

In equation (14) priori estimate of error covariance for Kthtime is calculated using posteriori estimate of error covarianceat (K-1)th time instant.

In the corrector stage priori estimated values of state vectorand error covariance for Kth instant are used together with themeasurement value (z) of battery terminal voltage at Kthinstant to calculate the posteriori estimated values of statevector and error covariance at Kth instant.

First of all Kalman Gain at Kth instant (KK) is computed- T - T -1

K = P (CP + R)K K KC C (15)

- -x = x +K (z - Cx )K K K K K (16)

-P = (I - K )PK K KC (17)

C.Proposed Algorithm for Kalman filter

Kalman filter is implemented on the microcontroller asrecursive algorithm. Following are the initializations that aredone.

Ccb(Farads)

Ccs(Farads)

Rt(Ohm)

Rs(Ohm)

Re(Ohm)

575.9171 0.7984 0.75 0.9 0.9

IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

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647

Figure 5(a): Experimental Setup for Battery State Estimation

Sz = 4.5063e - 7 ,

10.0008 0 0

Sw = 0 10.0008 0

0 0 0.88

,

1 0 0

P = 0 1 000 0 1

and

9.48

X = 9.309.39

.

For microcontroller programming, functions for multiply(MUL) and transpose (TRAN) of 3X3 matrix were generated.The appropriate function calls are made from main predictionprogram on requirement. The algorithm for Multiplication ofmatrices is as shown and is denoted by MUL function.1) Algorithm for MUL function

Input: A1 & B1 = Matrix inputsra & ca = No. of rows & columns in A1cb = No. of columns in B1

Output: ans= A1*B1Step 1: STARTStep 2: for (i=0;i<ra;i++)Step 3: for (j=0;j<cb;j++)Step 4: *(ans+(3*i)+(j))=0Step 5: for(k=0;k<ca;k++)Step 6: *(ans+(3*i)+(j))+=((*(A1+(i*ca)+(k)))

*(*( B1+(cb*k)+(j))))Step 7: STOP

2) The algorithm for evaluation of transpose of matrix is asshown and is denoted by TRAN function

Input: A1 = Matrix Inputsra & ca = No. of rows & columns in A1

Output: ans= transpose of matrix A1

3) The algorithm pertaining to the Kalman filter for batterystate estimation

Input: Ccb = Bulk Capacitance ValueCcs = Surface Capacitance ValueRe = End ResistanceRt = Internal Resistance.MinV= Knee voltage.

Output: Time ‘t’ when predicted terminal voltage of batteryfalls below knee voltage .

Step 1 : STARTStep 2 : Declare A[3][3], B[3][3], C[1][3]= {0,0,1},

eye[3][3]={1,0,0,0,1,0,0,0,1}, X[3][1]={9.48,9.3,9.39}, X_p1[3][3],X_p2[3][3], X_p3[3][3], X_p4[3][3], X_p5[3][3], X_p6[3][3],X_p7[3][3], Xhat[3][1], Sz=4.5063e-007, Sw[3][3]={10.0008,0,0,0,10.0008,0 ,0,0,0.88}, p[3][3]={1,0,0,0,1,0,0,0,1},u=X[3]/101, t, p, y, inn, S, K[3][1], LED=0, a1[3][3]={(-1/Ccb),(1/Ccb),0, (1/Ccs),(-1/Ccs),0, (1/Ccs)-(1/Ccb),0,(1/Ccb)-(1/Ccs)}, b1[3][1]={(1/Ccb),(1/Ccs),(1/Ccs)+(Rt/(Ccs*Re))-(Rt/(Ccb*Re))}, dt=5

Step 3: For i=0 to 3 Step 1Step 4: For j=0 to 3 Step 1Step 5: A[i][j]= (a1[i][j]*(1/(2*Re))*dt)+eye[i][j]Step 6: For i=0 to 3 Step 1Step 7: B[i][0]=(b1[i][0]*dt)/2

Step 8: Xhat[i][0]=X[i][0]Step 9: While (Xhat[0][0]>=MinV)Step 10: MUL(A,3,3,Xhat,1,X_p1)Step 11: MUL(B,3,1,u,1,X_p2)Step 12: For i=0 to 3 Step 1Step 13: Xhat[i][0]=X_p1[i][0]+X_p2[i][0]Step 14: y=receive_dmm()Step 15: MUL(C,1,3,Xhat,1,X_p1)Step 16: inn=y-X_p1[0][0]MUL(C,1,3,P,3,X_p1)Step 17: MUL(C,1,3,P,3,X_p1)Step 18: TRAN(C,1,3,X_p2)Step 19: MUL(X_p1,1,3,X_p2,3,X_p3)Step 20: S=X_p3[0][0]Step 21: S+=SzStep 22: MUL(P,3,3,X_p2,3,X_p1)

(Optimization: x_p2 has transpose of C value and s is 1*1matrix so inv(s)=1/s)

Step 23: For i=0 to 3 Step 1Step 24: K[i][0]=(X_p1[i][0])/SStep 25: For i=0 to 3 Step 1Step 26: Xhat[i][0]+=(K[i][0]*inn)Step 27: MUL(A,3,3,P,3,X_p1)Step 28: TRAN(A,3,3,X_p3);Step 29: MUL(X_p1,3,3,X_p3,3,X_p6)Step 30: MUL(A,3,3,K,1,X_p4)

(Optimization: P*C'*inv(s) is replaced by K)Step 31: MUL(P,3,3,X_p3,3,X_p2)

(Optimization: x_p2 has transpose of C value and s is 1*1matrix so inv(s)=1/s)

Step 32: MUL(C,1,3,X_p2,3,X_p5)Step 33: MUL(X_p4,3,3,X_p5,3,X_p7)Step 34: for (i=0;i<3;i++)Step 35: for (j=0;j<3;j++)Step 36: P[i][j]=X_p6[i][j]-X_p7[i][j]+Sw[i][j];Step 37: STOP

VII. EXPERIMENT

The experiment consists of running the algorithm given inprevious section through a single chip microcontroller. Fewoptimizations were done in the code in order to reduce codecomplexity and subsequent reduction in time for real timeapplication. These optimizations were performed in step 22,30 and 31 as mentioned in the algorithm. In step 14receive_dmm() function waits for 5 sec and then receive thevalue from digital multimeter through serial communication.Subsequently the hex code was generated and the code wasfinally emulated in P89C51RD2BN microcontroller with11.0592 MHz crystal oscillator. The experimental set up for

Step 1: STARTStep 2: for (i=0;i<ra;i++)Step 3: for (j=0;j<ca;j++)Step 4: *(ans+(3*j)+(i))=0Step 5: *(ans+(3*j)+(i))=*(A1+(ca*i)+(j));Step 6: STOP

IEEE-International Conference On Advances In Engineering, Science And Management (ICAESM -2012) March 30, 31, 2012

ISBN: 978-81-909042-2-3 ©2012 IEEE

648

Figure 5(b): Schematic of Experimental Setup

microcontroller based battery state estimation is as shown inFigure 5(a).

The algorithm has been coded in Keil C compiler. Thealgorithm before running need to be initialized with initialvalue of state vector (Voc,Vsc,Vt). For this purpose as shownin figure 5(b), the switch is turned off to measure Voc, thenswitch is closed to measure Vt and finally Vsc is calculatedusing Vsc= (2 * Vt) -Voc.

VIII. EXPERIMENTAL RESULTS

The real time discharge profile and the estimated dischargeprofile of battery using microcontroller under a constant loadof 101 ohm as are shown in Figure 6 and 7.

Figure 7 shows the performance of Kalman Filter. Att=2710 sec Kalman filter predicts the value of battery terminalvoltage to be 8.323V when battery is considered to beconnected to the constant load of 101 ohm, shown by bluecolor continuous line in graph. As in this work Kalman filterpredicts the value 5 sec ahead of real time so we havecompared this predicted value of battery terminal voltage withthe battery terminal voltage at t=2715 sec obtained bydischarging the battery for 2715 sec against the constant loadof 101 ohm, which is shown by red color dashed line in graph.At t=2715 sec the battery terminal voltage was found to be8.33V which is very close to the predicted value of 8.323V.

This shows that Kalman filter prediction can be very muchhelpful in predicting the battery state and avoiding thedischarge of NiMH battery beyond knee voltage of discharge.

IX. CONCLUSION

It can be very well understood from the experimental resultsthat the Kalman filter based algorithm is a good choice to be

implemented on microcontroller for battery state estimationproblem in real time. The maximum percentage error inestimation is 14.762% and the minimum error being 4.9X10-5

%. The mean error is 0.3246%, the variance being 1.0587. Outof 2705 seconds of estimation, the error stays within 2% of itsmaximum value for 1705 seconds. Thus there is a goodagreement between predicted and real time values. The workhas immense applications in modern portable devices, batterypowered vehicle mobile robot where size and powerconsumptions are the major limitations.. The process is able toalarm a battery user regarding its deteriorating state of charge.It can prevent accidental damage or inconveniences caused toa system due to power fail almost automatically.

REFERENCES

[1] “Total battery production statistics in 2010 (Calender year),” BatteryAssociation of Japan, 2011.

[2] S. Yasuoka, Y.Magari, T.Murata, T. Tanaka, J.I shida, H. Nakamura, T.Nohma, M. Kihara, Y. Baba, H. Teraoka, “Development of highcapacity nickel-metal hydride batteries using super lattice hydrogen–absorbing alloys,” Journal of Power Sources, Vol. 156, pp. 662-666,2006.

[3] B. S. Bhangu, P. Bentley, D. A. Stone, and C. M. Bingham, "Nonlinearobservers for predicting state-of-charge and state-of-health If lead-acidbatteries for hybrid-electric vehicles," IEEE Trans. Veh. Technol., vol.54, pp. 783-794, May. 2005.

[4] M. Doyle, T. F. Fuller, and J. Newman, “Modeling of galvanostaticcharge and discharge of the lithium/polymer/insertion cell,” Journal ofthe Electrochemical Society, vol. 140, no. 6, pp. 1526–1533, 1993.

[5] S. C. Hageman, “Simple PSpice models let you simulate commonbattery types,” Electronic Design News, vol. 38, pp. 117 – 129, 1993.

[6] D. Rakhmatov and S. Vrudhula, “An analytical high-level battery modelfor use in energy management of portable electronic systems,” inProceedings of the International Conference on Computer Aided Design(ICCAD’01), pp. 488–493, 2001.

[7] J. Manwell and J. McGowan, “Lead acid battery storage model forhybrid energy systems,” Solar Energy, vol. 50, pp. 399–405, 1993.

[8] V. H. Johnson, A. A. Pesaran, and T. Sack, “Temperature-dependentbattery models for high-power lithium-ion batteries,” presented at theProc. EVS 17, Montreal, PQ, Canada, 2000.

[9] M.R. Jongerden and B.R. Haverkort, "Battery Modeling" University ofTwente, Centre for Telematics and Information Technology, TechnicalReport TR-CTIT-08-01, 2008.

Figure 7: Estimated and predicted time

Figure 6: Real time discharge profile of battery