Electrical Battery Model for Dynamic Simulationsof Hybrid Electric Vehicles

Ari Hentunen∗, Teemu Lehmuspelto † and Jussi Suomela∗∗Aalto University School of Electrical Engineering, Espoo, Finland

†Aalto University School of Engineering, Espoo, Finland

Email: ari.hentunen@aalto.fi

Abstract—This paper presents an electrical battery model forlithium-ion (Li-ion) batteries that can be used for dynamic simula-tions of hybrid electric non-road mobile machinery (NRMM) andother vehicles. Although the model has been developed mainlyfor large vehicle batteries with Li-ion based chemistries, it can beused for other battery chemistries as well. The parameters can beextracted from simple measurement sets. The model calculatese.g. state-of-charge, terminal voltage, and open-circuit voltage.In this paper, the model structure and parameter extractionare explained in detail, and a model for a 25.9 V lithium-ionpolymer battery module with 40 Ah cells is presented. Parametersare extracted from experimental measurements and the modelis validated by making another experiment with more realistichybrid electric NRMM current profile.

I. INTRODUCTION

Due to tightening legislation and regulations for the exhaust

emissions of the diesel engines used in non-road mobilemachinery (NRMM) [1], [2], the original equipment manu-facturers (OEMs) have shown rising interest to electrify the

drivetrain. Electrification is going on strong in the areas of

passenger vehicles, light-duty vehicles, and heavy-duty vehi-

cles as well. Common for all electric vehicles (EV) and hybridelectric vehicles (HEV) is that a large electrochemical battery

is usually used as electrical energy storage (ES). Lithium-ion(Li-ion) chemistries offer superb properties such as high power

rating, high energy density, and high cycle life. Therefore, Li-

ion batteries are under intensive research at the moment and

are likely to be the choice of EV manufacturers [3].

In the early stages of research and development process

of vehicle electrification, simulations are usually used as a

tool to evaluate concept studies and to validate early design

goals. Fairly simple models are usually enough at that stage—

models need not be very accurate, instead, ease of modeling

and fast execution time of the simulation are more important.

As the development process goes on, more accurate models of

subsystems are needed to validate the component selection and

sizing as well as to provide important information for vehicle’s

control algorithms. It is important to calculate the state-of-charge (SOC), open-circuit voltage (OCV), power losses, and

terminal voltage accurately also for arbitrary current profiles.

An accurate and fast simulation model with good runtime

prediction and voltage response characteristics is needed.

The electrical battery model proposed in [4] is very intuitive,

offers good runtime prediction and voltage response charac-

teristics as well as SOC and OCV calculation, and is fairly

straightforward to parameterize. The parameter extraction is

done by making few discharge cycles with certain load current

profiles. Also ageing and ambient temperature can be taken

into account in an intuitive manner. The model was developed

for small batteries in portable electronics, and thus, it does

not take into account all aspects of large HEV batteries and

their use. However, with minor additions and modifications

it is well suitable for large HEV batteries as well. Most

notably, HEV batteries are typically high power batteries,

which can be loaded with very high currents. The current

rate affects model parameters as well, and this effect must

be taken into account in the model. Another difference is

that also charge process must be considered in HEVs. The

charge characteristics differ from discharge characteristics, and

therefore, another parameter set and experiment set must be

made.

In this paper, the proposed battery model structure and

parameter extraction are explained, and a model for a 25.9 V

lithium-ion polymer battery comprised of seven 40 Ah cells is

presented. Model parameters are extracted from experimental

measurements. The model is validated using a battery current

profile that is similar to an underground mining loader’s typical

duty cycle by comparing simulation results with experimental

results.

II. MODEL

The electrical equivalent circuit of the battery model is

shown in Fig. 1. The SOC is determined in the left-hand

side and it is represented as the voltage uSOC, which varies

between 0 and 1 as SOC varies between 0 and 100 %. The

usable capacity—which is affected by e.g. cycle number,

calendar life, and temperature—of a battery is transformed

into a capacitor with a capacitance Ccap in such a way that

its voltage goes from 0 to 1 as the battery is charged from

empty to full charge. The self-discharge property of a battery

is modeled with self-discharge resistance Rsd. The controlled

current source jb is directly controlled by the battery current

ib, i.e., jb = ib. The right-hand side determines the OCV

uOCV and terminal voltage ub, which differs from OCV under

loading.

A. Usable capacity

The voltage of the capacitor Ccap represents the whole

charge that is stored in the battery. The equivalent capacitance

978-1-61284-247-9/11/$26.00 ©2011 IEEE

Rsd Ccap

+

uSOC

−

jb−

+ uOCV

RsRτh

Cτh

Rτm

Cτm

Rτs

Cτs

ib

+

ub

−

Fig. 1. Battery model.

is defined as

Ccap = 3600× C × f1(N)× f2(T ) (1)

where C is the nominal capacity in Ah, N is cycle number, Tis temperature, f1(N) is cycle-dependent correction factor, and

f2(T ) is temperature-dependent correction factor. Function f2is time-dependent, and also f1 changes with time as the cycle

number is increased, although the change in f1 is very small

during short simulations.

B. Open-circuit voltage

The nonlinear relation between SOC and OCV is repre-

sented as a voltage-controlled voltage source uOCV(uSOC).The relation between SOC and OCV can be obtained from

experimental tests.

C. Transient response

Under loading the terminal voltage differs from the OCV.

In the model, resistor Rs represents battery’s internal dc

resistance. The following RC network represents the time

constants that are related to the relaxation effect in the

battery’s voltage response to stepwise current excitation. A

comprehensive analysis of the selection of proper number of

RC networks and their parameterization can be found in [5].

In [6], three RC networks was proposed for HEV batteries.

In the proposed model, three time constants was found

to be a good compromise between accuracy and complexity.

The time constants are in the areas of hours, minutes, and

seconds. Therefore, the resistances and capacitances are named

correspondingly as Rτs, Cτs

, Rτm, Cτm

, Rτh, and Cτh

. All

resistances and capacitances in the right-hand side are also

functions of SOC, current rate, calendar life, cycle life, and

so on, i.e., those are not constant values. If a simpler model is

needed, the shortest time constant branch of the RC network

can be neglected.

D. Curve fitting methodologies

The open-circuit voltage as well as the resistors and capac-

itors of the transient response RC network are functions of

SOC. The easiest method to implement their characteristics

is by representing them with look-up tables. However, better

results can be obtained by utilizing curve fitting techniques.

For example, the OCV can be represented with a polynomial

f(x) = p0 + p1 x+ p2 x2 + . . .+ pn xn (2)

where n is the degree of the polynomial, constants p0 to pn

are the corresponding constants from the fitted curve, and x

is the variable, i.e., SOC. Also an exponential fit is may be

useful:

f(x) = a1 eb1·x + a2 e

b2·x + . . .+ an ebn·x (3)

where n is the number of exponential terms, an and bn are

the corresponding constants from the fitted curve, and x is the

variable, e.g., SOC

III. MODEL EXTRACTION

The true capacity of even a new battery may differ from

the nominal value given in the specification. Moreover, when

two or more cells are connected in series, the total available

capacity is affected also by the charge balance as well as

the differences of the cells. The weakest cell with the lowest

capacity determines the maximum available capacity of the

battery. In order to be able to utilize the whole capacity, the

cells need also to be in balance. A battery management system(BMS) is usually used to balance the cells and to monitor

the cells and the battery system. Without a BMS the battery

will eventually get unbalanced, because the cells have different

leakage currents that discharge the cells slowly with different

rate [7].

Because the actual capacity may differ from the nominal

capacity, the actual capacity of a battery needs to be measured

first by discharging a full battery at the nominal temperature

with a low rate—e.g. 0.5C—in such a way that the temperature

would stay constant. Discharging should be stopped when the

terminal voltage reaches the cut-off voltage. If the battery con-

sists of several cells, the discharging should be stopped when

the first cell reaches the cut-off voltage. However, because the

terminal voltage is affected by the internal resistance voltage

drop and relaxation effect, there is still some charge left in

the battery that can be extracted when the relaxation effect is

over. Therefore, the discharging should be continued after a

pause period and finished with a very low current. The actual

measured capacity can then be used as a nominal capacity of

a battery in the equivalent capacitance calculation in (1).

The cycle-dependent correction factor f1 in (1) can be

implemented e.g. as a look-up table or a polynomial function,

if the relation between the cycle number and the capacity is

known accurately. If there is no exact information available

other than cycle life at 80 % DOD, a linear polynomial can

be used as a first approximation:

f1(N) = 1−0.2

Nnom×N (4)

The next question is related to the determination of a cycle.

In hybrid vehicles, the battery is usually never fully charged or

discharged. Instead, the loading profile consists of repetitive

short-time charging and discharging pulses. Consequently, the

SOC is usually kept in a small area around some operating

point, e.g. 50 %. These shallow cycles need to be converted to

full cycles for the function f1. If there is no better knowledge

available, it is possible to convert the nominal cycle life

value to ampere-hours and to integrate the charge or discharge

0 3600 7200 10800 14400 18000 216000

5

10

15

20

25

30

35

40

45

Time [s]

Cur

rent

[A] |

Tem

pera

ture

[C]

CurrentTemperature

(a) Measured discharge current and temperature.

0 3600 7200 10800 14400 18000 21600

20

22

24

26

28

30

Time [s]

Bat

tery

vol

tage

[V]

Measurement

(b) Battery voltage.

Fig. 2. Experimental discharge cycle. Full battery is discharged in pulsesof 10 % of the battery’s capacity with 1C current. After each 10 % dischargepulse, a pause is performed.

current to get the actual cumulative ampere-hour value of the

battery.

Function f2(T ) can be determined from experiments at dif-

ferent temperatures, or from manufacturer’s datasheet, which

usually includes charge and discharge curves with relative

capacity at different temperatures. Remarkable here is that f2can also be slightly larger than one, which is often true at

elevated temperature.

Parameters of the right-hand side circuit of Fig. 1 can be

extracted by making a set of pulse discharge experiments with

several discharge rates. In each experiment, a full battery is

discharged in pulses of 10 % of the battery’s capacity with

constant current. After each 10 % discharge pulse, a pause is

performed. The lenght of the pause period can be selected

arbitrarily. With longer pause periods more accurate OCV

values can be obtained. However, the duration of the test

becomes easily very long with long pause periods.

An example discharge current profile and voltage response

at 1C rate are shown in Fig. 2. From the experimental data,

expressions for the right-hand side parameters of the model

can be obtained.

At the time instants when a new discharge pulse is begin-

ning, the direct voltage drop is due to the series resistance Rs.

That is:

Rs =ΔU

ΔI(5)

The resistance can be calculated at either or both edge of each

discharge pulse. Then, a lookup table or curve fitting can be

utilized in order to express the resistance as a function of SOC.

After each discharge pulse, the voltage continues to increase

for a long time, first at higher rate and after a few minutes

at slower and decaying rate. This is known as a relaxation

effect. Eventually, the voltage reaches a steady value, i.e. OCV.

However, because it takes hours to achieve the steady-state

OCV, it is more convenient to estimate the OCV based on

shorter voltage response. In the experiments, OCV is estimated

by extrapolating the voltage curve of 30 minutes pause periods.

Rapid test methdods for OCV estimation are described in more

detail in [8].

IV. EXPERIMENTS AND MODEL VALIDATION

A. Test equipment

A commercial battery module from Kokam [9] was used in

the experiments. The battery consists of seven series-connected

SLPB 100216216H lithium polymer cells. The specification of

the battery is shown in Table I. A photo of the battery is shown

in Fig. 3.

TABLE ISPECIFICATION OF THE BATTERY CELL AND MODULE.

Property Unit Cell Module

Nominal capacity Ah 40 40Nominal voltage V 3.7 25.9Maximum voltage V 4.2 29.4Cut-off voltage V 2.7 18.9Maximum charge current A 80 80Continuous discharge current A 200 200Peak discharge current A 400 400Nominal temperature ◦C 25 25Max temperature during charge ◦C 40 40Max temperature during discharge ◦C 60 60Cycle life @ 80 % DOD Cycles 1 200 1 200

Load current was made with a water-cooled programmable

dc electronic load, model PLW12K-120-1200 from Amrel,

which has maximum current, voltage, and power ratings

of 1 200 A, 120 V, and 12 kW, respectively. A Powerfinn

PAP3200 was used as a power supply. The power supply

can be used as a controlled voltage or current source with

output voltage area of 0–36 V and current area of 0–127 A.

Its maximum output power is 3.2 kW at 24 V. A Hioki 3390

power analyzer with a Hioki 9278 current clamp was used

to measure current, voltage, power, ampere-hours, and watt-

hours.

All equipment except the power analyzer is controlled with

a dSPACE MicroAutoBox (MABX) DS 1401/1505/1507 rapid

control prototyping electronic control unit (ECU). Model-

based software development is utilized to produce code for the

ECU. Models are made with MATLAB/Simulink/Stateflow.

The ECU is connected to the host computer via high speed

Fig. 3. Photo of the battery under test. Additional temperature sensors havebeen attached inside the module.

Fig. 4. Battery test setup.

link. All measurements as well as other signals can be

monitored online from the host computer through dSPACE

ControlDesk software.

All relevant data were logged with 50 ms sampling interval.

Data from ControlDesk and power analyzer were merged and

postprocessed afterwards to make a unified data structure

that has all relevant information for model extraction and

validation.

B. Model extraction

The capacity of the battery was measured to be approxi-

mately 44.5 Ah at room temperature and slightly more than

that at elevated temperatures. Manufacturer has also some

relative capacity information for colder temperatures. A look-

up table was made for f2(T ). The cycle-dependent correction

factor f1(N) was ignored.

0 3600 7200 10800 14400 18000 216000

5

10

15

20

25

30

35

40

45

Time [s]

Cur

rent

[A] |

Tem

pera

ture

[C]

CurrentTemperature

(a) Measured discharge current and temperature.

0 3600 7200 10800 14400 18000 21600

20

22

24

26

28

30

Time [s]

Bat

tery

vol

tage

[V]

MeasurementSimulation

(b) Battery voltage.

Fig. 5. Experimental discharge cycle. Full battery is discharged in pulsesof 10 % of the battery’s capacity with 1C current. After each 10 % dischargepulse, a 30 min pause is performed.

Figure 5 shows the measured current pulses, battery hot spot

temperature, and voltage response. Based on this experiment,

OCV-curve, dc resistance Rs, and RC-network resistance and

capacitance values were extracted at each SOC level. The

measured current and temperature was then used as an input

for the battery plant model, and the resulting simulated voltage

response is also shown in 5(b). Because battery’s charging

characteristics are a bit different from discharge characteris-

tics, also a similar charging experiment was made to extract

the parameters during charging. However, the charging was

finalized with constant-voltage charging instead of constant-

current charging.

These experiments were repeated for different current rates

to get a relationship between the current rate and resistance

values. However, temperature was not constant during all these

experiments, so there is some uncertainty of which part of the

change is due to the current rate and which part due to the

increased temperature.

Figure 6 shows the measured and simulated voltage re-

sponse of a constant-current discharge experiment at 1C rate.

The measured and simulated curves are almost identical.

0 500 1000 1500 2000 2500 3000 3500 4000

20

22

24

26

28

30

Time [s]

Bat

tery

vol

tage

[V]

MeasurementSimulation

Fig. 6. Constant-current discharge experiment at 1C rate, measured andsimulated voltage.

C. Model validation

Since there are no standard drive cycles for NRMM, a

battery current profile is formed based on a measured power

profile of an underground mining loader [10]. The mining

loader has a hydrostatic driveline and a hydrostatic implement.

An illustration of the duty cycle of the loader and the measured

power profile are shown in Fig. 7. The net power is a sum

of traction pump power and implement pump power. It is

assumed that there are 14 series connected battery modules

similar to the tested module, i.e., the nominal voltage of the

battery is 362 V. One of many possible electrification schemes

is to downsize the internal combustion engine (ICE) and to

include a battery as an energy buffer. Because of simplicity, a

series-hybrid vehicle with a battery in the dc link is considered

here to form a battery current profile from the measured

power profile. The ICE is assumed to provide constant 25 kW

power, thus shifting the power profile of Fig. 7(b) downwards

25 kW. The battery current is then calculated in real-time in

the MABX ECU based on the measured module voltage that

is multiplied by the number of series-connected modules.

The results of the experiment and simulation are shown

in Fig. 8. The charger was disabled in the middle of the

experiment, because the battery temperature increased above

the maximum allowed charging temperature. As can be seen

from the figure, the simulated voltage response follows very

closely to the measured voltage in the whole operating area,

except at the very end of the test. The error at the end is

related to the steep voltage curve near SOC 0 %, where even

a very small error in SOC prediction causes visible error in

the voltage.

V. CONCLUSION

A versatile battery model for dynamic simulations of EVs

and HEVs was presented. A lithium-ion polymer battery was

used in the experimental tests, and the model parameters were

extracted from a set of simple experimental measurements.

The model was validated by using a current profile that was

formed from a measured power profile of an underground

mining loader’s typical duty cycle.

(a) Duty cycle.

0 50 100 150 200 250 300 350 400 450−40

−20

0

20

40

60

80

Time [s]

Pow

er [k

W]

(b) Measured net power of traction and implement hydraulics.

Fig. 7. Duty cycle of an underground mining loader. Total length of thecycle is 680 m.

The model predicts SOC, OCV, and terminal voltage re-

sponse during discharging as well as charging with good

accuracy. It also takes temperature and current rate effects into

account in fairly easy and intuitive manner. Calendar life and

cycle life effects can also be easily included into the model.

Future work will include a thermal model, which calculates

the temperature from the ambient temperature and power loss

information. Also different lithium-ion battery chemistries will

be tested and modeled in near future.

ACKNOWLEDGMENT

This study has been carried in HybLab project funded by

the Multidisciplinary Institute of Digitalization and Energy(MIDE) of Aalto University.

REFERENCES

[1] “Directive 97/68/EC of the European Parliament and of the Councilof 16 December 1997 on the approximation of the laws of theMember States relating to measures against the emission of gaseousand particulate pollutants from the internal combustion engines tobe installed in non-road mobile machinery,” Journal of the EuropeanUnion L 59 of February 1998, page 1 et seq. [Online]. Available: http://ec.europa.eu/enterprise/sectors/mechanical/non-road-mobile-machinery

[2] “Directive 2004/26/ECof the European Parliament and of the Councilof 21 April 2004 amending Directive 97/68/EC on the approximation ofthe laws of the Member States relating to measures against the emissionof gaseous and particulate pollutants from internal combustion enginesto be installed in non-road mobile machinery,” Journal of the European

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−100

−50

0

50

100

150

200

Time [s]

Cur

rent

[A] |

Tem

pera

ture

[C]

CurrentTemperature

(a) Measured battery current and temperature.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

20

22

24

26

28

30

Time [s]

Bat

tery

vol

tage

[V]

MeasurementSimulation

(b) Battery voltage.

Fig. 8. Model validation with experimental current profile of an undergroundmining loader work cycle.

Union L 146 of April 2004, page 1 et seq. [Online]. Available: http://ec.europa.eu/enterprise/sectors/mechanical/non-road-mobile-machinery

[3] M. Armand and J.-M. Tarascon, “Building better batteries,” Nature, vol.451, pp. 652–657, 2008.

[4] M. Chen and G. A. Rincon-Mora, “Accurate electrical battery modelcapable of predicting runtime and I-V performance,” IEEE Trans. EnergyConvers., vol. 21, no. 2, pp. 504–511, 2006.

[5] H. Zhang and M.-Y. Chow, “Comprehensive dynamic battery modelingfor PHEV applications,” in Proc. IEEE Power and Energy SocietyGeneral Meeting, Minneapolis, USA, Jul. 2010.

[6] R. C. Kroeze and P. T. Krein, “Electrical battery model for use indynamic electric vehicle simulations,” in Proc. IEEE Power Electron.Specialists Conf., Rhodes, Greece, Jun. 2008.

[7] D. Andrea, Battery Management Systems for Large Lithium-Ion BatteryPacks. Norwood, MA: Artech House, 2010.

[8] S. Abu-Sharkh and D. Doerffel, “Rapid test and non-linear modelcharacterisation of solid-state lithium-ion batteries,” J. Power Sources,vol. 130, pp. 266–274, 2004.

[9] Kokam homepage. [Online]. Available: http://www.kokam.com/[10] T. Lehmuspelto, M. Heiska, A. Leivo, and A. Hentunen, “Hybridization

of a mobile work machine,” World Electric Vehicle Journal, vol. 3,2009. [Online]. Available: http://www.evs24.org/wevajournal/