## Paper.dvi

**Proceedings of the ASME 2010 International Design Engineering Technical Conferences &**
**Computers and Information in Engineering Conference**
**August 15-18, 2010, Montreal, Quebec, Canada**
**SYMBOLIC MATH-BASED BATTERY MODELING**
**FOR ELECTRIC VEHICLE SIMULATION**
**Aden N. Seaman**
**John McPhee**
Department of Systems Design Engineering
Department of Systems Design Engineering
University of Waterloo
University of Waterloo
Waterloo, Ontario, Canada. N2L 3G1
Waterloo, Ontario, Canada. N2L 3G1
the system is described using the physics-based equations that

*We present results of a math-based model of a battery elec-*
govern the behaviour of its components. These mathematical

*tric vehicle (BEV) designed in MapleSim*1

*. This model has the*
equations are processed symbolically before finally being solved

*benefits of being described in a physically consistent way us-*
numerically to generate output data. This approach makes it eas-

*ing acausal system components. We used a battery model by*
ier for designers to specify component behaviour, and constrains

*Chen and Rin´con-Mora to develop a math-based model of a com-*
them to describe components in a more physically-consistent lan-

*plete battery pack, and developed simple power controller, mo-*
guage. This makes it easier to swap or modify components and

*tor/generator, terrain, and drive-cycle models to test the vehicle*
simplifies the description of the system [1].

*under various conditions. The resulting differential equations*
The Modelica [2] description language has been used by

*are simplified symbolically and then simulated numerically to*
many authors [3–7] to model hybrid electric vehicle systems

*give results that are physically consistent and clearly show the*
acausally, mostly using the Dymola [8] simulation environment.

*tight coupling between the battery and longitudinal vehicle dy-*
We have chosen to use MapleSim [9] from MapleSoft as
our simulation environment, as this allows us to access the un-derlying mathematical equations which govern the system beingsimulated.

This approach yields a simplified equation-based description
Vehicle modeling is a complicated and challenging task. Au-
of the system which can be simulated efficiently. The equations
tomotive companies release several new vehicles each year, and
can also be used in real-time simulation for hardware-in-the-loop
all of these need to be simulated and tested before they are actu-
(HIL) applications, and can be used in sensitivity analysis and
ally manufactured.

system optimization [10, 11].

With the push towards cleaner and more energy-efficient
vehicles, powertrains are incorporating motors, generators,
In this paper we present the results of a battery electric ve-
continuously-variable transmissions, energy storage devices
hicle (BEV) we have modeled using math-based modeling tech-
such as batteries and fuel-cells, and traditional internal combus-
niques in MapleSim. See Fig. 1 for a block diagram of the overall
tion engines (ICEs).

BEV system. This is the beginning of a more complex math-
One of the techniques that can ease the growing complex-
based hybrid electric vehicle (HEV) model we aim to develop
ity of vehicle modeling is acausal math-based modeling in which
using symbolic mathematics.

We have incorporated a lithium-ion electric-circuit battery
model by Chen and Rin´con-Mora [12] into the BEV system. We
1Maple and MapleSim are trademarks of MapleSoft
modified the battery equations to simulate a battery pack com-
3.1 of the Partnership for a New Generation of Vehicles (PNGV)
posed of series and parallel combinations of single cells. In order
Battery Test Manual [16]; the Kalman filtering techniques of
to connect the battery pack to a motor we had to develop a power
Piller

*et al. *[17]; the electrical circuit model of Chen and Rin´con-
controller model as part of the system integration. We further
Mora [12]; and the impedance model of Nelson

*et al. *[18]. These
incorporated a simple one-dimensional vehicle model that drives
different techniques have their strengths and weaknesses and lim-
on an inclined plane, a terrain model that controls the incline,
ited ranges of application.

and a drive cycle model that controls the vehicle's desired speed.

There is a great interest in using lithium-ion batteries in
By varying the drive cycle and terrain model, we tested the
electric vehicles, as they are light and have a higher power-to-
BEV under various driving conditions.

weight and power-to-size ratio than Lead-Acid or Nickel-basedbatteries. Great demands are placed on vehicle batteries as thedriver accelerates and brakes regeneratively, putting the batteriesthrough periods of high current draw and recharge. Dependingon the driving environment, the batteries can also be subjected tolarge temperature variations, which can have a significant effecton the battery's performance and lifetime.

Thus we needed to model a lithium-ion battery chemistry
over a wide state-of-charge (SOC) range, under widely-varyingcurrents, for various temperatures. Since we would eventuallylike to model this vehicle in a hardware-in-the-loop (HIL) sys-tem, we needed a model that was not computationally expensive,and we did not require a high-fidelity model.

These requirements led us to the electrical circuit model of
Chen and Rin´con-Mora. We implemented their components inMapleSim and used a custom function block to represent the non-

**FIGURE 1**.

BLOCK DIAGRAM OF OVERALL BEV MODEL
linear relationship between the state of charge and the electricalcomponents (Equations 2 to 6 in their paper). See Fig. 2 for ablock diagram of the battery.

**System Modeling and Simulation**
The technique we decided to use was math-based model-
ing using MapleSim as the simulation environment, which has agraphical interface for interconnecting system components. Thesystem model is then processed by the Maple mathematics en-gine, and finally the differential-algebraic equations (DAEs) de-scribing the system are simulated numerically to produce out-put data. For 3D multibody simulation it uses the DynaFlex-Pro engine, which uses linear graph-theory for system simula-tion [1, 11].

**FIGURE 2**.

BLOCK DIAGRAM OF SINGLE-CELL BATTERY
One of the most important components of an electric vehicle
– either BEV or HEV – is the battery. There are many ways ofmodeling different battery chemistries depending on the fidelityneeded and the battery parameters of interest. See the article by
Since their model is of a single cell, we modified their equa-
Rao

*et al. *[13] for an overview of some of the techniques. Gen-
tions to simulate a battery of cells in parallel and series. The
erally, with increasing model accuracy comes increased compu-
Chen and Rin´con-Mora battery can be divided into two linear
circuits with a non-linear coupling between them. See Fig. 2 for
Some modeling techniques that we reviewed were: the lead-
labels of these different circuits. One circuit is a large capaci-
acid model of Salameh

*et al. *[14]; the mathematical lithium-ion
tor in parallel with a resistor that models the charge state of the
model of Rong and Pedram [15] that incorporates state-of-health
battery and self-discharge. This can be called the "capacity cir-
and temperature effects; the lumped-parameter model in section
cuit". Another circuit is a voltage source in series with a resistor-
capacitor network that models the time response of the battery.

power going back into the battery during regenerative braking.

This can be called the "time response circuit".

Generally, boost or buck converters are used depending on
To adapt their single cell model to simulate an entire bat-
whether the output voltage is higher or lower, respectively, than
tery pack, let Nparallel be the number of cells in a parallel pack,
the input voltage [20]. By varying the duty cycle of a high-
and let Nseries be the number of parallel packs placed in series
frequency switching circuit, the output voltage and thus current
to make the whole battery. The open circuit voltage in the time
and power can be controlled.

response circuit is multiplied by Nseries. The current flowing in
Instead of modeling the high frequency circuit in MapleSim,
the time response circuit is divided by Nparallel when it flows in
we decided to use a simple approximation that can serve as both
the capacity circuit. The resistors in the time response circuit are
a boost or buck converter with power flowing from the battery
multiplied by Nseries/Nparallel and the capacitors are multiplied by
to the motor, or vice-versa. Figure 3 is a picture of the power
controller block diagram. Although the current model has a fixed
A single cell of the battery model has an open-circuit voltage
converter efficiency of 100%, a more realistic efficiency model
of 3.3 V and a capacity of 837.5 mAh at a 1 A discharge rate
such as the one used by Hellgren [3] can be incorporated.

starting at 100% state of charge. By placing 8 cells in parallel,and 74 of these parallel packs in series, a 244.2 V, 6.7 Ah batterypack was created. This pack is comparable to that in a 2007Toyota Camry hybrid [19].

The Chen and Rin´con-Mora battery model is simple enough
to simulate in a short amount of time while being complexenough to provide the following: variations in the open circuitvoltage with SOC; transient effects of charge depletion and re-covery and their dependence on SOC; and the variation in batterycapacity with discharge current. Furthermore, since it is an elec-trical circuit model it can easily be incorporated into the electricalsystem of the BEV model and is amenable to being representedusing math-based modeling techniques.

**FIGURE 3**.

BLOCK DIAGRAM OF POWER CONTROLLER
One of the downsides of this model is that no temperature ef-
fects of any kind are modeled, although Chen and Rin´con-Morastate it would not be difficult to include them. In an electric ve-hicle the temperature will vary with external environmental con-
Using a signal-driven current source in the output loop, the
ditions, with heating of the battery due to internal losses, and
output voltage is measured and the output power is calculated.

with endo- and exothermic chemical reactions. The only model
The input current is adjusted by a PID controller so that the input
we encountered that explicitly included temperature dependence
power matches the output power. This circuit works both for
was the mathematical model of Rong and Pedram [15], but their
positive or negative current, which determines the direction of
model assumes a constant discharge current and thus is not suit-
power flow. This model avoids the divide-by-zero problem of a
able for our BEV system.

simple algebraic power converter when the output voltage and
The Chen and Rin´con-Mora model can also be overcharged
current goes to zero, and adapts to changing input and output
and does not consider the increasing resistance of the battery as
impedance. However it does not take into consideration physical
it nears a full charge. Furthermore the variations of the battery's
limitations of components such as the battery's maximum charge
state-of-health (SOH) with time and charge cycles is not mod-
or discharge rates, and voltage and current limits of the motor,
eled. These downsides are acceptable given that in future mod-
wires, or power electronics.

eling the vehicle's control system will limit maximum batterycharge, and although in this paper we are not interested in model-ing temperature or state-of-health, they should not be too difficult
to incorporate.

The electrical motor used in the vehicle model is the Model-
ica DC permanent magnet motor, which includes internal resis-tance, inductance, and rotor inertia [21].

Its mechanical and electrical behaviour are modelled by
The next important component of an electric vehicle is a
Equations 1 and 2 where

*Ja *is the armature inertia, θ (

*t*) is the
power converter that acts as an interface between the battery and
armature rotation angle,

*Vnom*,

*Inom*, and

*fnom *are the nominal mo-
the drive motor/generator. This component controls the amount
tor voltage, current, and rotational frequency, respectively. τ(

*t*)
of power going to the motor during driving, and the amount of
is the shaft torque, and

*La *and

*Ra *are the armature inductance and
resistance, respectively. Finally

*V *(

*t*), and

*I*(

*t*) are the voltage and
per wheel revolution. The wheels have the same diameter as the
current at the motor terminals, respectively.

P215/60VR16.0 tires on the Camry.

Equation 3 describes the relationship between the rotation
and torque of the motor shaft. τ(

*t*) is the torque seen at the motor
shaft,

*m *is the vehicle's mass,

*R *is the drive tire radius, ρ is the

*nom *−

*RaInom*)

*I*(

*t*)
− τ(

*t*) = 0
gear ratio from the motor to the tire, θ (

*t*) is the motor shaft's
rotational displacement,

*g *is the gravitational constant, and φ (

*t*)is the terrain inclination angle.

Table 2 lists the values used for these parameters.

30(

*Vnom *−

*RaInom*) ˙θ(

*t*)

*aI*(

*t *) +

*RaI*(

*t *) −

*V *(

*t*) +
(

*t*) +

*g *sin(φ (

*t*))
We chose to use the physical parameters of the LEM-200
Model D127 DC permanent magnet motor from L.M.C. Ltd [22].

However we modified the rated current and voltage of the motorto be more compatible with our battery voltage. This would ef-
VEHICLE MODEL PARAMETERS
fectively require re-winding the motor with different wire andchanging its magnets.

The parameters used for the motor are presented in Table 1.

Note that the peak current and power of the motor are twice therated value.

MOTOR MODEL PARAMETERS
The only type of braking included in this model is regener-
ative braking where the current to the motor is reversed and the
battery is charged with the kinetic energy of the vehicle. We did
not take into consideration recharge current limits of the battery.

To this vehicle model we attached a simple terrain model.

A time-dependent lookup table controlled the inclination of the
terrain on which the vehicle traveled. This allowed us to simulate
the vehicle's performance on flat and hilly terrain.

The drive cycle is a time-dependent lookup table of the vehi-
cle's desired speed. A PID controller compares the desired speedto the actual speed and drives the input of the power controller totransfer power to the motor, or to extract power from the motoruntil the vehicle's speed matches the desired speed.

See Fig. 1 for the block diagram of the overall BEV model.

The vehicle model we used was very simple. Its physi-
cal parameters were based on the 2007 Toyota Camry hybrid.

Since we were concerned only with the performance of the pow-
After MapleSim converts the vehicle model into differential
ertrain components, we did not concern ourselves with vehicle
equations, it simplifies and reduces the system of equations sym-
suspension or steering. We used a one-dimensional model of a
bolically. Then using this reduced equation set it solves them
frictionless cart on an incline under the force of gravity. The
numerically to produce the final output data.

drive motor is connected to one of the slipless wheels of the cart
MapleSim simulated our system with its non-stiff solver,
through a fixed transmission with a ratio of 9 motor revolutions
which uses a Fehlberg fourth-fifth order Runge-Kutta method
with degree four interpolant. We used an adaptive time-step
DRIVE CYCLE AND TERRAIN MODEL PARAME-
with absolute and relative error tolerances of 1e-7, and turned on
MapleSim's native code generation ability which runs the sim-ulation faster. The model was simulated on a 3 GHz Intel Core2 Duo using MapleSim version 3 for Linux. It was set to simu-
late over a 30 second time interval, and took 10 seconds of actual
time to complete.

Figure 6 plots the battery's state of charge versus time. Re-
call that this model is without rolling resistance. One can see thatthe hard acceleration drive cycle ends up with a lower final state
of charge than the gentle cycle. This difference is due to ohmiclosses in the motor windings and chemical losses in the battery.

The second test we did was to drive the vehicle up and down
a hill. The battery should lose energy going uphill as the vehi-
cle gains gravitational potential energy, and gain energy going
downhill as the vehicle loses potential energy. See the hill cycle

**FIGURE 4**.

MODELED-VS-ACTUAL [12] BATTERY UNDER
curve of Fig. 5 for a plot of the drive cycle speed with time. The
PULSED CONSTANT-CURRENT DISCHARGE
terrain cycle is very simple: at t=9.5 s the vehicle encounters thehill, then it drives up or down an 8◦ incline before returning toflat terrain at t=20.5 s.

Figure 4 is a comparison between the MapleSim model and
Figure 7 plots the battery's state of charge versus time for
an actual cell for a pulsed current discharge of a single battery
this test. In both cases the battery loses energy as it accelerates
cell. The actual cell data was extracted from Fig. 5 of Chenand Rin´con-Mora's paper. Like the model in their paper, ourmodel does not include a self-discharge resistor. An initial SOC
of 98% gives a close match to the experimental results, tracking
them very well until the battery capacity is almost exhausted.

Our model requires one discharge cycle more than the actual to
see a rapid collapse in the battery terminal voltage.

Using our vehicle model we performed two simple and intu-
itive tests. Table 3 lists the parameters used in the drive cycles.

The first test we did was to simulate the vehicle driving un-
Hard & Hill cycles
der hard and gentle accelerations on flat terrain. Battery and in-
ternal combustion engine vehicles are more efficient if gentle ac-celeration is used compared to hard acceleration, due to internal
losses. The initial accelerations of the hard and gentle cycles are
different, but the maximum speed and rate of deceleration are the
same. See the hard and gentle curves of Fig. 5 for a plot of the

**FIGURE 5**.

DRIVE CYCLE: SPEED-VS-TIME FOR HARD, GEN-
drive cycle speed with time.

TLE, AND HILL CYCLES
the vehicle, transferring energy from the battery to the vehicle's
kinetic energy.

A comparison can be made between the MapleSim results
In the uphill case the state of charge decreases. The drive
and approximate calculations based on energy conservation. The
controller applies more power to the motor to match the vehicle's
points of comparison on the hard and gentle acceleration cycles
speed to the desired speed, and the battery's energy is put into the
are before the vehicle starts moving, and after it has settled at
vehicle's gravitational potential energy.

its maximum speed, just before regenerative braking. Since thevehicle moves without rolling resistance on flat terrain, only the
In the downhill case the state of charge increases. The drive
kinetic energy of the vehicle and the resistance losses in the mo-
controller applies the regenerative "brakes" to keep the vehicle's
tor and battery need to be considered.

speed constant, and the vehicle's gravitational potential energy is
See Table 4 for a comparison between the approximate
transferred to the battery.

theoretical calculations based on energy conservation and the
Finally the vehicle encounters a flat spot and uses regenera-
MapleSim results for the hard and gentle acceleration cycles for
tive braking to come to a halt, transferring the vehicle's kinetic
the following quantities: J, the energy transferred into the vehi-
energy to the battery.

cle; P, the average power during acceleration; ∆SOC, the changein the battery's state of charge taking into considering losses inthe motor and battery. See Appendix A for the steps used in thehard drive cycle calculation.

COMPARISON BETWEEN MAPLESIM AND AP-
PROXIMATE THEORETICAL CALCULATIONS
State of Charge (%)

**FIGURE 6**.

STATE OF CHARGE FOR HARD AND GENTLE AC-
CELERATIONS ON FLAT TERRAIN
∆SOCgentle 1.055%
The MapleSim results compare favourably with the approx-
imate theoretical results. The small amount of difference is notsurprising considering the simplicity of the approximate theoret-
ical equations used.

We modeled a simple battery electric vehicle with math-
based methods using MapleSim. This technique reduces devel-
State Of Charge (%)
opment time and brings the system representation closer to the
physics of the system.

Using a math-based model of a complete battery pack based
on the battery model of Chen and Rin´con-Mora, a simple power
controller model, and a standard Modelica DC motor we were

**FIGURE 7**.

STATE OF CHARGE FOR UPHILL AND DOWNHILL
able to put together a BEV powertrain and connect it to a simple
vehicle dynamics model.

By applying different terrain conditions and driving cycles,
High-performance multi-domain modeling
two different scenarios were tested to compare the performance
and simulation.

of our vehicle model to what one would expect from an actual
vehicle. In both cases the results agreed with intuition and with
[10] Jalali, K., Uchida, T., McPhee, J., and Lambert, S., 2009.

approximate theoretical calculations.

"Integrated stability control system for electric vehicles
The underlying mathematical equations describing the sys-
with in-wheel motors using soft computing techniques".

tem can be used for sensitivity analysis, optimization, or for use

*SAE Int. J. Passeng. Cars - Electron. Electr. Syst.,*
in real-time HIL simulation.

pp. 109–119.

Future work will include adding an internal combustion en-
[11] Vogt, H., Schmitke, C., Jalali, K., and McPhee, J., 2008.

gine as a range-extender, increasing the fidelity of the power con-
"Unified modelling and real-time simulation of an electric
troller and motor model, and adding more complex vehicle, ter-
vehicle".

*Int. J. Veh. Auto. Sys., ***6**, pp. 288–307.

rain, and drive cycle models to the system.

[12] Chen, M., and Rin´con-Mora, G. A., 2006. "Accurate elec-
trical battery model capable of predicting runtime and I-V

performance".

*IEEE Transactions on Energy Conversion,*

**21**(2), June, pp. 504–511.

[13] Rao, R., Vrudhula, S., and Rakhmatov, D. N., 2003. "Bat-
We would like to thank Toyota, MapleSoft, and the Natural
tery modeling for energy-aware system design".

*Computer,*
Sciences and Engineering Research Council of Canada for fund-

**36**(12), December, pp. 77–87.

ing and support.

[14] Salameh, Z. M., Casacca, M. A., and Lynch, W. A., 1992.

"A mathematical model for lead-acid batteries".

*Trans. Energy Conv., ***7**(1), March, pp. 93–98.

[15] Rong, P., and Pedram, M., 2006. "An analytical model for
[1] Schmitke, C., Morency, K., and McPhee, J., 2008. "Using
predicting the remaining battery capacity for lithium-ion
graph theory and symbolic computing to generate efficient
batteries".

*IEEE Transactions on Very Large Scale Inter-*
models for multibody vehicle dynamics".

*IMechE J. Multi-*
*gration (VLSI) Systems, ***14**(5), May, pp. 441–451.

*body Dynamics, ***222**, pp. 339–352.

[16] US DEPARTMENT OF ENERGY, 2001.

[2] Modelica association. http://www.modelica.org/.

*tery Test Manual*, 3rd ed., February.

[3] Hellgren, J., 2002. "Modelling of hybrid electric vehicles
in Modelica for virtual prototyping". In Proceedings of the
2nd International Modelica Conference, pp. 247–256. See
[17] Piller, S., Perrin, M., and Jossen, A., 2001. "Methods for
also URL http://www.modelica.org/events/
state-of-charge determination and their applications".

*Jour-*
*nal of Power Sources, ***96**, pp. 113–120.

[4] Laine, L., and Andreasson, J., 2003.

[18] Nelson, P., Amine, K., Rousseau, A., and Yomoto, H.,
of generic hybrid electric vehicles".

2007. "Advanced lithium-ion batteries for plug-in hybrid
of the 3rd International Modelica Conference, pp. 87–
electric vehicles".

In Proceedings from the 23rd Elec-
See also URL http://www.modelica.org/
tric Vehicle Symposium. See also URL http://www.

[5] Wall´en, J., 2004. "Modelling of components for conven-
[19] U.S. D.O.E. advanced vehicle testing activity: 2007 Toy-
tional car and hybrid electric vehicle in Modelica". MS
ota Camry hybrid electric vehicle. http://avt.inl.

Thesis, Link ¨opings universistet, May.

[6] Zhou, Y. L., 2005. "Modeling and simulation of hybrid
[20] Bauman, J., 2008. "Advances in fuel cell vehicle design".

electric vehicles". MS Thesis, University of Victoria.

PhD Thesis, University of Waterloo.

[7] Simic, D., and B¨auml, T., 2008.

[21] Modelica standard library. http://www.modelica.

of hybrid electric vehicles using the VehicleInter-
faces and the SmartElectricDrives libraries".

[22] L.M.C. Ltd. DC permanent magnet motor technical details,
Proceedings of the 6th International Modelica Con-
Multi-engineering and
model simulation.

**Appendix A: Hard Driving Cycle Calculation**
Here the battery losses do not need to be taken into consid-
The average power, total energy, and change in state of
eration because the theoretical and MapleSim results are based
charge will be compared between the MapleSim model and first
only on the energy going into the motor. The MapleSim results
principle calculations based on the conservation of energy. This
do include losses due to the resistance of the motor armature
is for the hard acceleration drive cycle conducted on flat terrain.

which the theoretical results do not incorporate, but these losses
The maximum vehicle velocity is V
are negligibly small as will be demonstrated in the next section.

max = 9 m/s and the
time during which it accelerates is ∆t = 5.6 s, which is an ac-celeration of a = 1.607 m/s2. The vehicle mass is m = 1613 kg.

**State of Charge**
The state of charge calculation is significantly more compli-

**Powers and Energies**
cated, as the battery losses need to be taken into consideration.

The time response circuit of the Chen and Rin´con-Mora battery
model contains a resistor, and two parallel resistor-capacitor cir-
max = 65.326 kJ.

The average power during ac-
cuits in series. It will be shown that during the period of ac-
celeration is Pavg = J∆ = 11.66 kW. Since the vehicle's velocity
celeration the resistances of the resistor-capacitor circuits can be
is linearly proportional to time, the vehicle's energy at any
time can be written as J(t) = 1 ma2t2. The instantaneous power
The current flowing through a parallel resistor-capacitor cir-
needed by the vehicle is then P(t) = dJ(t) = ma2t. Thus during
cuit is related to the voltage across them by the following equa-
the 5.6 seconds of acceleration, the required power is linearly
proportional to time.

*dV *(

*t*)

*V *(

*t*)
=

*I*(

*t*)
The power flowing out of the battery is expressed by
P(t) = V(t)I(t). If we assume that the battery terminal voltage isconstant, then I(t) = P(t)/V = ma2t/V = At where A is a con-
stant and the current is linearly proportional to time.

Substituting this into Equation 4 and solving for V(t) with
the initial condition V(0) = 0 yields
Output Power (kW) 5

*V *(

*t*) =

*RA t *+

*RC e RC *− 1
Now to find the equivalent resistance of the circuit at any
time we need to evaluate R
. Since I(t) = At then

**FIGURE 8**.

MAPLESIM AND THEORETICAL POWER TO MO-
TOR DURING HARD CYCLE ACCELERATION
dI(t) = Adt. Implicitly differentiating Equation 5 with respect totime, collecting the Adt term, then dividing both sides by Adtgives:
See Fig. 8 for the theoretical and MapleSim's calculated
power going into the motor during the hard cycle's accelera-tion period. In accordance with theory, the simulated power is

*Req*(

*t*) =
=

*R *1 −

*eRC*
seen to rise linearly with time over a period of 5.6 seconds up
to a maximum value of 23.8 kW. The initial delay and finaltrailoff in the MapleSim results are due to the PID controller
At t = 0 the equivalent resistance of the parallel circuit is
that is driving the vehicle.

In MapleSim the average power
zero. As t → ∞ the equivalent resistance approaches the value of
is Pavg = 23.8/2 = 11.9 kW. The total energy can simply be
the resistor, R.

calculated by the area under the triangle-shaped curve to give
Thus the equivalent resistance depends on the time over
J = 23.8 × 5.6/2 = 66.64 kJ.

which the current flows through the circuit. The time constant
These theoretical and simulated values appear in Table 4.

of the circuit is simply RC. For our battery pack of 8 parallel
cells and 74 packs in series at 80% state of charge the time con-
The power losses due to the various resistors are
stants for the short time response, and long time response circuits
Plosses(t) = I(t)2 [Rseries + Rmotor]. The energy losses can be cal-
are 32.85 s and 223.03 s, respectively.

Since the vehicle only accelerates for 5.6 seconds, both of
these time constants are significantly larger than the accelera-tion time, and thus the resistance of both parallel circuits can
be ignored in favour of the battery's series resistance, which is

*Jlosses *=

*J loss *+

*J loss *=
series = 0.668 Ω.

When analysing the voltage-vs-current be-
haviour of the battery using MapleSim, this conclusion is jus-
= [

*Rseries *+

*Rmotor*]
So instead of being constant, a better approximation
of the battery's terminal voltage is V(t) = Voc − RseriesI(t),
This yields J

*loss *= 12.45 kJ and J

*loss *= 0.30 kJ. One can
see that the motor losses are 41.5 times smaller than the battery
P(t) = ma2t = [Voc − RseriesI(t)]I(t).

Solving this equation
losses, and over 200 times smaller than the overall vehicle en-
ergy. This is why they can be ignored in the calculation of powersand energies.

*I*(

*t*) =
Jbattery = 6.5 Ah × 290 V × 3600 s/h = 6786 kJ.

The total energy spent by the battery in accelerating the
The negative root is taken since

*I*(0) = 0.

Joutput = Jvehicle + Jlosses = 65.326 + 12.754 kJ.

tuting in the numerical parameters of the model gives us
I(t) = 210.756 − 77.781 7.336 − t for t = 0 to 5.6 s.

∆SOC = Joutput/Jbattery × 100% = 78.08/6786 × 100% = 1.15%.

This is the value that appears in Table 4.

Source: http://maplesoft.lt/Whitepapers/BatteryEV_ASME2010.pdf

Fact sheet BT Wholesale American Share Fund ARSN: 087 594 509 About the Fund Performance The BT Wholesale American Share Fund (Fund) is an actively managed portfolio of North American shares. The Fund invests primarily in United States and Canadian shares. The Fund may also hold cash and may use derivatives.

Glasgow Assessment and Management of Alcohol If you would like further information or advice on the alcohol screening and withdrawal management guideline(GMAWS) please contact your local acute addiction liaison nurse service; Glasgow Hospitals and Vale of Leven: 0141 211 2835 Royal Alexandra Hospital: 0141 314 4472 Inverclyde Royal Hospital: 01475 715 353