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 MapleSim1. 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