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Hybrid Electric Vehicle Analysis |
Introduction | Requirements & Installation | Input File Editors | Running HEVA | Results | |
Appendix: Energy Balances Appendix: State-of-Charge Calculations |
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For more information, contact Dale Stalnaker at the NASA Lewis Research Center. |
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This appendix describes equations in the source code that perform energy balance calculations. The calculations are performed during each time interval. Most variables in this section are referred to by the variable names used in the source code.
All power values are expressed as Kilowatts (KW). Energy removed from the storage system is defined as a positive power gain. The fuel consumption during a time interval is calculated as the total power loss multiplied by the time increment and is defined as negative.
The following equation calculates power loss due to aerodynamic drag, represented by the variable Paero(Watts).
Afrontal = Vehicle frontal area (m2)
CD = coefficient of drag of vehicle
V = vehicle velocity (m/s)
air = atmospheric density, kg/m3
The following equation calculates loss required to overcome rolling resistance, represented by Proll.
Proll = MGr.Veh g (R0 + R1 V + R2 V2 + R3V3) V
MGr.Veh. = gross vehicle mass (kg).
g = acceleration due to gravity (m/s2)
R0, 1, 2 = rolling resistance coefficients
The following equation calculates power loss due to road inclination, represented by the variable Pincl (Watts).
where 
incl. = road inclination angle
(degrees from horizontal, converted to radians in equation)
The following equation calculates the power requirement for vehicle acceleration, represented by the variable Paccel.
Paccel = Vave MGr.Veh. a ,
in which Vave= average velocity = ½ (V2 + V1), and
a = acceleration =
V/t (m/s2).
The power loss due to transmission inefficiencies is estimated by dividing the power required to move the vehicle (sum of power losses due to aerodynamic drag, rolling resistance, road inclination, and acceleration) by the transmission efficiency. The transmission efficiency is determined from the drive train efficiency data and the torque data. First, the torque converter output turbine speed (rpm) and torque (N-m) must be determined.
d = vehicle tire diameter (m)
G = gear ratio.
= wheel rotation rate (RPM)
After these values are calculated, the torque data input table (output torque as a function of output speed) is interpolated to find the speed ratio corresponding to the output speed - torque combination. The drive train efficiency is interpolated from the drive train efficiency table as a function of the speed ratio. The power required from the engine.
A-6. Power losses due to Engine InefficiencyThe engine efficiency is defined as the ratio of the power produced by the engine divided by the energy consumed. The amount of fuel consumed at any time is inversely proportional to the engine’s efficiency. The engine efficiency model is currently interpolated or extrapolated using a table of engine efficiency vs. percent rated engine power in the engine parameters input file. The efficiency is plotted in Figure A-1 as a function of the rated engine power, based on fuel economy data reported by Cleveland RTA. |
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The following equation calculates the power gain from regenerative braking.
Pregen = -e regen MGr.Veh. a V
a = acceleration, (m/s2)
e regen = Regenerative braking efficiency
The following equation calculates the power from a solar array.
Psolar = (Solar Flux) As.a. es.a.
Solar flux is the flux incident on the solar array, expressed in Watts/m2.
As.a.= solar array area (m2)
es.a. = solar array efficiency
Fuel consumption is determined using the following equation:
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Pengine = power required from engine (W) t = time interval (seconds)
Hfuel = heat value of fuel used in engine (J/kg) |
fuel = density of fuel (kg/m3)
eengine = engine efficiency ealternator = alternator efficiency (only applicable for hybrid configuration) |
For a conventional vehicle, the power required from the engine (Pengine) is equal to the sum of all power losses.
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Pparasitic = Parasitic Losses Pmove = Total power to move vehicle Paero = Power to overcome Aerodynamic Drag |
Paccel = Power due to acceleration Prolling = Power to overcome Rolling Resistance etrans = Transmission Efficiency |
For a hybrid vehicle, there are additional sources such the energy storage system and the regenerative brakes. For hybrid vehicles, energy is estimated as the integral of power and time. The energy losses and gains are added to the state of charge of the storage system. The engine power level is adjusted using the APU power control file.
