Solar Car Power Management

Power Usage Equations

Paul Vincent Craven
All Rights Reserved

The estimation of total power consumption uses standard physics equations. Every x meters along the data stream we calculate how much energy is used to move the car. We cannot do calculations between every point since the GPS data is not entirely accurate. We need to make sure that the car has traveled farther than the error tolerance. Performing calculations every fifty feet seems to work fairly well. Non-differentiated GPS data would need spacing every third of a mile or so, causing the estimation to trim many hills and corners. This would lead to under-estimation of power needed.

Below are the equations that are used to calculate the total amount of energy to be used. They are courtesy of Jeff Shapiro, from the NREL.




drag coefficient


frontal surface area


density of air, 1.2 kg/m3


vehicle velocity


opposing wind velocity

atan(%grade/100) where grade is the road slope


vehicle mass


force of gravity, 9.8 m/s2


first coefficient of rolling resistance, 0.06


second coefficient of rolling resistance, 0.0502


number of wheels


horizontal acceleration of the car





Table 1: Definition of equation variables

Equation 2: Air Resistance

This equation gives the opposing force on the car due to air resistance.

Equation 3: Rolling Resistance #1

This equation is the rolling resistance of the tires and it is due to the inelastic nature of tires. It's also dependent on tire pressure and road conditions, etc. This is the term people use when referring to rolling resistance. The is there because the resistance is proportional to the normal force on the tire and the normal force is .

Equation 4: Rolling Resistance #2

This second rolling resistance equation is related to the bearing friction. It is small enough to be insignificant for many calculations.

Equation 5: Gravity Resistance

This is the energy required to overcome gravity. Note that we can gain energy when going down hill. If we have a net gain in energy, we can assume one of three things: the car will increase its speed, the car will gain energy through the regen feature on the motor, or the energy will be converted to heat through the brakes. Regen is the ability of the motor to actually generate power when braking or going down hill. This power is then used to recharge the batteries.

Equation 6: Inertial Resistance

Our calculations assumed the car was going a constant speed the entire race, eliminating the need for the inertial resistance part of the equation. For races with many starts and stops, this would become an important factor.

Part of above the force calculation only needs to be done once when we estimate our total power usage over a GPS charted path. However, those force equations that change given a change in theta (the grade of the road) need to be calculated between each GPS point above our threshold of error tolerance.

Equation 7: Total Energy

Adding all of the force equations gives us a final figure of the force needed to move the car, f. Next, a calculation is needed to find the time it will take us to traverse the distance. This is trivial as we are given our speed, and can easily calculate the distance between two GPS points. The total energy used is the final force, times the amount of time it is applied. Standard conversions may necessary to get this into the desired units, depending on what unit of measurement was used for the equations' parameters.

Having the vehicle weight, and a good data-logger measuring time and current, one should be able to find CdA, Crr_a, and Crr_b using a third-order polynomial fit. However, Crr_b may be too small to accurately measure. This polynomial fit would be the first thing to do with a working data logger on the solar car.

[Figure 15 was lost somehow. It just showed a screenshot of a dialog box, all the equations you need are listed above.]

Figure 15: Calculation of total power consumption

The solar car's motor is not 100% efficient. Some of the energy sent to the motor is changed into heat due to friction in the motor. Therefore it is necessary to find the efficiency factor of the motor. The motor's efficiency varies according to the speed (RPM) and opposition to turning. For Sunrayce '95 we used a value of eighty percent as our efficiency rating, which probably put us close to what the actual rate was. For Sunrayce '97, it would be recommended to have a student borrow the needed equipment and make charts as to the actual power in vs. power out.

Because the UMR Solar Car Team did not have a functioning on-board computer for the solar car by the time this thesis was completed, the accuracy of the estimation of total power consumption is not known. If the on-board computer would have been functioning, we would have been able to obtain this measurement by logging the motor current and voltage once or twice every second.