#### Anti_Climax

##### Well-known member

I own a 2012 Focus BEV that just recently had the battery replaced by the dealer. Seems they were not able to get the original 23kWh pack so I was given the upgrade to the 33.5kWh battery.

In the month since I got it back I have put more than 2000 miles on it (you read that correctly) and have taken the opportunity to directly validate several aspects of the car's energy usage. Nerdy physics equations follow.

Most recently, I performed coastdown tests on a relatively flat/level/straight stretch of road that allowed me to coast in neutral from top speed at cruise to a stop while capturing my speedometer value at intervals. I ran the stretch 6 times (3 each way to minimize grade/wind effects) and plugged the average interval speeds into a formula that combines the vehicle mass (measured on a truck stop scale with me inside), drag coefficient (published as .295), aerodynamic cross section (published estimate of 2.25m^2, to be independently validated later), air density and gravity to work out the coefficient of rolling resistance (0.016±0.0002) and overall power (instantaneous Watts) needed to maintain any given speed.

This is the equation for anyone else that would like to do something similar:

Code:

```
(Coefficient of Drag [0.295] x Cross Section [2.25m^2] x 0.5 x Density of Air [1.2956 kg/m^3] x Velocity [m/s]^2) + (Coefficient of Rolling Resistance [0.016] x Mass [1740kg] x Gravity [9.794
m/s^2]) = Force in Newtons
```

The first term covers the force needed to overcome air resistance at speed and increases in an exponential manner. The second term covers the force needed to overcome the total resistance of the drivetrain and tires and is fairly linear.

Power in Watts is equal to Force in Newtons we found multiplied by velocity in meters per second, so multiplying that whole equation again by the velocity gives you instantaneous power in Watts for that velocity.

Adding another term that includes the product of the acceleration due to gravity and the "Sine of the Inverse Tangent" of the road grade (percentage) will account for the additional power needed to move the mass of the vehicle against gravity on that incline. Similarly, the velocity factor of the drag term could be written as the sum of vehicle velocity and windspeed to account for head/tail winds.

Taken on it's own, it's nice to be able to quantify just how much more energy is used at different speeds and grades but using it in the real world can be even more tedious. So I went a step further and used Excel to fit a cubic polynomial line to the power versus speed curve I've gotten and added those coefficients to a custom vehicle entry on https://www.evtripplanner.com

With those values, it can now closely estimate the actual energy use for trips that cover varying speeds and road grades. Should anyone else wish to do the same, the values I obtained are a cubic coefficient of 0.43, a linear coefficient of 276.264 and zeros for the square coefficient and constant. Be sure to use the stock curb weight (1644kg for mine) so the payload value can be adjusted accordingly. I do not yet have figures for regenerative efficiency (high or low) but intend to find those next.

Keep in mind that several deviations from my values are possible:

- Air mass varies with temp, humidity and altitude.

Local gravity varies with altitude and latitude.

Drag and cross section will be impacted if you have luggage racks, bike racks, roof storage, open windows or damage to the underpaneling - I know mine has broken on one corner and is "wired" into place.

Your actual energy use will be impacted by head and tailwinds.

Rolling resistance will vary with tires and tire pressure.

Others may find slightly different values for the equation above but I would not anticipate them to vary by more than a few percent if at all.

Some other observations I've made:

After your battery reads 0%, you have 1kWh of energy available before it cuts you off. Within a few days of getting my new pack, I did a full rundown and found that I got 30.2kWh of energy out at 0%. My previous 23kW pack had around 15kWh to 0% so I'm now getting double the range.

Driving at top speed on flat road uses more than 30kW. The same speed on a 4% grade almost 60kW. The motor is rated for 107kW so unless a thermal cutoff kicks in, the car should be able to maintain 85mph on an 11% grade but would deplete a new 24kW battery in about 7 and a half minutes.

I've also been able to validate the amount of energy it takes to fully charge the battery from 0% and found that the charging efficiency is around 87% - though that figure can be negatively impacted by cooling cycles during charging.

Based on the values from the equation above, locking the cruise control on the lowest speed possible and driving on a level road can give you more than 200 miles on a fully charged 35kWh battery - though it will take you around 11 hours of driving to do it. I found a 2 mile circular residential drive with no stops and have considered doing 100+ laps to validate but I can't bring myself to drive that long.

Should someone wish to get the most miles out of the car over multiple charges (think 24 hour LeMans) the max possible appears to be whatever speed will discharge the battery in the same amount of time it takes to recharge. If you are able to pump 29A into the car at 264V (240V+10%), you'll be charging at ~7.6kW but getting back ~6.6kW with the observed efficiency. That rate of energy use corresponds to around ~37mph on flat ground. Since half your time will be spent charging and half discharging, you're average speed over time will be half that value for as long as you care to continue. Improving the rate or efficiency of charging or discharging will push this figure up, though it's unlikely to be improved significantly. Keeping the battery in a range of charge that allows for the best combined charge and discharge characteristics might push it a little.

So there you have it - a bunch of really nerdy number work that I hope helps everyone get the most out of their EV. Enjoy.

*An addendum, there is another component of this equation that will come into play at much lower speeds due to the increasing effects of static friction and low speed motor efficiency (which is better in a permanent magnet motor like this, but still very poor at the lowest speeds). This will not take that into account. I'll look at quantifying those effects along with regeneration efficiencies.