Exhaustive quantitative Power and Energy values at speed

Hello all,

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:
For our puroses, this force value is equivalent to Watt*Seconds/Meter and can be directly converted to Watt*Hours/Mile as desired.

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.

Good stuff! I glossed over the math for now but wanted to congratulate you on the free battery upgrade!

Strangest part, I had asked about paying the difference to upgrade when the warranty replacement came up and they said they couldn't do it.

Afterwards no one even mentioned it, didn't realize until I looked at the range the next morning. I knew the old one had degraded a bit so I expected an increased range figure, but not double.

Sure enough I checked my charge time and it had been pulled juice for close to 5 hours when it had never taken more than 4 on the old pack.

it's a 33.5kWh pack not 35kWh. What happened with your original pack the necessitated it's replacement?

Prior to it going in, it threw the wrench icon and continued to drive and charge for a few days before it finally decided to pull the "Stop Now Safely" bit in my driveway with a full battery. The manual says to contact service if the fault didn't clear, so it was going to get sent off soon anyway.

Didn't hear from them for a week because they didn't have anything to tell me prior to declaring they couldn't further isolate the cause of the fault and were replacing the pack. Took another three weeks for it to show up and get installed.

Anti_Climax said:

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.

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If they are using a newer FFE with CCS they will be charging considerably faster.

Correct. And in that case the best possible speed will still correspond to whichever matches battery drain time to recharge time. A 50kW fast charger could likely keep recharging the battery in less time than it could be drained at full speed on flat ground.

I'm disappointed it doesn't seem to be an option to add fast charging to my older model at all

Anti_Climax said:

I'm disappointed it doesn't seem to be an option to add fast charging to my older model at all

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Sure it always is an option. It's just how much time money and effort are you willing to put in. Sefs added a CHAdeMO port under the hood and made his own "super charger" by ganging up salvaged FFE chargers.

As the daily driver for more than just myself, I have to stick with factory/dealer mods.

After it's out of warranty it's likely to get hacked on mercilessly though.

Probably easier to add CHAdeMO to a vehicle than CCS sadly, as CCS's protocol for communication looks like a massive committee-designed clusterf*** depending on obscure hardware (Homelink GreenPHY) whereas CHAdeMO uses CAN... available natively on a number of different microcontrollers, or addable to any system using an SPI-bus CAN controller (e.g. MCP2515 - https://www.microchip.com/wwwproducts/en/en010406 )

Well yeah, congratulations on the battery replacement. Here's to hoping we all have such good luck! (although being without the car for a month is not a good thing).

I love the derivation - scientific approach to energy use. Thank you so much for the explanation. That's so darn cool.

A thought - road grade isn't included in the equation for force. So that variable isn't taken into account.

It is nice to see, the one massive variable you have control over that affects energy use the most - speed. As speed increases energy use squares - or goes up logarithmic - a lot. Air density, mass, gravity all have a much smaller effect and are usually more or less constant. So many people talk about colder air being more dense and rain having a significant effect on range. From the equation, and if you plugged in real numbers, you'd see the effect is minor. Battery temperature is the real culprit here. We can't do much about coefficient of drag, other than don't put roof racks on, close the windows, etc.

It would be interesting to understand how much slope or incline has on energy use. Sadly I'll bet it is a function of speed, horsepower of the motor, and efficiency of the motor, so probably impossible to estimate. Again, there is little you can do about inclines. I'm a bit wary of how evtriplanner accounts for altitude changes. The equation they have might work perfectly for a Tesla, but will fall apart for any other car.

The second biggest use of power in the car is heating the battery. And in the FFE (because the car has a lot of air leaks - intrusions through the firewall) cabin heating is next. If you were looking for a grand equation to cover energy use - that would have to be included somehow.

Awesome, amazing job. Thank you so much for sharing. I enjoyed thinking about all this.

The equation we have here is the baseline energy required to maintain a given speed accounting for the expected load from rolling resistance and aerodynamic drag. Comparing the Wh/Mi it gives versus actual measurements at cruise on level ground will give me an idea of the true efficiency. Since it is a permanent magnet synchronous motor it should be pretty efficient at normal speeds.

Believe me I am not stopping here - I'm going to get this all on a site for folks to play with.

As for road grade, it's not included in the equation but I noted how it can be added. Since the rolling resistance shouldn't change significantly for normal grades, you need only add another term that adds the force needed to counteract the vector component of gravitational acceleration multiplied by the mass of the car


The effect appears to be fairly linear in the real world, corresponding to about double the energy to climb a 4% grade and freely coast down in neutral - at least according to the DoE study I saw.

The impact of needing to cool the battery starts to get into complex feedback and might be where I'd have to just say it uses "more" but the energy loading for heating and cooling with closed windows should not be too complex to quantify. Ben, the admin for EVTripPlanner, calculated the heating and cooling for Teslas based on the window area as it seems little thermal energy transfers through the rest of the car. If nothing else, removing as many unknowns as possible will allow for more accurate estimates of whatever else is left.

I will say that the coastdown tests were done at around 33F and instead of getting 30.2 kWh to empty I got 29.8. Not a huge drop but I'd imagine it gets much worse below that.

Ben actually added my values to the vehicle selection so you can use it to plan Focus trips. Be mindful that the Motor efficiency, heating/cooling, battery temp and regen figures have not been tested so they should be considered - but I do hope I'll be able to get them in there soon. In the meantime, the output it has is fairly reasonable and fairly close to what I've seen in my normal driving.

One other thing to consider is the energy used to get to speed. The smoother and longer you can stretch out your increases in speed, the less energy you'll waste. Reality often won't allow you to glide from 25 to 65 but the difference in energy use is definitely measurable.

Wow, thanks for that addition.

I mentioned that issue with cabin heating the FFE versus Tesla from personal experience. Tesla requires very little energy to heat the interior. I've driven long distances in really cold weather. Turning off the interior heat would normally end up with a car getting cold really fast, not so in a Tesla. Because they don't have to cut the car up with all kinds of mechanical stuff sticking through the firewall, the car is incredibly tight. Outside air at maybe 0, going 70 MPH, I turned off the heat and the car remained at about the same temperature for a good 10 minutes.

The FFE is a whole different story. There are a lot of air leaks, so it would take more energy to heat the cabin, or at least maintain warmth. When you start with an ICE car, all that engine heat can be used to warm the inside of the car for free. So you don't necessarily care about a really tight car.

Battery heating would be pretty similar between the two cars.

Don't think you really have to concern yourself with cooling nearly as much. It is heat that uses a ton of energy. Cooling a lot less.

Being in Arizona, it's quite the opposite for me.

It's January and just today I got to choose between rolling down the windows or kicking on the AC.

Maybe it would be beneficial for someone to try and catalog the various "leaks".

Even if you're not doing it for energy savings with heating and cooling, the noise reduction would be appealing.

Thicker door seals, firewall fitment... Anything else?

Thought of your analysis Anti_Climax yesterday.

Check this out:
https://chargedevs.com/features/a-closer-look-at-energy-consumption-in-evs/

I can see why.

Though the note that power use goes up as the cube of speed is not right. The v and v-squared parts are summed rather than multiplied so it's still a second order function.

I know this is a REALLY old thread, but getting a copy of the service info for my car also got me a nice SVG cross sectional view of the car that I was able to use to make a new estimate of the area instead of taking 2.25m^2 as truth.

I'll spare you the details, but I got a number far closer to 1.8m^2 than 2.25m^2. Based on what I'm seeing, that old value was closer to a straight height*width rather than a real cross section. I may have to re-run my energy numbers with this new value. Of course now I have the OBDlink so I can directly capture speed data instead of estimating it from the dial over time.

This is all very interesting, but does not really explain a phenomena that I encounter occasionally. Most days with temperate weather, 75-80f, I can drive around town and get about 200-205 watt/hours per mile. There are occasions though that this goes to 230-240 watt/hours per miles for about 15 minutes, then starts to drop again. I have thought that perhaps the battery is being heated, but there is never a significant difference in temperature. Perhaps the battery is being cooled? In the FFE, there is really no way to know what happens to that disappearing energy from the interface. I find this happens a couple of times a month.

It's only meant to catalog the energy requirements versus speed.

There's a view that shows climate, accessory, wh/mi and range/SoC on a single screen. That might give some clues.

The battery heater doesn't kick on unless the coolant going into the battery is at or below 40F - and it's unlikely to happen after you've been driving for a bit.

The coolant loop bypasses the radiator unless it's needed and the motor and DC/DC converter will be heating it while you drive.

If the coolant hits 97F the car will cycle the AC compressor and pipe it through a chiller on the battery loop, but that will not show on the "climate" gauge, just on the accessory gauge. That would be my guess.