Theoretical Estimate of EV range "consumed" at various speeds and grades

Ok, this started out as a reply in another thread discussing possibility of using google maps elevation data to optimize EV routes, but got a little out of control so I'm giving it its own thread in case anyone finds it interesting Its very first order, very theoretical, and based on lots of simplifying assumptions, but it may be interesting anyway

One way to get a first order estimate of the impact of a certain grade on EV range is to use an EV Calculator such as this one, which happens to have a 2004 Prius model built in. No doubt the model accuracy could be improved, but I'm just trying to get a quick estimate of relative performance.
EV Calculator

The parameters I entered are:
Vehicle: 2004 Prius
Motor: Warp 9
Battery: Thundersky
Controller: Zilla 1k
Adjustments/Voltage: 80 (4.4kW Prius Pack over 55 Ah Thundersky capacity)
Adjustments/Lbs Removed: 343 (yields final weight = original weight of 2890 lbs)
Adjustments/Misc Added Lbs: 0
Adjustments/Incline = Wind = 0

For a baseline this yields the following flat driving characteristics, which look somewhat reasonable:
Speed -- HP Required -- Wh/mi -- Range
20mph ----- 3.43 ------ 149.7 -- 24.5 mi
40mph ----- 9.10 ------ 170.3 -- 17.2 mi
60mph ---- 19.26 ------ 280.7 -- 11.1 mi

Setting incline under Adjustments to various percent grades gives an idea of the additional power required, for example here is the 40mph case:
Incline -- HP Required -- Wh/mi -- Range
0 ----------- 9.10 ------ 170.3 -- 17.2 mi
1 ---------- 12.49 ------ 273.0 -- 12.0 mi
2 ---------- 15.87 ------ 346.0 -- 9.1 mi
4 ---------- 22.65 ------ 489.5 -- 6.1 mi
8 ---------- 36.19 ------ 767.3 -- 3.5 mi

The HP calculation is likely to be the most accurate, as I believe its only based on the vehicle characteristics. The Wh/mi and Range bring the motor, battery and controller into the picture which add more variables that are not very well modeled in this example.

Looking at the delta, it seems like the HP requirement at 40mph goes up quite linearly with percent grade.
From 0 to 1%, its 3.39 hp, 1% to 2% its 3.38 hp, 2% to 4% its 6.78 hp or 3.39 hp per % grade, and from 4% to 8% its 13.54 hp or 3.385 hp per % grade. This should work both ways, meaning a 1% downhill grade should use 3.38 hp less power, and a ~3% downhill grade you could coast indefinitely down at 40mph assuming you are gliding with no regen resistance. This agrees with the calculator output if you put in negative incline numbers.

So estimated power required at 40mph = 9.1 + (%grade)*3.39 hp, or expressed as a percent, power required = 100% + 37.3%*(% grade) where '+' % grade = uphill, and '-' equals downhill.

At a given speed/rpm of MG2, power is proportional to torque, which is proportional to current, which is proportional to electrical W, which is proportional to Wh/mi, so you could also say Wh/mi = 100% + 37.3%*(% grade). The calculated Wh/mi is a little different than this, but I think we can chalk that up to the second order effects of the modeled motor/battery/inverter characteristics which do not match the actual motor/battery/inverter.

This is equivalent to saying that each Mile on a given uphill grade at 40mph uses 1mi + .37mi * (% grade) of EV range, or on a downhill grade 1mi - 0.37mi * (% grade).

From the google elevation data for a given leg, you could calculate % grade as the altitude change divided by the distance over which it changes. From that you should be able to calculate the affect of various routes on your EV range.

Note that this effect is speed dependent, because even thought the incline force is constant to the first order, it varies as a % of total force required (due to air resistance, etc). To scale vs. speed, we can use the EV calculator to calculate the new % hp factor for each speed by using the flat and 1% grade values for each speed.

Speed -- Flat HP -- 1% Grade HP -- % increase
20mph --- 3.43 ------- 5.12 -------- 49.3%
30mph --- 5.84 ------- 8.38 -------- 43.5 %
40mph --- 9.10 ------ 12.49 -------- 37.3%
50mph -- 13.48 ------ 17.71 -------- 31.4%
60mph -- 19.26 ------ 24.34 -------- 26.4%
70mph -- 26.73 ------ 32.66 -------- 22.2%

For each speed, you can substitute in the new % for the .37 mi in the formula above. For example at 60mph for a 1% grade, each mile would burn about 1.26 miles of EV range. Now that makes it seem like its better to drive uphill faster, but bear in mind you would expect to have less EV range at a higher speed to begin with. You can also estimate that from this data in the same way although you also have to compensate for the fact that at a different speed you cover a different distance in a given time. If your EV range is estimated based on an average of 40mph, you could estimate the impact of various speeds as follows:

Speed -- Flat HP -- % time inc - EV mile
20mph --- 3.43 ------- 100% ---- -31.2%
30mph --- 5.84 ------- 33% ----- -14.6%
40mph --- 9.10 -------- 0 % ------ 0.00%
50mph -- 13.48 ------ -20% ------ 18.5%
60mph -- 19.26 ------ -33.3% ---- 41.2%
70mph -- 26.73 ------ -42.9% ---- 67.7%

So for example you could say that driving a mile at 60mph uses 1.41 miles of EV range compared to driving the same distance at40mph. Putting those two together, you could say that driving up a 1% grade at 60mph would use 1 + .26 + .41 = 1.67 mi of EV range relative to 40mph flat driving, whereas driving 40mph on the same grade would have only used 1.37 mi.

So to summarize all that and acknowledging that this is all just a rough theoretical estimate wiyth lots of assumptions and simplifications which may help make some sense of real world observations:

EV miles "penalty" for various speeds relative to 40mph, IE how many "extra" miles of EV range are consumed (or gained) per mile at the following speeds:
Speed -- penalty per mile
20mph -------- -0.31 mi
30mph -------- -0.15 mi
40mph --------- 0.00 mi (reference)
50mph --------- 0.19 mi
60mph --------- 0.41 mi
70mph --------- 0.67 mi

Additional EV Miles "penalty" for each percent uphill grade, or "bonus" for each percent downhill grade:
Speed -- penalty/bonus per mile
20mph -------- 0.49 mi
30mph -------- 0.43 mi
40mph -------- 0.37 mi
50mph -------- 0.31 mi
60mph -------- 0.26 mi
70mph -------- 0.22 mi

So to calculate the effective EV miles used on a given percent grade at a given speed, you'd use the formula:

EV miles consumed per mile = 1 + speed penalty + grade penalty (or - for downhill "bonus")

For example, a 4% uphill grade at 60mph would use 1 + 0.41 + 4*0.26 = 2.45 miles of EV range per mile. A downhill grade of 3% at 40mph would use 1 + 0 - 3*0.37 = -.11 mi, meaning you could essentially glide down this hill without using any EV range.

Rob

 

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Ok, that's an interesting data point. kWh used is probably a more reliable way to look at it than actually trying to use EV range differences. The calculations above use a fixed flat 40mph EV mile as the reference which is useful for comparison, but may be difficult to relate to the complex calculation the PiP is using to dynamically estimate EV range. Note that this calculation assumes pure EV drive, as this is the total power required. This is likely not possible on a stock PiP for all speeds/inclines.

All that being said, here's a wag at testing the calculations above against this data.

800' over 10 miles is about a 1.5% average grade. As the equation is linear, an average grade calculation should be accurate as long as it doesn't encompass both rises and drops.

- If average speed is 60mph, you'd expect to be burning about 1 + 0.26 * 1.5 (grade) = 1.39 times as much power on the uphill relative to flat, and 1 - 0.26 *1.5 = 0.61 time as much power on the downhill relative to flat. 1.39 / 0.61 = 2.27 times as much power uphill vs. downhill, compared to the observed 6.6kWh / 3.3 kWh = 2.0.
- At 50mph it would be 1 + 0.31 * 1.5 = 1.465 and 1 - 0.31 * 1.5 = 0.535, for a ratio of 1.465 / 0.535 = 2.74.
- At 40mph it would be 1 + 0.37 * 1.5 = 1.555 and 1 - 0.37 * 1.5 = 0.445, for a ratio of 1.555 / 0.445 = 3.49.

From the description though, it sounds like the ICE is running at least during the uphill drive. From the observation that changes in speed result in changing mpgs, but not kWh, it sounds like the MG2 may be maxed out even at the lowest speed tested. In that case the ICE would be basically making up the rest of the power ratio difference. Theoretically with the mpg data we could estimate total power to see how it correlates, but an easier case would be one where both the downhill and uphill drive can be completed without the ICE.

Rob