Back to the Power Curve then!
I've been looking over the EEStory post, thanks EVNow, and think he's got a good approach, all be it lacking wind speed and a/c / heater power. Seems to me, wind speed is applied with a ws * cos f term where f is the angle of the wind direction relative to the vehicle, 0-degrees being a head-wind and 180-degrees being a tail wind, and ws is the wind speed. Of course, a direct side-on strong wind will force you to steer into it slightly to counter-act the lateral force so take f relative to the correction, not to the parallel of the travel lane.
The heater / cooler would be a discrete function of temperature but I'm not 100% on this. Obviously, at least A/C operates on the principle of PV=nRT by manipulating Pressure and Volume to change Temperature in the unit, taking in warm air and putting out chilled. The question is, is this proportional to the temperature outside. I think this is rather complicated when you consider solar-heating via incident sunlight which, as we all know, can make a cabin temperature well above the outside ambient. For someone who drives West each day around 3pm-4pm, that solar heat can't even be defeated by my A/C at full blast in my ICE. Of course, on a cloudy day, this effect is much diminished, so you really have 2 factors when it comes to A/C: external temperature and solar input. I would guess that A/C power is asymptotically proportional to these 2 factors since there will be a maximum power rating for any A/C unit. It may therefore be easiest to just consider the worst case for both A/C and Heating (which I guess will be heating coils + the fan) and plug those numbers in as a constant.
Have I missed anything significant?
So, putting it all together, we have:
Ptractive = M/1000 * v * (a + g * sin ?) + (M * g * Cr + ?/2 * (v + ws * cos f)**2 * A * Ca) * v/1000 + (ACP + HCP)/1000
Where:
Ptractive = tractive power (kilowatts)
M = vehicle mass (kg)
v = vehicle velocity (m/sec)
a = vehicle acceleration (m/s**2) = ?v/?t
g = gravity (m/s**2) ~= 9.81 m/s**2
? = road grade angle
Cr = rolling resistance coefficient
? = mass density of air (kg/m3), depends on temperature and altitude, 1.225kg/m3 is common
A = cross-sectional area (m2)
Ca = aerodynamic coefficient
ws = Ambient Wind Speed
f = wind direction, relative to straight ahead
ACP = Power used to run the A/C & fan full blast, IFF too hot, in Watts
HCP = Power used to run the Heater & fan full blast IFF too cold, in Watts
(Chief part of equation thanks to EEcclesiastical on the EEStory forum; I can't seem to join that forum to thank him so I thank him here!)
All that said, the think that I bring out of this is that a) both v and a are properly represented as factors and b) those factors are of the form c1 * a * v + c2 * v + c3 * v**2 + c4 * v**3 + c5, where:
c1 = M/1000 in kg
c2 = (M * g * (Cr + sin ?) + ?/2 * ws**2 * cos**2 f * A * Ca) / 1000 in Newtons
c3 = ? * ws * cos f * A * Ca / 1000 in kg-Hertz
c4 = ?/2 * A * Ca /1000 in kg/m
c5 = (ACP + HCP)/1000 in Watts
Now, what we really want to know is range, which as I showed before is equivalent to v * E / P. The question is, to go from power to range is it as I've expressed or is there a dP/dv derivative I need to take. Certainly c1 v ?v / ?t indicates to me there some kind of calculus that should be applied. Like, could I replace this with c1 ? v**2 / 2 ? t?
Again, I think computing 1/range is easier than computing range because 1/range is proportional to power: 1/range = P / (E * v). So that's why I'm thinking derivative:
What if 1/range is actually (dP / dv) / E, where E is of course our constant old friend the 24 kWh battery.
Honestly, I like the idea of the derivative of continuous power function but there is a problem with that: the ACP and HCP terms would drop and we
know those effect range! So it probably looks something like:
1/range = (c1 * a + c2 + c3 * v + c4 * v**2 + c5 / v) / E
So there you have it: an equation of
five variables! Which means I need
5 data points to evaluate. Theoretically, evnow's numbers do have A/C / Heating runs so I could pull one of those to get one more constant, though it doesn't sound like any of them are A/C + Cruise so it would be hard to evaluate in terms of this equation. Interestingly, the c1 term is 0 for the 37/38 mph runs since they're essentially cruising and I could estimate acceleration with the EPA LA4 by simply assuming ?v * frequency = acceleration, where for LA4, frequency (sample rate) is 1 Hz so it's simply the difference (v[n] - v[n-1]) in m/s**2. One more cruise or the HWFET numbers would allow us, I think, to get this power curve and corresponding range function very accurate if we ignore c5. With c5, we'd need an A/C (and Heater) result with cruise control, or we could just fill those in from spec.
