Trying to figure out Power Curve

[MikeD]

TimeHorse: For your convenience I have "eyeballed" the following Tesla "Range vs (Constant) Speed" graph data points (R, v):

2.5, 175
005, 285
010, 375
015, 407
020, 408
025, 394
030, 368
035, 338
040, 311
045, 286
050, 262
055, 239
060, 218
065, 199
070, 181
075, 166
080, 152
085, 139
090, 128
095, 118
100, 110
105, 102
110, 095
115, 089
120, 084
125, 079

If you want to use only 3 data points, it would seem reasonable to select them across the entire graph, say (5, 285), (65, 199), and (125, 79) or maybe (30, 368), (65, 199), and (100, 110).

Needless to say if the resulting model function does not reasonably closely model all the data points, then a different (better) model needs to be tried, and if more complex (i.e. more terms w/ coefficients) then clearly more data points will be needed to determine the model function.

 

[LEAFer]

Actually ... if you go HERE: http://www.teslamotors.com/blog4/?p=70
Scroll down to the end of the blog (just before the first comments), then you can click on the word "here" in the sentence "Download the excel data files here"

 

[garygid]

I agree about chopping some digits off the long numbers ...
100.00000000433258986543335557890954312334 percent.

 

[MikeD]

TimeHorse, looking at the details provided in the Tesla Excel spreadsheet data, the Drivetrain Wh/mi below ~20 mph looks like it may be difficult to model accurately with the kind of basic functions you have been using. Although it would be interesting to know if the LEAF has similar issues at low speeds (relevant in the situation of being stuck in very low speed traffic), most people probably are most interested in range for speeds beyond 20 mph. So given that and in the interest of keeping models reasonably simple, I withdraw my request for a (probably overly complex) model function for the entire Tesla "Range vs Speed" graph.

The reality is that before this year is over, graphs like this and more will likely be available based on actual measured test data. Thanks for your exploration of the theoretical issues involved in determining EV power usage and range for the LEAF.

 

[TimeHorse]

TimeHorse, looking at the details provided in the Tesla Excel spreadsheet data, the Drivetrain Wh/mi below ~20 mph looks like it may be difficult to model accurately with the kind of basic functions you have been using. Although it would be interesting to know if the LEAF has similar issues at low speeds (relevant in the situation of being stuck in very low speed traffic), most people probably are most interested in range for speeds beyond 20 mph. So given that and in the interest of keeping models reasonably simple, I withdraw my request for a (probably overly complex) model function for the entire Tesla "Range vs Speed" graph.

The reality is that before this year is over, graphs like this and more will likely be available based on actual measured test data. Thanks for your exploration of the theoretical issues involved in determining EV power usage and range for the LEAF.

I decided to try anyway; I took 4 samples (30, 60, 90 and 16 mph) and applied a cubic function in v divided by v, so you end up with

R = 1/(a*v**2 + b*v + c + d/v)

And with the numbers (which I report with a relative error I shall discuss in a moment):

a = 6.55E-7 hr**2 per mi**3
b = 8.11E-6 hr per mi**2
c = 1.60E-3 per mi
d = 8.85E-3 per hr

(See, much smaller!)

Now, the reason I report such small numbers is, as you can see from my copy of the Tesla numbers, I calculate the Range Curve within 2.7 mi standard deviation. Thus, the results could only be considered accurate with a relative error of about 2% and so is only accurate to about 3 digits.

The worst results are indeed in the 2-45 mph range so the question is, can we get more accurate, and should we?

Right now, thanks to the Student's

endless stream of nonsense and FUD

I am very confident that the cubic equation is the right model for us. Basically, if Power is cubic, then Range becomes a cubic / v, where the 1/v term is your radio, your lights, your A/C, and all the other components that are effectively "always on". Of course, there's no A/C here so the d constant represents the power of the non-A/C components on the Roadster, which is a term sorely missing from my calculations (the acceleration, regenerative breaking being the other corrupted computations: how can the acceleration component be negative!?)

Anyway, I think we're on the right track, we just need better data and numbers.

 

[TimeHorse]

Wow - I guess Leaf would have a similar range equation ?

No, I don't think so. You could fit a power function to any polynomial with enough known points, and this Tesla equation doesn't take acceleration into account since the spreadsheet and curves on their site don't discuss it at all. Matching it to LA4 could give us a good idea if we assume no regenerative breaking, though, or if we know the rb ratio. I still stand by my 8-constant evaluation for the LEAF.

 

[evnow]

I still stand by my 8-constant evaluation for the LEAF.

You mean this one ? So, only for LA4 we really know "v and ?(v**2)" now ...

Once each velocity variable is computed, you get a linear equation with the 8 unknowns c1..c8. If you have a series of sample runs, where v and ?(v**2) are known, you now have a series of linear equations that you can solve using matrix algebra. This is more or less what I have done.

 

[TimeHorse]

You mean this one ? So, only for LA4 we really know "v and ?(v**2)" now ...

From LA4 we have an equation involving c1 - c5 and c6 only by guessing what the Regenerative Breaking efficiency is, as Student is doing. LA4 isn't enough to figure out all 6 variables, but with 4 - 5 other samples, even if those samples are all fixed-speed and with no A/C, no Heater, negligible wind speed and road grade, then we could come close to the no-A/C, no Heater range of the LEAF. Since the fixed-speed tests have no acceleration, c5 and c6 cannot be computed from them. It's not safe to take a test with A/C as one of the 5 used to compute c1 - c5 (+ c6) since then you have a c7 component and would need 1 more reading, be it with A/C or not.

BTW, here's the source of the Forbes numbers. Not much information there on how they were computed, though, alas.

 

[evnow]

BTW, here's the source of the Forbes numbers. Not much information there on how they were computed, though, alas.

Those are included in my range table (though I've indicated Edmunds as the source for most). They all come from Nissan.

 

[TimeHorse]

And back to the equations again.

NOTE: This forum does not support superscript and subscripted BB code so I've just included the markup where necessary. I wish, personally, that there was the tex tag so I could show you how things should really look.

So I've been working with some folks on the Physics Forum to try and narrow down the correct parameters to evaluate the Nissan LEAF's LA4 test results.

Once again, I return to the 4 components of the primary forces on a car:

½ ? A Cd * ½(un + un + 1) ½(un² + un + 1²)

Where:
? = mass density of air, depends on temperature and altitude, 1.225kg/m3 is common
v = vehicle velocity
ws = wind velocity
f = wind direction, relative to straight ahead
A = cross-sectional area
Cd = The drag (aerodynamic) coefficient

and:
u = v + ws cos f (the velocity of the car relative to the wind)

In LA4, AFAICT, ws is 0 so u = v.

Crr Nf ½(un + un + 1)

Where:
m = vehicle mass
g = gravity ~= 9.81 m/s**2
Crr = rolling resistance coefficient

and:
Nf = m g (the normal force of the car against the road)

½ m ? (un + 1² - un²), where un + 1² > un²

Where:
? = The sample rate (1 Hz for LA4, 10 Hz for HWFET)

½ m ? rb (un + 1² - un²), where un + 1² < un²

Where:
rb = Efficiency of Regenerative Breaking

m g ? h

Where:
? = angle of road grade

And:
? h = ½(un + un + 1) sin ? (height times sample frequency)

In LA4, ? is 0 AFAICT, so the road grade term can probably be ignored.

And of course there are the constant Power functions, such as console, head lights, A/C and Heater.

Right now, this is my best estimate of the real-world result of running an LA4 test. As you can see, it averages the velocity between samples in most cases and with acceleration takes the difference between the square of the velocity and in drag takes the average square of the velocity times the average velocity.

Since Paverage = ?W / ?t and ? = 1 / ?t, we can compute the work (energy) for each cycle by dividing the average Power by ?. If we sum for each Paverage divided by ? across the entire test suite, we can get the total energy used to perform the test, and if we know how far the tests goes (LA4 is 7 8107 / 18000 mi), we know therefore know that the ratio between the test suite range and the estimated total range based on that test suite is equal to the ratio between the amount of energy used during the test suite and the total battery capacity of the car.

?W = E * LLA4 / RLA4

Where:

E = the Total Battery Capacity (24 kWh for the LEAF)
LLA4 = The Length of LA4 (7 8107 / 18000 mi)
RLA4 = The estimated Range of the Car under the LA4 cycle (100 mi for the LEAF)

It seems to me this, then, is how we should go about evaluating the LEAF in real terms. I will work with Student for these calculations and provide him with the details of this analysis when I've verified it with the folks on the Physics Forum, but I think these are the numbers we want.

 

[evnow]

I don't know whether this is just idealized ... or real & to scale.

This is as if, the higher speeds don't have much effect on range when using AC, since you would use less total energy for AC that compensates for more power used for driving.

http://www.nissan-zeroemission.com/EN/LIBRARY/PDF/PANEL/07.pdf

 

[TimeHorse]

I don't know whether this is just idealized ... or real & to scale.

This is as if, the higher speeds don't have much effect on range when using AC, since you would use less total energy for AC that compensates for more power used for driving.

That looks about right. Remember, drag is the monster; it is v**3 relative to the instant power required to move the car, which generally translates into a quadratic curve in range. I'm working on some new equations at the moment, based on some of Student's work, and I evaluate things pretty much like that chart and am finding some interesting results. Namely, the base power may be as much as 500 W for the console head lamps while running during the LA4 EPA test. This is a preliminary result, mind, but we should all expect there to be some non-zero base load on the LEAF.

The interesting thing is in the D.C. Auto Show interview, Mark Perry states that the load for A/C is much less than the load for the heater since the heater is just a space-heating coil (as we all expected) and requires much more energy to run than the A/C. In my charts, I'm going to allow Power per K, or Power per Centigrade Degree differential between the cabin and the outside (A/C when negative, Heater when positive). This seems the best simple extrapolation, though I would not be surprised if the climate power to temperature grade ratio was non-linear -- Blackbody radiation is, after all, temperature to the 4th!

Also, I'm using a slightly different power function. I'm using the Linear one I just described and a Quadratic one that's too complicated to describe. The Quadratic assumes that the velocity of the car in LA4 is given by a quadratic curve from the previous point (sample) to the next, including the current point. Then, this curve is integrated from -1/2w to +1/2w (half the sample distance) for each point. The result is slightly less than linear (about 1% difference) and arises mainly AFAICT from the fact that going to or from 0 velocity results in a negative velocity measured at the half-way point between the two 0 samples. You can see this when I post my EPA spreadsheet hopefully later today.

 

[evnow]

What surprised me aboput the graph above is that AC power is so high - equal driving power only around 40 mph.

 

[LEAFer]

What surprised me aboput the graph above is that AC power is so high - equal driving power only around 40 mph.

I think you meant ... ... 40 km/h not 40 mph. (So that's 24.8 mph.)

Edit: and based on the graph's resolution on my screen A/C=DrivePower equilibrium probably closer to the 30 km/h data (18.6 mph).

 

[garygid]

I suspect that Karen has a quite useful model of PHEV energy usage in her driving range simulator, probably still available in the ApteraForum.

 

[TimeHorse]

What surprised me aboput the graph above is that AC power is so high - equal driving power only around 40 mph.

That's A/C & Heater combined, though. As I read it, maybe 20% - 40% of the green line represents A/C power; the rest represents Heater power. It's a bit weird to have munged them together, but it's either the sum of heat and cool or the average, which is just the sum divided by 2. Either way, I think the main component is the heater which is why it's not scheduled for NY / NJ / New England any time soon.

 

[TimeHorse]

Here are my current results for the Nissan LEAF based this time on Student's calculus with a better fit for the forces on the car than the Riemann sum he performs.

https://spreadsheets.google.com/ccc?key=0AsiXrrqlvUM4dHR3QU81bVFtb052Z1kwLWhQWTczd0E&hl=en

That said, there is little difference between Riemann and Linear extrapolation; I measured something like 0.06% difference and the Quadratic Extrapolation has its own quirks as I've already mentioned. That done, I think actually the right curve to estimate the space between the points will be a Spline / Bézier so I'll probably copy and recreate that spreadsheet with only the very complex Cubic results. As you can see, with the numbers I've used the LA4 estimate comes within half a per thousandth of the Linear calculation and about a percent with the Quadratic. Then, using Student's method of estimated the HWFET fuel economy, I get within about 3% of the correct result with CAFE, Combined of his 367, but with Motortrend's 350 mpg result, the HWFET numbers are off by about 15%.

Like I said, I hope to get better numbers with a Spline solution but it will take some time even with Sage to get this all calculated.

 

[evnow]

So, according to these calculations, Highway range would be 105 miles - actually better than LA4 !?

 

[EVDRIVER]

Here are my current results for the Nissan LEAF based this time on Student's calculus with a better fit for the forces on the car than the Riemann sum he performs.

https://spreadsheets.google.com/ccc?key=0AsiXrrqlvUM4dHR3QU81bVFtb052Z1kwLWhQWTczd0E&hl=en

That said, there is little difference between Riemann and Linear extrapolation; I measured something like 0.06% difference and the Quadratic Extrapolation has its own quirks as I've already mentioned. That done, I think actually the right curve to estimate the space between the points will be a Spline / Bézier so I'll probably copy and recreate that spreadsheet with only the very complex Cubic results. As you can see, with the numbers I've used the LA4 estimate comes within half a per thousandth of the Linear calculation and about a percent with the Quadratic. Then, using Student's method of estimated the HWFET fuel economy, I get within about 3% of the correct result with CAFE, Combined of his 367, but with Motortrend's 350 mpg result, the HWFET numbers are off by about 15%.

Like I said, I hope to get better numbers with a Spline solution but it will take some time even with Sage to get this all calculated.

Try the RWC method. That's the real world calculation, why are people crunching numbers, No offense but this is a futile exercise because you don't have the correct data points to work with.

 

[garygid]

Crunching for practice, for a time when real LEAF numbers can be obtained.

Exercise:
Given a LEAF to "measure", what "street" tests would you run to get the data you need, but with a near-minimum of testing?