Trying to figure out Power Curve

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TimeHorse

Well-known member
Joined
May 13, 2010
Messages
999
In this thread I started a discussion of how to calculate the power curve of the Nissan Leaf in terms of velocity / wheel angular rotation similar to the chart shown in that post from Tesla.

I am pretty confident that the power curve is a quadratic in terms of the angular velocity of the wheel and came up with the following equations:

Code:
37,949.797 930 052 3 rad^2/sec^2 * a + 37,949.797 930 052 3 rad/sec * b + c - 6,437,136 Joules = 0

Where a, b and c are the constants of the Power Curve I wish to calculate such that:

Code:
P(w)[instant] = a * w^2 + b * w + c

Where w is the angular velocity of the wheel at any given time. The problem is, I can't solve this equation without some more information. It's not necessary to do the US06 or HWFET test to fill in the gaps, though those would do it. Instead, IMHO, the easiest way to do this calculation is to set the LEAF up on a test bed, mount a dynamometer on it's wheel and run it at 2 consistent speeds for 5 minutes straight and then measure the Energy used to perform this test. Take the near-ideal speed of 20 mph / 32 km/h for the first test and 45 mph / 72 km/h for the second and please report back here so I can do a proper calculation of the expected battery life.

Okay, I've stopped my drug-induced dream and am moving back to reality. So, is there anything we can do to figure out a, b and c given all the information we currently have available? Can anyone see a way of solving this problem?

The reason I am trying to get this information is because I want to calculate the expected energy requirements and thus range based on the EPA US06 and HWFET test vectors and to do this I need to calculate the instant power for each of the velocities specified in the test suite so that I can sum them to get the total energy drain. That would allow me to compute the useful range of the LEAF under more realistic circumstances that an average of 19.6 mph in stop-and-go traffic with only 1:45 seconds or so spend on an actual highway in EPA LA4. Since my commute is mostly highway driving, the LA4 numbers aren't adequate for my understand of how the LEAF will fit into my life and I really need to know these numbers before I can buy this wonderful car.
 
What is wrong with using Andy's numbers from the reliable data thread?

http://www.mynissanleaf.com/viewtopic.php?f=8&t=352

range1.jpg
 
palmermd said:
What is wrong with using Andy's numbers from the reliable data thread?

range1.jpg

I read Andy's Thread. It's actually pretty good and if we did have the 45 mph and 60 mph ranges from Nissan I could probably get a very accurate reading for the Power Curve for the Nissan Leaf so I could compute the expected range for an arbitrary speed. I don't know how I would compute a log of my instantaneous speed so that I could build my own table to get a very accurate estimate of whether the LEAF can get me home after work but that seems next to impossible without some major equipment.

In fact, I was thinking of taking those numbers and figuring out what their power curves look like, but this time in terms of speed not angular velocity of the wheel and without having a known energy value. I'm going to put these numbers into a Google Spreadsheet and see if I can get out the power function for each of them.
 
TimeHorse said:
palmermd said:
What is wrong with using Andy's numbers from the reliable data thread?


I read Andy's Thread. It's actually pretty good and if we did have the 45 mph and 60 mph ranges from Nissan I could probably get a very accurate reading for the Power Curve for the Nissan Leaf so I could compute the expected range for an arbitrary speed. I don't know how I would compute a log of my instantaneous speed so that I could build my own table to get a very accurate estimate of whether the LEAF can get me home after work but that seems next to impossible without some major equipment.

In fact, I was thinking of taking those numbers and figuring out what their power curves look like, but this time in terms of speed not angular velocity of the wheel and without having a known energy value. I'm going to put these numbers into a Google Spreadsheet and see if I can get out the power function for each of them.


you have WAY too many unknowns to calculate this. If you are going to just guess at numbers, then the table above is more than enough to just guess. We KNOW that the S10 and the Rav4 were similar to the LEAF in the LA4 test. We KNOW their 45mph consumption and their 60mph consumption. There seems to be about a 30% loss from 45-60. If we assume another 30% loss from 60-75 then you end up with 55 miles range for the LEAF at 75mph. It will probably be a bit more of a loss so lets just say 50 miles range. Now you have 3 data points and just fit a curve to these and you'll be close enough. Much closer than you will get trying to do detailed calculations with 10's if not 100's of unknowns. (Tire rolling resistance, bearing losses, etc...)
 
palmermd said:
you have WAY too many unknowns to calculate this. If you are going to just guess at numbers, then the table above is more than enough to just guess. We KNOW that the S10 and the Rav4 were similar to the LEAF in the LA4 test. We KNOW their 45mph consumption and their 60mph consumption. There seems to be about a 30% loss from 45-60. If we assume another 30% loss from 60-75 then you end up with 55 miles range for the LEAF at 75mph. It will probably be a bit more of a loss so lets just say 50 miles range. Now you have 3 data points and just fit a curve to these and you'll be close enough. Much closer than you will get trying to do detailed calculations with 10's if not 100's of unknowns. (Tire rolling resistance, bearing losses, etc...)

It's not as complicated as all that. You start with a basic assumption: there exists some function P(v) such that for any given v, the power required to drive at that v is given by P(v). How do we calculate P(v)? Well, we know that the most significant factor in a car's power drain is drag or air resistance. This is not the case at low speeds, but dominates as speeds increase fairly rapidly. Now Drag is proportional to the square of the velocity. So we know that P(v) is proportional to v^2. And since we have a quadratic expression, we know that P(v) = av^2 + bv + c. So all we really need to calculate is a, b and c. Another way of looking at this is as a linear algebra problem: plug in 3 know velocity / power pairs and you can solve for a, b and c.

In theory we should be able to do this with Andy's tables, since we have 3 velocities (45pmh, 60mph and LA4) and 3 powers in the form of energy in the form of distance. So theoretically, it should be calculable for the other cars.

I'm not trying to get an absolutely correct curve, but I think the Quadratic is a very good approximation of the power curve and so should work very nicely.

But, the thing to remember is that curve is not linear. The power required to drive and the energy used by driving and thus the available range when driving 75mph will not simply be the x + (x-y) for x, y ranges at 45 and 60, respectively; it will more likely look like x + (x-y)^2 or some such, because, as I said before, the power / energy / range function is quadratic.
 
1. Use v (velocity) instead of w, it is easier to understand and work with.

2. at v=0, the power consumption = 0, so C=0.

3. The LA4 test is not a good data point, since it is not at any "average" speed.

4. We will have to wait for more data to get the A and B in
P = A*V?2 + B*V
 
TimeHorse said:
And since we have a quadratic expression, we know that P(v) = av^2 + bv + c. So all we really need to calculate is a, b and c. Another way of looking at this is as a linear algebra problem: plug in 3 know velocity / power pairs and you can solve for a, b and c.

So, what is it for Tesla ? We should be able to validate this equation using the graph they have published.
 
garygid said:
1. Use v (velocity) instead of w, it is easier to understand and work with.

Well, technically it should be little-omega because it really comes down to tire rotation, but since there's a constant conversion factor between angular velocity and linear speed, based on the wheel+tire radius, I'll do a first cut with velocity. Though I'll often convert mph to m/s (SI Units) to make the math easier.

garygid said:
2. at v=0, the power consumption = 0, so C=0.

I don't think that's a safe assumption from a mathematical point of view. It sounds weird, I know, but from an mathematical point of view we may not end up with 0-velocity = 0-power. We'll have to handle that as a special case. I think it's a consequence of choosing a second-order equation to represent things. If you look carefully at Tesla's graph:

display_data.php


The curve never hits the Y-axis, but if you extrapolate toward it, there would still be some positive power usage. That's because it's a mathematical model. It's a very good approximation for things at 'practical' velocities up to very small 2-3 mph, but less than that I think any mathematical model is going to break down and we'll just the function is correct for speeds above 2 mph but less than that the power is 0 or negligible. Most people aren't spending large periods of time at (0, 3] mph (down to, but not including 0) so we could to all intents and purposes wing these numbers and 0 would be set to 0 as a special case.

I'll probably just assume P(v) for v >= 3mph, P(3)*v/3 for 0 < v < 3 and 0 for v = 0.

garygid said:
3. The LA4 test is not a good data point, since it is not at any "average" speed.

LA4 is not an easy test point, to be sure, but the test is well-defined at a 1-second interval which gives us the speed for each moment of the test and this can be summed to get the energy such that:

Code:
P(v)[instant] * 1 second = E(v)[instant]

Here, we don't take into account the discontinuity between the speed changes and treat everything as a sum since our sample set is discrete and not continuous:

Code:
E[total] = sum(E(v)[instant])

And since range is related energy usage by a constant (though we don't know what that constant is for any of the cars) the same holds true for the range calculation. The trick with the LA4 numbers is how they relate to the fixed-speed numbers. One way of looking at it is LA4 is a 1370 second test that goes 7.45 mi, and the other tests are tests that go a distance of their given range when run at the given speed, or for (range / speed) seconds.

garygid said:
4. We will have to wait for more data to get the A and B in
P = A*V?2 + B*V

For the Nissan Leaf, I agree. I am unable to find better information for this. I think just having the Leaf run the 45mph and 60mph tests would be enough and so I'm trying to prove this by calculating it for our sample cars.
 
Preliminary Results for other EVs:

The thing is, when you think about it, the P(v)[instant] function is pretty easy to understand: the faster you go the more power you need. Energy is also pretty simple: the longer you apply that power, the more energy you need. Distance, however, is not so. In fact, the less power you use the farther you go. So where as P(v)[instant] and E(v)[instant] are proportional to the square of v, range is inversely proportional to E(v)[instant] and thus P(v)[instant].

With that in mind, the curve I've calculated is more of a perRange calculator. It returns a result in 1/mi units. To get the range, take 1/perRange(v). I'm not sure yet how perRange can be applied to HWFET or US06 though since the units are wacky.

Car c b a
==============
MiniE 2009 Li 0.009047892441973 -0.000203542250306 0.000003262459189
GM EV1 (NiMH) 0.011322370019654 -0.000348887994312 0.000004399322139
Chevy S-10 (NiMH) 0.016362903701009 -0.000525841207901 0.00000738613259
Toyota RAV4 (NiMH) 0.014322159326115 -0.000337522146135 0.000004880725828
Solectria Force 0.017806016459982 -0.000558308534978 0.000008276907776
Ford Ranger (NiMH) 0.019417767703272 -0.000656042998219 0.000009283861427
Chrysler Epic Minivan (NiMH) 0.019879514555274 -0.000623232274862 0.000008286025798
GM EV1 (PbA) 0.021705752081388 -0.000747817612853 0.000009551847013
Toyota RAV4 (PbA) 0.021470350349737 -0.000654344101743 0.000010019953168
Ford Ranger (PbA) 0.02379492643136 -0.000766029009648 0.000010954992032
Baker EV100 Pickup 0.029909060664314 -0.001313261414528 0.000022482751739
Solectria Pickup 0.030123092894218 -0.001073347609568 0.000015088289956
Chevy S-10 (PbA) 0.036449834546514 -0.00123448301367 0.000017608983931

So, for instance for the MiniE, you have:

Code:
perRange(v)[MiniE] = v^2 * 0.000003262459189 hr^2/mi^3 + v * -0.000203542250306 hr/mi^2 + 0.009047892441973

So, if we calculate perRange(45mph)[MiniE], we get 0.006494971036024/mi. 1/perRange then gives us just under 154 mi. Any very very very lucky Mini-E users out there want to put that to the test? 154 mi at 45mph?
 
There are at least 5 more important factors to consider in calculating the power used for a given driving segment, other than the constant-speed velocity-related components.

1. Mass of the car and "cargo" needs to be treated for "acceleration" (changes in V).
2. Change in potential energy (going up and down hills) are important.
3. Account for Energy lost in braking.
4. Net usable Energy regained due to Regen, if any.
5. Energy lost due to battery loading beyond the constant-V current demands.

The result is that one cannot use the LA4 speed profile alone and get an "accurate" energy usage for the cycle.

One must also know how the dynamometer was programmed to simulate hills (some of the cycle is not "flat", I believe), mass (one can guess at a constant mass for car and "cargo"), wind effects (one might assume no-wind conditions), etc.

Load at V=0 is primarily radio, electronics, lights, A/C, heater. They can be "little" (all off) or "significant" when all on, but are usually small (except when stopped or coasting) compared to the power used for moving.
 
garygid said:
One must also know how the dynamometer was programmed to simulate hills (some of the cycle is not "flat", I believe), mass (one can guess at a constant mass for car and "cargo"), wind effects (one might assume no-wind conditions), etc.

The EPA LA4 test doesn't have any data points for terrain differences; from my reading, the test is on flat laboratory conditions since that's the only safe way to get the exact speed curve required by the test. But I'd be willing to guess the fixed-speed numbers are equally 'laboratory' numbers: I doubt anyone flew out to Montana, and just started driving in a straight line to see when the battery went out at a given speed. Of course, they'd have to measure wind resistance at the higher speeds because that's the largest power component, but that could be simulated with fans in a laboratory.

All the components you list are factors in the calculation but I think we need to figure this out in systematic fashion rather than trying to solve it all at once. Once we have the 'base' power function, we can analyze conditions such as relative weight gain, a/c at a given temperature, and the various other components. Granted, Acceleration will be the most difficult, but again let's first make sure we have the math right for the base function before we worry about adding linear and constant factors to it. The only major quadratic factor AFAICT is drag though Drive-train also seems to be, but is another vehicle constant. This can clearly be seen in Tesla's breakdown of the power curve components in their tests:

display_data.php
 
In the above, only the aero part is a quadratic equation. We don't know about drive train.

May be something like w(dt) = av+b/v+c.

That is why I was saying, using Tesla's curves, we should first check the validity of the quadratic equation.
 
evnow said:
In the above, only the aero part is a quadratic equation. We don't know about drive train.

May be something like w(dt) = av+b/v+c.

That is why I was saying, using Tesla's curves, we should first check the validity of the quadratic equation.

The reason the curve has a slope change at a non-zero point in those graphs is because they're not measuring power, they're measuring energy/distance vs. velocity. This is like taking the y-axis of the Power curve and dividing it by the x-axis and then plotting the new graph. That's why you see a 1/v term in this chart. It's equivalent to the Power curve, it's just the P(v)/v, which is a*v + b + c/v. We could easily solve this equation rather than the straight quadratic, to be sure, but I think it makes more sense to take that chart and transform it back to a pure Power function. When you do that, the reciprocal terms become constant and the linear ones become quadratic.

Anyway, since technically E = Integral[P(t) dt], we could expect the Energy curve to realistically be a cubic function if Power is quadratic. But the thing to remember is that the higher powers dominate with higher velocities but we're not worried about mach-1 velocities here so for the most part it's doubtful we need anything above cubic for our approximation, and maybe less but certainly at least quadratic because of drag.
 
And then you change the temperature, the slope, accessories, pack age, precipitation, wind, or any such factor and your equation blows up. ;) Heck, even the above charts are oversimplifications -- for example, tire rolling drag drops as the tires heat up over time.
 
KarenRei said:
And then you change the temperature, the slope, accessories, pack age, precipitation, wind, or any such factor and your equation blows up. ;) Heck, even the above charts are oversimplifications -- for example, tire rolling drag drops as the tires heat up over time.

I agree those are all factors, ones that even EPA doesn't account for when you look at MPG numbers, but the thing is, are we to go into this totally blind, and not even try to guess what the realistic range of the Leaf is, or do we just waive our hand, buy it and prey we can get home in the evening on day 1 of ownership?

Remember, 100mi is at the LA04 EPA driving cycle. This does briefly reach highway speeds but has a lot of stop and go, and an average speed of 19.5 mph. Of all factors, drag, with it's proportionality to v**2, is the biggest factor when driving at speed. It's one of the Tesla's 2 big secrets (the other is the lack of weight of an engine leads to more difficult handling in curves). That's not to say we can know exactly what the Leaf can do. And even my equation requires 2 more numbers: the 45mph and 60mph tests, for instance. But at least it's a start. At least we can try and get a ballpark figure. Is it 20 mi under US06? 50 mi under US06? 80 mi under US06? This is what many consumers really need to know. Range in an of itself isn't a concern to most people except in so far as it is well within the bounds of the 2-way trip to and from the office. If one's commute is 67 or so miles, round trip, will the car suffice. Clearly, weather and age will be factors. But at the very least, shouldn't we be allowed to know what it can do under California New conditions, and then worry about D.C. heat and chill? Shouldn't we try and break the problem down into simpler parts? Like, separate the power curve from the battery capacity; do the power curve under ideal battery conditions then calculate the battery drain due to weather and age. That's all I'm saying.
 
Yes, I believe you are asking the right questions.
With a "longer" commute, getting home from the round trip with 5 or 10 miles to spare, what would be the conditions to watch out for that would likely require us to get some additional e-fuel before we get home.

Will it be a cold, rainy day with a headwind that will use the heater, headlights, and windshield wipers?

Armed with more accurate expectations, we might plan to charge at work that day, even though we normally could make it all the way home.

Realizing that, we might want to plan ahead and arrange for that occasional at-work charging with our employer. Would it take just a long extension cord, or significant "construction" and actually installing some outdoor, "waterproof", 120v GFI sockets near some special parking places?

Sometimes that takes several months, even if the "boss" is willing.

Plan B:
Locate and test (when you do not NEED it) an e-fuel point near work, or part-way home. Perhaps drop buy occasionally to check that they are working. This might even be a willing friend's house.
 
evnow said:
Now that we have some numbers, wher is TimeHorse ?

I'm here; Show me the numbers! :)

Actually, I've been playing with numbers a lot recently: http://aecn.timehorse.com/2010/05/playing-with-numbers.html
 
gas ("g" = one gallon):
trip-distance / trip-gallons = mpg
or, tank-distance / tank-gallons = mpg
and then, cost (as $-per-mile) = $-per-gallon / mpg

For EV (using 1 "e" (edie) = 1 kWh of e-fuel):
trip-distance / trip-edies = mpe
or, tank-distance (the range) / tank-edies = mpe
and then, cost (as $-per-mile) = $-per-edie / mpe

So, for example:
300 mi / 10 gallons = 30 mpg
$3.00 per gallon / 30 = $0.10 per mile

80 miles / 24 edie = 3.33 mpe
$0.333 per edie (kWh) / 3.33 = $0.10 per mile

Of course, your numbers will vary.
 
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