Regen vs. Coasting

[stevezzzz]

I must say I'm absolutely impressed, flabbergasted, really, at the brilliance in this thread, and the elegance of various people's solutions. Not to mention depressed by realizing there was a time, around fifty years ago, when I probably would have understood the equations. It stands to reason that Tesla owners are, in general, better educated that average drivers.

Still, like hometheatermaven, I'm thinking specifically and simply. Coming home I go down a long hill, about a mile and a half, with a stop sign at the bottom. So there's no timing to avoid stopping. In that case, it seems intuitive (because, as implied, I no longer have the gray matter to behave otherwise) that in that case it makes sense to use regen. Given a set distance and a predictable stop, getting from top to bottom without regen will leave the battery with less energy than using it. My question as been whether it makes more sense to have low regen vs high: because with high regen there's a need to apply some go pedal to avoid slowing to nearly a stop. I don't want to exceed the 40 mph limit. So, for a constant speed, constant slope, constant distance, predictable stop at the end, is it not the case that coasting is the least efficient? And wouldn't it also be the case that if low regen allowed traveling at the constant speed till the bottom and if high regen required applying a few electrons to maintain speed, that the best scenario would be low regen?

I haven't yet found a hill so steep that Standard regen isn't enough braking to allow you to set the cruise control and hold any speed you want. Just for grins, I've let regen slow me almost to a stop on some terrifyingly steep hills; it's one of the marvels of the Tesla drivetrain that you can literally set the cruise control at the bottom of a big mountain pass and never touch the go pedal again as you go up one side and down the other; if the speed varies by as much as 1mph, I'd be surprised. There's no benefit to setting Low regen because, Standard or Low notwithstanding, the car will apply exactly as much regen as is required to maintain speed. You can do the same thing manually (pedally?), of course, but the point is this: regen braking is infinitely variable from no regen to full regen, it's not an all-or-nothing proposition.

 

[ItsNotAboutTheMoney]

I must say I'm absolutely impressed, flabbergasted, really, at the brilliance in this thread, and the elegance of various people's solutions. Not to mention depressed by realizing there was a time, around fifty years ago, when I probably would have understood the equations. It stands to reason that Tesla owners are, in general, better educated that average drivers.

Still, like hometheatermaven, I'm thinking specifically and simply. Coming home I go down a long hill, about a mile and a half, with a stop sign at the bottom. So there's no timing to avoid stopping. In that case, it seems intuitive (because, as implied, I no longer have the gray matter to behave otherwise) that in that case it makes sense to use regen. Given a set distance and a predictable stop, getting from top to bottom without regen will leave the battery with less energy than using it. My question as been whether it makes more sense to have low regen vs high: because with high regen there's a need to apply some go pedal to avoid slowing to nearly a stop. I don't want to exceed the 40 mph limit. So, for a constant speed, constant slope, constant distance, predictable stop at the end, is it not the case that coasting is the least efficient? And wouldn't it also be the case that if low regen allowed traveling at the constant speed till the bottom and if high regen required applying a few electrons to maintain speed, that the best scenario would be low regen?

Ah, yes, but putting my hypermiling hat on again, my question is: Why did you expend so much energy getting to the top of the hill at high speed, when you can crest at low speed and glide to the bottom just using potential energy? This is the essence of Driving With Load: it is moee efficient to use less power going up the hill, allowing the car to slow down uphill and then allowing gravity to assist on the way down. In other words: it's best to use as little regen as possible to achieve your target.

 

[Jeff Miller]

I haven't yet found a hill so steep that Standard regen isn't enough braking to allow you to set the cruise control and hold any speed you want. Just for grins, I've let regen slow me almost to a stop on some terrifyingly steep hills; it's one of the marvels of the Tesla drivetrain that you can literally set the cruise control at the bottom of a big mountain pass and never touch the go pedal again as you go up one side and down the other; if the speed varies by as much as 1mph, I'd be surprised. There's no benefit to setting Low regen because, Standard or Low notwithstanding, the car will apply exactly as much regen as is required to maintain speed. You can do the same thing manually (pedally?), of course, but the point is this: regen braking is infinitely variable from no regen to full regen, it's not an all-or-nothing proposition.

We unfortunately don't have hills here in Illinois so I have zero experience driving them with the Model S. But what you're saying if I understand correctly is that the regen calibrates itself to maintain constant speed when you are using cruise control.

So for your I-70 Eisenhower over-pass example, which has a grade of around 6%, the force of gravity is m*g*sin(.06) = 1240 N, the force of friction at 65mph is (using Todd and JohnQ's numbers for the Model S) -av^2 - b = 640 N, so the regen force will set itself to 1240 - 640 = 600 N to keep the speed constant, leaving a power of F*v = 600*29 = 17kw to charge the battery.

 

[stevezzzz]

We unfortunately don't have hills here in Illinois so I have zero experience driving them with the Model S. But what you're saying if I understand correctly is that the regen calibrates itself to maintain constant speed when you are using cruise control.

So for your I-70 Eisenhower over-pass example, which has a grade of around 6%, the force of gravity is m*g*sin(.06) = 1240 N, the force of friction at 65mph is (using Todd and JohnQ's numbers for the Model S) -av^2 - b = 640 N, so the regen force will set itself to 1240 - 640 = 600 N to keep the speed constant, leaving a power of F*v = 600*29 = 17kw to charge the battery.

That sounds about right. Fourteen miles descending at 65 mph takes a little over 13 minutes. 13/60 x 17 = 3.68 kWh, which is somewhat more regen gain than I observe, but it's in the ballpark.

 

[hcsharp]

I probably should have a plot, but the idea is basically

case 1. start ppppppppppppppppppppppppppppppppppppppp end (p means under power, ~25kw, Todd's assumptions)
case 2. start pppppppppppppppppppppppccccccccccccccccccc end (c means coast)
case 3. start ppppppppppppppppppppppppppppppppppprrrrrr end (r means regen)
.......................................................|<--526m-->|


In case 3, you end up driving around 526m further at your original speed vi than you do in case 2. To maintain that speed over this distance you need to overcome the frictional forces -a*vi^2 - b = 770 N over this distance. In case 2, you're just coasting over this distance so not spending any extra energy from the battery.

That's not the problem we're trying to solve. In the OP the question is whether he should use regen to slow down immediately from 75 to 55, then continue at 55 with far less wind drag losses, vs coasting with no regen from 75 down to 55. The distance D is the same in both cases, but not the time. D is defined by the how far it takes to coast from 75 to 55. So the cases should be:

case 1. (same as you have. this case is just for reference i think.)
case 2. start ccccccccccccccccccccccccccccccccccccccccccccc end
case 3. start rrrrrrrrpppppppppppppppppppppppppppppppppppppppp end (p=power but far less required for 55 than 75)

I also agree that regen should be closer to 60kW, which it usually is when applied in full at 75mph. I suspect the regen case will be more efficient in that scenario, and very close if not better even at only 30kW. Hard to say just yet. Coasting is likely to be more efficient at higher altitudes, regen will probably win in cold temperatures.

 

[wycolo]

> We unfortunately don't have hills here in Illinois so I have zero experience driving them with the Model S. [Jeff Miller]

Thats probably why I run into so many flatlanders up here on the Rocky Mt passes during the warm months. I'm 940 miles to the Mississippi River. Check out I-72 over the Bluffs & Illinois River, also some nice hills down around Carbondale, IL.


> Fourteen miles descending at 65 mph takes a little over 13 minutes. [stevezzzz]

Mucho cojones!! Too mucho for me anyway, esp knowing you are in CC. I do it in 'variable regen' because I'm always in flux. Besides for years I only did this descent in 'diesel + trailer load' mode - in 2nd gear over in the right lane with the semis. Using brakes!: diesels have no throttle therefore NO manifold vacuum, NO engine braking. You are in COAST going down steep hills (back to topic).
--

 

[Jeff Miller]

That's not the problem we're trying to solve. In the OP the question is whether he should use regen to slow down immediately from 75 to 55, then continue at 55 with far less wind drag losses, vs coasting with no regen from 75 down to 55. The distance D is the same in both cases, but not the time. D is defined by the how far it takes to coast from 75 to 55. So the cases should be:

case 1. (same as you have. this case is just for reference i think.)
case 2. start ccccccccccccccccccccccccccccccccccccccccccccc end
case 3. start rrrrrrrrpppppppppppppppppppppppppppppppppppppppp end (p=power but far less required for 55 than 75)

I also agree that regen should be closer to 60kW, which it usually is when applied in full at 75mph. I suspect the regen case will be more efficient in that scenario, and very close if not better even at only 30kW. Hard to say just yet. Coasting is likely to be more efficient at higher altitudes, regen will probably win in cold temperatures.

"the question is whether he should use regen to slow down immediately from 75 to 55"

What does "immediate" mean in this context? Start slowing down using regen at the same distance that you would have started coasting if you were going to coast? What's special about that distance? Obviously, the longer you drive at 55, the less energy you use.

So doing it this way, you get the following with 60kw regen:

Coasting:
Coasting distance: 851 m
Energy cost to coast over this distance: 0

Regen:
Start regening at distance 851m from the end.
Time to regen from 75 to 55: 6.92 seconds.
Distance to regen: 202 m
Maximum possible gain from regen (100% efficiency): 6.92*60,000/3600 = 115 wh
Cost to drive the remaining 851-202 = 649m is (-avf^2 -b)*649/3600 = 529*649/3600 = 95wh.

So in this case, assuming 100% regen, you are better of regening - you pick up a net of 20wh.
Assuming 85% efficency on the regen, the actual energy gain is 85*115 = 98 wh.
Net of the 95 wh to drive the remaining distance, you save 3 wh by regening vs coasting.

But, this savings is mostly due to driving a longer distance at 55 rather than 75.
You're getting where you want to go more slowly in the regen case so
you spend less energy which is what we probably all expect...

Time to coast: 29.6 s

Time to regen: 6.9 s
Time to drive 55 after regen: 649/vf = 26.4 s
Total regen + drive 55 time = 33.3 s

A fairer comparison would require imposing the constraint of equal times.

 

[Todd Burch]

SOLUTION

OK guys, I spent a few hours this afternoon calculating this out using numeric methods. My time deltas in calculating these numbers were anywhere from about 0.01s to 0.05s, so the numbers should be reasonably good. (Note--I've already shown very good agreement with analytical solutions for the coasting case, discussed above).

First, let me share the results. I believe they are very interesting.

Interesting Result #1: Coasting is Not Always More Efficient
I will share some numbers that I've calculated, but I think I've proven that coasting is not always more efficient. I will show you several examples in which, given that you cover the same distance, using regen you'll have more energy in your pack than if you coasted (as I suspected initially).

Interesting Result #2: This is Complicated
Also as suspected, the answer is not cut and dry. It requires this sort of calculation to figure out whether you're better off using regen. However, we may be able to come up with some "Hypermile" rules of thumb using my data.

Interesting Result #3: The faster your final target speed, the larger the differential between your starting and final speed must be to get a regen benefit.

For example, if your target speed is 40 mph, you must start out going at least about 67.5 mph to get a benefit from regen. Any slower than 67.5 mph and it's better to coast. Any faster, and you're (much) better off using regen.
If your target speed is 55 mph, you must start out going at least 84 mph to get a benefit from regen.
If your target speed is 75 mph, you must start out going at least about 106 mph to get a benefit from regen.

Interesting Result #3
The improvement in regen over coasting is probably not going to be very noticable in the real world unless you're going from highway speeds down to surface streets (like 35 mph)...and even then we're gaining a fraction of a mile of range. You probably don't do this much--so you probably won't actually notice much improvement unless you're in a hypermiling competition

Real World Conclusion
In all cases where regen puts out about 60 kw (about 40 MPH and above-ish), your initial speed must be somewhere in the ballpark of about 30 mph faster than your target speed to get a benefit from regen. This is a decent rule of thumb to use.

So, if you're going 70 mph and you're dropping to 40 mph, USE REGEN.
If you're going 75 mph and you're dropping to 55 mph (as in my initial post), COAST.
If you're going highway speed and you're dropping down to a 35 mph surface street, USE REGEN.

This is particularly interesting because there are some experienced hypermilers here for whom this changes their game plan!

OK, so there are the results. Obviously, these rules of thumb change with higher altitudes (which favor coasting), colder temperatures (which favor using regen), etc. But these are good starting estimates.
Note that it's possible that something like mass doesn't even matter. I didn't take the numerics far enough to look into this, but I'll provide my spreadsheet so someone here can. (Gotta tend to my kids!) Also, feel free to experiment with different air densities, masses, regen efficiencies, etc. to see how it affects things.

Here's the techincal stuff. I'll try to be brief.
Assumptions:
-Constant regen of 60kW going into the battery (so target speed of 40 mph or higher)
-Regen efficiency of 85%
-Power production efficiency of 92% (opposite of regen)
-I Included drag, rolling resistance, and drivetrain losses. (Estimating the drivetrain losses).
-Ignored other power uses (HVAC, accessories, etc). Except for HVAC, they are minimal compared to the energies we're talking about here. You could take it a bit further to include HVAC based on the difference in times for regen vs. coasting--but I suspect it's not too drastic of a difference.

Basic Methodology

2 Cases:

Coasting: Go from speed A to B by coasting only. This takes a certain amount of distance. There is no net change to the energy in the battery.

Regen: Apply full regen at Speed A. When speed B is reached (which is at a shorter distance than when coasting), maintain speed until you cover the same distance covered in the coasting case. Take the energy gain from the regen and the energy loss from maintaining speed for the remaining distance to get a net change in battery energy.

In general, the numeric method is as follows:
1. Use numerical methods to calculate drag force, rolling resistance, and drivetrain resistance at the starting speed.
2. Use the sum of these forces and the mass of the car to calculate instantaneous acceleration.
3. Use the instantaneous acceleration to calculate a new speed at a new time t + dt, where dt is small (0.01-0.05 sec) Use the speed and the dt to calculate the distance covered in this small time dt.
4. Lather, rinse, repeat.

Here are some numbers. Note in all cases than for coasting, the energy change in the battery is zero--so if regen gives a gain of energy as listed below, it's better than coasting.
65 to 55 MPH: Regen LOSES about 30,000 Joules of energy.
85 to 55 MPH: Regen GAINS about 8,000 Joules of energy.
105 to 55 MPH: Regen GAINS about 150,000 Joules of energy.

50 to 40 MPH: Regen LOSES about 24,000 Joules of energy.
60 to 40 MPH: Regen LOSES about 22,500 Joules of energy.
70 to 40 MPH: Regen GAINS about 10,000 Joules of energy.
90 to 40 MPH: Regen GAINS about 155,000 Joules of energy (about 43 Wh, or about 0.15 mi of range).

Here's my spreadsheet for those who want to fiddle:
http://www.sendspace.com/file/r4l4t7

OK, there ya go guys. Don't know if this makes me a hero, a nerd, or both . Anyway, gotta get back to being a father. Question answered!

 

[hcsharp]

"the question is whether he should use regen to slow down immediately from 75 to 55"

What does "immediate" mean in this context? Start slowing down using regen at the same distance that you would have started coasting if you were going to coast?

Yes. That's how the problem was stated in the OP. You start regen at d=0 meters, and start coasting at d=0 meters in the coasting case.

...So in this case, assuming 100% regen, you are better of regening - you pick up a net of 20wh.
Assuming 85% efficency on the regen, the actual energy gain is 85*115 = 98 wh.
Net of the 95 wh to drive the remaining distance, you save 3 wh by regening vs coasting.

But, this savings is mostly due to driving a longer distance at 55 rather than 75.
You're getting where you want to go more slowly in the regen case so
you spend less energy which is what we probably all expect...

Not necessarily. Most people who posted were under the impression that coasting always wins. So I would say most people probably didn't expect that. And obviously the coasting case will win at lower speeds or higher altitude, so the answer is not obvious.

A fairer comparison would require imposing the constraint of equal times.

You may be right but it's a subjective discussion for another day. That's not the problem JohnQ, Todd, and myself are trying to solve. While the equal-time problem may be interesting to some people, I can't foresee a scenario at any speed where it would be beneficial to use regen.

 

[Todd Burch]

By the way, some enterprising mind could also model how the regen tapers off at lower speeds and then determine how things play out in those cases. I have to move on though . I suspect that below around 35 mph, the inefficiencies of power production and regen overcome the benefits and for those lower speeds, coasting always wins.

 

[brianman]

For legal purposes: THIS IS A THOUGHT EXPERIMENT; I'M NOT ASKING ANYONE TO TRY THIS.

Suppose you are coming down a steep cliff and you punch it to 130mph. At this point, you leave the accelerator pedal all the way on the floor. Suppose it's a 45 degree decline. What happens at this point? Does the car accelerate well past 130mph, or does the Model S engage regen as a safety measure or some-such?

 

[Jeff Miller]

Not necessarily. Most people who posted were under the impression that coasting always wins. So I would say most people probably didn't expect that. And obviously the coasting case will win at lower speeds or higher altitude, so the answer is not obvious.

I think that may be because many people think of the "regen case" as I did in my first calculations above: you only start regening when you need to in order to come to your final speed at the right place, rather than starting to regen way early (at the point where you would have to start coasting if you were going to coast; again, what's special about this point? Why not do the whole trip at 55 and use way less energy?). The "equal distance" approach seems a bit arbitrary. For example, in Todd's example, you would use exactly the same amount of energy if you drove the first 651m of your trip at 55, the remainder of your trip before regen at 75, and then regened just in time to get to 55 at the end of the ramp as you would if you drove 75 the whole way up until the point where you would start coasting if you were going to coast, regening at that point instead, and then driving on the remaining 651m at 55. So you can equally well say, "if you drive the first part of your trip slowly, at 55, then if you drive the rest at 75, then regen at the very end, you use a bit less energy than if you drive 75 the whole way and coast at the end". But that just seems like an odd comparison to make - if you drive part of the trip at a lower speed, you expect to use less energy, it's not directly related to coasting or regening per se.

I looked at the equal time case, seeing if coasting for a while then regening afterward could ever be done in the same total trip time as just coasting (assuming that when you finish regen you are at the right speed). The answer is you can't - adding regen always makes the trip time shorter, so is always less efficient (again assuming that you regen "just in time").

 

[Todd Burch]

For legal purposes: THIS IS A THOUGHT EXPERIMENT; I'M NOT ASKING ANYONE TO TRY THIS.

Suppose you are coming down a steep cliff and you punch it to 130mph. At this point, you leave the accelerator pedal all the way on the floor. Suppose it's a 45 degree decline. What happens at this point? Does the car accelerate well past 130mph, or does the Model S engage regen as a safety measure or some-such?

My guess is that regen goes to full. At 45 degrees (that's STEEP! The steepest road in the world is 19 degrees), that wouldn't be anywhere near enough, so I suspect at that point the parking brakes begin to engage or the car says "all yours buddy" and shifts to neutral .

- - - Updated - - -

I think that may be because many people think of the "regen case" as I did in my first calculations above: you only start regening when you need to in order to come to your final speed at the right place, rather than starting to regen way early (at the point where you would have to start coasting if you were going to coast; again, what's special about this point? Why not do the whole trip at 55 and use way less energy?). The "equal distance" approach seems a bit arbitrary. For example, in Todd's example, you would use exactly the same amount of energy if you drove the first 651m of your trip at 55, the remainder of your trip before regen at 75, and then regened just in time to get to 55 at the end of the ramp as you would if you drove 75 the whole way up until the point where you would start coasting if you were going to coast, regening at that point instead, and then driving on the remaining 651m at 55. So you can equally well say, "if you drive the first part of your trip slowly, at 55, then if you drive the rest at 75, then regen at the very end, you use a bit less energy than if you drive 75 the whole way and coast at the end". But that just seems like an odd comparison to make - if you drive part of the trip at a lower speed, you expect to use less energy, it's not directly related to coasting or regening per se.

I looked at the equal time case, seeing if coasting for a while then regening afterward could ever be done in the same total trip time as just coasting (assuming that when you finish regen you are at the right speed). The answer is you can't - adding regen always makes the trip time shorter, so is always less efficient (again assuming that you regen "just in time").

It's worthy of note that sometimes driving slower is dangerous. Like when you're on a 4-lane divided freeway (2 lanes each direction) where the speed limit is 70 MPH, there's an exit from the left lane, and all traffic is going 75 (exactly my case). Not only is driving slower in the left lane dangerous and illegal...I don't want to be THAT guy.

 

[hcsharp]

Interesting Result #3: The faster your final target speed, the larger the differential between your starting and final speed must be to get a regen benefit.

For example, if your target speed is 40 mph, you must start out going at least about 67.5 mph to get a benefit from regen. Any slower than 67.5 mph and it's better to coast. Any faster, and you're (much) better off using regen.
If your target speed is 55 mph, you must start out going at least 84 mph to get a benefit from regen.
If your target speed is 75 mph, you must start out going at least about 106 mph to get a benefit from regen.

This result surprises me. I can't think of any good reasons why this would be the case. I can see how a smaller spread would diminish the regen advantage, but I can't see why the spread would have to be greater for higher ending speeds.

Thanks Todd for your analysis.

 

[Todd Burch]

This result surprises me. I can't think of any good reasons why this would be the case. I can see how a smaller spread would diminish the regen advantage, but I can't see why the spread would have to be greater for higher ending speeds.

Thanks Todd for your analysis.

No problem . As for why the spread increases as the target speed increases, it has to do with the fact that after the regen is complete, you have to hold the target speed to get to the location where the coasting case reached the target speed.

For a given, say, 20 mph spread:

From 60-40 mph, you have to hold 40 MPH for 31 seconds after regen to get to the location where coasting reached the target speed.
For 95-75 mph, you have to hold 75 for 13.12 seconds. Holding 75 mph for 13.12 seconds requires much more energy than holding 40 MPH for 31 seconds (about half the time, but 3.5 times the drag force).

Also, since regen is limited to 60 kW max, the faster you go, the more insignificant of a difference the regen can make on your overall speed--so it has less impact on drag (60 kW of regen at 70 mph is much less deceleration than 60 kW at 45 mph).

 

[hcsharp]

No problem . As for why the spread increases as the target speed increases, it has to do with the fact that after the regen is complete, you have to hold the target speed to get to the location where the coasting case reached the target speed.

For a given, say, 20 mph spread:

From 60-40 mph, you have to hold 40 MPH for 31 seconds after regen to get to the location where coasting reached the target speed.
For 95-75 mph, you have to hold 75 for 13.12 seconds. Holding 75 mph for 13.12 seconds requires much more energy than holding 40 MPH for 31 seconds (about half the time, but 3.5 times the drag force).

Actually holding the final speed for longer gives the regen case a greater advantage. The fact that this period shortens at higher speeds is why it needs a bigger spread to make it equal out. Since the energy losses increase quadratically with speed, my intuition was telling me it didn't need as big of a spread to create a larger energy difference at higher speeds.

Also, since regen is limited to 60 kW max, the faster you go, the more insignificant of a difference the regen can make on your overall speed--so it has less impact on drag (60 kW of regen at 70 mph is much less deceleration than 60 kW at 45 mph).

I haven't looked at your spreadsheet yet but I suspect this is the biggest reason. This is partly why the post-regen holding time gets shorter. The coasting case decelerates considerably faster at higher speeds but the regen case decelerates only slightly faster. It shortens the time you can get a benefit out of driving slower. I look forward to looking at the spreadsheet.

I've learned a lot from this. I doubt if I'll change the way I'm driving as a result, but it's nice to know what's going on.

 

[simonog]

As (originally - a long time ago!) a physicist, this thread is fascinating. May I offer a simple way of looking at the title question?

the fundamental law which applies is the conservation of energy. In a world of no friction, no air resistance and 100% battery efficiency (ie no losses storing and retrieving electricity), it wouldn't matter what combination of regen or coasting you use - for a fixed speed of the car at the destination point (if stopping, it would have to be by regen as there's no friction for brakes to work).

of the factors, on a model S the rolling friction should be irrelevant for the discussion - with so few moving parts compared to an ICE the factors which matter are air resistance and battery efficiency.

the air resistance follows a square law ie it increases with the square of the speed. It also depends on altitude, the air being thinner higher up.

Battery efficiency depends on temperature, rates of storing and retrieving, chemistry and other factors which include the dynamo (regen unit) efficiency and the motor efficiency: one unit of electricity taken from the battery (loss) and fed to the motor produces motion (and loss, mainly heat), and recovered by the dynamo (more loss) and returned to the battery (and still more loss) will be materially reduced.

I would therefore suggest it's a simple trade off of electrical system efficiency (battery etc) versus wind resistance. Any suggestions on the easiest way to take some practical measurements to let us see what the real world graph of air resistance against speed and battery / electrical system overall efficiency against rate of retrieval/ recovery of energy look like?

 

[djp]

Any suggestions on the easiest way to take some practical measurements to let us see what the real world graph of air resistance against speed and battery / electrical system overall efficiency against rate of retrieval/ recovery of energy look like?

This is a good read, from back in the day when Tesla had a technical blog (complete with downloadable Excel files).

Roadster Efficiency and Range | Blog | Tesla Motors

 

[stevezzzz]

> Fourteen miles descending at 65 mph takes a little over 13 minutes. [stevezzzz]

Mucho cojones!! Too mucho for me anyway, esp knowing you are in CC. I do it in 'variable regen' because I'm always in flux. Besides for years I only did this descent in 'diesel + trailer load' mode - in 2nd gear over in the right lane with the semis. Using brakes!: diesels have no throttle therefore NO manifold vacuum, NO engine braking. You are in COAST going down steep hills (back to topic).
--

No cojones required, wycolo, and no braking, either: with a 6% grade the S uses much less than 60kW of regen to hold 65mph. So I just set the cruise control, stay in the middle lane and enjoy the scenery. People whiz by me in the left lane with their brake lights flashing intermittently and big rigs occupy the right lane going 25 in low gear; but all is serene in the middle.

- - - Updated - - -

SOLUTION

OK guys, I spent a few hours this afternoon calculating this out using numeric methods. My time deltas in calculating these numbers were anywhere from about 0.01s to 0.05s, so the numbers should be reasonably good. (Note--I've already shown very good agreement with analytical solutions for the coasting case, discussed above).
.
.
.
Here are some numbers. Note in all cases than for coasting, the energy change in the battery is zero--so if regen gives a gain of energy as listed below, it's better than coasting.
65 to 55 MPH: Regen LOSES about 30,000 Joules of energy.
85 to 55 MPH: Regen GAINS about 8,000 Joules of energy.
105 to 55 MPH: Regen GAINS about 150,000 Joules of energy.

50 to 40 MPH: Regen LOSES about 24,000 Joules of energy.
60 to 40 MPH: Regen LOSES about 22,500 Joules of energy.
70 to 40 MPH: Regen GAINS about 10,000 Joules of energy.
90 to 40 MPH: Regen GAINS about 155,000 Joules of energy (about 43 Wh, or about 0.15 mi of range).

Here's my spreadsheet for those who want to fiddle:
http://www.sendspace.com/file/r4l4t7

OK, there ya go guys. Don't know if this makes me a hero, a nerd, or both . Anyway, gotta get back to being a father. Question answered!

You're my nerdly hero, Todd! Thanks for the analysis; I especially like it when answers to interesting questions are counterintuitive.

 

[Todd Burch]

Actually holding the final speed for longer gives the regen case a greater advantage. The fact that this period shortens at higher speeds is why it needs a bigger spread to make it equal out. Since the energy losses increase quadratically with speed, my intuition was telling me it didn't need as big of a spread to create a larger energy difference at higher speeds.

I agree, and your statement does not contradict what I said (at least I don't think!). My point was that at higher target speeds, the speed you have to hold is higher, and therefore the energy usage increases significantly.

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of the factors, on a model S the rolling friction should be irrelevant for the discussion - with so few moving parts compared to an ICE the factors which matter are air resistance and battery efficiency.

I originally thought so as well--however rolling resistance and drivetrain losses make up for about half(!) of the force that the motor must counteract to maintain 55 mph! I thought this was nuts, but when you consider the Model S's low drag coefficient, high mass, and sticky tires, it begins to make sense!

I would therefore suggest it's a simple trade off of electrical system efficiency (battery etc) versus wind resistance. Any suggestions on the easiest way to take some practical measurements to let us see what the real world graph of air resistance against speed and battery / electrical system overall efficiency against rate of retrieval/ recovery of energy look like?

I was also thinking this, but soon after plugging in my numbers and determining that at many speeds drag isn't as significant as I thought, things got more complicated.

As for getting actual numbers, I think it would be tricky given the resolution of the data on the Model S readout. We're not talking about more than about a tenth of a mile savings in range, unless you have very dramatic drops in speed (100 mph to 20 mph, for instance) or you do it over and over again.