# Forums

## Optimal speed

## Optimal speed

Submitted by Teslamodels4me on April 25, 2012

Does anyone know the optimal speed to travel 1000 miles? Assume that recharging is done with a supercharger.

Also use the numbers:

45 mph = 378 miles

50 mph = 350 miles

55 mph = 321 miles

60 mph = 295 miles

65 mph = 271 miles

Assume that all road conditions are perfect, no others cars on the road, and that there is a working supercharger available with no awaiting.

These are numbers I got from a computer at the Tesla store.

If you have numbers for high speeds please add those numbers.

The question is at which speed will take the car go 1000 miles in the least amount of time?

BYT| April 25, 2012The slower you go, the more miles you can get out of the battery, they less often you need to recharge and since charging can take at least 30 minutes on a 300 mile pack, then the mph becomes less of a factor between 45 mph and 65 mph. That extra 20 miles per hour cost you 100 miles from your battery. 20 miles vs. over a quarter of your entire battery pack?

mikeadams| April 25, 2012For the first two options 45 and 50mph you would have to charge up twice on the road. For the options 55, 60 and 65mph you would have to charge up three times. If you have to charge twice, then you might as well go 55mp or if you have to charge three times then you might as well go 65mph. So the best option is one of those two.

So now lets look at those two options. If you could go 50mph continuously, it would take 20hours. So you just need to add your two charging times to that (about 45min? x2 or 1.5hr) for a total of 21.5hrs. If you could go 65mph continuously, it would take 15h 23min. If you add three charging times to that (about 2h 15min) you get 17hr 37min. So under ideal conditions, the faster speed still gets you there faster. However there may still be a certain point somewhere over 65mph that due to the increased drag and additional recharges it would cause it to take longer to get there.

jerry3| April 25, 2012And it's possible you might be able to use the terrain to go even further on a charge even though the average speed would be the same.

Brian H| April 25, 2012jerry3;

I don't think so. Unless you creep uphill, and speed downhill. ;)

Kinda reminds me of the old story of a spinster driving her Model A at a breakneck 30mph down the main drag. When pulled over, she explained she was almost out of gas so was hurrying to get to the gas pump before she ran out ...

?:/

Brian H| April 25, 2012I can see that when the first Ses are delivered, there's going to have to be a thread for hypermilers! :D

mikeadams| April 25, 2012Small correction... where I said '55mp' in the first paragraph, I meant 50mph. 50mph since it was the faster of the 'two recharge' options.

jerry3| April 25, 2012Works well enough in my 2004 Prius:-)

DATE__________ODO____INC_____AVG

--- Trip to NE starts here (warm weather)

08/13/10____111690____625____59.8 (3.9)

08/14/10____112308____618____60.0 (3.9)

08/20/10____112972____663____64.2 (3.7)

08/22/10____113411____438____58.9 (4.0)

08/31/10____113922____510____61.8 (3.8)

--- Trip to NE ends here

--- Trip to NE starts here (cold weather)

01/07/12____128603____481____56.6 (4.2)

-- 13 F here

01/12/12____129042____438____52.7 (4.5)

01/15/12____129420____378____50.3 (4.7)

01/20/12____129094____481____56.2 (4.2)

--- Trip to NE ends here

JohnEC| April 25, 2012I’ve thought about this a bit, but I keep coming up with the same problem. I don’t think I’ll be able control myself well enough to go slower even if it gets me there sooner. It’ll just be too much fun to go fast. I think I’ll just have to resign myself to more stops for charging (and highway patrol!).

Teslamodels4me| April 25, 2012Write.mikeadams thanks for the info, so far 65 mph is the optimal speed does anyone have numbers for speeds over 65 mph?

cerjor| April 25, 2012I discussed this in the thread "Time for a long distance trip." My conclusion was that for a 1600 mile trip, it made little difference whether you drive at 50, 60, or 70 mph. The total time for the trip was about the same. At 70 you spend less time driving and more time charging. At 50 it's just the opposite. Look at that thread for the details.

jbunn| April 26, 2012Power needed to push the car increaces as the cube of its velocity. I'm not convinced of the free lunch

Volker.Berlin| April 26, 2012I discussed this in the thread "Time for a long distance trip." (cerjor)

Here is the link:

http://www.teslamotors.com/forum/forums/time-long-distance-trip

Timo| April 26, 2012I made some time ago same calculation as cerjor, and was quite surprised how little the speed actually means when you need to use about a hour to recharge fully. The time saved with faster speed is countered by additional time needed to recharge and vise versa.

ggr| April 26, 2012Square of wind velocity, not cube.

BruceR| April 26, 2012ggr has it correct. Thank you for pointing that out for those who may not be familiar with the formulas. Air drag = 1/2 * p * V^2 * A * Cd......

BruceR| April 26, 2012While we really don't know the exact numbers for the Model S we have some good aproximations. If we use a power form fit for the data given for range and assume we can get 80% of the range in a 45 minute supercharge we can then make some estimates.

MPH-Hours to travel 1000miles-Avg MPH

40-26.50-37.7

45-24.77-40.4

50-22.25-44.9

55-20.43-48.9

60-18.92-52.9

65-18.38-54.4

70-17.29-57.9

75-16.33-61.2

80-17.00-58.8

Now if those superchargers are not available in exactly the spot you need them or if you have to rely on "standard" chargers, this trip is going to take you a whole lot longer .....

So somewhere around 70 to 75 is the predicted minimum time. This is a good thing because as JohnEC pointed out, we'll be traveling that fast anyway! ;-)

Leofingal| April 26, 2012My guess is that if you exclude supercharging (say you didn't get an S that supports supercharging), you may find yourself taking the slower, windier (not like breeze, but curvy) trips through the non expressways to get places. This might be pretty refreshing actually. This will extend your usable range, and enable you to find new and interesting places to drive. You could plan your trips around interesting stops/restaurants off the beaten path, and discover a whole new world around your normal routes. I think this might be fun.

Brian H| April 26, 2012BruceR;

Here's your table using the <pre> tag:

MPH Hours to travel Avg MPH

1000miles

40 26.50 37.7

45 24.77 40.4

50 22.25 44.9

55 20.43 48.9

60 18.92 52.9

65 18.38 54.4

70 17.29 57.9

75 16.33 61.2

80 17.00 58.8

"Pre"serves your spaces; don't use tabs, tho'.

EdG| April 27, 2012Here's my graph (if posted correctly) of distance vs. time for some continuous driving.

Assumptions: 85kWh battery, starts fully charged; simulation granularity is 15 minutes; when battery gets low, there's a fast charger right there; you spend 3/4 hour charging at 90kW.

I did a best fit on the curve shown at http://www.teslamotors.com/goelectric/efficiency and am using the formula: kW used = (.0254 R*R + .7 R + 103) where R is the rate of speed in MPH.

To get there the fastest, go faster. Unless you're in the cross-over areas.

BYT| April 27, 2012That's what I get for guesstimating it rather then actually crunching numbers. I also figured for much longer charge times.

foto| April 27, 2012@ EdG, I assume that the plateau on each line (avg speed) is the charging period. Would you actually be able to go a little over 300 miles at 70mph before charging? Have I misread the graph?

Larry Chanin| April 27, 2012EdG,

Very interesting graph, but I'm a little confused. The original poster showed that at 55 mph the Model S has a range of 321 miles. (I assume that is with aero wheels.) Your graph seems to show a range of over 300 miles without a charge at 70 mph. The Roadster, upon which your data is based, can't achieve that sort of range either at that speed.

Am I misinterpreting what the graph is displaying?

Thanks.

Larry

EdG| April 27, 2012Good catch. I'm going to reprogram the simulation to stop for recharging at some reasonable number other than 0. I'll have it stop when getting below 10 kWh and re-post.

If anyone else sees an issue here, please let me know - I'd rather fix them all at once!

EdG| April 27, 2012Stop whenever below 10kWh

jerry3| April 27, 2012I'm missing something, because if I'm reading the chart correctly, you're saying that you can go about 350 miles at 60 mph before the first charge. That would be nice, but it's higher than the 300 miles at 55 mph (or even 320 with aero wheels).

Volker.Berlin| April 27, 2012Also, AFAIK 30-45 min of charging on the supercharger gains around 50% (i.e., ca. 40 kWh) on the 300-mile-battery. Charging to full 100% takes considerably longer than just twice that.

EdG| April 27, 2012My charging rate is likely off. I assumed the full wattage is added, without loss, to the battery. So the charging rate is probably too fast. I picked 45 minutes of charging because Elon said it would charge to about 80% in that much time. My simulation shows close to that, so either there is little loss (??) or the wattage of the high speed charging unit is rated by results, not the source (????). I don't know. I just used the numbers I have. On the optimistic side, it's also possible that the car will really perform to this range, but the various numerical claims made have been very conservative.

Any tweaks to these assumptions that sound closer to reality can be added.

After said tweaks, I can do the same for a high speed 60kWh battery and lower speed charging for all sizes, if desired.

jerry3| April 27, 2012In many ways it sort of doesn't matter because a 500-700 mile trip basically takes the entire day. The main difference is that you start earlier if it's going to take 14 hours. And having a forced stop for an hour every 250-300 miles is probably something that will really improve highway safety.

Crow| April 27, 2012I thought using the supercharger more than once per day would unnecessarily degrade the battery. I read somewhere that TM recommends not using it more than that.

flar| April 27, 2012Drag increases as the square of velocity, but the higher velocity requires the work to be done faster so the power increases as the cube of velocity. So, both statements are correct depending on the question you are asking.

From http://en.wikipedia.org/wiki/Drag_(physics)#Power

Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Exerting four times the force over a fixed distance produces four times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, four times the work done in half the time requires eight times the power.

So, power does increase with the cube of velocity.

But, I think we are measuring charge consumed per mile rather than per time period. So I think the charge consumed per mile goes up with the square of the velocity and only when you look at the time it took do you see the cubed value for the power requirement (in other words, the engine has to have cubically higher power to achieve that speed against the wind, but its consumption per mile is only quadratically higher). Or, is my math off?

flar| April 27, 2012So much for my attempt to use the quoting tags. The third paragraph that started with "Note that" and ended with "eight times the power" was from the wikipedia article. All other paragraphs were mine...

Brian H| April 28, 2012EdG;

You're missing the point. It's not the charging time that's the problem, it's the range. You've got impossible ranges for the 55MPH+ speeds. E.g.: at 70mph you'll only get about 250 miles per charge, even draining the battery fully, IIRC, not 300. IOW, you're going to need many more charging stops.

Per the ranges specified in the OP, you have each speed band going about as far as the next slower one actually does. So your 70mph line is actually 60mph, etc.

EdG| April 28, 2012I got the point. My point is I'm using the graph given by TM for energy used per mile. And I'm using the wattage given (in full, no losses) for the supercharger. And I'm assuming an 85kWh battery that loses exactly the amount of energy the graph says the car is using.

The rest is calculated, not fantasized.

So, the question backs off to: Where is the simulation wrong? Is that curve for some other car? Does it assume you're going downhill, or that there's no wind or hills? Or, as I said, is it possible that this simulation is real and the numbers you quote are to ensure everyone is happy when they get their cars?

I'm thinking that the battery charging should have some percentage loss and that the energy used in the graph may be the energy output of the motor, not the energy loss to the battery. If that's the case, what assumptions should I make on those losses?

I could just fiddle with the simulation until I get the numbers you quoted, and then chalk it up to discharge and charging losses. Should I do that?

Crow| April 28, 2012I think where the analysis goesvwrong is that you should Supercharge once per 24 hours only. The second charge stop should overnight, approximately 10 hours. 40 mph is much faster over a 24 hour period by a big margin.

digitaltim| April 28, 2012I'm just interested in the real world test of my one way 200 mile, 2.5 hour commute from Baltimore to NJ/NYC...guess I'll be driving at a more reasonable (responsible) pace to save some juice...

jbunn| April 28, 2012Ed,

The simulation might be wrong regarding charge depth, and time to full charge. Looks like your assuming an hour to charge, that might be too short. Changing the sim from 60 to 90 for a full charge changes the simulation.

ddruz| April 28, 2012Having had the opportunity to play extensively with the range simulator at Santana Row last week, remember that climate control + outside temperature can significantly affect driving range. Windows up or down also make a difference as does day or night driving. Speaking from personal experience in driving a Leaf the last 7 months I can attest that climate control is often as critical to range as speed in an EV. Hills are not far behind. Variables in addition to optimal speed and charging times may figure more prominently than expected. It's really a very interesting problem.

jerry3| April 28, 2012Certainly using the climate control properly makes a big difference in the Prius and I don't expect it to be any different in the Model S--especially on trips.

Brian H| April 29, 2012EdG;

The point remains your graph shows 300mi at 70mph (e.g.) Not possible.

Brian H| April 29, 2012@EdG;

" kW used = (.0254 R*R + .7 R + 103) where R is the rate of speed in MPH. "

kW is a power measure. If R = 70, then the formula generates the answer 276.46kW . You must mean kWh for the whole 1000mile trip. So that means 276Wh/mi, or 307.5 mi. running down the battery from 85kWh to 0. But I don't know if you can do that even with Range mode, can you? Certainly not on a routine basis ...

Brian H| April 29, 2012Hm. I see you're actually doing a time (12hrs) comparison, and have the second 70mph leg at about 260mi, which is an 85% charge run down to 0. The last leg is much shorter, about 170 mi.

So working backwards from the total mileage, your 730 miles would also break down as, say, 250+240+240, which is do-able with about 80% charges.

Similarly, the 80mph plot shows 3 charges, about 790 miles, and perhaps 2/3 of the previous charge remaining at the end. The formula says that's a total of 321.56kWh, or ~322Wh/mi. Using 80% full charges (68kWh), that gets you 211mi/charge, which is plenty to accommodate 4 legs totalling ~790 miles.

So your analyses work out, as long as you don't mind ending with a fully depleted battery, even accepting an 80% charge ceiling. The extra stop at 80mph is more than made up for by reduced road time in this (or any sufficiently long trip) scenario.

EdG| April 29, 2012I restricted the plot to 12 hours. I was not trying to calculate an optimal speed and recharging plan for a 12 hour trip. Normally, I expect one would plot a trip for distance and determine how many recharging stops would be feasible given the location of chargers, how to limit speed to make it to each, and overall time for the trip.

The formula was one to match the figure on the Tesla page, and I erred. As in the original Tesla plot, it's in watts, not kilowatts. kW used = (.0254 R*R + .7 R + 103)/1000. Good catch.

The value of the plot is simply to try to determine whether it's faster to go faster or whether you get there faster by reducing your speed. The way I read it, if the assumptions were perfect (not), is that except for those cross-over areas, it's faster to go faster.

As for the unreasonably good performance shown in the plots, let me quote from http://www.teslamotors.com/forum/forums/300-mile-range-based-55mph#new

Brian H | April 29, 2012 newDallas;

Supposedly, the beta "real-world" users were finding the range estimates were conservative; they were doing up to 15% better.

EdG| April 29, 2012Sorry... The italics representing a quote was supposed to go to the end of the post.

Teslamodels4me| April 29, 2012@EdG

Where did you find the information on the 15% boost in range?

Timo| April 30, 2012EdG, could you do that same graph based on Roadster efficiency chart in blog http://www.teslamotors.com/blog/roadster-efficiency-and-range?

That's really close to real-world figures and you get that in excel files (there is a link at the bottom of the blog entry).

IIRC I used that when I calculated basically that same and my results were less good for going faster, though difference was small (and IIRC I also got the result that faster was in fact faster with 80% charging in about a hour with supercharger).

EdG| April 30, 2012@Timo: Note that the power used versus speed curve on the page you reference starts out at near (0,0). The curve at the page http://www.teslamotors.com/goelectric/efficiency shows non-zero power usage at near zero speed for all but "AERO" and "HVAC".

[All other power usage is fairly constant except for transmission losses, which dissipate. I don't know why HVAC tapers to zero as speed decreases while other things remain constant. I would think HVAC would be a strong non-zero while standing still.]

So, because your Roadster curve is near zero at tiny speeds while the other curve is not, I'm not sure we're comparing similar situations.

Though not necessary for my simulation, I did a curve fit on the spreadsheet from the Roadster page (https://www.teslamotors.com/display_data.php?data_name=range_table) to get a polynomial.

Interestingly, I needed a cubic to get a near match, where a quadratic was good for the simulation I posted earlier:

kW (yes, kW) = .5531 + R*.07835 + R*R*.0007955 + R*R*R*.00003331 (where R is in MPH).

So before posting a graph which is so pretty that people are convinced of its veracity, I'm looking for comments on the difference between the two data sets so we get a better handle on things.

Also, I'm leaning toward using posting of 95% battery charging efficiency (referencing your earlier post regarding regen) (do I need to drop multiply by 95% for PEM, too? Is that used in supercharging?) so the 90kW input is lowered to 85kW or 81kW, respectively. Help?

EdG| April 30, 2012Well, I should have read further into BrianH's comment. I think I may have confused some units along the way. wH/mile versus watts or something like that...

So, until I get everything straightened out, with units and calculations checked, stay tuned...

nickjhowe| April 30, 2012Has/can anyone guesstimated the range at 75-80mph, assuming full charge at start and 10% at end? I just did a 250 mile run up the turnpike in my Range Rover and would love to know how much slower (if at all) I'd need to go in the "300 mile" Model S.

ddruz| May 1, 2012nickjhowe, Will you be using climate control? Are your windows up or down? What is the outside temperature? Are you driving during the day or at night? Are you driving on flat ground or up and/or down hills? Though at 75-80 mph speed may be the largest determining factor every one of these other variables will count as well. They are not insignificant. I speak as a Leaf driver who has real world experience learning to maximize mileage in an EV. I'd expect Roadster owners would also concur that these other factors are things to be considered. It's truly an interesting mix of variable to consider if you are trying to maximize speed for a given distance you wish to drive.

EdG| May 1, 2012Re: previous plot

What I had been doing was multiplying the number of miles traveled per 1/4 hour by the number of kWh/mile as calculated by my formula (which matches the plot on the TM page) to get the number of kWh used for that time period. I subtracted that from the amount left in the battery.

Example: For 40 mph traveling starting at a full 85kWh charge, the charge after 15 minutes is ( 85 kWh - 10 miles * .171 kWh/mile ) = 83.3 kWh remaining.

When the battery charge reads less than 8 kWh on any 1/4 hour interval, my simulator stops the car and charges it for 3/4 hour. Using a 90kW charger adds 22.5 kWh per 1/4 hour. Once the 3/4 hour is up, the car speeds off again. Under these assumptions, the car will leave the charger with about 75kWh.

[Note: The previous graph assumed no charging losses - my next graph will use 95% * 90/4 for power added to the battery, but that only changes the plot above the first charge point.]

Perhaps the plot at http://www.teslamotors.com/goelectric/efficiency is too theoretical or optimistic? Note that it suggests that at 55 mph the S will use about 220 Wh/mile, or 1/.220 = ~ 4.5 miles per kWh.

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