"Mini-max" problem

[mbender]

Ok, this is a little academic, but it can and has (for me, and maybe others) had "real life" applications.

Let's suppose you need to make a 180-mile (one-way) trip, and there are Level-2 chargers at 10-mile intervals all the way to the destination. Because there are side roads paralleling a highway, let's say you can drive at any speed, along any part of the route, from 25-75mph.

The question is, what should your (one's) driving strategy be to get to the destination in the least amount of time? Assume uniform driving conditions throughout the route and identical chargers.

[ Edited original route distance from 120 to 180 after first reply. ]

 

[Nubo]

Assuming a full-capacity battery, drive at a steady 35 mph and avoid charging altogether.

 

[mbender]

Assuming a full-capacity battery, drive at a steady 35 mph and avoid charging altogether.

That's 3:25:43 according to my calculations. Can anyone do better?

But that sort of avoids the crux of the problem (I thought 120 was beyond true max range at any speed). So I'm going to edit the original to be 180 instead...

 

[iamwjh]

Assuming a full-capacity battery, drive at a steady 35 mph and avoid charging altogether.

EXACTLY, simple math, not that academic if you can read a table. http://bit.ly/leafrangechart

Drive 35mph and make it there in about 3.5 hours. Or drive at 70 mph, stop and charge and make the trip in about 5.4 hours.

Again, not that academic, just ironically opposed to the common ICE response of maximize speed to minimize time, which unfortunately does not account for the EV concepts of energy efficiency and charge time.

 

[iamwjh]

Assuming a full-capacity battery, drive at a steady 35 mph and avoid charging altogether.

That's 3:25:43 according to my calculations. Can anyone do better?

But that sort of avoids the crux of the problem (I thought 120 was beyond true max range at any speed). So I'm going to edit the original to be 180 instead...

I don't like loaded questions and games of changing the questions. If you already know the fricken answer, then WTF are you asking the question?????

It's nice to work together and help each other with stuff, but I can't stand playing childish games....

So, I'm going to edit my answer to "IDFC."

 

[smkettner]

180 miles I would take the Subaru

If I had to do it in the LEAF I would set the cruise at 30/35 and stop after getting first LBW.
Charge to lower of 80/90% or 5 miles beyond destination.
Assuming charging at destination.
Good luck on calculating time as I really do not know the range.

Great idea for a car rally/race if a stretch of road could be found with 19 charging stations 10 miles apart.
Cars with similar capacity bars would be in separate catagories.

 

[DNAinaGoodWay]

Easy, take the Tesla, do a gentle 60 avg and it's 3 hours.

 

[mbender]

loaded questions [yada-yada] questions. [yada-] fricken [yada...] WTF are you asking the question?????

[...] can't stand playing childish games....

[...] "IDFC."

Excellent! Go play in a different sandbox then. No one is stopping you and we don't FC either!

And for the record, I don't know the answer to the general (and thus "academic") question. I do know now though by looking at the chart that at the very least, driving 40mph would get you to the original destination 25 minutes quicker. So 35mph was not correct regardless.

 

[apvbguy]

Easy, take the Tesla, do a gentle 60 avg and it's 3 hours.

best answer!!
until this type of range can be offered at a price the average person can afford the EV will remain a niche vehicle, hopefully as the technology gets better prices will come down and then the EV could one day become the mainstream means of transportation

 

[Nubo]

loaded questions [yada-yada] questions. [yada-] fricken [yada...] WTF are you asking the question?????

[...] can't stand playing childish games....

[...] "IDFC."

Excellent! Go play in a different sandbox then. No one is stopping you and we don't FC either!

And for the record, I don't know the answer to the general (and thus "academic") question. I do know now though by looking at the chart that at the very least, driving 40mph would get you to the original destination 25 minutes quicker. So 35mph was not correct regardless.

I added a bit of cushion for the real world.

Good luck.

 

[iamwjh]

why is anyone surprised that after electing clowns to hold office that a circus has broken out?

EXCELLENT !

 

[DaveEV]

How fast can you charge? Are the stations 208 or 240V and is your LEAF 16 or 30A capable?

Anyway, here are some calcs making the following assumptions:

* 208V (commercial charging tends to be), let's just compare 16A/30A (3.3 kW, 6.2 kW).
* efficiency listed in the range chart for given speeds is correct
* Add data points for 25 / 30 mpg which where we will assume 6.8 mi/kWh for 30 mph and 7.3 mi/kWh for 25 mph.
* Assume 85% charging efficiency, 16A charging = 2.8 kW, 30A charging = 5.3 kW
* Assume 21 kWh usable energy in pack
* All numbers are rounded down

16A charging:
25 mph: 7.3 mi/kWh, 7.2 hr drive, 153 mi / charge, 20 mi/h charge rate, 27 mi charging, 1.4 hr charging = 8.6 hours
30 mph: 6.8 mi/kWh, 6.0 hr drive, 142 mi / charge, 19 mi/h charge rate, 38 mi charging, 2 hr charging = 8.0 hours
35 mph: 6.3 mi/kWh, 5.1 hr drive, 132 mi / charge, 18 mi/h charge rate, 48 mi charging, 2.7 hr charging = 7.8 hours
40 mph: 5.9 mi/kWh, 4.5 hr drive, 123 mi / charge, 16 mi/h charge rate, 57 mi charging, 3.5 hr charging = 8.0 hours
45 mph: 5.2 mi/kWh, 4.0 hr drive, 109 mi / charge, 14 mi/h charge rate, 71 mi charging, 5.0 hr charging = 9.0 hours

Looks like 35 mph is the fastest for this distance and charge rate.

30A charging:
30 mph: 6.8 mi/kWh, 6.0 hr drive, 142 mi / charge, 36 mi/h charge rate, 38 mi charging, 1.0 hr charging = 7.0 hours
35 mph: 6.3 mi/kWh, 5.1 hr drive, 132 mi / charge, 33 mi/h charge rate, 48 mi charging, 1.4 hr charging = 6.5 hours
40 mph: 5.9 mi/kWh, 4.5 hr drive, 123 mi / charge, 31 mi/h charge rate, 57 mi charging, 1.8 hr charging = 6.3 hours
45 mph: 5.2 mi/kWh, 4.0 hr drive, 109 mi / charge, 27 mi/h charge rate, 71 mi charging, 2.6 hr charging = 6.6 hours
50 mph: 4.6 mi/kWh, 3.6 hr drive, 96 mi / charge, 24 mi/h charge rate, 84 mi charging, 3.5 hr charging = 7.1 hours

Looks like 40 mph is the fastest for this distance and charge rate.

Faster charging lets you get there a bit faster, but +- 10 mph around the sweet spot doesn't make a huge difference in overall time.

 

[mbender]

Anyway, here are some calcs making the following assumptions:
[...]
Faster charging lets you get there a bit faster, but +- 10 mph around the sweet spot doesn't make a huge difference in overall time.

Thanks for the detailed analysis and calculations, drees. I learned from it and it made me look deeper into the general case.

And from what I can tell, it really is a classic "minimax" problem where the ideal velocity ends up being dependent solely on the Wattage of the charger (and notably independent of the number of miles in the trip). I also assume:

• a fixed inverse relationship between speed (v) and energy consumption in m/kWh (say r), which is roughly true: I used v*r=230,
• and (to make it independent of original charge and total miles), I now assume that the batteries are empty to begin.

So if I'm doing it right, here are the governing equations, where N = trip miles, which drops out, and W=charging rate in kW:

Total Time TT(v) = Charging time T1(v) + Driving Time T2(v)
TT(v) = N / (W*230/v) + N / v, or
TT(v) = (N/W*230)*v + N/v, then
TT'(v) = N/W*230 - N/v^2 (calculus! and set = 0 to find the v which minimizes TT)

and you get... v = sqrt(230*W), which yields the following "table":

Charging rate (W)
=> Optimal speed (v)
3.0kW => ~26mph
6.0kW => ~37mph
10kW => ~48mph
15kW => ~58mph
20kW => ~68mph
30kW => ~83mph
40kW => ~96mph
Quick chargers => as fast as your LEAF (and law enforcement) will let you go!

The only thing of note or a little counter-intuitive is that if you must charge at 3kW or less, a lower speed (~25mph or less) will actually reduce your total time. But this implication may be exaggerated by the v*r=230 assumption. Otherwise, between 35 and 40 is your optimum speed with 6kW Level II chargers, and pedal-to-the-metal if you can quick-charge