Time Optimized Traveling Speed: the Final Graph

[SageBrush]

Someone periodically asks this question. The answer used to be "ymmv depending on charge rate," but in the Supercharger era the answer is "just a mite slower than a speeding ticket." Even so, I've been tempted to graph the relationship for my own amusement even after Saghost rightly pointed out that it boils down to a minima problem easily solved with calculus.

The spreadsheet.

The graph:

 

[xyeahtony]

the majority of model 3 owners are american. can you show us a mph equivalent chart

 

[ilg]

the majority of model 3 owners are american. can you show us a mph equivalent chart

Just replace the labels on the X-axis with 62, 74.4, 86.8, 99.2, and 111.6 Good enough?

(posted by an arithmetic-aware American -- it's really not hard)

 

[pandam3]

Just replace the labels on the X-axis with 62, 74.4, 86.8, 99.2, and 111.6 Good enough?

(posted by an arithmetic-aware American -- it's really not hard)

Yeah us ‘Muricans are a lazy bunch... any way to stick this in the graph to visually see it?

Also so what does the graph mean? What is the final optimum speed to travel at? I don’t read graphs and charts anymore...

 

[SageBrush]

the majority of model 3 owners are american. can you show us a mph equivalent chart

100 kph = 62 mph

Or let me help you.

 

[SageBrush]

Also so what does the graph mean?

Well, here is one application:
If you know your average charging speed, the graph tells you the speed to travel to minimize travel time.

Or you could interpret the other way:
At a set travel speed the graph tells you the minimum charging rate you need so that a slower travel speed does not save you travel time.

 

[SageBrush]

I should have added minor grid lines to the graph ...

 

[xyeahtony]

All the passive aggressive replies to my request, anyone can google. kind of defeats the purpose of using a graph to quickly display information. Didn't realize my response was so outrageous, seeing how everyone's odometers are in MPH, all street signs are in MPH, the cars range is rated in MILES, but hey whatever. Someone asks for help interpreting information and it brings out the most *****bag reply, lmgtfy.

 

[Arpe]

I do not understand the numbers in "Aero Wh/km at sea level".

The Wh/km seems way too low at the start, at 90 kph i would guess on 120-130 Wh/km. On your tables it says 57.2 Wh/km at 90 kph.

What does the column represent?

 

[TJtv]

Instead of labeling the x-axis:KPH and y-axis:KW, they should be labeled x-axis: Driving speed(KPH) and y-axis:Charge Rate(kW).

Knowing that the y-axis is Charge Rate is very important to understanding what this graph is trying to say. And it's not at all obvious that's what you're plotting with only labeling it with KW.

 

[lklundin]

Someone periodically asks this question

Yes, for a fleet manager an accurate answer to this question can make or break the company - especially if the fleet is one of Tesla Semis.

So thanks for addressing this topic in a systematic way.

In the spreadsheet, where do the numbers in column C ('Aero Wh/km at sea level') come from?
And do I read that column correctly, that e.g. at sea level at a speed of 140 km/h the consumption is 138.5 Wh/km (i.e. 7.22 km/kWh) ?

In that case the values seem quite optimistic when compared to evtripplanner.com/, where a LR Model 3 at that speed at a outside temperature of 22.2C and an inside temperature of 21.1C and a payload of 400 kg (and an altitude somewhere between 0.0 - 0.4 km) has a consumption around 200 Wh/km (i.e. 5km/kWh).

 

[SageBrush]

I do not understand the numbers in "Aero Wh/km at sea level".
What does the column represent?

And do I read that column correctly, that e.g. at sea level at a speed of 140 km/h the consumption is 138.5 Wh/km (i.e. 7.22 km/kWh) ?

Just the Aero fraction of the total consumption.
Aero force is 0.5 * rho * Cd * A * V * V
Newtons / 3.6 = Wh/Km

0.5 * Rho * Cd * A / 3.6 = 0.5 * 1.225 * 0.23 * 2.34 / 3.6 = 0.09156875
The C column function then is 0.09156875 * v * v

Do you see an error here ?

Left unsaid, but perhaps I should clarify that I was only interested in the additional energy/Km the car consumes as the speed increases. I presumed that all forces and fixed power consumptions stay the same except the Aero drag.

 

[SageBrush]

Consolidated post

 

[SageBrush]

Instead of labeling the x-axis:KPH and y-axis:KW, they should be labeled x-axis: Driving speed(KPH) and y-axis:Charge Rate(kW).

Good thought, thanks. Alas, editing is not allowed one hour after posting.

 

[Daniel in SD]

The plot is sort of silly (but cool!). Basically it says you can drive as fast as you want if you're supercharging. Most public chargers are around 6kw so if you're not supercharging (why???) you should drive as slow as you safely can, probably 55mph.

 

[SageBrush]

Basically it says you can drive as fast as you want if you're supercharging.

Unfortunately, it takes *way* too much time for cops to hand out speeding tickets. ;-)

Our cars have a 48 Amp OBC, so we might find ourselves charge hopping at say 11.5 kW. A bit surprising to me, optimized traveling speed even at that very good L2 rate is only 96 kph.

 

[SageBrush]

Tangential to this thread and forum, I'll probably get around to a plot for the 2018 Nissan LEAF.
It has a much higher CdA and in hot weather DC charging on trips rather quickly gets capped at 22 kW (for those curious, Google "rapidgate.")

The graph may have practical application for those car owners.
Addendum: Google says Cd is 0.29. I did not find frontal area for the 2018 LEAF but it is mostly a re-skinned prior generation that is 2.45 m*m. So the relative CdA of the 2018 LEAF is (2.45/2.34)*(29/24) = 1.265x that of the Model 3. Normalized to the Model 3 graph above a 22 kW charge rate would be read at the 22/1.265 = 17.4 kW data point which is between 105 and 110 kph.

 

[SageBrush]

0.5 * Rho * Cd * A / 3.6 = 0.5 * 1.225 * 0.23 * 2.34 / 3.6 = 0.09156875
The C column function then is 0.09156875 * v * v

I amended for spreadsheet for clarity.

 

[lklundin]

Just the Aero fraction of the total consumption.
Aero force is 0.5 * rho * Cd * A * V * V
Newtons / 3.6 = Wh/Km

0.5 * Rho * Cd * A / 3.6 = 0.5 * 1.225 * 0.23 * 2.34 / 3.6 = 0.09156875
The C column function then is 0.09156875 * v * v

Do you see an error here ?

Left unsaid, but perhaps I should clarify that I was only interested in the additional energy/Km the car consumes as the speed increases. I presumed that all forces and fixed power consumptions stay the same except the Aero drag.

Last night I thought about this interesting problem and came up with a relatively simple solution that probably holds reasonably well:
https://www.eso.org/~llundin/optispeed.pdf

We seem to be in agreement on the general assumptions and your plot is in agreement with my result to the extent that the power required to maintain the optimal speed increases more than linearly.

 

[SageBrush]

Last night I thought about this interesting problem and came up with a relatively simple solution that probably holds reasonably well:
https://www.eso.org/~llundin/optispeed.pdf

Saghost pointed out in an earlier discussion we are looking for the time minima in the f(driving_time + charging_time).
I don't think he explicitly said so, but I presume he simply solved the first derivative for zero.