Time Optimized Traveling Speed: the Final Graph

[lklundin]

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.

Yes, I have done exactly the same. Since last night, I amended my model of the optimal driving speed to take into account the power consumption that is independent of the speed. If this consumption is significant (maybe heating in cold weather, of the cabin but maybe also of the battery), then this actually increases the optimal driving speed (since that in turn reduces the amount of time this power is spent).

Edit: @mongo, maybe as a technically inclined member, you would have some interest in this thread?

 

[5_+JqckQttqck]

I think a screen showing the Wh consumption of each vehicle system would help with summer driving model vs winter driving model. Hint hint to Tesla engineers - you have these variables, make it into a spiffy infotainment screen for us nerds

 

[Shizzrock]

I have no idea what this thread is about

 

[asmite]

This thread is about someone sharing a very useful idea but opted to use a communication method that is not suitable for the audience. The end result is the useful idea is overshadowed by the distractions...

 

[lklundin]

I have no idea what this thread is about

The idea is to try and figure out what the ideal driving speed is, if one wants to cover a distance as quickly as possible, but including the time it takes to charge the car.

If you drive too slow, you will get very far on each charge, but will spend too much time driving, if you drive too fast, you will spend too much time charging (because your consumption is so much higher).

It turns out that depending on several factors (mostly on the charging power), there is an ideal speed that achieves a combined minimum of time spent driving and time spent charging.

Basically, the higher the charging power you have, the faster it pays off to drive. It follows from this that you want to make relatively frequent and relatively short charging stops, with a pretty much empty battery (to ensure a high charging power), when you want to minimize your traveling time. This driving style is not optimal for conserving the health of your battery.

To get an idea, you can just go to EV Trip Planner and try out different speeds for the same trip, and watch how the total traveling time (incl. supercharger visits) changes.

 

[usofrob]

I feel like there should be multiple lines for how many additional minutes it takes you off your path to get to the charger.
I can kind of estimate that by just lowering the charger rate accordingly.

 

[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.

Actually, any power use that is directly proportional to the speed has no effect on the optimal driving speed. Imagine your car has such a power use and you cut your speed in half. Then you will cut that power use in half. But since you have also just doubled your driving time, the energy consumption is unchanged.

The car also has some power consumption that is independent of the speed (e.g. lights, infotainment, seat heating). It may seem counter-intuitive at first, but an increase in this type of power use actually increases your optimal driving speed. This is so because increasing your speed reduces your driving time, thus reducing the amount of time the speed-independent power usage occurs. However, unless the charging power is very small, the effect of the speed-independent power use on the optimal driving speed is negligible.

What remains is the power use that increases with the square and cube (and possibly higher orders) of the speed.

The optimal speed given a specific charging power in the above plot seems a little high to me.

I went to EV Trip Planner to get their consumption numbers for the LR Model 3. I chose a flat road at sea level, with ambient and cabin temperature equal (at 21C), no wind, no payload (although that should only have an effect if altitude is gained or lost, or many starts and stops are made). Using different speed multipliers, I received its estimate of the consumption per distance (which is really a force, namely the force that the car needs to exert on the road in order to maintain its speed). With speed in unit km/h and driving force in unit Wh/km I got these numbers:

v F
55 79
66 87
77 97
88 110
99 126
110 144
121 164
132 186
143 211
154 238
165 267
176 298
182 315
187 332
193 349
204 386
209 405
220 445

Since the drag power (from the drag equation) grows with the speed cubed, I fitted the above values to a cubic polynomial, with these coefficients (using SI-units, i.e. m/s for the speed and W for the power):
0 328.93
1 211.893
2 -3.2902
3 0.424175
The first coefficient is the constant term, i.e. a speed-independent power use of 329 W (negligible with even a small charging power), and the whole polynomial approximates the driving power, i.e. the power required to maintain the given speed.

For various charging speeds I then set out to solve equation (1) in my optispeed-paper:
https://www.eso.org/~llundin/optispeed.pdf
With the driving power being a cubic polynomial, the equation (1) also becomes a cubic polynomial.

For varying charging speeds I found not only the optimal driving speed but also the effective speed (i.e. the speed when including also the charging time), the driving power and the driving force.

Charging Power 3 kW: Vcar= 61.8 km/h Veff= 22.8 km/h Pcar= 5.1 kW F= 83 Wh/km
Charging Power 11 kW: Vcar= 90.3 km/h Veff= 46.7 km/h Pcar=10.3 kW F=114 Wh/km
Charging Power 22 kW: Vcar=111.9 km/h Veff= 64.0 km/h Pcar=16.5 kW F=147 Wh/km
Charging Power 50 kW: Vcar=145.2 km/h Veff= 89.2 km/h Pcar=31.4 kW F=216 Wh/km
Charging Power 117 kW: Vcar=190.9 km/h Veff=122.3 km/h Pcar=65.6 kW F=344 Wh/km
Charging Power 120 kW: Vcar=192.5 km/h Veff=123.5 km/h Pcar=67.1 kW F=349 Wh/km
Charging Power 180 kW: Vcar=219.6 km/h Veff=142.6 km/h Pcar=97.3 kW F=443 Wh/km

I verified the approach by going back to EV Trip Planner and by trying out various driving speeds, I found that the optimal driving speed for charging at 120 kW as determined by the above is consistent with the results found manually on EV Trip Planner.

As can be seen, using the numbers from evtripplanner.com, the optimal driving speed while charging at 50 kW is about 145 km/h. The above plot has that speed closer to 155 km/h.

The speeds I calculated above already seem quite high, but if realistic then the supercharging at 180 kW of the Model 3 will be very handy here in Germany (once the Superchargers become upgraded). But even the charging speed of 117 kW that current Model 3 users are reporting will allow for a useful effective traveling speed.

PS. The fitting of the polynomial and the solving of the differential equation was done with www.eso.org/cpl/,
a tax-payer funded open-source library for astronomy related tasks.

 

[Arpe]

Thanks for sharing @lklundin

It is great to know that in order to reach your destination as fast as possible, using a supercharger, we have to keep up with the speed limits and preferable a little above without speeding.

Germans on the Autobahn may want to cruise at 180-190 km/h

Just make sure that you have enough battery left to make it to the next supercharger!

 

[ℬête Noire]

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.

I'm pleasantly surprised it is that high. That's pretty good since the area I'm most likely to be relying on L2, way out past the land of Superchargers (for now, anyway), the speed limits are mostly 90 and 100. I'm not sure how much it impacts how I'd drive but it certainly puts my mind at ease about not costing time driving at the speed limit.

P.S. Wind however could be a factor. If you don't slow in the face of significant headwind you're going to take a beating on range.

 

[ℬête Noire]

It is great to know that in order to reach your destination as fast as possible, using a supercharger, we have to keep up with the speed limits and preferable a little above without speeding.

Germans on the Autobahn may want to cruise at 180-190 km/h

Just make sure that you have enough battery left to make it to the next supercharger!

Keep in mind our Supercharging rate isn't actually 120kW though, not even if you only take it up to 80% SOC. You need to average your rate over the entire time you're charging, so functional charging rate could easily be less than 50kW. Remember that the higher you go in SOC the longer it's spending at that given lower rate. Fortunately there's the high plateau for the first 50% of SOC, on the other hand it gets really ugly after 80% SOC. If you are depleting your battery enough that you need more than 80% to jump you're probably costing yourself time.

 

[lklundin]

Keep in mind our Supercharging rate isn't actually 120kW though, not even if you only take it up to 80% SOC.

Actually, the Model 3 is rated for charging up to 180 kW, so it can charge at close to 120 kW up to a pretty high SoC, some users have reported 117 kW up to about 40% SoC.

 

[Johann Koeber]

Thank you for the valuable discussion here.

When driving fast in Germany, another factor needs to be considered. The calculations show the optimum steady state speed. In practice, the faster you go the more often you have to slow down and accelerate due to traffic. Recuperation is not 100 % effecient (I would estimate 50 % max). Thus in real world application it is probably advisable to go slower than calculated.

OTOH with the many construction zones we have, there are many stretches of the Autobahn limited to 80 km/h. This enables a higher speed on those stretches that are not limited.

Finally in a real trip, we start with SOC 90%or 100%, plan to arrive with 10% or 20%. To optimize total travel time we need to optimize the legs depending on available charging options along the way.

So the question of optimal trip planning is a lot more complex (and non-linear) in practice.

 

[lklundin]

I'm pleasantly surprised it is that high. That's pretty good since the area I'm most likely to be relying on L2, way out past the land of Superchargers (for now, anyway).

Yes, me too.

Since most BEV-drivers realize that during no-sleep road-tripping it is preferable to keep the charging speed close to the rated power where they are plugged in, I created a plot that for selected charging speeds shows how the effective speed depends on the driving + charging speed:

https://www.eso.org/~llundin/optispeedplot.pdf

The plot shows that independent of the charging speed, the effectiveness around the optimum is pretty flat for a quite wide range of speeds.
This means that the effective speed does not decrease drastically if one drives a bit slower than the speed that is optimal for the given charging power.

 

[ℬête Noire]

Actually, the Model 3 is rated for charging up to 180 kW, so it can charge at close to 120 kW up to a pretty high SoC, some users have reported 117 kW up to about 40% SoC.

Yeah, I needed to add a "* With existing SC equipment."

When the next generation of SC comes out, yes that'll shake it up if those are on your route. Not sure how fast they'll roll out that new generation of equipment, though? They'll likely put it at the currently swamped SCs, the "flagship" Kettleman, etc. but I'd be very surprised if we'll see many converted outside CA for some time. More likely they'll focus on filling in grey pins.

 

[lklundin]

I needed to add a "* With existing SC equipment."

Actually, with the existing 120 kW Superchargers, the Model 3's rating for 180 kW still gives it the stated advantage, since that allows it to charge at 120 kW up to a quite high SoC, significantly higher than that of the Model S/X.

But it is true that with Superchargers able to provide 180 kW, the Model 3 will be able to charge a good deal faster.

 

[ℬête Noire]

...the Model 3 is rated for charging up to 180 kW...

BTW has this now been hard-confirmed? I saw the speculation on 180KW (that seemed pretty sound) and I've seen numbers as high as 210KW. Has Tesla made any official or semi-offical confirmation of it?

 

[ℬête Noire]

the existing 120 kW Superchargers, the Model 3's rating for 180 kW still gives it the stated advantage,

Yeah, it's assumed that's what is underlying the LR's ability to maintain near-120KW charge rates all the way to 50% SOC, what I referred to as the plateau.

 

[lklundin]

BTW has this now been hard-confirmed? I saw the speculation on 180KW (that seemed pretty sound) and I've seen numbers as high as 210KW. Has Tesla made any official or semi-offical confirmation of it?

Model 3 owners have posted photos (of the EPS filing) where the charging is rated for 525 A and the battery voltage is at 350 V, that is about 180 kW.

The 210 kW apparently assumes a voltage of 400 V, I believe that is a misunderstanding.

 

[OPRCE]

For comparison, does anyone know the maximum rated charging rate for an S100D and/or whether the max rate is likely to be raised above the current 120kW when the generation 3 Superchargers arrive?

 

[lklundin]

Here is a plot of the power use, interpolated from RWD Model 3 numbers from evtripplanner.com:

https://www.eso.org/~llundin/poweruse.pdf