Chewing up time savings at the Supercharger

[ThosEM]

Ok, here's my take on this question, based on a modest amount of analysis, but assuming superchargers are always sufficiently close together to avoid running out:

The total trip time is the sum of the driving time plus the charging time:

T = D/V + Dr/Vc

where Dr is the rated distance used for the trip (to be restored by charging) and Vc is the average rate of charging (in rated distance units).

Let's assume that Dr = D * V/Vr, where Vr is the speed at which one gets rated range. This says the energy consumed (rated range units) is proportional to the speed driven, which is pretty consistent with published Range vs speed plots.

So now we have:

T = D/V + D*V/(Vr*Vc)

If you plot T vs. V here, or do a bit of calculus, you’ll find that the minimum time for the trip occurs for:

Vopt^2 = Vr*Vc

Too slow and the driving time increases and dominates; too fast and the charging time increases and dominates. Hit it right and you have the minimum trip time.

If Vr is 100 kph (it might be a bit higher in good weather) and Vc is 300 kph (typical average for a supercharger session starting out at 600 kph and tapering down), this says the minimum trip time is obtained by driving at an optimal average speed of:

Vopt = √(100*300) = 175 kph or 108 mph

On the other hand, with L2 charging at a rate of perhaps 45 kph / 30 mph:

Vopt = √(100*45) = 67 kph or 42 mph

Bottom line: your charging rate does determine the optimal (minimum time) speed for road trips, but if you drive that fast with superchargers, you'll spend as much time charging as you do driving, which will not feel optimal and probably get you a speeding ticket. Or if traveling with L2 charging, your gains from driving faster than optimal will indeed be eaten up charging and more so. With moderate CHAdeMO charging, you'll need to spend about the same time charging as driving and it won't feel optimal.

A good compromise is to speed up as much as possible as the next supercharger approaches.

 

[mspohr]

A good compromise is to speed up as much as possible as the next supercharger approaches.

I think this is the best advice.

 

[Owner]

Data's up....Driving Charging Time | Tesla Owner

Conclusion... no reason to drive over 75... generalizing from the Harris Ranch to Tejon Ranch journey in good weather arriving with an empty battery...

 

[mspohr]

Data's up....Driving Charging Time | Tesla Owner

Conclusion... no reason to drive over 75... generalizing from the Harris Ranch to Tejon Ranch journey in good weather arriving with an empty battery...

But... it looks like no real penalty for driving up to 85 and only a slight penalty for 90... so between 75 and 85 is good.

 

[liuping]

But... it looks like no real penalty for driving up to 85 and only a slight penalty for 90... so between 75 and 85 is good.

For me, this means go with the flow of traffic more or less and not worry about it.

 

[Owner]

The difference in going faster is just marginally slower. Speed up only slows you down by a couple of minutes. You could in essence think of the curve as a hockey puck. Its pretty linear in angle down to 75, than almost basically flat after that if you arrive at a reasonably low state of charge.

I think driving faster than 75 in good conditions will likely more cost you speeding tickets before you'll notice the 2 or 3 minute overall time loss.

Also, your just wasting energy in the ether if that is a concern.

But... it looks like no real penalty for driving up to 85 and only a slight penalty for 90... so between 75 and 85 is good.

 

[Zapped]

For me, this means go with the flow of traffic more or less and not worry about it.

+ 1

 

[WSE51]

Owner has done a great service by showing the top speed after which all time saved going faster is lost in the taper at the supercharger (let's call this the Optimal Speed). There are a few other factors I would like to mention, some of which have already been reported by others:

1) The Optimal Speed will change depending upon the wind, temperature, payload and elevation changes. The greater the ratio of Rated Miles to actual Miles on a particular route, the lower the Optimal Speed

2) If you know you will be stopping to eat at the next Supercharger and that it will take more minutes to eat than to charge the Rated Miles you'd typically need, then might as well drive faster.

3) If you are close enough to the Supercharger to be confident that you will not need your safety buffer of Rated Miles, you could drive faster and figure that you will be recharging that incremental energy at the early part of the charging curve, before the taper is a factor, and your driving time saved will be much greater than incremental charging time.

4) Road or traffic conditions could make Optimal Speed unpleasant or unsafe.

5) For me, perceived time waiting at the Supercharger is longer than the same time driving. Also, for the last 5 mph below Optimal Speed, the great majority of time saved going faster is lost charging, so rather than driving faster which may involve actively changing lanes more often to pass more cars, I'd rather drive more relaxed, especially on a long trip where I am visiting 4-5 superchargers a day.

6) Even though driving faster than the Optimal Speed may not cost much overall driving time, it does cost more energy. Its just that I don't pay for it, Tesla does. At the margin, even though its not a huge factor, I'd prefer saving Telsa a bit of money ... and until the Superchargers are all solar powered ... save the environment a bit more carbon emissions from the electric plant supplying that Supercharger. Finally, there is the small factor that your battery pack has a finite life. Driving faster and recharging more, yet not saving overall time, does increase battery cycles without any time benefit.

 

[ThosEM]

Apologies! My earlier analysis here was seriously flawed in the assumption that energy use and charging time increase linearly with speed driven. In fact it's the square of the speed driven because the pressure of the wind goes like the square. The conclusions change dramatically, so here's the corrected story:

The total trip time is the sum of the driving time plus the charging time:

T = D/V + Dr/Vc

where Dr is the rated distance used for the trip (to be restored by charging) and Vc is the average rate of charging (in rated distance units).

Let's assume that Dr = D * (V/Vr)^2, where Vr is the speed at which one gets rated range. This says the energy consumed (or rated range) is proportional to the square of the speed driven, which is (!) consistent with published range vs speed analysis, a very nice example of which that set me straight is: http://www.solarjourneyusa.com/EVdistanceAnalysis5.php

So now we have:

T = D/V + D*V^2/(Vr^2*Vc)

If you plot T vs. V here and-or do a bit of calculus, you’ll find that the minimum time for the trip occurs for this optimal speed:

Vopt^3 = (Vr^2*Vc)/2

Too slow and the driving time increases and dominates; too fast and the charging time increases and dominates. Hit it right and you have the minimum.

If Vr is 100 kph and Vc is 300 kph (typical average for a session starting out at 600 kph and tapering down), this says the minimum trip time is obtained by driving at an average speed of

Vopt = 3√(100^2*300/2) = 114 kph or 71 mph

This means that rushing to the supercharger will consume more time in charging than is saved by the speed increase, under conditions where aerodynamics are the dominant factor.

Ok, here's my take on this question, based on a modest amount of analysis, but assuming superchargers are always sufficiently close together to avoid running out:

The total trip time is the sum of the driving time plus the charging time:

T = D/V + Dr/Vc

where Dr is the rated distance used for the trip (to be restored by charging) and Vc is the average rate of charging (in rated distance units).

Let's assume that Dr = D * V/Vr, where Vr is the speed at which one gets rated range. This says the energy consumed (rated range units) is proportional to the speed driven, which is pretty consistent with published Range vs speed plots.

So now we have:

T = D/V + D*V/(Vr*Vc)

If you plot T vs. V here, or do a bit of calculus, you’ll find that the minimum time for the trip occurs for:

Vopt^2 = Vr*Vc

Too slow and the driving time increases and dominates; too fast and the charging time increases and dominates. Hit it right and you have the minimum trip time.

If Vr is 100 kph (it might be a bit higher in good weather) and Vc is 300 kph (typical average for a supercharger session starting out at 600 kph and tapering down), this says the minimum trip time is obtained by driving at an optimal average speed of:

Vopt = √(100*300) = 175 kph or 108 mph

On the other hand, with L2 charging at a rate of perhaps 45 kph / 30 mph:

Vopt = √(100*45) = 67 kph or 42 mph

Bottom line: your charging rate does determine the optimal (minimum time) speed for road trips, but if you drive that fast with superchargers, you'll spend as much time charging as you do driving, which will not feel optimal and probably get you a speeding ticket. Or if traveling with L2 charging, your gains from driving faster than optimal will indeed be eaten up charging and more so. With moderate CHAdeMO charging, you'll need to spend about the same time charging as driving and it won't feel optimal.

A good compromise is to speed up as much as possible as the next supercharger approaches.

 

[ThosEM]

The OP reminded me that my analysis did not account for the tapering of the charging rate as the battery fills. So I have added that effect and of course it does make a big difference. Here’s the new story, which should be getting closer to final:

Let’s take the charging speed Vc to start at an initial value Vco and drop linearly to zero as the battery reaches full. Then Vc decreases as Dr increases from driving faster, which eventually negates the time savings.

Let Vc = Vco*(1-Dr/2Dm) so the dependence is linear, for lack of more specific data on that and for simplicity.

Then Dr/Vc = Dr/(Vco*(1-Dr/2Dm))

and with Dr = D*(V/Vr)^2

T = D/V + D*(V/Vr)^2/(Vco*(1-D/2Dm*(V/Vr)^2)/2)

Setting the derivative of this to zero gives a fifth order polynomial and I don’t go higher than the quadratic equation, so I decided to plot this rather than calculate the minimum trip time. Here’s the result for three different distances (or fractions of a full battery needed):

So if we want to drive fast, it stands to reason we'll need to target a supercharger that is significantly closer than rated range. If we push the rated range, we'll need to slow down to not much faster than the speed that gives rated range.

If we increase the trip as a fraction of rated range, to say 95% of it, the optimal speed is essentially the speed that gives rated range, of course. But the trip takes about 1.3 times as long as the driving part owing to the charging part, which becomes pretty long since a full recharge is needed (assuming one keeps doing that).

As we target a distance that is a smaller fraction of the rated range, the optimal speed increases, and the total trip time including recharge approaches the driving time only at the (lower) speed that gives rated range. If we push the speed up above optimal, the recharge time lengthens, and if we push it too far, we don't make it to the destination.

 

[ThosEM]

Data's up....Driving Charging Time | Tesla Owner

Conclusion... no reason to drive over 75... generalizing from the Harris Ranch to Tejon Ranch journey in good weather arriving with an empty battery...

View attachment 65579

I digitized these data points and superposed them on my analysis plot, below. The result suggests that the data came from a trip distance of about half of rated range. Actually the best fit is for a trip of 43% of 265 miles, or 114 miles. The actual trip from which the data came was quoted as 116.2 miles. Not bad! There are two other parameters assumed as the rated range, teken to be 265 mi, and speed for rated range, taken to be 65 mph. Those could vary in specific cases, of course.

The proof is in the pudding ;=) I feel pretty confident of this model at this point.

 

[yobigd20]

Tesla should turn the center console into a touchscreen gaming center while you're plugged into the supercharger. It's not like you're going anywhere. Only activated as you're plugged in and charging. They could put a bunch of chrome app games on there. That would be kinda fun to kill time.

 

[ThosEM]

View attachment 65807

A bit of interpretation of these curves might be helpful in applying them to trips.

Here is an example: Let's say we are driving north from Woodbridge VA to Newark DE superchargers along I-95. a distance of 124 miles or about 50% of Rated Range (using 250 miles, rated range may vary). So looking at the 50% curve, you can see that the speed that gives the fastest overall time is 1.3 V/Vr ... or 30% faster than the speed at which we would get the rated consumption and rated range. That speed depends on weather and other conditions, so we consult EVPlanner for this trip in freezing weather, and find that a speed factor of 0.65 would get us there in the same number of rated miles as the actual distance. So the speed for rated range is 0.65 of the traffic speed on I-95 (which is about 75 mph; EVTripPlanner makes us guess at this), or 49 mph. And the optimal trip time speed is 30% higher than that, or 49 x 1.3 = 64 mph. Going faster than 64 mph will get us to the Newark supercharger faster, but we will lose more time in recharging than we gained by going faster than 64 mph.

Another example shows how elevation impacts these calculations. On the St. George Utah to Beaver Utah route there is an elevation increase of nearly 4000 ft, and according to EVPlanner the speed at which actual distance equals rated miles is only 40 mph (guessing again at the typical speed on that route). That route is also about 50% of 250 rated miles so uses the same curve, and the optimal speed for that route would be 30% faster than 40 mph, or only about 52 mph.

Another important case is when we drive nearly the rated range. Looking at the 95% curve, we see that the optimal speed is equal to the speed at which we get rated range. In spring/fall conditions with no need for HVAC, the drive from Portland ME to Milford CT is just about 250 mi. I find that in warm (but not hot) weather, I get rated range driving at most 65 mph. So when driving the full rated range, driving faster than this will not improve your total travel time, but in this case it will prevent you from getting to your destination.

I hope this helps in planning your trips. The tricky part is determining what speed would use rated miles at the same rate as actual distance. Knowing that, perhaps from recent driving experience, the curves will tell us how much faster to drive for minimum total trip time. They show that higher speeds are only a time advantage for shorter hops between superchargers.

 

[Cottonwood]

Another example shows how elevation impacts these calculations. On the St. George Utah to Beaver Utah route there is an elevation increase of nearly 4000 ft, and according to EVPlanner the speed at which actual distance equals rated miles is only 40 mph (guessing again at the typical speed on that route). That route is also about 50% of 250 rated miles so uses the same curve, and the optimal speed for that route would be 30% faster than 40 mph, or only about 52 mph.

I beg to differ with you on using these curves. The analysis and charts that you cite are based on polynomial approximations of charge rate and rated miles used. The biggest component of rated miles used in the analysis assumes a V[SUP]2[/SUP] increase of usage with speed. The extra rated miles used for elevation gain are for increases in potential energy from the start to the finish and those extra rated miles are not a function of speed. A reasonable approximation for that extra energy needed or gained is 6 rated miles per 1,000 ft, or in this case (3,000, not 4,000 ft), about 18 extra rated miles to overcome the altitude increase.

Because this large fraction of energy usage from Las Vegas to St George does not vary with speed, in reality, the optimum speed for that leg is far greater.

The following table is from EVTripPlanner with an 85 on 19's, normal temps, normal load. The charge times are from supercharge data from cross-country road trip and assume that the trip is done with 40 rated miles margin. The charge time are then from 40 rated miles to 40 + rated miles needed. From these results, the the optimum speed is around 80 mph on I-15, but is pretty broad optimum from 75 to 95 mph.

Speed Multiplier	Time	I-15 Speed	Rated Miles Used	Charge Time	Total Time
0.6	2:26	46	97	0:19	2:45
0.8	1:50	61	114	0:23	2:13
1.0	1:28	76	138	0:31	1:58
1.1	1:20	83	152	0:35	1:55
1.2	1:13	91	167	0:43	1:56

 

[Sunnyday]

The calcs here are great for maximizing drive time and minimizing charge time, but here's another consideration to layer on that might help your time utilization. If I'm on a long roadtrip, I'll plan a full charge at the same time I want to have a meal, and I'll do shorter charges elsewhere to get me to the next SC and next meal. I'll plan my departure time with the breakfast/lunch/dinner meal stops in mind, and I plan my meals to happen while charging. Early in my ownership experience I made the mistake of doing meal stops away from SCs, and this caused the total trip times to balloon. I also made the mistake of overcharging outside of meal times which meant that the car hit full charge halfway through the meal stops (wasted charge time). The non-meal SC stops, I work. In my job, I always have work to do, and all I need is my laptop and my wireless modem. I view SC stops as a time to get work done if I'm not eating. The more work I get done while charging, the less work I have to do that evening or the next day, which means I'm earning free time on the other end.

 

[WSE51]

The new firmware version 6.1 makes implementing the speed pacing much easier as the software constantly adjusts its prediction of your finishing State of Charge to your driving style and the upcoming elevation changes.

 

[snellenr]

The new firmware version 6.1 makes implementing the speed pacing much easier as the software constantly adjusts its prediction of your finishing State of Charge to your driving style and the upcoming elevation changes.

With firmware 6.1, does the predicted State of Charge in navigation update as you're charging if the next destination is active? For example: I pull into the Huntsville supercharger, start charging, and select Corsicana in navigation -- if the state of charge prediction initially shows a deficit, will it eventually change to show a surplus as charging continues?

I realize that I could cancel and re-select the destination -- but it'd be cool if the firmware took care of that little housekeeping chore...

 

[swaltner]

Yes, the predictive algorithm is active during charging sessions. You get the car's prediction on energy consumption while your charging session is active,