4 replies to this topic

[l2k2]

There was the Canadian grand prix last weekend. If I had to describe the race with one phrase it would be: "lift-and-coast, from start to finish". It seems, that the teams have found out that lift-and-coast strategy is best for saving fuel when racing. But, I started to wonder whether it is really so.

 

As an engineer, I built a small and simple model to test this.

 

First, we must define some parameters. On a really long straight, with low downforce setup, an F1 car can reach 360 kph (100 m/s); an F1 car weights 750 kg (with driver and half-a-tank of gasoline); and, an F1 car has a peak power output of approximately 550 kW (approximately 740 hp, in race trim). We also assume that shifting is instantaneous and power curve is relatively flat. That means that we can use peak power all the time.

 

Peak braking de-acceleration for an F1 car is approximately 5 g. One can estimate that 0.5 g is due to air resistance, and 4.5 g's due to the brakes. The tyre friction coefficient for a racing tyre is approximately 1.5, so, peak effective mass is 3 times the actual mass. Downforce is this minus 1. However, when accelerating, only 80% of this effective mass can be translated to rear axis.

 

With these, we can build a (very) simple model for an F1 car driving the last straight of Circuit Gilles Villeneuve; which we assume to be a flat 1000 meter straight with initial velocity of 60 kph and final velocity of 80 kph.

 

First, some feasibility testing. How fast can an F1 car change its velocity as a function of its velocity:

How about the forces that are involved:

The results seam reasonable: one can floor the accelerator around 130 kph; 0-to-100-kph in 1.9 s; and 0-to-200-kph in 3.7 s. However, we slightly underestimated the contribution from air resistance to braking, as the peak de-acceleration is 5.25 g's instead of 5 g. (If I would be doing this for real, I would use real tabulated data from, for example, wind tunnel. However, I do not have access to such a data.)

 

Then, the meat. With full power, the straight takes 14.302 s, with top speed of 342 kph and a fuel usage of 329 g. If we lift one second before we would have braked, it takes 0.051 s more (+0.36%) but we save 28 grams of fuel (-8.4%). The top speed is 337 kph and we start braking at 311 kph. To match this time increase, we could alternatively cut the power by 1.55%. This is equivalent of the old "short-shifting strategy". Now, the top speed is 340 kph (and we start braking at that speed). However, we only save 3.2 g (-0.97%) of fuel (less fuel is saved than 1.55% because now we need to accelerate for a longer time).

However, when looking at the velocity graph, one may wonder whether there would exist an even better way to save fuel. For saving the brakes, lift-and-coast is very likely the optimal strategy. For gathering kinetic energy it is not. But, more about that later.

 

PS. Crosses denote the instant of lifting, the instant of starting to brake, and the instant of getting to the final chicane.

 

PPS. A simplified and commented version of the code producing the results shown so far is available in GitHub: https://github.com/L...ster/montreal.m.

[munks]

Good analysis. However, if you had the choice to lift early at different points around the track, you would want to do so on the shorter straights, not the longest straights. The lower speeds at the end of the shorter straights would result in less time lost for the same amount of fuel saved. I think.

Edited by munks, 12 June 2015 - 16:58.

[mariner]

I am stil trying to grasp all this due to my poor maths but are you sure the peak braking G of 4.5 can be held right down to 80kph given it is so aero dependent?

 

I have had " lift and coast" explained to me by Hugh Chamberlain who has run many, many Le mans etc endurance races. Apparently it was standard in group C days, and the drivers had to be told to stay in top gear when braking to avoid using fuel to spin up the drive train at each downshift.

[l2k2]

I am stil trying to grasp all this due to my poor maths but are you sure the peak braking G of 4.5 can be held right down to 80kph given it is so aero dependent?

 

I have had " lift and coast" explained to me by Hugh Chamberlain who has run many, many Le mans etc endurance races. Apparently it was standard in group C days, and the drivers had to be told to stay in top gear when braking to avoid using fuel to spin up the drive train at each downshift.

 

Hi, you are correct about the braking. It is aero dependent because of two reasons: air resistance and downforce, both of which increase with speed. As you can see from the first figure, the peak (de-) acceleration changes with speed: at 80 kph its -2.2 g, at 200 kph -3.4 g, and at top speed -5.25 g. (In reality things are not always this simple, because at high speed one needs more braking power to realize even the same deacceleration*. However, the carbon brakes are quite good at generating that power.)

 

* Partially because of this, it is much easier to lock the brakes at slower speeds. This becomes too obvious at least with road bicycles and high speed decents.

[l2k2]

Finally, I found some time to continue posting to this the thread.

 

First, the unfortunate results. With the assumptions made in the opening post: at most 50 ms more time, and save as much fuel as possible; the optimal way to race is to lift-and-coast. In order to verify this, I implemented a genetic optimization algorithm. The algorithm of choice was differential evolution, a simple but effective algorithm that is well described in, for example, Wikipedia. Also, a heavily commented example of how to implement a naive algorithm in my github public repository. The actual algorithm had similar 'perfect traction control' as the example in opening post. In addition one must limit the values for throttle input between 0 and 1, for obvious reasons. (Yes, I admit this was a way overkill method for the problem.)

 

Then, the slightly surprising results. If we break fuel flow limit rule: that is, if we assume we can convert fuel to power at the same efficiency, and are not limited to 100 kg per hour, we can do significantly better. 

 

Indeed, underneath one can find the throttle input graph, and the velocity graphs for a 550 kW formula and a 600 kW formula (fuel flow limit of 109 kg per hour). The latter can lift almost three seconds before braking without being out-paced by the former. And, due to this, it uses 49 grams less fuel (15% less)!

 

 

 

In racing, more power = better fuel efficiency? At least from this point of view, the lower power limit for the race in the Formula E is a bit questionable.

 

PS. Only the "more power = better efficiency" only works up to a limit, when limited just by the traction, lift-and-coast is no longer the best way.