## Sunday, July 14, 2019

### The problem of refuelling

FIA President Jean Todt has floated the idea of re-introducing refuelling to Formula 1, largely it seems to reduce the running-weight of the cars. According to Andrew Benson's BBC report, "Todt said he had been warned that the reintroduction of refuelling would likely lead to teams' race strategies being too similar to each other but countered that that was a product of there being too much simulation in F1."

Quite. So let's have a quick look at why refuelling doesn't necessarily make race-strategy more interesting. Suppose we have the following fairly typical parameter values:

Tyre-deg: 0.05 sec/lap
Fuel-effect: 0.033 sec/kg
Fuel-consumption: 1.5 kg/lap

Suppose that the deterministically optimal first pit-stop would be after 20 laps. That requires a fuel-load of 30kg. Suppose that the second stint would also require a fuel-load of 30kg. The lap-time penalty for 30kg of fuel would be 30*0.033 = 1 sec.

After 20 laps, the cumulative tyre degradation would be 20*0.05 = 1 sec.

Suppose two cars with identical zero-fuel-load lap-times are racing each other. As we approach the first pit-stop window, there would be no benefit of the car behind trying to pit first to undercut the car ahead: the penalty of taking on 30kg of fuel cancels out the advantage of switching to a fresh set of tyres. The conventional undercut logic, which permits overtaking between equally matched cars, would be lost.

With this set of parameter values, there would also be no benefit to the car behind running longer: the cumulative tyre deg cancels out the benefit of lapping on almost empty fuel tanks. However, one might suspect that this result only follows from choosing a special set of parameter values, so let's assume that the tyre-deg is lower, at 0.03 sec/lap. Surely this would tip the balance in favour of running longer?

Well, suppose the car behind is planning to run 3 laps further in the race, to lap 23. That requires a starting fuel-load of 23*1.5 = 34.5kg. That's an extra 4.5kg which the car behind needs to carry around for the first 20 laps of the race. With a fuel-effect of 0.033 sec/kg, that's a lap-time penalty of 0.033*4.5 = 0.15 secs on every lap of the first 20 laps. So, if we assume both cars are running in free air, the car behind would have lost 3 secs of cumulative time after 20 laps (assuming the extra weight didn't also increase the tyre-deg).

With the assumed tyre-deg of 0.03 secs/lap, the cumulative deg after 20 laps would be 0.6 secs. Which is less than the 1 sec penalty for taking on 30kg of fuel. Hence, the car running to lap 23 would make up 0.4 secs/lap on the car which has pitted on lap 20. Over 3 laps, that would be 1.2 secs.

Would the car behind be able to overcut the car ahead? Unfortunately not. That extra fuel-weight has already cost it 3 secs over the first 20 laps of the race. The 1.2 secs regained still leaves a net loss of 1.8 secs when it finally pits on lap 23.

Obviously, if both cars were running in traffic, and the car ahead was unable to exploit its superior potential lap-time over the first 20 laps, then the overcut might still work.

In summary, however, we can see why refuelling pushes strategies towards the deterministic optima: if you try to overcut an opponent, the greater fuel-weight necessary for that is counter-productive; conversely, if there's any benefit to be had from undercutting an opponent, that benefit would be even greater in the absence of refuelling.