F1 Fuel Saving in 2014

F1 this year is going to be very different from any previous season. The technical regulations open the doors to turbos connected to electric motors, 120kW KERS motors, 4MJ batteries, and wastegates. The ways in which the 2014 power unit can be used are myriad and it will be very difficult for the TV commentary teams to understand the implications themselves, let alone explain them to viewers! However, there are some things we can work from very basic information available online.

Aside from all of the turbo, MGU-H, battery stuff that makes it more complicated for us to understand, there are two parts of the 2014 rules which are very straightforward, from the technical regulations:

5.1.4 Fuel mass flow must not exceed 100kg/h.

And from the sporting regulations:

29.5 No car is permitted to consume more than 100kg of fuel, from the time at which the signal to start the race is given to the time each car crosses the Line after the end-of-race...

First off, let's observe that if fuel flow is limited to 100kg/h, then we know the fuel flow of every car at full throttle (100kg/h!). Let's take an extreme example and look at Monza, the high speed circuit of F1. The race in 2012 was won by Lewis Hamilton in a total time of  1:19:41.22. Monza is 53laps, so that's an average laptime of 90.21s. Depending on which website you get your F1 stats from, a lap of Monza is about 75% full throttle. The remaining 25% is divided into braking (at zero throttle) and accelerating from zero to full throttle. If we say that the split between braking and part throttle acceleration is even, and that the average throttle in the part throttle regions is 50%, then we can approximately represent all the time that's not full throttle by saying that a quarter of that time is full throttle, and three quarters of it is zero throttle. Adding this throttle usage to the 75% full throttle portion we get to 81.3% of the lap at full throttle as an approximation including the part throttle regions. Now, we should translate the FIA's fuel rate into a proper unit:

100kg/h / 3600s/h = 0.0278kg/s

If we multiply our three numbers so far together, we should get fuel use per lap:

90.21s/lap x 81.3% x 0.0278kg/s = 2.039kg/lap

Which is 2.039kg/lap x 53laps/race = 108.06kg/race. Which was fine in 2012, but it's over budget for 2014 by a little over 8kg! So, clearly you can't just smash round the track at full throttle like you did in the good old bad old use-as-much-fuel-as-you-like days. The question is, what's the best way of saving fuel without losing time? As any well informed motorsport fan knows, the most laptime-efficient way to say fuel is to completely lift off the throttle at the end of each straight and coast for a bit before hitting the brakes for the next corner, called 'lift-off', or 'lift and coast'. The reason is that saving fuel at the start of a straight means that you accelerate less and therefore go slower for the whole straight, whereas lifting at the end of the straight doesn't make you slower further down the track because you were about to brake anyway. How much lift-off are we talking about here? Well, if we lift-off at the ends of the straights then we're swapping time spent at full  fuel-flow for time spent at zero fuel-flow, so we save 0.0278kg/s during lift-off. To save our 8.06kg per race we need:

8.06kg / 0.0278kg/s = 289.9s/race

289.9s/race / 53laps/race = 5.47s/lap of lift-off!

No, your eyes do not deceive you (I encourage you to check my maths): this season, at Monza, drivers could have to lift-off the throttle at the ends of straights for an average of nearly 5 1/2 seconds per lap! If we distribute this time mainly over the four longer 'straights' (the Curva Grande is effectively a straight) and a little on the two shorter sections leading into each Lesmo then we might get something like this for an average lift-off schedule:

1.24s on the main straight into the first chicane
1.02s after Curva Grande into the second chicane
0.54s before the first Lesmo
0.54s before the second Lesmo
1.02s into Ascari
1.02s into Parabolica

As you can see, it's not exactly going to be the traditional 'last of the late brakers' scenario into the first chicane, or into any of the corners for that matter. Even if we're overestimating the amount of lift-off by a factor of two, we're still talking about half a second at the end of each long straight.

Of course drivers and teams might decide that lifting-off at the end of the main straight is too costly in strategic terms because it's a prime spot to overtake, or be overtaken. In which case they may not lift there, but they will then have to lift even more elsewhere to save the extra fuel.

There will be races where fuel saving is not an issue, Monaco for example is so slow that it's likely that no fuel saving will be necessary. Monza is an extreme example of a high-speed track where the fuel limit will have a big effect. The other factor that would eliminate the need for fuel saving is the safety car. F1 cars use (relatively) so little fuel when behind the safety car that as few as 4 laps behind a safety car could save enough fuel to complete the race without lifting-off any more.

I can't wait to see how this season plays out, and what strategic effects these radical fuel saving regulations will have.

"Going Downstairs" in Professional Darts

A week ago I flicked on the TV and caught the tail end of a quarter final in the World Darts Championship. One of the pundits (with lots of gold chains round his neck) was saying "if you're not hitting twenties, go for 19s, and if that doesn't work try 18s, even 17s if you have to!". The message was that if you start missing the treble 20 then you should switch it up and go for lower numbers until you found a treble you could hit more reliably. This got me thinking, do darts players actually do better if they switch to aiming at treble 19 after missing a couple of treble 20s? To do so they would surely have to get more accurate after switching to 19s than they were when aiming at 20s, to compensate for the lower score for each dart. I mentioned this to Dave Millican at work and he suggested that 19 might be a safer target for a player with an accuracy problem because the segments either side of it aren't such low scores as those surrounding the 20. This is a good point, the 20 is flanked by 5 and 1, whereas the 19 has the slightly friendlier 7 and 3 for company. Although the same logic doesn't support what the pundit was saying about going for 18 if 19 wasn't working for you, as the 18 has the 1 and the 4 either side which is worse than the 20!

A quick search on Google Scholar revealed a very interesting paper written by researchers at Stanford who have looked into this question and found some interesting results. They calculated the optimal place to aim on the dartboard for players of varying levels of skill. Skillful players consistently land their darts close to the point that they're aiming at, so when aiming at a fixed point their darts land in tight groups; in other words, the standard deviation of the distance between the landing point and the aiming point is small. Rubbish players' darts land all over the place; they have a high standard deviation. Tibshirani, Price, and Taylor found the best place for any player to aim at by maximising the integral of the score over the area in which their darts are likely to land, weighted by the likelihood that their darts land there, for any given aiming point. As you might expect, they found that a very good player, with a standard deviation of only a few millimetres, should aim at the treble 20 for maximum points. A rubbish player, with a very high standard deviation, should aim very near the centre of the board, to minimise the chance of missing the board altogether! The interesting bit is what happens in between very good and rubbish. As standard deviation increases from zero, the optimum aiming point moves slightly up and to the left, to favour hitting the 5 instead of the 1 on the occasional wild dart. Then, when standard deviation increases beyond 16.4mm, the optimum aiming point jumps to treble 19! As the standard deviation further increases, perhaps after the third pint, the optimal aiming point curves upwards and then around to the right until it settles just to the left of the bullseye.

Figure 1: movement of the optimum aiming point as a player's standard deviation increases (reproduced with permission from A Statistician Plays Darts , Tibshirani, Price, Taylor - JRSS Series A, Vol. 174, No. 1, 213-226, 2011)

So it seems that our pundit was right in one case, it is worth "going downstairs" for the treble 19 if your aim on treble 20s degrades past a certain point. But in terms of maximising expected score, it's not worth switching to 18 if your aim on 19 isn't good, nor to 17 if your aim on 18 isn't good. Of course there may be psychological factors, and a certain bias due to a player's dart distribution being skewed at an angle and therefore making her better at hitting trebles at one angle than another (this is also investigated in the paper).

In terms of whether darts players do the right thing during matches (statistically speaking) there is still a question mark over switching to treble 19. Players obviously only switch to 19 if they start missing treble 20s, in other words their internal estimate of their standard deviation rises and they react by switching to 19. The question is how does this internal estimate compare to the true value, and how close to the ideal threshold of 16.4mm do they switch between 19 and 20? This is much more difficult to answer than our first question. To do so we would need dart by dart position data for a player over many many games, including many occasions where the player switched to 19, which most players don't do that often. Perhaps I should write a video analysis algorithm to watch and datafy TV footage of darts and then crunch the numbers to see which players' mental estimates are closest to the truth. My guess would be that some players tend to switch too soon because of one random dart going awry when in fact this is not sufficient evidence that the underlying standard deviation has actually changed. On the other hand, maybe the player has a good idea of not only what his aim is like at the moment, but whether he's feeling more tense and therefore about to get worse, enabling him to preemptively switch to 19 before hitting the 1!