Faster than Hamilton! Optimizing Formula 1 strategies

Transcription

1 Faster than Hamilton! Optimizing Formula 1 strategies December 3, 2018 Contents 1 Introduction 2 2 Lap time modelling Hypothesis Example Optimum strategy Pit-stops Example Pitting on lap Pitting on lap Pitting on lap Conclusion Questions 5 5 Let s compete! Build your racing team Sign drivers Develop your car Example Questions Lap time calculator Championship rules List of circuits Competition timeline

2 MATH Faster than Hamilton! 2018/19 Introduction We make many decisions every day, some of them on important matters. There often exists an optimal decision but the situation is often too complicated for us to figure it out. In these instances, it is good to rely on intuition and experience, both of which can easily be gained from modelling. Such situations arise in the professional setting where strategic choices with nontrivial consequences have to be made. They are at the core of professions such as stock trader or company founder but are also hidden within unsuspicious tasks for other professions such as the allocation of a budget, the distribution of tasks amongst people and the planning of your day. Here is an example from a student situation: a bad student is about to take a 15 credit and a 10 credit exams in a couple of days and hasn t started revising. How much time should they devote to revising for each exam? In this project, we consider a simple model of an extreme case: the Formula 1 Strategy Engineer. In Formula 1, each driver must complete a certain number of laps as quickly as they can and can stop to put on fresher tyres and refuel the car (this is a general rule, not necessarily applicable in the current Formula 1 regulations). While planning their race, they have to take into account a large number of parameters which will impact their performance, such as tyre wear and fuel load [1]. As you will see, the problem is too complicated to handle for the driver who has the demanding task to drive the car to its limits, so Formula 1 teams rely on Strategy Engineers. They are the brain of the team on race day. Ruth Buscombe (Figure 1) was a Strategy Engineer for the Formula 1 team Sauber and commented on her work in a video interview [2]. Figure 1: Ruth Buscombe is a Formula 1 Strategy Engineer for the Sauber F1 Team. After [2]. Since all decisions depend on the level of performance of the car, the most important element in a Strategy Engineer s work is a good lap time modeller. We introduce this tool in Section 2. The task of the Strategy Engineer being to decide when pit-stops have to be made to change tyres and refuel so as to optimize the overall pace of the car, we introduce the concept of pit-stop and optimum strategy in Section 3. Section 4 provides a list of questions for your MATH3001 project. Lastly, in Section 5, we introduce a competition to test your understanding of race strategy. Will you beat your lecturer? Lap time modelling Hypothesis It is a good first approximation to assume that the lap time is affected by tyre wear and fuel load: the more a set of tyres is used, the less performant it becomes; the more fuel onboard, the heavier and slower 2

3 the car becomes. Neglecting any nonlinear coupling, we can write: t = t b p t p f, (1) where t is the lap time, t b is the base time, corresponding to the perfect/fastest possible lap, p t is the time penalty due to tyre wear and p f is the time penalty due to the fuel load onboard. Coming up with the laws associated with these quantities is the job of the Strategy Engineer and involves collecting data on past races and collapsing them onto simple formulae. 2.2 Example Let us assume that: t b = 80s, (2) p t (l) = T l, (3) p f (l, l max ) = F (l max l), (4) where T > 0 is the additive time penalty per lap due to tyre wear, F > 0 is the additive time penalty per lap due to the weight of fuel onboard, l is the number of laps already completed and l max is the number of laps of fuel onboard at the start of the stint. In this example, the lap time (1) becomes: t(l, l max ) = 80 F l max (T F )l. (5) If we further assume that the tyre and fuel penalties account for 1 second per lap each, T = F = 1s, we have: t(l max ) = 80 l max. (6) For a 4 lap race, the total race time is: l=4 t race = t(l, 4) = 4(80 4) = 336s, (7) l=1 so that the driver completes the race in 336s. 3 Optimum strategy 3.1 Pit-stops During a pit-stop, the driver stops the car to change tyres and refuel. This allows the car to be lighter (it does not have to carry the fuel load necessary to complete the race from the start) and to wear fresher, and thus faster, tyres. Both of these account for an increased performance and there is, thus, a tradeoff between the time wasted pitting and the time gained by running a lighter car on fresher tyres. Calculating the optimum number of pit-stops and when to make them is an optimization problem that can win races. Figure 3.1 shows the 2016 Abu Dhabi Grand Prix race report by tyre manufacturer Pirelli. The drivers were allowed to use three tyre compounds, from hardest (slowest) to softest (fastest): Soft, Supersoft and Ultrasoft (sigh). Several strategies were used during the race: Hamilton and Rosberg opted for a two stop strategy using Ultrasoft, then Soft and finally Soft tyres; Vettel and Alonso made the second stint longer and used Supersoft tyres in their last stint; Verstappen opted for a one stop strategy with Supersoft and Soft tyres; etc. 3

4 Figure 2: Pirelli race report from the 2016 Abu Dhabi Grand Prix showing the pit stop strategy used by each driver. Soft tyres are shown in yellow and are the hardest compound, followed by Supersoft tyres, in red and Ultrasoft tyres, in purple. After [6]. 3.2 Example Let us continue on our 4 lap race example and assume that a pit-stop costs t p = 7s and only consider one pit-stop strategies. Since the race lasts 4 laps, there are only three possibilities for the pit-stop: lap 1, lap 2 and lap 3. Let us calculate the race time for each of these strategies Pitting on lap 1 If we decide to pit on lap 1, the total race time becomes: t race = t stint (1) t p t stint (3), (8) where t stint (l) is the stint time corresponding to a stint of l laps. Here, we have: l=1 t stint (1) = t(l, 1) = 81s, (9) l=1 l=3 t stint (3) = t(l, 3) = 249s, (10) l=1 4

5 such that the first stint (until the pit stop) lasts 81s and the second stint (between the pit-stop and the end of the race) lasts 249s. The race time is then t race = = 337s, which is longer than the 336s obtained from the zero pit-stop strategy Pitting on lap 2 If we decide to pit on lap 2, the total race time becomes: with: t race = t stint (2) t p t stint (2) = 2 t stint (2) t p, (11) l=2 t stint (2) = t(l, 2) = 164s (12) l=1 so that the total race time is t race = = 335s, which is shorter than the 336s obtained from the zero pit-stop strategy Pitting on lap 3 Due to obvious symmetry reasons, this calculation is the same as that in Section and we obtain 337s, which is slower than the zero pit-stop strategy by 1s Conclusion On this specific example, the one stop strategy is the fastest provided that the pit-stop is made on lap 2. The driver will then complete the race in 335s. The zero pit-stop strategy is 1s slower while the worst strategies were one pit-stop strategies with the pit-stop either on lap 1 or on lap 3. 4 Questions (Data1) Obtain a database of lap times and find the corresponding value for T and F. Hint: You can find such databases online, e.g. [3, 4]. (Data2) Are the linear laws (3) and (4) good approximations for the tyre wear and fuel load effects? If not, can you come up with a better fit? (Data3) Are there other effects that would modify equation (1)? (Strat1) The discrete summations are an inefficient way to calculate the race time (equation (7)) or stint time (equations (9) (12)). Could you come up with a better formulation for these calculations? Hint: Some help can be obtained in [5]. (Strat2) How can we model the duration of a pit-stop t p? (Strat3) Assume you enter a 10 lap race and consider the lap time model you obtained in Questions (Data1) (Data3) and the pit-stop duration you obtained in Question (Strat2). What is the best strategy, i.e., how many pit-stops should you make and when? (Strat4) How sensitive is this strategy to changes in the parameter values (pit-stop duration, tyre wear effect, fuel load effect)? (Strat5) Same question as Questions (Strat3) and (Strat4) but for an N lap race, N > 10. 5

6 5 Let s compete! This is more interesting than real F1 (Ati Sharma, Associate Professor of Aerodynamics at the University of Southampton) We are now going to test your understanding of the pit-stop optimization problem by setting up a competition. The competition will take place in three steps: (i) you will build your racing team by hiring drivers and investing in your car, (ii) you will decide on the best race strategy based on your understanding of the pit-stop optimization problem and (iii) a ten race season will be run (and livestreamed) where you can follow your performance. 5.1 Build your racing team You are at the head of a new Formula 1 team and you need to hire two drivers and develop your car. To this purpose, the team sponsors have given you a budget of 10ı (pronouned like the imaginary unit this is an imaginary game!). You will not be allowed to take a loan or go bankrupt and are allowed to spend it with a precision of one decimal place Sign drivers The first thing you need to do is to invest some of this budget into marketing to attract the best drivers. The more you invest in marketing, the more you attract drivers, but this is a competitive market: your competitors will do the same. Let m i be the marketing investment of team i and n be the total number of competing teams. Any given driver will be wanting to sign a contract with team i with probability: p i = mh i n j=1 m h j, (13) where h = 2 is the Hill exponent. There are various drivers of different ability. The driving level is assessed from 0 (worst) to 5 (best) and impacts linearly the performance on track Develop your car Besides completing your driver lineup, you need to invest the rest of your budget in your car. There are two components to the car: the chassis and the engine. Some circuits tend to favour cars with a strong chassis while some others favour those with a powerful engine. Each ı that you invest in the chassis and he engine will produce a performance equivalent to one and impact the performance on track linearly with the same rate as the driving level of your drivers Example Team A, Team B and Team C have 10ı of budget and there are 6 drivers on the market: Archibald (driving: 4.8 ), Beulah (driving: 3.9 ), Cecil (driving: 3 ), Dagobert (driving: 2.1 ), Earl (driving: 1 ) and Fritz (driving: 0.2 ). We consider the case h = 2 in this example. Team A decides to spend 4ı on marketing, Team B 1ı and Team C to not spend anything. Drivers have thus a probability p a to be attracted to Team A, p b to be attracted to Team B and p c to be attracted to Team C, where: p a = 16 17, p b = 1 17, p c = 0. (14) 6

7 A number between 0 and 1 is then drawn from a uniform distribution for each driver, starting from the most desirable (highest driving rating) one. If this number is smaller than p a, then the driver signs with Team A, otherwise, if it is smaller than p a p b, they sign with Team B, otherwise they sign with Team C. In this example, Archibald signs with Team A, Beulah with Team B and Cecil with Team A. At this stage, Team A has a full lineup and does no longer seek to attract drivers. Remain Team B and Team C with updated probabilities: p b = 1, p c = 0, (15) calculated in the same way as before but without the contribution from Team A. Dagobert signs with Team B. Only Team C has free seats, so Earl and Fritz take them. After the driver signing stage, Team A has 6ı left, Team B has 9.0ı left and Team C has 10ı left. Further to this, Team A decides to spend 4ı on developping their engine and 2ı on their chassis, Team B decides to spend 7ı on their chassis and 2ı on their engine and Team C decides to invest 5ı on their chassis and the same amount on their engine. The team strengths before the season is summarized in table 1. Team B and C end up stronger than Team A despite having less talented drivers as a result of Team Chassis Engine Driver Driving Overall Team A 2 4 Archibald Cecil 3 9 Team B 7 2 Beulah Dagobert Team C 5 5 Earl 1 11 Fritz Table 1: Summary of the team lineup and strength from the example used in Section 5. their more cost effective marketing plan Questions (Proba1) Let us consider two teams, Team A and Team B, looking for one driver each. There are two drivers on the market: driver 1 with a 1 driving level and driver 2 with a 0 driving level. The probabilistic attraction law for each driver is given in equation (13) for h = 1. What is the optimum marketing investment for each of the teams? (Proba2) Generalize your result to any h and for any driving level d 1 and d 2? (Proba3) What happens when three teams compete for 3 drivers? (Proba4) What happens when n teams compete for n drivers? (Proba5) Same question as Questions (Proba1) (Proba4) but when the teams need two drivers each and for twice as many drivers as there are teams. 5.2 Lap time calculator To spice up the competition, we introduce some new rules: DNF: as this is motorsport, it is possible to suffer a mechanical failure and abandon the race. DNF stands for Did Not Finish [the race]. This occurs with a probability of 1/7 for any driver and for any race. 7

8 Set-up: to simulate the effect of free practice sessions, a random uniform variable between 0.5 and 0.5 will be drawn for each team and for each circuit, then a similar one for each driver on each circuit. The sum of these variables is expressed in (it is between 1 and 1 and two drivers from the same team cannot have a result more than 1 apart) and constitutes the set-up modifier to the performance of the driver. Traffic: it takes more than just being faster to overtake. Cars cannot get closer than a minimum gap from the car ahead and it takes a certain performance gap to successfully overtake a car. This quantity is circuit specific. Drag Reduction System: any driver following another one within less than 1s will be allowed to decrease the angle of attack of their rear wing during the straight line. This has as an effect to reduce the drag applied to the car, thus to improve top speed and to provide a gain of 0.3s per lap. Humanness: a random function has been added to the laptime calculator so that drivers display realistic pace irregularities. The laptime is calculated in the following way: t = t b t perf p t p f R, (16) where t b is the base time, t perf is the base performance proportional to the number of of your driver (driving chassis engine set-up, with suitable bonuses depending on the circuit as explained further below; each accounts for an improvement of 0.15s per lap), p t the tyre wear penalty, p f is the fuel load penalty and R a random function. These quantities are explained further below. There are three different tyre compounds with different tyre wear effect: Soft, Medium and Hard tyres. These tyres change from one circuit to another and you can race with whichever you want and change tyre compound at each pit-stop. The tyre laws are explained further below. During a pit-stop, your pit crew will change your tyres as well as refuel your car. During a pit-stop, the driver loses time in two different ways. The first one is by getting into the pit lane which has a speed limitation. The second one is during the action of the mechanics. Changing tyres typically takes a couple of seconds (F1 mechanics are highly skilled!) and it takes 1s for the team to pump enough fuel into the car for a 2 lap autonomy. As a result, a pit-stop costs: t p = t pb 0.5l max, (17) where t pb is the time wasted due to the pit lane limitation and l max represents the number of laps in the next stint. The quantity t pb is circuit specific and the full specified pit-stop law is given for each circuit further below. 5.3 Championship rules The season consists in 10 races on circuits with different characteristics. The list of these circuits and characteristics is reported further below. Each race consists in one lap of qualification to determine the starting grid and a full length race of a duration of about 90min. The qualifications will take place over one lap. There will only be one car on the circuit at any time and the order in which the drivers do their qualification lap is random. The lap will be calculated in the best conditions (fresh tyres of the same compound as the tyres used at the start of the race and minimal fuel load). Each driver will start the race on a brand new set of tyres of the same compound as the tyres used during the qualifications. At the end of the race, drivers and teams score points. The number of point scoring drivers depends on the number of drivers registered for the championship: 8

9 Less than or 12 drivers: 10pts, 6pts, 4pts, 3pts, 2pts, 1pt. Less than or 16 drivers: 10pts, 8pts, 6pts, 5pts, 4pts, 3pts, 2pts, 1pt. Less than or 20 drivers: 20pts, 16pts, 13pts, 10pts, 8pts, 6pts, 4pts, 3pts, 2pts, 1pt. More than 20 drivers: 25pts, 20pts, 16pts, 13pts, 11pts, 10pts, 9pts, 8pts, 7pts, 6pts, 5pts, 4pts, 3pts, 2pts, 1pt. Championship tie-breakers will be made by comparing the best race result, then the second bests result, etc. The number of team points corresponds to the sum of the points of their drivers. 5.4 List of circuits 1. Australia (58 laps, t b = 84s) Balanced circuit Overtake delta: 0.8s p f = 0.08(l max l) t p = l max Soft: p t = l 2 Medium: p t = 0.02 l 0.2 Hard: p t = Spain (66 laps, t b = 80s) Balanced circuit Overtake delta: 0.8s p f = 0.09(l max l) t p = l max Soft: p t = exp(0.06 l) 1 Medium: p t = 0.04 l 0.25 Hard: p t = 0.02 l Monaco (78 laps, t b = 74s) Favours driver Overtake delta: 1.0s p f = 0.06(l max l) t p = l max Soft: p t = 0.08 l Medium: p t = 0.12 l 0.4 Hard: p t = Belgium (44 laps, t b = 105s) Favours chassis and engine Overtake delta: 0.8s p f = 0.13(l max l) t p = l max Soft: p t = exp(0.1 l) 1 Medium: p t = l Hard: p t = 0.04 l 0.7 Every race is unique: some favour specific elements of your team (Driving, Chassis or Engine), some provide easier overtaking, and some consist in more laps than others. When a circuit favours a specific 9

10 5. Italy (53 laps, t b = 85s) Favours engine Overtake delta: 0.7s p f = 0.11(l max l) t p = l max Soft: p t = exp(0.07 l) 1 Medium: p t = 0.14 l 0.35 Hard: p t = Hungary (70 laps, t b = 80s) Favours chassis Overtake delta: 1.0s p f = 0.08(l max l) t p = l max Soft: p t = exp(0.07 l) 1 Medium: p t = 0.05 l 0.35 Hard: p t = United Kingdom (52 laps, t b = 90s) Favours driver and engine Overtake delta: 0.8s p f = 0.11(l max l) t p = l max Soft: p t = exp(0.08 l) 1 Medium: p t = l 0.4 Hard: p t = 0.03 l Japan (53 laps, t b = 90s) Favours driver and chassis Overtake delta: 0.8s p f = 0.10(l max l) t p = l max Soft: p t = exp(0.09 l) 1 Medium: p t = l Hard: p t = 1 9. Canada (70 laps, t b = 74s) Balanced circuit Overtake delta: 0.9s p f = 0.08(l max l) t p = l max Soft: p t = l 2 Medium: p t = 0.13 l 0.1 Hard: p t = Brazil (71 laps, t b = 72s) Balanced circuit Overtake delta: 0.7s p f = 0.08(l max l) t p = l max Soft: p t = l 2 Medium: p t = 0.03 l 0.3 Hard: p t = 0.02 l

11 characteristic, this element counts double is the performance calculation. For instance, at Monaco, the drivers of Table 1 will have the following performance: Archibald (Team A) 15.6, Cecil (Team A) 12, Beulah (Team B) 16.8, Dagobert (Team B) 13.2, Earl (Team C) 12 and Fritz (Team C) How close can one follow a car and how little performance gap is needed to overtake is what is indicated by the Overtaking delta variable. 5.5 Competition timeline All interaction will be made via ing: c.m.l.beaume@leeds.ac.uk. Failure to meet a deadline or formatting will result in penalties. First, you need to declare your intention to participate by Monday 12 November 2018 by sending an to the above address. The driver market will be announced shortly after and will open the marketing phase. During this phase, you will need to indicate how many ı you want to spend in marketing (you have a 10ı budget and are allowed to spend non-integer values of ı only one decimal place is allowed), as well as in the chassis and engine development. The deadline for the marketing phase is Monday 21 January 2019 (indicative date). The driver signing phase will be livestreamed shortly after and you will be notified of its time and link. Your race strategies will have to be submitted by to c.m.l.beaume@leeds.ac.uk by Monday 18 March 2019 (indicative date). To submit your strategy, you need to specify the track, the driver, their starting tyres (on which they will qualify and do their first stint) and when they pit as well as the new tyres. Team A would submit their strategies by sending: 1. A u s t r a l i a A r c h i b a l d S 10 M 30 H 60 C e c i l H 30 H Spain A r c h i b a l d M 20 H 50 M 70 C e c i l M 20 M 40 M 60 S The best way to send this is within the text of your (do not send spreadsheets or attachments). Each Grand Prix will consist in 3 lignes: the Grand Prix name and one line per driver. The Grand Prix have to be presented in the order given in Section 5.4. Separate each tyre and lap indications by a space and terminate the strategy by the number of laps comprising the race. Please respect this format! In this example and for the Australian Grand Prix, Team A will have Archibald qualify on soft tyres and race with them until lap 10 where he will pit for medium tyres, race 20 laps with them and pit on lap 30 for hard tyres which he will keep until the end of the race. Races will be run and livestreamed on every week day from Monday 25 March 2019 to Friday 5 April 2019 and posted online at References [1] Formula 1. Race strategy. strategy.html. [2] Formula 1. Ruth buscombe on the role of the f1 strategy engineer. Youtube,

12 [3] J. Bouchet and E. Wojcikiewicz. Formula 1 race data [4] K. Collantine and D. Rencken. Racefans independent motorsport coverage [5] W. Mulholland. Formula one race strategy. Royal Academy of Engineering. [6] Pirelli. Pit stop summary from the 2016 abu dhabi grand prix. 12