Racing Simulation of a Formula 1 Vehicle With Kinetic Energy Recovery System

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SAE TECHNICALPAPER SERIES 2008-01-2964

Racing Simulation of a Formula 1 Vehicle with Kinetic Energy Recovery System

Aldo SorniottiUniversity of Surrey

Massimiliano CurtoPolitecnico di Torino

Motorsports Engineering ConferenceConcord, North Carolina

December 2-4, 2008

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ABSTRACT

This paper deals with the development of a Lap Time Simulator in order to carry out a first approximateevaluation of the potential benefits related to the adoption of the Kinetic Energy Recovery System (KERS). KERS will be introduced in the 2009 Formula 1 Season. This system will be able to store energy during braking and then use it in order to supply an extra acceleration during traction. Different technologies (e.g. electrical, hydraulic and mechanical) could be applied in order to achieve this target. The lap time simulator developed by the authors permits to investigate the advantages both in terms of fuel consumption reduction and the improvement of the lap time.

INTRODUCTION

Starting from 2009 FIA allows the adoption of a Kinetic Energy Recovery System, characterized by 60 kW of maximum output power measured at the driven wheels. The energy released by KERS may not exceed 400 kJ during a single lap.

According to the new Formula 1 regulations, there is no particular specification or limitation related to the kind of technology which could be adopted for the development of KERS. From a theoretical viewpoint, KERS could be based on the following technologies:

• Mechanical technology based on the adoption of flywheels;

• Electric hybrid technology based on the adoption of supercapacitors;

• Hydraulic technology based on the adoption of a hydraulic motor/pump connected to an accumulator.

This work deals with the investigation, through lap time simulation, of the new opportunities (in terms of the overall vehicle performance) offered by the implementation of this system, both in terms of lap time and overall fuel consumption. In particular, a lap time simulator has been adopted in order to simulate the performance of different vehicle layouts and to evaluate the basic performance improvement related to the new system (which will be activated by the driver in the most suitable areas of the tracks – a controlled activation is not permitted).

The basic hypotheses of the adopted simulation software are discussed. Then the layouts related to the different options for the development of KERS are briefly presented. In particular, the lap time simulator presented in this paper has been developed in the context of a research activity focused on the comparison between the electrical layout and the mechanical layout based on the adoption of a flywheel as energy storage device. The paper deals with the simulation models which have been developed during this preliminary analysis. The simulation results of the paper are related to a vehicle equipped with a KERS system based on the adoption of electrical components.

Firstly, Dynamic Programming has been adopted (and the methodology is explained) for the optimization of vehicle performance in terms of fuel consumption, for an established trajectory in terms of vehicle speed. Some qualitative and quantitative results related to possible fuel saving strategies are analyzed in detail. Secondly, a brute force method is presented for lap time optimization (the limitations due to the adoption of

2008-01-2964

Racing Simulation of a Formula 1 Vehicle with Kinetic Energy Recovery System

Aldo Sorniotti University of Surrey

Massimiliano Curto Politecnico di Torino

Copyright © 2008 SAE International


dynamic programming are discussed). The principles for KERS optimal control are derived through basic theory (energy balance equations applied to simple case studies) and lap time simulation.

Finally, the lap time simulation software has been further developed in order to simulate the entire race, including the pit-stop laps (comprehensive of the braking phase along the pit lane, the refueling phase and the subsequent re-acceleration). As the simulator considers the whole race, a pit-stop strategy optimization algorithm is provided and discussed. A comparison, in terms of the overall performance during the race, between the fuel saving strategies and the lap time optimization strategies is dealt with in the final section of the article.

The conclusion of the paper deals with the future developments of the activity, related to the enhancement of the presented lap time simulator and the possible opportunities offered by an extended adoption of KERS (in terms of values of power and energy released per lap).

FORMULA 1 LAP TIME SIMULATOR

The main aim of the developed simulator is to compute comparative lap times so that the KERS parameters and control can be optimized. The simulator has been implemented in MATLAB®.

The Lap Time Simulation (LTS) package of the University of Surrey has been adopted. This software package includes steady state Lap Time Simulators characterized by increasing levels of complexity, in terms of degrees of freedom of the vehicle during the simulation process. Within this specific activity, the basic level of vehicle dynamics model has been adopted, in order to reduce the computation time of the procedure. The target is a first approximation of the real benefits related to the adoption of KERS (and not a detailed quantitative analysis on a specific vehicle, which would require the most sophisticated simulators of the package).

VEHICLE DYNAMICS MODEL – The adopted LTS allows the optimization of vehicle performance by simulating a F1 car negotiating the circuit in any of its possible setup combinations.

In order to simulate a full lap, the circuit is discretized into segments (typically 1 meter long). For each segment, vehicle lateral acceleration is evaluated as a function of the longitudinal speed and the path radius. The cornering force required to maintain this lateral acceleration at a given level of bank angle of the track can be computed. By using a friction ellipse approach, tire force available in longitudinal direction is found [5]. This value is limited by the maximum traction force due to the engine (or it is not limited, if the traction force due to the engine is larger than the available force according to the friction ellipse).

If the friction ellipse is not satisfied, vehicle velocity is reduced (and as a consequence the values of the lateral forces) by steps until the computed tire forces are within the friction ellipse. This procedure ensures that the total combined lateral and longitudinal force generated by the tire is consistent with the maximum available grip. This approach considers the vehicle at its limits along the track, so that the tires are working near their critical condition in terms of grip. The friction ellipse formulation considers the different tire behavior in traction or braking. However, it does not take into account the temperature effect on the grip limit.

The simulator computes the braking forces by assuming them to be kept at the maximum possible value by the driver (according to the friction level and the friction ellipse approach). The model considers aerodynamic forces in the form of experimental data as functions of vehicle velocity.

In order to evaluate vehicle behavior along the whole circuit and the lap time, the vehicle has to negotiate the entire track four times in different conditions. These are:

1. During the first lap, the initial speed at the starting line of the track is evaluated. In the first lap, the vehicle is characterized by a null initial speed, which increases according to the layout of the track and vehicle characteristics. The final value of vehicle velocity computed during this first lap is adopted as starting point for the second lap;

2. Vehicle dynamics during traction is computed (second lap). This is a repetition of the first lap, with an initial condition in terms of vehicle velocity;

3. Vehicle behavior during braking is computed (this lap is a reverse lap, third lap);

4. A merge between the results of the second and third laps is carried out;

5. Vehicle performance estimated at point 4. is corrected by considering KERS contribution to the longitudinal forces during traction and braking (fourth lap).

Surrey’s Advanced Vehicle Analysis Group (SAVAG) has developed more sophisticated LTS software (in terms of detailed simulation of vehicle motion), which will be adopted for the second approximation analysis of KERS performance. The most sophisticated software developed by SAVAG includes the simulation of the full motion of vehicle sprung mass and unsprung masses, the simulation of suspension non-linear elasto-kinematics (including the effects related to the generation of jacking forces, roll centre motions and suspension compliances), and tire thermal effects (variation of tire cornering stiffness, tire longitudinal slip stiffness and longitudinal/lateral friction coefficients as functions of the tire temperature level estimated by the software).

POWERTRAIN MODEL – A map-based engine model is implemented into the software. It permits to achieve an output in terms of engine power/torque and engine fuel


consumption/efficiency as functions of engine speed and throttle position.

The gearbox model takes into account the time lag during gearshifts. Both sequential and non-sequential models are implemented (for general race applications). The gearbox is automatically actuated by the simulation software as a function of engine speed values. Upshift is actuated when engine speed reaches a threshold; downshift is actuated when engine speed falls below a lower threshold. In particular, the model computes the gear ratios to be adopted along the circuit within the second lap (according to the procedure described in the former section). The gearbox strategy allows the vehicle to have the best gear at the beginning of each traction zone. Thanks to this strategy the maximum acceleration is ensured.

BRAKING SYSTEM MODEL - The simulator provides a simplified braking system model. It allows the evaluation of KERS influence on brake forces and their distribution. Figure 1 shows the braking system schematic. The model computes, for the considered section of the track, a reference value of total braking power (Pbraking). The power related to engine brake effects (Pbraking,ICE) is subtracted to Pbraking, so that the total brake power P’brakingrelated to the conventional brakes and KERS is obtained. If KERS can give origin, in the actual working conditions of the vehicle on the track, to a brake power PKERS,MAXlevel which is higher than P’braking, the dissipative brake system should not be used by the driver. Otherwise the required levels of front and rear brake pressures (pfrontand prear) are computed as a function of the actual brake distribution factor BD (due to the hydraulic and mechanical layout of the brake system) and as a function of vehicle parameters, like velocity V and the size of the braking system components.

Figure 1 – Basic principle of braking system model with KERS

The most sophisticated level of the SAVAG LTS software is capable of considering the variation of brake pads friction coefficients as functions of the estimated temperature of the friction surfaces, brake pressure and brake disc speed.

KERS MODEL – The simulator considers three different layouts for the capacitor based system (Figure 2) and one layout for the mechanical system (Figure 3). The efficiency of the different components of the powertrain has been considered. Also the effect of the inertia of the components of the powertrain has been simulated.

Figure 2 – Electrical layouts for KERS included in the implemented LTS (for the abbreviations, refer to the explanation in the final section of the paper)

Figure 3 – Schematic of the mechanical solution for KERS adopted within the LTS

BASIC CALCULATIONS ABOUT KERS

On the basis of the typical time histories of vehicle speeds along the circuits in previous races, it is possible to demonstrate that in all the tracks currently adopted within the Formula 1 Championship (Silverstone is the track which implies the lowest level of regenerated energy), the energy level which can be stored by KERS through the regeneration of the rear braking forcesgreatly exceeds the level of 400 kJ (during a single lap) specified by the rules.

Also brake powers can easily exceed the level of 60 kW specified for KERS. In fact, in Formula 1 the duration of brake maneuvers is usually very short, but peak values of some thousands kW (nearly 4000 kW, Figure 4) canbe dissipated within the brake system.

A simple procedure to evaluate the power flows generated during braking is provided. Through equation


(1), the power due to the aerodynamic drag force is computed:

3

21 VCS�P DRAGAIRDRAG ⋅⋅⋅⋅= (1)

AIR� is air density, S is vehicle front surface, DRAGC is

the aerodynamic drag coefficient, V is vehicle velocity. The drag contribution related to the rolling resistance coefficient of the tires is:

( ) VgmVKfP RESISTANCEROLLINGTIRE ⋅⋅⋅⋅+= 20_, (2)

0f and K are the factors which contribute to the computation of the rolling resistance coefficient (which is a polynomial function of vehicle velocity), m is vehicle mass, g is gravity. Tire rolling resistance can be associated to the fact that the vertical force between the tire and the road surface is applied to the front part of the contact patch (and not to the centre). Then, the total vertical force on the four tires can be calculated – through equation (3) – and finally the power dissipated within brakes (4).

2

21 VCS�gmF DOWNAIRZ ⋅⋅⋅⋅+⋅= (3)

( ) V�FP ZefficiencyBRAKING ⋅⋅=%100, (4)

ZF is the total vertical force between the four tires and

the road pavement, DOWNC is the downforce

aerodynamic coefficient, efficiencyBRAKINGP %100, is the maximum brake power which can be generated by the friction contact between the tires and the road, � is the friction coefficient between the tires and the road surface. The power related to equation (4) – it is the braking power in case of a braking system characterized by a 100% efficiency level – is due to the effect of engine brake, KERS and dissipative brake. After the first two contributions have been computed, the repartition factor (Figure 1) between the front and rear brakes can be adopted for the computation of the power levels on the two axles, by considering the limitations related to the real efficiency of the brake system. The total power related to the decelerationa of the vehicle is equal to:

SYSBRAKINGKERSBRAKEENGINE

GRADIENTALLONGITUDINRESISTANCEROLLINGTIRE

DRAGeqONDECELERATIVEHICLE

PPP

PP

PVamP

,_

__,

_

+++

+++

+=⋅⋅=(5)

ONDECELERATIVEHICLEP _ is the power contribution related to vehicle deceleration, a is vehicle deceleration,

GRADIENT_ALLONGITUDINP is the power contribution related

to the longitudinal slope of the road, BRAKEENGINEP _ is the power contribution related to engine friction torque,

KERSP is the power contribution related to KERS

activation, SYS,BRAKINGP is the power contribution related to the intervention of the hydraulic brakes. The equivalent mass eqm of the vehicle is equal to:

�+=i

iieq R

�Jmm 2

2 (6)

iJ is the moment of inertia of the ith component of the

driveline, i� is the gear ratio – referred to the vehicle – of the ith component of the driveline, R is wheel radius.

Figure 4 – Power Flows vs. Vehicle Speed: brake power (total value and contribution of the rear brakes), power related to the aerodynamic drag, maximum power regenerated by KERS, total power related to vehicle deceleration

Figure 4 plots the main power contributions related to equations (1)-(5) vs. vehicle longitudinal speed. The vehicle data adopted for these calculations are referred to a race car of a few years ago. It is possible to observe that the power level of the rear brakes exceeds the maximum level (60kW) which can be recovered by KERS for vehicle velocities below 50 km/h. This means that the impact of the KERS system is not very significant from the viewpoint of the brake distribution between the front and the rear axles (as KERS contribution will be a minority of the total required brake power, for typical values of vehicle velocity). As a consequence, vehicle dynamics during braking maneuvers should be marginally affected by the new system (even if a proper detailed analysis needs to be carried out).

Figure 5 summarizes the results of basic calculations for a brake maneuver carried out from an initial velocity of 300 km/h, with a final velocity of 100 km/h, for two levels of friction coefficient between the road surface and the tire (1.3 and 1.75). The calculations show that a low deceleration during the brake maneuver (for the same values of initial and final velocities) implies a higher level


of recovered energy, due to the longer duration of the maneuver (KERS is continuously recovering the maximum energy according to the Formula 1 rules, due to the very low value of maximum KERS power in comparison with the total brake power on the rear axle).

Figure 5 – Analysis of achievable braking forces and recovered energies for a Formula 1 vehicle involved in an extreme brake maneuver (initial speed equal to 300 km/h and final speed equal to 100 km/h), for values of the friction coefficient (‘mu’ in the Figure) between the tires and the road surface equal to 1.35 and 1.7

FUEL CONSUMPTION OPTIMIZATION (FCO)

At the moment KERS is conceived as a system in order to reduce the lap time without increasing vehicle fuel consumption. The same system could be potentially adopted, within a race strategy, in order to decrease fuel consumption by keeping a reference velocity profile of the vehicle along the track. If in the future the adoption of KERS will be extended and/or new rules about fuel consumption limitation will be implemented, KERS could become useful in order to decrease vehicle fuel consumption, rather than to increase the overall performance. As a consequence, this section presents an algorithm which permits:

• To find the ideal velocity profile of the vehicle without the adoption of the power boost related to KERS;

• To minimize fuel consumption through the optimized adoption of KERS along the track.

The FCO routine has been implemented within the presented LTS in order to determine the ideal KERS utilization within a single lap from the viewpoint of fuel consumption reduction.

FCO has been designed by supposing that the vehicle is traveling at the longitudinal speed that it would have by using the ICE only. The new regulations specify KERS as a device in order to give origin to a power-boost and do not specify a fuel consumption limit. The best way of using it according to the 2009 rules will consist of optimizing the performance of the vehicle from the viewpoint of the lap time, without paying any attention to

the overall energy efficiency of the vehicle. Under this hypothesis, software like FCO, which does not imply a lap time improvement, would not have any benefit. However FIA has mentioned [2] the future intention of limiting the overall fuel consumption of the vehicle. In addition to this, within a complete analysis and optimization of the race strategy, this software can be useful in order to evaluate the possible benefits related to the selection of the number of pit stops.

Figure 6 shows the schematic of the input/output variables of FCO. It computes the optimal power distribution between KERS and ICE along each traction zone of the track, as a function of the required wheel torque Twheel, the inserted gear, current vehicle velocity and the cumulated energy level Erecovered already used by KERS along the single lap.

Dynamic Programming (DP) guarantees the optimal control to obtain the lowest fuel consumption [6]. It helps FCO to consider all KERS energy release possibilities, maintaining the computational complexity in an acceptable range. The simulator evaluates vehicle maximum speed according to the limitations related to the grip condition and maximum engine power. For each simulation segment, the routine defines the operating point on the engine map. FCO uses this information and evaluates the best power split between ICE and KERS (Optimal Control). Engine power evaluated by the simulator is decreased by an amount equal to the specified KERS power. Then a new engine operating point and the related fuel consumption are computed.

Figure 6 – FCO input/output variables

DYNAMIC PROGRAMMING (DP) IMPLEMENTATION – KERS control problem involves the determination of the ICE and KERS power flow profiles. KERS optimal control defines the sequence of optimal power splits (at each time instant) that minimizes the fuel consumption over a given circuit. Formulated in this way, the problem can be handled with DP, i.e. through a procedure – capable of taking decisions at every single stage – that optimizes a global cost function. DP application has a global objective, the lowest overall fuel consumption, but also a global constraint. In fact, one of the most desirable features of KERS is to be charge sustaining: the quantity of energy in the rechargeable source should be the same before and after the lap. However, in the particular


application this point is not a problem, due to the limitations related to the current rules (400 kJ can be recharged for sure).

A method such as DP is extremely appropriate to find the solution. In order to explain the algorithm, a generic point of the track is now considered (Figure 7 – Point A). It is characterized by a spatial coordinate distance = 3 (from the starting line of the track) and a generic energy level En = 50% (En is the energy which can be still produced by KERS during the lap). The transition between this point and the spatial coordinate 4 can be carried out according to different strategies in terms of traction power flow through KERS.

Two limit strategies can be defined: the first one does not imply any power release (PKERS = 0% in Figure 7), whereas the second one implies the maximum rate of energy release (PKERS = 100% in Figure 7). The algorithm is capable of considering all the intermediate transitions (e.g. PKERS = 50%) between the two limit conditions. An energy discretization interval defined by the user is adopted. The same procedure is repeated for all the energy levels at each spatial coordinate.

Figure 7 – Schematic of the Dynamic Programming routine

Several potential policies (according to the defined parameters) reach the same energy level (e.g. 25%) at a specific distance (distance 4 in Figure 7). The best policy is the one characterized by the lowest fuel consumption cumulated until this point. In general, this means that a clear and unique condition to compare the different policies at the same energy level can be defined by the next formula:

if FCk(xi;En) < FCm(xi;En) � FCk(xi;En) is the best policy

DP automatically eliminates the worse policy and saves, for each admissible energy level at each spatial coordinate, the vector containing the information (in terms of cumulated fuel consumption and power level supplied by the KERS) about the best policies. In the last point of the track (distance = 6 in Figure 7), the best policy is obtained. Finally, by adopting a recursive rule

from the track end to its starting point, the best policy is reconstructed.

Figure 8 – Influence of energy level discretization on the FCO Accuracy and the simulation time (Core2Duo 2.13GHz – 1.96 Gb of RAM)

Figure 9 – Each track is divided by the software into traction zones (T1, T2, …, Ti) and braking zones (B)

Energy discretization value affects the optimization procedure and also simulation time. Figure 8 shows the relationship between the simulation time and the energy discretization unit (energy lattice) for a typical case study. The diagram also points out the influence of discretization on the optimization accuracy. FC(i) is the fuel consumption value obtained for the ith level of energy discretization. The best fuel consumption (FCbest) refers to the value obtained using FCO with the best resolution in terms of possible energy levels.

FCO RESULTS – The performance of the routine is analyzed on a typical circuit. The track (Figure 9) is divided (by the software) into the different traction and braking zones automatically computed by the LTS according to an ideal driving strategy. Figure 10 plots vehicle longitudinal speed along the track. Figure 11 shows engine operating points within the engine map along the third traction zone, for both the

Accuracy

Simulation time


conditions of KERS switched on and off. The data are not referred to a real F1 vehicle, this is a demonstration of the capabilities of the software. It is possible to point out that in conditions of low torque KERS tends to be deactivated since the specific fuel consumption surface decreases for higher engine torque values.

Figure 10 – Vehicle speed along the track - dry pavement (B: braking zones, Ti: traction zones)

According to the particular example of engine map data, KERS has to be activated by a small amount of power, nearly continuously along the entire track. This is the ideal control for the KERS along the track. For a real driver it would be very difficult to follow the optimal control, as it would imply a smooth modulation of the system (however it is a useful suggestion of what should be done). By observing Figure 11, it is also possible to affirm that the engine operating points are (for a significant duration of the considered traction zone) constrained by the friction limit between the tires and track surface and not by the engine power.

Figure 11 – Engine map with engine operating points along the third traction zone (the difference in terms of engine operating points with and without KERS is marginal – for example look at the area close to the ellipse)

Figure 12 shows fuel consumption (expressed in liter per 100 kilometers) along the track. KERS controlled by FCO allows optimizing fuel consumption. During an entire lap on the considered track, FCO allows a fuel consumption

reduction equal to 7.4% without increasing the lap time (this value is referred to the data of the specific case study, an actual current Formula 1 vehicle is thought to achieve fuel consumption reduction by 3-5% on the same track). The specific example of track is one of the longest in Formula 1, then by considering that on all the circuits KERS is able to recover 400kJ, it is possible to state that FCO would possibly have better performance (an higher fuel consumption reduction in percentage) on other circuits.

Figure 12 – Fuel consumption [l/100km] along the considered track with and without KERS (controlled by FCO)

CONCLUSIONS ABOUT FCO – FCO guarantees the optimal control in terms of fuel consumption optimization. Thanks to this software it is possible to state that the fuel saving due to FCO is significant. FCO should become a key point in order to compute the optimal control if future rules will be imposed in order to limit fuel consumption during a race.

LAP TIME OPTIMIZATION (LTO)

The aims of this section are:

• The discussion of the reason why Dynamic Programming cannot be adopted within Lap Time Optimization;

• The description of the implemented software, based on the minimization of the lap time on an assigned track, through an effective adoption of the KERS;

• An overall analysis of the parts of the tracks where the power boost related to KERS can give origin to the most effective result in terms of lap time reduction.

LTO aims at optimizing the lap time by using KERS energy to give an extra power boost to the vehicle. DP cannot be adopted within LTO. Let us consider the case of an ideal section of a track consisting of two straights linked by two bends characterized by the same curvature radius. Let us suppose that the first straight is 1000 m

KERS Off

KERS On


long, whereas the second straight is 2000 m long. As the curvature of the bends before each straight is the same, the initial velocity of the vehicle on each straight will be the same (the bends can be neglected in the analysis). In a very first approximation, the velocity of the two vehicles along the two bends can be assumed equal. Let us compare the performance of two alternative control algorithms for KERS, the first one characterized by the full utilization of KERS power during the first part of the first straight, the second one characterized by the full utilization of KERS power during the first part of the second straight. For example, if KERS can give origin to its power boost for a distance equal to 300 m (after which the energy available for a single lap is over), after 1300 m (the bends are neglected) the two alternative control strategies would be characterized by the same residual energy level for KERS (zero energy level).According to a DP algorithm based on the time required for traveling from the starting point to the considered coordinate of the track, the first algorithm would be the better, as the intervention of the KERS would give origin to a benefit along the entire first straight (1000 m long, the velocity benefit related to KERS power boost is effective also after its 300 m long intervention) whereas the second algorithm would have given origin (for the coordinate x = 1300 m) to a benefit limited to the first 300 m of the second straight. However, from a real viewpoint of the laptime, the second algorithm would be better, as it would give origin to a benefit extended to the last 2000 m of the second straight (it is better to achieve an increase of vehicle velocity for a length equal to 2000 m rather than 1000 m). A DP algorithm based on time could be implemented only if the comparison for the definition of the optimal policy were carried out for a given value of energy level (as in FCO) and also vehicle velocity (this is the additional factor). As a consequence, for each spatial coordinate of the circuit, a set of possible energy levels and vehicle velocities should be considered. The result is a significant increase of the complexity of the procedure, which becomes a bidimensional optimization procedure.

As a consequence, a brute force method (brute force method or exhaustive search is a trivial but very general problem-solving technique, that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem stated) has been implemented. The algorithm has different steps:

1. LTO identifies the different traction zones (zones along which a positive engine torque needs to be applied);

2. LTO orders the different zones according to their lengths, as the length of the traction zone is the main index for a potentially high benefit of KERS in terms of lap time (as a function of what discussed within the previous trivial example which demonstrated the problems related the implementation of DP within LTO);

3. LTO asks the user how many traction zones he wants to consider within this analysis. The software automatically considers the zones starting from the longest one;

4. LTO starts an iterative algorithm that evaluates the lap time against all the possible energy release distributions between the different traction zones. The result is a simulation campaign whose dimension depends on the number of considered traction zones and on the energy unit discretization adopted for the distribution of KERS activations in the different traction zones;

5. LTO chooses the best strategy and carries out a final simulation by considering this energy profile.

ANALYSIS OF THE RESULTS – In this section the effect of the main parameters having an influence on the lap time is analyzed.

Activation point and power rate – This section shows the optimal activation point for a sample oval track. KERS activation point refers to the point where KERS starts releasing its energy. In particular, this section analyzes the influence of the distance between the activation point and the initial point of the considered traction zone (i.e. the initial point of each straight in the specific example). Different simulations have been carried out to identify the best point to discharge KERS energy. The lap times obtained by releasing KERS energy at different distances from the beginning of the traction zone are plotted in Figure 13. KERS achieves the best performance if activated at the beginning of each straight.

Figure 13 – Activation point analysis in terms of lap time (LT) and lap time variation (LT(i)-LTmin)) (50% of KERS energy released within each straight section)

By considering the results attached to this section, it is demonstrated that the best KERS activation point is the first point of the traction zone. In the same way, it has been proved that the optimal control is obtained if the KERS energy is released at the maximum possible rate (Maximum Power).

Initial Velocity Influence on the optimal control – This analysis aims to demonstrate that the KERS energy discharging strategy is more convenient on a traction zone where the initial velocity (the velocity at the first


point) is lower rather than higher. Figure 14 is related to a track characterized by two straight sections having the same length. The first straight is characterized by a lower initial velocity than the second one. This is due to the layout of the bends which are located at the entrance of each straight section.

Figure 14 – Traction zone: initial speed analysis (velocity profiles according to two different strategies are compared and the related time difference in terms of time – DeltaTime – is plotted)

The result is related to two different effects. Firstly, the non conservative forces are lower if the speed along the straight section is lower. Secondly, and most importantly, if we consider kinetic energy conservation, if the longitudinal speed is lower, it is possible to achieve an extra acceleration. By neglecting the effect of the non conservative forces (this approximation is not realistic at all – it is now adopted just to explain the basic concept), it is:

( )22

21� initialfinaleq VVmE −⋅⋅= (7)

E� is the energy boost due to KERS activation, initialVis vehicle velocity at the beginning of KERS activation,

finalV is vehicle velocity after KERS activation. (7) gives origin to:

2�2initial

eqfinal V

mEV +⋅= (8)

The speed gain due to the energy released by KERS strongly depends on the initial speed. For example, if the initial speed of the vehicle is equal to 100 km/h, at the end of the energy release from KERS the velocity is equal to 165 km/h (+65%), whereas if the initial velocity is equal to 200 km/h the final velocity at the end of KERS activation would be 240 km/h (+20%, under the unrealistic hypothesis – but the statement remains true anyway – of neglecting the non-conservative forces).

Longitudinal Slope Analysis influence on the optimal control - This section shows the results obtained by

analyzing the effect of the longitudinal gradient of the track. This analysis studies if the KERS energy discharge is more convenient on a positive longitudinal gradient zone rather than a zone where the longitudinal gradient is negative. By convention, the longitudinal gradient is considered positive when the vertical coordinate is growing.

Figure 15 – Longitudinal gradient analysis on the simplified track adopted for Figure 14

By observing the results obtained through the implemented models, the longitudinal gradient has an influence on the optimal control. If the pilot wants to make the maximum use of the KERS energy, he has to discharge more energy when the longitudinal slope is positive rather than negative.

Indeed, at a positive longitudinal gradient corresponds a lower velocity and, consistently with what stated in the previous paragraph, it is clear that in this zone it is better to release KERS energy. The significance of the gradient on the activation of the KERS system appears to be much less in comparison with the weight related to the length of the traction zone or the initial speed.

LTO CONCLUSIONS – This section shows the results obtained through the adoption of the developed software for the simulation of a lap along a real track. Chart 1 shows the distribution (between the different traction zones of the track of Figure 9) of the energy discharged by the KERS system, according to the best input defined by the LTO routine. By considering the results of Chart 1 and Figure 16, it is possible to state that the two most important parameters that influence LTO are the length of the traction zone and its initial speed.

Zone 1st 2nd 3rd 4th 5th 6th 7th 8th

EKERS[%]

0 0 40 0 20 10 30 0

Vstart[km/h]

282 108 85 134 162 222 241 277

Length [m]

570 20 1091 95 569 1092 1231 894

Chart 1 – Example of results in terms of optimal distribution of KERS energy utilization along the different traction zones (numbered as functions of their length) of a track


Figure 16 – Analysis of the results related to the adoption of the LTO along a track: distribution of the energy released in the different traction zones of the track (with their respective length and initial velocity). KERS should be activated within the traction zones characterized by a low initial velocity and a significant length

PIT STOP OPTIMIZER (PSO)

The aim of this routine developed within the LTS consists of selecting the best race strategy in order to minimize the overall race time. The routine considers the influence of the pit stops (for refueling and substituting the tires) and optimizes their distribution during the race. It also specifies the ideal amount of fuel which should be delivered to the vehicle at the beginning of the competition and during each pit stop. The algorithm consists of the following steps:

1. Implementation of series of simulations of single laps (for different values of vehicle mass) along the considered track, in conditions of optimal strategy in terms of lap time (LTO) or fuel consumption (FCO). The results are saved in a vector which contains the lap times (and the required additional information selected by the user in order to achieve a full picture of vehicle performance) computed for the different values of vehicle mass depending on the amount of fuel in the tank;

2. Definition of the possible strategies (in terms of pit stop distribution during the race) in order to complete the race (through the implementation of a brute force algorithm). This section of the routines generates all the possible alternatives which will be evaluated within point 3;

3. Evaluation of the total race time and total fuel consumption for each single strategy, by combining the results obtained at point 1. and the options defined at point 2.;

4. Detection of the two best strategies, the first one characterized by one pit stop and the second one characterized by two pit stops.

Figure 17 – Layout (top view) of the pit stop lane on a typical track

After each lap, the mass value used for the computation of the cumulative race time at point 3. is reduced by the fuel consumption related to the previous lap. This loop continues until vehicle mass is lower then the minimum value (imposed by the user at the beginning of the simulation), which implies a pit stop. For each value of vehicle mass, PSO carries out two simulations (step 1.), the first one related to a lap without refueling and the second one considering a pit stop lap, including its braking phase, refueling time and re-acceleration (the reference value of vehicle mass for the pit stop lap is referred to the condition of refueled vehicle, as the software imposes the pit stop when the vehicle is characterized by the minimum level of fuel selected by the user – any strategy considered within PSO is characterized by the same level of mass in the lap immediately before the pit stop).

PIT STOP LAP SIMULATION – Referring to Figure 17, two different vehicle trajectories are possible within one lap, with or without a pit stop:

��

++−++−BBoxABBoxALapPitStopBOABOALapNormal

:: (9)

Point B (Figure 17 is only a qualitative example) is the initial coordinate of the last traction zone before the vehicle enters the pit stop lane. Point A is the coordinate of the track at which the vehicle enters the main track after the pit stop. From point A to point B vehicle path will be the same for a normal lap and a pit stop lap.

Due to refueling, vehicle mass during the pit stop lap has two different values (smaller before the pit stop):

MMMassMinimumABoxHMMassHighBoxA

__

�

� (10)

The following part of this section considers the approximations adopted within the simulator in order to reduce the complexity of the algorithm for the computation of the pit stop lap. In particular, a simulation


of two laps is analyzed: the pit stop lap and the first lap after the pit stop (in conditions of increased vehicle weight). Formulas (11) express the values of vehicle masses which characterize the vehicle (the actual vehicle and the simulated vehicle) within the different sections of the track during the two laps (according to what happens during the race and according to the simplified calculations of the simulator):

��

�

��

�

�

++++++

++++++

HMHMHM

MMMMMM

HMHMHM

MMMMMM

BBoxABBoxABOABOA

NDESCRIPTIOPSO

BOABBoxABBoxABOA

NDESCRIPTIOREAL

:

:

(11)

The subscripts are referred to the values of mass along the different sections of the track. By subtracting the second expression to the first one, the error related to the simplifications of the simulator is obtained:

( ) ( )HMMMMMHM BBoxBBoxBOBO -- +=� (12)

This approximation of the simulator is composed by two contributions. The first contribution is related to the braking phase during the section BBox of the track:

HMMM BBoxBBox tt � (13)

where t is the time related to the considered section of the track (specified by the subscript). The braking phase in the pit stop lane is not significantly influenced by the value of vehicle mass. As a consequence, this simplification is acceptable.

The second approximation is related to the section BO of the track:

MMHM BOBO tt � (14)

Along the track, vehicle mass strongly influences vehicle performance. As consequence, the times to cover section BO of the track for the different mass values are saved. Thus the pit stop lap time MMPSLLT , is modified by considering the correct vehicle mass for this section.

( ) PTttLTLTMMHM BOBOMMPSLPSL ++= -, (15)

PT is the pit stop time (required for refueling and possible tire substitution). Vehicle velocity during the pit stop lap includes the first deceleration in order to reach the speed level required along the pit stop lane, the constant velocity section (pit lane before the pit stop), the pit stop (refueling phase at zero velocity) and the re-

acceleration phase after the pit stop (vehicle re-acceleration includes a first re-acceleration along the pit stop lane – from 0 to 100 km/h –, a constant velocity phase and a second acceleration at the end of the pit stop lane).

PSO RESULTS – PSO computes the best pit stop strategies (in case of one and two pit stops) considering the optimization of the performance in terms of fuel consumption and lap time for the entire race. This section shows examples of results for the following cases (these simulations are shown in order to deal with the potential of the software, not for a quantitative analysis of the results):

• Fuel consumption optimization. In this case step 1. of the routine of the PSO is carried out by the FCO. The global optimization strategy of the PSO is based on lap time minimization also in this case;

• Lap time optimization. In this case step 1. of the routine of the PSO is carried out by the LTO.

Figures 18-20 plot the results in terms of relative variation of the race time RT(i) (race consisting of 25 laps) according to the i-th race strategy in comparison with the best scenario (RTBEST-RT). Figure 18 (referred to a single pit stop scenario) shows that it is more convenient to start the race with a significant fuel level in the tank. In addition Figure 19 confirms this statement in case of a 2 pit stop strategy.

This conclusion has been confirmed by adopting PSO in case vehicle performances are optimized by the FCO and in case the vehicle is not equipped with KERS. Figure 20 shows the correlation between the pit stop strategy and the total amount of fuel consumed during the race (RFC).

Figure 18 – PSO results in case of 1 pit-stop on the considered track. Lap times are optimized by LTO


Figure 19 – PSO results in case of 2 pit-stops on the considered track. Lap times are optimized by LTO

Figure 20 – PSO results in case of 2 pit-stops on the considered track. Fuel consumption RFC is optimized by FCO

The target of this analysis is the definition of the best solution for a whole race. In other words, this analysis has to distinguish which optimization between the FCO and LTO allows the vehicle to obtain the lower race time. PSO guarantees the best pit stop strategy between those simulated by LTO and FCO.

Figure 21 – Race time considering different optimization routines (FCO and LTO) and pit stop strategies on the considered track

By considering the results in Figure 21, it is beyond doubt that the best strategy in order to obtain the best race time is the lap time optimization (by LTO). This strategy allows saving about 17 sec (over a total race time of more than 1 hour) in comparison with a vehicle having the same mass without KERS and 14 sec considering the FCO strategy. The fact that the optimized strategy based on the adoption of the FCO permits a better performance than the vehicle without KERS is significant. The FCO-based strategy (Figure 22) allows significant fuel savings in comparison with the LTO based strategy.

Figure 22 – Fuel consumed during the race considering different optimization routines (FCO and LTO) on the considered track

CONCLUSION

A simulation program for Lap Time Simulation for different options of KERS plants has been developed. It permits to optimize lap time (LTO routine) or fuel consumption (FCO routine) either for a single lap or for the entire race, including the pit stop phase (PSO). As a function of the results achieved during the activity, the following conclusions can be drafted:

• All tracks currently adopted within the Formula 1 Championship permit the regeneration of more than 400 kJ (maximum value of regenerated energy within one lap according to the 2009 rules). As a consequence, there is significant potential for a more extensive adoption of KERS.

• The effect of KERS activation on the front to rear brake force distribution is not very significant (due to the current limitation of the system in terms of power).

• KERS management for fuel consumption minimization appears complex (to be managed by the driver) in terms of time history of required KERS power along the lap.

• In terms of lap time optimization, it is better to use KERS power during the longest traction zones of the circuit (and the KERS should be discharged at the maximum rate).


• In terms of lap time optimization, it is better to use KERS power for the traction zones characterized by the lowest initial velocity.

• The influence of the longitudinal slope of the road in terms of the overall performance of the vehicle equipped with KERS is negligible.

• The adoption of KERS can provoke a non negligible benefit in terms of fuel consumption.

• The optimum distribution of the pit stops and amount of fuel for each pit stop can be computed through the adoption of the PSO routine.

Future steps will consist of repeating the same analyses by adopting the most sophisticated vehicle dynamics model within the LTS software of the University of Surrey.

REFERENCES

1. FIA Formula One Technical Regulation 2009 - document update 22/12/06.

2. M. Mosley, 2006 British Grand Prix, Press Conference, June 9, 2006.

3. Flybrid Flywheel Hybrid System Passes First Crash Test; Developing for Road Cars as Well. Green Car Congress. October 28, 2007, http://www.greencarcongress.com/2007/10/flybrid-flywhee.html.

4. Flybrid F1 Kinetic Energy Recovery System Voted Engine Innovation of the Year, Green Car Congress, November 9, 2007.

5. Milliken W., Milliken D.L., Race Car Vehicle Dynamics, Ed. SAE International, ISBN 1-56091-5269, 1995.

6. Brahma A., Guezennec Y., Rizzoni G., Dynamic Optimization of Mechanical/Electrical Power Flow in Parallel Hybrid Electric Vehicles, Proceedings of the 5th International Symposium in Advanced Vehicle Control, Ann Arbor, MI, USA.

7. Bertsekas D.P., Dynamic Programming and Optimal Control, 2nd Edition, Athena Scientific, 2001.

8. Thomas D. W., Segal D. J., Milliken D. L., Michalowicz J., Analysis and Correlation Using Lap Time Simulation – Dodge Stratus for the North American Touring Car Championship, SAE 962528.

9. Gadola M., Vetturi D., Cambiaghi D., Manzo L., A Tool for Lap Time Simulation, SAE 962529.

10. Gadola M., Vetturi D., Candelpergher A., Developments of a Method for Lap Time Simulation, SAE 2000-01-3562.

11. Law E. H., Morales J., Lap Time Simulation of Stock Cars on Super Speedways With Random Wind Gusts, SAE 2004-01-3509.

12. Siegler B., Crolla D., Lap Time Simulation for Racing Car Design, SAE 2002-01-0567.

13. Siegler B., Crolla D., Deakin A., Lap Time Simulation: Comparison of Steady State, Quasi-Static and Transient Racing Car Cornering Strategies, SAE 2000-01-3563.

ABBREVIATIONS

EM: Electric Motor SCap: Super Capacitor CONTR: Controller DIFF: Differential CVT: Continuous variable transmission ICE: Internal Combustion Engine RT: Refueling time LT: Lap time PSL: Pit Stop Lap PSO: Pit Stop Optimization LTO: Lap Time Optimization FCO: Fuel Consumption Optimization

CONTACT

Aldo Sorniotti Faculty of Engineering and Physical Sciences (FEPS)University of Surrey GU2 7XH Guildford United Kingdom Phone: +44 (0)1483 689688 Email: [email protected]


Racing Simulation of a Formula 1 Vehicle With Kinetic Energy Recovery System

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