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IFAC-PapersOnLine 49-11 (2016) 147–152

Optimal Control Policy Tuning and Implementation Optimal Control Policy Tuning and Implementation Optimal Control Policy Tuning and Implementation Optimal Control Policy Tuning and Implementation Optimal Control Policy Tuning and Implementation for a Hybrid Electric Race Car for a Hybrid Electric Race Car for a Hybrid Electric Race Car for a Hybrid Electric Race Car for a Hybrid Electric Race Car ∗∗ Carlo Bussi ∗∗ ∗∗ ∗∗ ∗∗ Mauro Salazar Fernando P. Grando ∗∗ Carlo ∗∗ ∗∗ Mauro Salazar Bussi P. Grando ∗∗ Fernando ∗∗ Mauro Salazar Carlo Bussi Fernando P. Grando ∗ Mauro Salazar Carlo Bussi Fernando P. Grando ∗ ∗ Carlo BussiH. ∗∗ Onder ∗∗ Christopher H. Onder ∗ Mauro Salazar Christopher Fernando P. Grando ∗ Christopher H. Onder Christopher Christopher H. H. Onder Onder ∗ ∗∗ Institute for Dynamic Systems and Control ETH Z¨ uurich, 8092 Z¨ uurich, for Dynamic Systems and Control ETH Z¨ 8092 Z¨ ∗∗ Institute Institute for Dynamic Systems and Control ETH Z¨ uurich, rich, 8092 Z¨ uurich, rich, for Dynamic Systems and Control ETH Z¨ rich, 8092 Z¨ ∗ Institute Switzerland (e-mail: [email protected]) Switzerland (e-mail: [email protected]) Institute for Dynamic Systems and Control ETH Z¨ u rich, 8092 Z¨ urich, rich, Switzerland (e-mail: [email protected]) ∗∗ Switzerland (e-mail: ∗∗ ERS and Ferrari ∗∗ ERS Design Design and Development, Development, Ferrari S.p.A., S.p.A., 41053, 41053, Maranello, Maranello, (MO), (MO), Switzerland (e-mail: [email protected]) [email protected]) ∗∗ ERS Design and Development, Ferrari S.p.A., 41053, Maranello, (MO), Development, Ferrari ∗∗ ERS Italy (e-mail: [email protected]) [email protected]) Italy (e-mail: ERS Design Design and and Development, Ferrari S.p.A., S.p.A., 41053, 41053, Maranello, Maranello, (MO), (MO), Italy (e-mail: [email protected]) Italy Italy (e-mail: (e-mail: [email protected]) [email protected])

Abstract: The Formula 1 is aa state-of-the-art hybrid electric vehicle. Its power unit is composed by Abstract: The Formula 11 car car is hybrid electric vehicle. Its power unit is composed by The Formula car is aa state-of-the-art state-of-the-art hybrid electric vehicle. Its power unit is composed by Abstract: The Formula 1 car is state-of-the-art hybrid electric vehicle. Its power unit is composed by aaAbstract: turbocharged gasoline engine and two electric motor/generator units connected to the traction system turbocharged gasoline engine and two electric motor/generator units connected to the traction system Abstract: The Formula 1 car is a state-of-the-art hybrid electric vehicle. Its power unit is composed by aand turbocharged gasoline engine and two electric motor/generator units connected to the traction system aand turbocharged gasoline engine and two electric motor/generator units connected to the traction system to the turbocharger. Such a powertrain offers an additional degree of freedom and therefore requires to the turbocharger. aa powertrain offers an additional degree freedom and therefore aand turbocharged gasoline Such engine and two electric motor/generator unitsof connected to the tractionrequires system to the turbocharger. Such powertrain offers an additional degree of freedom and therefore requires and to Such offers additional of and requires an energy management system. system. This supervisory supervisory controller hasdegree significant influence on the the vehicle’s vehicle’s an energy management This controller has aaa significant influence on and to the the turbocharger. turbocharger. Such aa powertrain powertrain offers an an additional degree of freedom freedom and therefore therefore requires an energy management system. This supervisory controller has significant influence on the vehicle’s an energy management system. This supervisory controller has a significant influence on the vehicle’s fuel consumption and on the achievable lap-time. Therefore, a thorough, systematic optimization of the fuel consumption and on the achievable lap-time. Therefore, a thorough, systematic optimization of the an energy management system. This supervisory controller has a significant influence on the vehicle’s fuel consumption and on the achievable lap-time. Therefore, a thorough, systematic optimization of the fuel consumption and on the achievable lap-time. Therefore, thorough, systematic optimization of the energy management system is a crucial prerequisite to win a race. The complexity of the system and energy management system is a crucial prerequisite to win a race. The complexity of the system and fuel consumption and on the achievable lap-time. Therefore, thorough, systematic optimization of the energy management system is a crucial prerequisite to win a race. The complexity of the system and energy management system is a crucial prerequisite to win a race. The complexity of the system and the strict regulations make the time-optimal energy management problem non-trivial to solve and an the strict regulations make the energy management non-trivial and an energy management system is atime-optimal crucial prerequisite to win a race.problem The complexity of to thesolve system and the strict regulations make the time-optimal energy management problem non-trivial to solve and an the strict regulations make time-optimal management problem non-trivial and an effective implementation of the its solution solution on the theenergy car difficult difficult to achieve. achieve. In Ebbesen, Ebbesen, S. et etto al.solve (2016) and effective implementation of its on car to In S. al. (2016) and the strict regulations make the time-optimal energy management problem non-trivial to solve and an effective implementation of its solution on the car difficult to achieve. In Ebbesen, S. et al. (2016) and effective implementation of its solution on the car difficult to achieve. In Ebbesen, S. et al. (2016) and Salazar, M. et al. (2016) a convex numerical solver was designed and the optimal strategies were derived Salazar, M. et al. (2016) a convex numerical solver was designed and the optimal strategies were derived effective implementation of its solution on the car difficult to achieve. In Ebbesen, S. et al. (2016) and Salazar, M. et al. (2016) aa convex numerical solver was designed and the optimal strategies were derived Salazar, M. al. numerical solver and the strategies derived analytically using aa non-smooth of Pontryagin’s Minimum respectively. Building on analytically using non-smooth version of Pontryagin’s Minimum Principle, respectively. Building on Salazar, M. et et al. (2016) (2016) a convex convexversion numerical solver was was designed designed andPrinciple, the optimal optimal strategies were were derived analytically using a non-smooth version of Pontryagin’s Minimum Principle, respectively. Building on analytically using a non-smooth version of Pontryagin’s Minimum Principle, respectively. Building on these results, we design a nonlinear program to tune an efficient version of the optimal control policy, in these results, we design aa nonlinear program to tune an efficient version of the optimal control policy, in analytically using a non-smooth version of Pontryagin’s Minimum Principle, respectively. Building on these results, we design nonlinear program to tune an efficient version of the optimal control policy, in these results, we design a nonlinear program to tune an efficient version of the optimal control policy, in order to precisely precisely matchavarious various boundary conditions onefficient the fuel fuelversion consumption and on on the the battery usage. order to match boundary conditions on the consumption and battery usage. these results, we design nonlinear program to tune an of the optimal control policy, in order to precisely match various boundary conditions on the fuel consumption and on the battery usage. order to precisely match various boundary conditions on the fuel consumption and on the battery usage. This allows a simple and robust implementation of the time-optimal control strategies on the vehicle, and This allows a simple and robust implementation of the time-optimal control strategies on the vehicle, and order to precisely match various boundary conditions on the fuel consumption and on the battery usage. This allows aa simple and robust implementation of the time-optimal control strategies on the feedforward vehicle, and This allows implementation of the control strategies on vehicle, guarantees with the FIA rules. A simulator is then used to test the obtained guarantees compatibility with the FIA rules. A simulator is then used to test the obtained This allowscompatibility a simple simple and and robust robust implementation of the time-optimal time-optimal control strategies on the the feedforward vehicle, and and guarantees compatibility with the FIA rules. A simulator is then used to test the obtained feedforward guarantees the rules. A is to test the feedforward controls on compatibility one race race lap lap in inwith Barcelona. The results stand comparison comparison with the optimal solutions obtained controls on one Barcelona. results stand with solutions obtained guarantees compatibility with the FIA FIAThe rules. A simulator simulator is then then used used to the testoptimal the obtained obtained feedforward controls on one race lap in Barcelona. The results stand comparison with the optimal solutions obtained controls one lap Barcelona. results comparison with with the on numerical solver and validate The the effectiveness effectiveness of this strategy. strategy. with the numerical validate the this controls on one race racesolver lap in inand Barcelona. The results stand stand of comparison with the the optimal optimal solutions solutions obtained obtained with the numerical solver and validate the effectiveness of this strategy. with the numerical solver and validate the effectiveness of this strategy. with the numerical solver and validate the effectiveness of this strategy. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Optimal Optimal Control, Control, Formula Formula 1, 1, Hybrid Hybrid Electric, Electric, Energy Energy Management, Management, Nonlinear Nonlinear Keywords: Keywords: Optimal Control, Formula 1,Robotics Hybrid Electric, Electric, Energy Management, Nonlinear Nonlinear Keywords: Optimal Control, Formula 1, Hybrid Energy Management, Programming, Intelligent Vehicles and Technology in Vehicles Programming, Intelligent Vehicles and Robotics Technology in Vehicles Keywords: Optimal Control, Formula 1, Hybrid Electric, Energy Management, Nonlinear Programming, Intelligent Vehicles and Robotics Technology in Vehicles Programming, Programming, Intelligent Intelligent Vehicles Vehicles and and Robotics Robotics Technology Technology in in Vehicles Vehicles

1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The effectiveness and the promising results obtained with hyThe effectiveness and the results obtained with The effectiveness effectiveness andterms the promising promising resultsand obtained with hyhyThe and the promising results obtained with brid technologies in of sustainability fuel efficiency brid technologies in of sustainability fuel efficiency The effectiveness andterms the promising resultsand obtained with hyhybrid technologies in terms of sustainability and fuel efficiency brid technologies in terms of sustainability and fuel efficiency have appealed to to the the Formula 1sustainability (F1) world. world. Starting Starting inefficiency the 2014 2014 have appealed (F1) the brid technologies in Formula terms of 1 and fuelin have appealed to the Formula 1 (F1) world. Starting in the 2014 have appealed to 11 (F1) Starting in 2014 season, aa downsized and turbocharged 1.6l V6 Internal season, and turbocharged 1.6l V6 Internal Comhave appealed to the the Formula Formula (F1) world. world. Starting in the theCom2014 season, a downsized downsized and turbocharged 1.6l V6 Internal Comseason, a downsized and turbocharged 1.6l V6 Internal Combustion Engine (ICE) was combined with two Motor/Generator bustion Engine (ICE) was combined with two Motor/Generator season, a downsized and turbocharged 1.6l V6 Internal Combustion(MGU). Engine The (ICE)first wasone combined with two Motor/Generator Motor/Generator bustion Engine (ICE) was combined with two Units is used to recover kinetic energy Units (MGU). is used to recover kinetic energy bustion Engine The (ICE)first wasone combined with two Motor/Generator Units (MGU). The first one is used to recover kinetic energy Units (MGU). The first one is used to recover energy from breaking (MGU-K) and the second one kinetic as an an electric electric from breaking (MGU-K) and the second one as Units (MGU). The first one is used to recover kinetic energy from breaking (MGU-K) and the second one as an electric from breaking (MGU-K) and the second one as an electric turbo-compound system (MGU-H), Algrain (2005); Katraˇ ssnik turbo-compound system (MGU-H), Algrain (2005); Katraˇ nik from breaking (MGU-K) and the second one as an electric turbo-compound system (MGU-H), Algrain (2005); (2005); Katraˇ nik turbo-compound system (MGU-H), Algrain Katraˇ sssnik et al. (2003); Wei et al. (2010). Moreover, the energy flows et al. (2003); Wei et al. (2010). Moreover, the energy flows turbo-compound system (MGU-H), Algrain (2005); Katraˇ nik et al. (2003); Wei et al. (2010). Moreover, the energy flows et al. Wei et the flows within this power power unit are(2010). subjectMoreover, to strict strict regulation regulation and the within this are subject to the et al. (2003); (2003); Wei unit et al. al. (2010). Moreover, the energy energyand flows within this power unit are subject to strict regulation and the within this unit are strict regulation and the overall fuel power consumption persubject race is isto limited to 100 kg kg gasoline, gasoline, overall fuel consumption per race limited to 100 within this power unit are subject to strict regulation and the overall fuel consumption per race is limited to 100 kg gasoline, overall fuel consumption per race is limited to 100 kg gasoline, 201 (2013). The authors Picarelli and Dempsey (2014) investi201 (2013). The authors Picarelli and Dempsey (2014) investioverall fuel consumption per race is limited to 100 kg gasoline, 201 (2013). (2013). The authors Picarelli and and Dempsey (2014) investi201 The authors Picarelli (2014) investigate different power unit configurations and present simulations gate different power unit configurations and present simulations 201 (2013). The authors Picarelli and Dempsey Dempsey (2014) investigate different power unit configurations and present simulations gate different power unit configurations and present simulations of thedifferent complete physical model. The The increasing increasing complexity of of the complete physical model. complexity of gate power unit configurations and present simulations of the complete physical model. The increasing complexity of of the complete physical model. The increasing complexity of the system and the strict regulations given by the F´ eed´ eeration Inthe system and the strict regulations given by the F´ d´ ration Inof the complete physical model. The increasing complexity of the system and the strict regulations given by the F´ e d´ e ration Inthe system and the strict regulations given by F´ eed´ eeration Internationale de l’Automobile (FIA) demand aa the systematic analyternationale de l’Automobile (FIA) demand systematic analythe system and the strict regulations given by the F´ d´ ration Internationale de l’Automobile l’Automobile (FIA) demand systematic analyternationale de (FIA) demand aaa systematic analysis of the time-optimal control problem, in order to understand sis of the time-optimal control problem, in order to understand ternationale de l’Automobile (FIA) demand systematic analysis of the time-optimal control problem, in order to understand sis of the time-optimal control problem, in order to understand and implement controlcontrol strategy that allows allows minimum lap-time and implement aaa control strategy that minimum lap-time sis of the time-optimal problem, in order to understand and implement control strategy that allows minimum lap-time and a control strategy that allows minimum lap-time to beimplement achieved. to achieved. and to be beimplement achieved. a control strategy that allows minimum lap-time to be achieved. to review be achieved. A of the the various various methodologies methodologies that that have have been been employed employed A of A review review of thetime-optimal various methodologies methodologies that have have been employed A review of the various that been employed to tackle the control problem of hybrid electric to tackle the time-optimal control problem of hybrid electric A review of the various methodologies that have been employed to tackle the time-optimal control problem of hybrid electric to tackle the time-optimal control problem of hybrid electric race cars can be found in Limebeer and Rao (2015). The racing race cars can be found in Limebeer and Rao (2015). The racing to tackle the time-optimal control problem of hybrid electric race cars can be found in Limebeer and Rao (2015). The racing race cars can be found in Limebeer and Rao (2015). The racing path and the power unit energy flows are optimized simultanepath and the unit flows optimized simultanerace cars can power be found inenergy Limebeer andare Rao (2015). The racing path and the power unit energy flows are optimized simultanepath and the power unit energy flows are optimized simultaneously for the sports series HEV using using the indirect method in Lot Lot ously for aaa sports HEV method in path and powerseries unit energy flowsthe areindirect optimized simultaneously for sports series HEV using the indirect method in Lot Lot ously for aa sports series HEV using the indirect method in and Evangelou (2013). Direct multiple shooting is employed and Evangelou (2013). Direct multiple shooting is employed ously for sports series HEV using the indirect method in Lot and Evangelou (2013). Direct multiple shooting is employed and and Evangelou Evangelou (2013). (2013). Direct Direct multiple multiple shooting shooting is is employed employed

in Casanova (2000) to solve the same problem for the F1 car. in (2000) to solve the problem for F1 in Casanova Casanova (2000) of to the solve the same same problem forinthe the F1 car. car. in Casanova (2000) to solve the same for the F1 car. The optimal control control power unit is isproblem addressed Limebeer The optimal of the power unit addressed in Limebeer in Casanova (2000) to solve the same problem for the F1 car. The optimal control of the power unit is addressed in Limebeer The optimal control of the power unit is addressed in Limebeer et al. (2014) together with the optimization of the race trajectory et al. (2014) together with the optimization of the race trajectory The optimal control of the power unit is addressed in Limebeer et al. al. (2014) together with with the optimization optimization of the race race trajectory et (2014) together the of the trajectory by using orthogonal collocation methods. Direct transcription by using orthogonal collocation methods. Direct transcription et al. (2014) together with the optimization of the race trajectory by using orthogonal collocation methods. Direct transcription by using orthogonal collocation methods. Direct transcription and Nonlinear Programming (NLP) are exploited exploited in Perantoni Perantoni and Nonlinear Programming (NLP) are in by using orthogonal collocation methods. Direct transcription and Nonlinear Programming (NLP) are exploited in Perantoni Nonlinear Programming (NLP) exploited in and Limebeer to also optimize the design parameters of and Limebeer (2014) to the design of Nonlinear (2014) Programming (NLP) are are in Perantoni Perantoni andcar. Limebeer (2014) to also also optimize optimize theexploited design parameters parameters of and Limebeer (2014) to also optimize the design parameters of the the car. and Limebeer (2014) to also optimize the design parameters of the car. the car. the contrast car. In to the presented methodologies, here the energy In the methodologies, here the energy In contrast contrast to toproblem the presented presented methodologies, here the driving energy In contrast to the presented methodologies, here the energy management was separated from the optimal management problem was separated from the optimal driving In contrast to the presented methodologies, here the energy management problem was separated from the optimal driving management problem was separated from the optimal driving problem. In Ebbesen, S. et al. (2016), the optimal control probproblem. In Ebbesen, S. et al. (2016), the optimal control probmanagement problem was separated from the optimal driving problem. In Ebbesen, Ebbesen, S. et et al. (2016), theand optimal controlracing probproblem. In S. (2016), the optimal control problem is addressed addressed for the the F1al. power unit, the optimal optimal lem is for F1 power unit, and the racing problem. In Ebbesen, S. et al. (2016), the optimal control problem is addressed for the F1 power unit, and the optimal racing lem is addressed for the F1 power unit, and the optimal racing path is precomputed and fed to the solver in form of maximum path is precomputed and fed to the solver in form of maximum lem is addressed for the F1 power unit, and the optimal racing path is isconstraint precomputed and fedslope. to the the solver solver in form form of maximum maximum path precomputed and fed to in of speed and track system is modeled in a speed and track The system is modeled in path isconstraint precomputed and fedslope. to theThe solver in form of maximum speed constraint and track slope. The system is modeled in aaa speed constraint and track slope. The system is modeled in convex form, which allows the time-optimal energy manageconvex form, the time-optimal managespeed constraint and allows track slope. The system energy is modeled in a convexproblem form, which which allows themeans time-optimal energy manageconvex form, allows the time-optimal energy management to be be solved solved by of convex convex optimization, ment problem to of optimization, convex form, which which allows by themeans time-optimal energy management problem to be solved by means of convex optimization, ment problem to by of Boyd and Vandenberghe (2004) and Rockafellar (1997), and Boyd and (2004) and (1997), and ment to be be solved solved by means means of convex convex optimization, optimization, Boyd problem and Vandenberghe Vandenberghe (2004) and Rockafellar Rockafellar (1997), and and Boyd and Vandenberghe (2004) and Rockafellar (1997), guaranteeing existence and uniqueness of the optimum guaranteeing existence and uniqueness of the optimum and Boyd and Vandenberghe (2004) and Rockafellar (1997), guaranteeing existence and uniqueness of the optimum and guaranteeing existence and uniqueness of optimum and computational times three of magnitude than the computational times three orders of magnitude lower than the guaranteeing existence andorders uniqueness of the the lower optimum and computational times three orders ofimplementation magnitude lower than the computational three orders magnitude lower the state of the the art art times methodologies. Theof of than the preprestate of methodologies. The implementation of the computational times three orders of magnitude lower than the state of the art methodologies. The implementation of the prestate of the art methodologies. The implementation of the presented strategies is altogether non-trivial, as the forward implesented strategies is altogether non-trivial, as the forward implestate of the art methodologies. The implementation of the presented strategies strategies is altogether altogether non-trivial, non-trivial, as the the forward forward implesented is as implementation of pre-computed trajectories in robustmentation of pre-computed input trajectories lacks in robustsented strategies is altogetherinput non-trivial, as thelacks forward implementation of pre-computed input trajectories lacks in robustmentation of input lacks robustness and adaptability. In Salazar, M. et al. (2016), the timeness and adaptability. In Salazar, M. et al. (2016), the timementation of pre-computed pre-computed input trajectories trajectories lacks in in robustness and adaptability. In Salazar, M. et al. (2016), the timeness and adaptability. Salazar, al. (2016), the timeoptimal control of the the In power unitM. is et treated analytically usoptimal control of power unit is treated analytically usness and adaptability. In Salazar, M. et al. (2016), the timeoptimal control of the power unit is treated analytically usoptimal control of the power unit is treated analytically using Pontryagin’s Minimum Principle (PMP) Pontryagin (1987) ing Pontryagin’s Minimum Principle (PMP) Pontryagin (1987) optimal control of the power unit is treated analytically using Pontryagin’s Minimum Principle (PMP) Pontryagin (1987) ing Pontryagin’s Minimum Principle (PMP) Pontryagin (1987) combined with non-smooth analysis, in order to understand the combined with non-smooth analysis, in order to understand the ing Pontryagin’s Minimum Principle (PMP) Pontryagin (1987) combined with non-smooth non-smooth analysis, in order to toway. understand the the combined with analysis, order understand optimal behavior and implement implement it in in in simple optimal and it aaa simple combined with non-smooth analysis, order toway. understand the optimal behavior behavior and implement implement it in in in simple way. optimal behavior and it a simple way. optimal behavior and implement it in a simple way.

Copyright 2016 IFAC IFAC 151 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 151 Copyright 2016 151 Copyright ©under 2016 IFAC IFAC 151Control. Peer review© of International Federation of Automatic Copyright © 2016 responsibility IFAC 151 10.1016/j.ifacol.2016.08.023

Time-optimal Energy Management of the Formula 1 Power Unit IFAC AAC 2016 148 June 19-23, 2016. Norrköping, Sweden

Pk,dc Eb

ES

Pb

MGU-K

Soren Ebbesen, Salazar and Bussi Mauro Salazar et al. /Mauro IFAC-PapersOnLine 49-11 Carlo (2016) 147–152

Paux Ph,dc

the braking power u = (Pf , Pk , Pbrk )T . The minimum lap-time energy management problem is then T min g(x, u)dt + h x(T )

Pk Pu

MGU-H

Ph

Pe0 Pe ICE

u

Pp

Pf Fig. the F1 power Bold linesunit indicate Fig. 1: 1. Schematic Schematic of representation of unit. the F1 power taken mechanical connections; arrows denote positive energy flow from Ebbesen, S. et al. (2016). Bold lines indicate mechanical connections and arrows denote positive energy flow directions. directions. The blocks represent the Energy Storage (ES), the MGU-K, the MGU-H and the ICE. This paper continues along the same lines. The optimal control policy obtained in Salazar, M. et al. (2016) depends on the co-state variables of the system, whose values are constant except for the first one, which is space-dependent and can be used as decision variable. If the terminal constraints on fuel and battery consumption are changed, the co-state variables also change, whereby the first one only slightly. Choosing a fixed first co-state as decision variable, a Nonlinear Program (NLP) is designed to tune the remaining co-state values, and thus the optimal control policy, to match the desired terminal conditions. Similar ideas are employed in Jiao and Shen (2014), Ahmad et al. (2013) and Hagura et al. (2013) to tune the energy management strategies for HEVs and to improve their fuel efficiency. The resulting control policy can be effectively implemented on the car with a 1-D look-up table, as shown in Section 4 with a forward simulator. The structure of this paper is as follows: the optimization problem is presented together with a summary of the results obtained in Ebbesen, S. et al. (2016) and Salazar, M. et al. (2016) in Section 2. Thereafter a NLP is designed in Section 3. Section 4 describes the control policy implementation in a forward simulator for one race lap on the Barcelona circuit and the obtained state trajectories are compared to the optimal ones. Conclusions and an overview of further research work are presented in Section 5.

=:Pe

Px,dc (Px ) = αx Px2 + Px for x = k, h

Carlo Bussi, Head of ERS Design and Development, Ferrari S.p.A., 41053, Maranello, (MO), Italy [email protected] 2.1 Optimization Problem

Pi (Pk , Ph ) = αb Pb (Pk , Ph ) + Pb (Pk , Ph ) Pb (Pk , Ph ) = Pk,dc (Pk ) + Ph,dc (Ph ) + Paux . The stage cost function of the constrained PMP is g(x, u) =1 + Ψ[−∞,0] (x1 − Ekin,max (x6 )) + Ψ[0,Pf ,max ] (Pf )+ + Ψ[Pk,min ,Pk,max ] (Pk ) + Ψ[0,PB,max ] (Pbrk ) ,

whereas the terminal cost function is h(x(T )) =Ψ{x1 (0)} (x1 (T )) + Ψ[−∞,E f ,max ] (x2 (T ))

+ Ψ[∆SOC,∞] (x3 (T )) + Ψ[−∞,4MJ] (x4 (T )) + Ψ[−2MJ,∞] (x5 (T )) + Ψ{S} (x6 (T )) ,

1

where λi are the co-state variables.

152

(4)

(5)

In Salazar, M. et al. (2016), it was shown that the last costate variable λ6 (t) is not needed to derive the optimal control policy. Therefore, the relevant co-state variables are λ1 (t), which is time-dependent, plus four other co-state values, which 2

i.e. the change in State of Charge over one lap

(3)

and ΨX (x) is the indicator function to the closed convex set X 2 . The Hamiltonian is then H(x, u, λ ) = 1 + λ1 cs,1 Pu (Pf , Pk )2 + cs,2 Pu (Pf , Pk ) − Pbrk − Pd (x1 , x6 ) + λ2 Pf − λ3 αb Pb (Pk , −ηh Pf )2 + Pb (Pk , −ηh Pf ) + λ4 max 0, Pk,dc (Pk ) + Ph,dc (−ηh Pf ) + λ5 min 0, Pk,dc (Pk ) + λ6 2x1 /m + Ψ[−∞,0] (x1 − Ekin,max (x6 )) + Ψ[0,Pf ,max ] (Pf ) + Ψ[Pk,min ,Pk,max ] (Pk ) + Ψ[0,PB,max ] (Pbrk ) ,

In order to express the optimization problem in a form compatible with PMP, the state variables are defined as x = (Ekin , E f , ∆Eb , EES2K , EK2ES , s)T , i.e. kinetic energy, fuel energy used, battery energy deployment 1 , energy transferred from the ES to the MGU-K and vice versa, and position on track. The inputs are represented by the fuel, the MGU-K and

(2)

2

2. BACKGROUND In this section, the time-optimal energy management problem for the F1 power unit shown in Figure 1 will be presented. Furthermore, the model derived and validated in Ebbesen, S. et al. (2016) and the results obtained in Salazar, M. et al. (2016) will be summarized. For the sake of clarity, the optimal solutions obtained numerically with the convex optimizer, Ebbesen, S. etSoren al. (2016), will be denoted as (·)o henceforth, whereas the Ebbesen, Institute for Dynamic Systems and Control ETH Zurich, onesZurich, derived analytically in Salazar, M. et al. (2016) with (·)∗ . 8092 Switzerland sebbese[email protected]

0

s.t. x˙1 = cs,1 Pu2 + cs,2 Pu − Pd (x1 , x6 ) − Pbrk x˙2 = Pf x˙3 = −Pi (Pk , −ηh Pf ) x˙4 = max 0, Pk,dc (Pk ) + Ph,dc (−ηh Pf ) 0, Pk,dc (Pk ) x˙5 = min x˙6 = 2x1 /m (1) Pf (t) ∈ [0, Pf ,max ] Pk (t) ∈ [Pk,min , Pk,max ] Pbrk (t) ∈ [0, PB,max ] x1 (t) ≤ Ekin,max (x6 (t)) x (T ) ∈ {x (0)} 1 1 x2 (T ) ≤ E f ,max x3 (T ) ≥ ∆SOC x 4 (T ) ≤ 4MJ x (T ) ≥ −2MJ 5 x6 (T ) = S , where the power of the power unit, the converter power, and the battery internal and terminal power flows are defined as Pu = ηe Pf − Pe0 +Pk

For detailed information refer to Vinter (2010) and Rockafellar (1997).

IFAC AAC 2016 June 19-23, 2016. Norrköping, Sweden

Mauro Salazar et al. / IFAC-PapersOnLine 49-11 (2016) 147–152

are constant, i.e. λ2 ≥ 0, λ3 ≤ 0, λ4 ≥ 0, λ5 ≤ 0. In the next subsection, a summary of how the first co-state variable λ1 (t) can be defined as a decision variable, whereby the constant costate variables λ2→5 are used to construct the optimal control policy as a function of λ1 (t).

149

0.8 ICE MGU-K

0.6

In Salazar, M. et al. (2016), the optimal control input trajectories were computed employing the non-smooth PMP by including zero into the subdifferential of the Hamiltonian (5) and solving with respect to the input u. As a result, the optimal policy for Pf∗ and Pk∗ was shown to be a non-smooth function of the constant co-state values λ2→5 and piecewise linear in λ1 . Thus, these optimal control inputs can be attained using a first order interpolation of the decision variable λ1 over the values listed in Table 1, since the values of λ2→5 are constant over an entire lap. Table 1. Switch points of λ1 with associated values of Pf∗ , Pk∗ and Pu∗ . λ1 λ f ,max λ f ,0 λk,max λk,h+ λk,h− λk,0+ λk,0− λk,min

Pf∗ Pf ,max Pf ,0 Pf ,kmax Pf ,kh+ Pf ,kh− Pf ,k0+ Pf ,k0− Pf ,kmin

Pk∗ Pk, f max Pk, f 0 Pk,max Pk,h (−ηh Pf ,kh+ ) Pk,h (−ηh Pf ,kh− ) Pk,0+ Pk,0− Pk,min

Pu∗ = Pe∗ + Pk∗ ηe Pf ,max − Pe,0 + Pk, f max ηe Pf ,0 − Pe,0 + Pk, f 0 ηe Pf ,kmax − Pe,0 + Pk,max ηe Pf ,kh+ − Pe,0 + Pk,h (−ηh Pf ,kh+ ) ηe Pf ,kh− − Pe,0 + Pk,h (−ηh Pf ,kh− ) ηe Pf ,k0+ − Pe,0 + Pk,0+ ηe Pf ,k0− − Pe,0 + Pk,0− ηe Pf ,kmin − Pe,0 + Pk,min

−0.2 −0.4 −0.4

11: 12: 13:

15: 16: 17: 18: 19: 20: 21:

0.8 0.6 Power Unit ICE MGU-K

0.4 0.2

22:

0

23:

−0.2 −0.4 −0.5

0

0.5

1

24:

1.5

−λ1

Fig. 2. Normalized power unit modes as a function of λ1 . Taken from Salazar, M. et al. (2016). Pk∗ are piecewise linear in λ1 , a piecewise linear relation with Pu∗ also exists, as shown in Figure 3, where a plot of the power split is presented. 2.3 Forward Simulator In order to simulate the Formula 1 car behavior, the dynamics can be integrated using the explicit Euler approximation 153

0

0.2

0.4

0.6

0.8

1

1.2

Pu [-]

o (0) Ekin (0) = Ekin E f (0) = ∆Eb (0) = EES2K (0) = EK2ES (0) = GL(0) = 0 for k = 0 : N do ∆t(k) = ∆s 2E m (k) kin

if GL(k) = 0 then Pu (k) = Pu∗ (λ1 ) else Pu (k) = Pu,max end if 10: Ekin (k + 1) = Pd Ekin (k) ∆t(k)

14:

1

−0.2

Fig. 3. Normalized power split as a function of Pu . Taken from Salazar, M. et al. (2016).

5: 6: 7: 8: 9:

1.2

0.2

0

1: 2: 3: 4:

In Figure 2 the optimal values of (λ1 , Puo ) are plotted together with the analytic results shown in Table 1 (λ1 , Pu∗ ). The latter are denoted by full dots connected by lines, to show the piecewise linearity of the optimal solution. As the single powers Pf∗ and

Pu , Pe , Pk [-]

Pe , Pk [-]

0.4

2.2 Optimal Control Policy

25: 26:

Ekin (k)

+

Pp Pu (k)

−

if Ekin (k + 1) > Ekin,max (k + 1) then Ekin (k + 1) =Ekin,max (k + 1) kin (k) P˜u (k) = Pp−1 Ekin (k+1)−E + Pd Ekin (k) ∆t(k) Pu (k) = max Pu,min , min Pu,max , P˜u (k)

GL(k + 1) = 1 else Pu (k) = Pu∗ (λ1 ) GL(k + 1) = 0 end if [Pf (k), Pk (k)] = PowerSplit(Pu (k)) E f (k + 1) = E f (k) + Pf (k)∆t(k) ∆Eb (k + 1) = ∆Eb (k) − Pi Pk (k), −ηh Pf (k) ∆t(k) EES2K (k + 1) = EES2K (k) + max 0, Pk,dc Pk (k) + Ph,dc − ηh Pf (k) ∆t(k) EK2ES (k + 1) = EK2ES (k) + min 0, Pk,dc Pk (k) ∆t(k) end for T = ∑Nk=0 ∆t(k)

Algorithm 1. Forward Simulator taken from Salazar, M. et al. (2016), where GL represents the grip-limited region flag.

in space-domain. To this end, the forward simulator designed in Salazar, M. et al. (2016) and shown in Algorithm 1 is used. During the design phase, a perfect driver was assumed and the FIA regulations regarding the freedom of control were implemented in a simplified way. Specifically, two control regions are defined: the grip-limited and the power-limited region. The first one is denoted by an active maximum kinetic energy constraint,

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which arises from the maximum grip of the car in the curves and which an expert driver would follow as long as the power unit would be capable to fulfill his thrust demand. In this region, the ECU can only split optimally the demanded Pu between the power unit components as in Figure 3. In power-limited regions the maximum velocity constraint is inactive and therefore an optimal driver would demand as much thrust as possible by completely pressing the gas pedal. In this case, the ECU can choose the optimal value of Pu as a combination of ICE and MGU-K power by interpolating λ1 (s) as in Figure 2. 3. NONLINEAR PROGRAM In this section, a NLP is designed to tune the values of λ2→5 , such that the boundary conditions regarding the amount of fuel and battery energy available over one lap are met when the optimal control policy presented in Section 2 is employed. Consider a race scenario, where EES2K < 4 MJ always holds. This means that the terminal constraint on x4 will always be inactive. Therefore we can assume λ4 = 0. The relation between co-states values and boundary conditions is modeled with an algebraic system with E f , Eb and EK2ES at the end of the lap as state variables and λ2 , λ3 and λ5 as inputs. To this end, we define T x := E f (T ), ∆Eb (T ), EK2ES (T ) (6) T λ := λ2 , λ3 , λ5 , where x = f (λ ) represents the energy levels at the end of one lap, which can be attained using the forward simulator presented in Algorithm 1. As this relation is not invertible, due to the non-smooth form of Algorithm 1, if we want to meet the desired energy target xd , we need to find the value of λd , such that xd = f (λd ) . (7) Equation (7) can be solved iteratively using xk = f (λk ) (8) λk+1 = λk + v(xk , xd ) , where we choose v(xk , xd ) = K∆xk , with ∆xk := xk − xd . Consider

xk+1 = f (λk+1 ) = f (λk + K∆xk ) ∂ f = f (λk ) + K∆xk + H.O.T. , ∂ λ λ =λk =xk

(9)

≈B

where B can be fitted using data from several laps as ∆X T = ∆ΛT BT , (10) which can be solved with a Least Error Square (LES) algorithm. Then we have ∆xk+1 = xk+1 − xd = xk − xd +BK∆xk + H.O.T. (11) =∆x k = I + BK ∆xk + H.O.T. . The local asymptotic stability of the origin of this system can be obtained neglecting the higher order terms and forcing the eigenvalues of I + BK to be inside the unit circle or using the Lyapunov Function (12) V (∆x) := ∆x2 = ∆xT ∆x , which evolves according to T V (∆xk+1 ) −V (∆xk ) := ∆xkT I + BK I + BK − I ∆xk . (13) 154

Choosing with α ≥ 0, we get

K = −αB−1 ,

(14)

V (∆xk+1 ) −V (∆xk ) = −α(2 − α)∆xk 2 < 0 ⇐⇒ α ∈ (0, 2) , (15) where the maximum contraction rate attained with α = 1 corresponds to a deadbeat behavior of the form ∆xk+1 = 0 + H.O.T. , (16) which can also be interpreted as an approximated Newton algorithm with full steps. 4. RESULTS In this part of the paper, the control policy presented in Section 2 will be tuned with the NLP designed in Section 3 and the obtained trajectories of the state and input variables will be compared to the ones obtained using the numerical solver presented in Ebbesen, S. et al. (2016). For the sake of consistency with the results presented in Ebbesen, S. et al. (2016) and Salazar, M. et al. (2016), a race lap in the circuit of Barcelona with 100% fuel load 3 and ∆SOC := ∆Eb (T ) = 2.0 MJ will be first considered and used to generate the nominal λ1∗ (s). Thereafter, the target for the battery state of energy will be lowered to ∆SOC = 1.0. MJ. In both cases, the co-state variables λ2→5 will Mauro Soren Ebbesen, be tuned to match the desired boundary conditions, whereby a precomputed λ1∗ (s) trajectory will be used. A schematic representation of the procedure is shown in Figure 4.

Time-optimal Energy Manageme

λ1∗ (s)

Pu∗ (−λ1 ) OCP

λ2→5

Pi∗ (Pu )

UPL

FS ∆xk

x(T ) xd (T ) −

Fig. 4. Structure of the NLP forF1 a fixed λ1∗ (s) Theindicate Fig. 1: Schematic of the power unit.trajectory. Bold lines Optimal Control Policy (OCP) is computed with the comechanical arrows denote positive energy flow state variablesconnections; λ2→5 , obtained with the Update Law (UPL), directions. and fed to the Forward Simulator (FS). The terminal value of the state variables is then compared with the desired one and used to update λ2→5 , if convergence is not achieved. 4.1 Control Policy Tuning with λ1∗ = λ1o As in Salazar, M. et al. (2016), the space-dependent co-state variable λ1∗ (s) is utilized as decision variable to interpolate the optimal control inputs with the piecewise linear control policy given in Figure 2 when the vehicle is in power-limited region. This variable is set equal to the one obtained with Ebbesen, S. et al. (2016), i.e. λ1∗ (s) = λ1o (s). Since in the grip-limited regions the power unit power Pu is determined by a perfect driver who follows the maximum kinetic energy curve, an optimal power split based on Figure 3 is performed. Taking the values of co-state variables from Ebbesen, S. et al. (2016) as initial guess, the NLP designed in Section 3 is used to tune 3 Since for a race 100 kg fuel are available, the nominal amount of fuel energy available for one lap is E f (T ) = 100Nkg·Hlhv . laps

Mauro Salazar et al. / IFAC-PapersOnLine 49-11 (2016) 147–152

151

0.02 0 −0.02

them, in order to meet the desired boundary conditions of 100% fuel load and ∆SOC = 2.0 MJ. A plot with the values of the co-state variables and the terminal states as a function of the iteration steps is shown in Figure 5. The algorithm converges within five iterations. In Figure 6 the optimal state trajectories

∆Pf dt [-]

IFAC AAC 2016 June 19-23, 2016. Norrköping, Sweden

−0.04

−0.53

4

0

4

100 99.98 99.96

2

−2

1.99 1.98

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s [m] 2 1 0 0

−2.02 −2.03

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2000

s [m]

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4

GL - optimal GL - forward

10

n

n

n

2

+

−1.99

x5 (T ) [MJ]

x3 (T ) [MJ]

100.02

0

0

n

2.01

3000

0

−1

n

n 100.04

x2 (T ) [%]

2

5000

−0.5

λo1

2

−0.46 −0.462

4000

−λ∗1

0

−0.458

[-]

−0.545

∆Pk dt [-]

0.0439

3000

0.044

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2000

0.5

∆t − ∆to [ms]

λ3

λ2

0.0441

1000

s [m]

−0.456

−0.535

0

ent ofFig. the Formula 1 Power Unit 5. Optimal (dashed) and forward NLP terminal states and co-states values (dotted).

5 0

0.0442

−0.454

λ5

0.0443

0

1000

2000

s [m]

Salazar and Carlo obtained in Bussi Ebbesen, S. et al. (2016) are compared to the

Ekin [MJ]

ones obtained with the implementation of tuned optimal control policy. The small differences in lap-time arise when the optimal 5

4.2 Control Policy Tuning with given λ1∗

0

Ef [MJ]

0

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2000

5000

3000

4000

5000

3000

4000

5000

3000

4000

5000

In the second part of the simulations, the value of ∆SOC is set to 1.0 MJ. Again, the values of the co-states obtained with the numerical solver are used as initial guess. However, to show the validity of the assumption that the first co-state is independent of the boundary conditions, λ1∗ (s) is set equal to the previously used λ1o (s), which was obtained with ∆SOC = 2.0 MJ. The NLP is employed to tune the optimal control policy for the given first co-state trajectory. The small difference in λ1∗ (s) due to different boundary conditions does not affect the performance of the NLP. On the contrary, convergence is achieved in less than 25 iterations. and a negligible lap-time difference of less than ten milliseconds is achieved.

0 1000

2000

s [m]

5 0

EES2K [MJ]

∆Eb [MJ]

4000

50 0

EK2ES [MJ]

3000

s [m]

100

−5

Fig. 7. Normalized cumulative difference between forward and optimal input trajectories along one lap, λ1∗ (s) = λ1o (s) and cumulative difference in lap-time together with the flags of the grip-limited regions.

0

1000

2000

s [m]

2 1 0 0

1000

2000

4.3 Discussion

s [m]

0 −1 −2

0

1000

2000

3000

4000

5000

s [m]

Fig. 6. Optimal (dashed) and forward (solid) state trajectories along one lap. In the first plot, the maximum kinetic energy curve (dotted) is plotted together with discrete markers denotig the grip-limited regions of the forward simulator. solution is in grip-limited region and the forward one in powerlimited or vice versa. This can also be seen in Figure 7, where the cumulative difference of time-steps results in a lap-time gap of less than ten milliseconds. 155

Using the values of the co-state variables obtained with Ebbesen, S. et al. (2016) as initial guess, it is possible to tune the optimal control policy derived in Salazar, M. et al. (2016) with an NLP, in order to meet the boundary conditions, i.e. fuel load, battery charge and K2ES energy, in a more realistic race scenario with a perfect driver and the FIA regulations regarding the freedom of control in power- and grip-limited regions. Moreover, the assumption of a boundary conditions independent λ1 is validated by tuning the control policy for ∆SOC = 1.0 MJ, whereby λ1∗ is taken as the numerical λ1o obtained with ∆SOC = 2.0 MJ. Therefore the time-optimal control policies for various energy budgets can be tuned on a given and fixed λ1∗ (s) trajectory, by finding the correct values for λ2→5 , which would in turn update Table 1 and reshape the optimal power curves depicted in Figures 2 and 3.

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5. CONCLUSION The time-optimal control policy for the Formula 1 power unit obtained in Salazar, M. et al. (2016) was further extended towards an online implementation. Taking advantage of the fact that the trajectory of λ1 (s) is not strongly dependent on the constant co-state variables λ2→5 , a fixed λ1∗ (s) trajectory was chosen with the numerical solver Ebbesen, S. et al. (2016) and the remaining co-state variables were tuned with the presented NLP to precisely meet the boundary conditions on the energy consumption. This approach has the great advantage of enabling a simple implementation on the vehicle, whereby only a unique λ1∗ (s) trajectory must be stored on-board, together with a look-up table containing various λ2→5 values. From such an implementation the optimal control input signals can be inferred for a desired set of boundary conditions and for both power-limited and grip-limited regions, where the input powers are interpolated from Figure 2 or the power Pu demanded by the driver is optimally split as in Figure 3, respectively. This way, the strategies for 10 different fuel loads and 10 ∆SOC levels could be saved in 10 × 10 × 3 = 300 numerical values. The methodology was tested with a forward simulator, and the resulting trajectories well reflect the optimal ones obtained with Ebbesen, S. et al. (2016) and confirm the effectiveness of this supervisory control algorithm. Further research could exploit this effective structure for real-time control applications using methodologies like MPC, Morari and H Lee (1999), or approaches inspired by ECMS, Amb¨uhl and Guzzella (2009). ACKNOWLEDGEMENTS We would like to thank Ferrari S.p.A. for supporting this project. Moreover, we would like to express our gratitude to Ilse New, Camillo Balerna and Philipp Elbert for their helpful comments and interesting advice during the proofreading phase. REFERENCES (2013). 2014 formula one technical regulations. Technical report, FIA. Ahmad, M.A., Baba, I., Azuma, S.i., and Sugie, T. (2013). Model free tuning of variable state of charge target of hybrid electric vehicles. In Advances in Automotive Control, volume 7, 789–793. Algrain, M. (2005). Controlling an electric turbo compound system for exhaust gas energy recovery in a diesel engine. In IEEE International Conference on Electro Information Technology, 6–pp. IEEE. Amb¨uhl, D. and Guzzella, L. (2009). Predictive reference signal generator for hybrid electric vehicles. Vehicular Technology, IEEE Transactions on, 58(9), 4730–4740. Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. Casanova, D. (2000). On minimum time vehicle manoeuvring: The theoretical optimal lap. Ebbesen, S., Salazar, M., Bussi, C., Elbert, P., and Onder, C. H. (2016). Time-optimal control strategies for a hybrid electric race car. Submitted to IEEE Transactions on Control Systems Technologies. Hagura, R., Kitayama, S., and Yasui, Y. (2013). Development of energy management of hybrid electric vehicle for improving fuel consumption via sequential approximate optimization. In Advances in Automotive Control, volume 7, 800–805. 156

Jiao, X. and Shen, T. (2014). Sdp policy iteration-based energy management strategy using traffic information for commuter hybrid electric vehicles. Energies, 7(7), 4648–4675. Katraˇsnik, T., Rodman, S., Trenc, F., Hribernik, A., and Medica, V. (2003). Improvement of the dynamic characteristic of an automotive engine by a turbocharger assisted by an electric motor. Journal of Engineering for Gas Turbines and Power, 125(2), 590–595. Limebeer, D.J. and Rao, A.V. (2015). Faster, higher and greener: Vehicular optimal control. IEEE Control Systems Magazine, 36–56. Limebeer, D., Perantoni, G., and Rao, A. (2014). Optimal control of formula one car energy recovery systems. International Journal of Control, 87(10), 2065–2080. Lot, R. and Evangelou, S. (2013). Lap time optimization of a sports series hybrid electric vehicle. In 2013 World Congress on Engineering. Morari, M. and H Lee, J. (1999). Model predictive control: past, present and future. Computers & Chemical Engineering, 23(4), 667–682. Perantoni, G. and Limebeer, D.J. (2014). Optimal control for a formula one car with variable parameters. Vehicle System Dynamics, 52(5), 653–678. Picarelli, A. and Dempsey, M. (2014). Simulating the complete 2014 hybrid electric formula 1 cars. Hybrid and Electric Vehicles Conference (HEVC). Pontryagin, L.S. (1987). Mathematical theory of optimal processes. CRC Press. Rockafellar, R.T. (1997). Convex Analysis. Princeton University Press. Salazar, M., Ebbesen, S., Bussi, C., and Onder, C. H. (2016). Supervisory feedforward optimal control of a high performance hybrid electric race car. Submitted to IEEE Transactions on Control Systems Technologies. Vinter, R. (2010). Optimal control. Springer Science & Business Media. Wei, W., Zhuge, W., Zhang, Y., and He, Y. (2010). Comparative study on electric turbo-compounding systems for gasoline engine exhaust energy recovery. In ASME Turbo Expo 2010: Power for Land, Sea, and Air, 531–539. American Society of Mechanical Engineers.

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