Regenerative Braking Strategy of a Formula SAE Electric Race Car Using Energetic Macroscopic Representation

Article Regenerative Braking Strategy of a Formula SAE Electric Race Car Using Energetic Macroscopic Representation

Andrés Camilo Henao-Muñoz 1,* , Paulo Pereirinha 1,2 and Alain Bouscayrol 3 1 Coimbra Polytechnic-ISEC, 3030-199 Coimbra, Portugal; [email protected] 2 INESC-Coimbra DEEC, 3030-790 Coimbra, Portugal 3 Univ. Lille, Centrale Lille, Arts et Métiers Paris Tech, HEI, EA 2697 – L2EP, F-59000 Lille, France; [email protected] * Correspondence: [email protected]

Received: 28 April 2020; Accepted: 2 June 2020; Published: 11 June 2020

Abstract: This paper presents a braking strategy analysis for a Formula SAE electric race car. The proposed braking strategy aims to increase the recovery energy by a relevant distribution of the braking forces between the rear and front wheels. A mathematical model of the car is presented, and a simulation is performed in Matlab-Simulink. The model is organized using the energetic macroscopic representation graphical formalism. A real racetrack driving cycle is considered. Three braking strategies are compared considering the energy recovery and the vehicle stability. The simulation results show that the proposed strategy enables higher energy recovery while avoiding locking on both rear and front wheels. As in such a race the driving range is ﬁxed, the reduction in energy consumption can be used to reduce the battery size. The battery weight can thus be decreased to improve the vehicle performance during competition.

Keywords: regenerative brake; race car; energetic macroscopic representation; EMR; car modeling; electric diﬀerential

1. Introduction Greenhouse gas (GHG) emissions and air pollution are major concerns related to the massive deployment of gasoline and diesel vehicles. The harmful effects of GHG and air pollution on human health should drastically change the trends in transportation evolution. Electric mobility could help in reducing air pollution in urban areas, where it is more critical. Furthermore, benefits such as low noise, high efficiency, and potentially smart interaction with the grid will increase the electric vehicle (EV) market. Following this tendency, the European Union has ambitious goals for low emission mobility aligned with the commitments acquired in the Paris agreement [1,2]. Furthermore, the path followed is similar around the world, even if statistics are different [3,4]. Regenerative braking systems (RBS) allow EVs to recover part of the kinetic energy usually dissipated in heat when traditional friction braking systems (FBS) are used. The main benefits of having an RBS are greater energetic efficiency, greater range, and reduced stress and heat dissipation requirements for FBS. However, usually, RBS alone cannot provide enough braking force to guarantee vehicle safety. So, RBS and FBS are used together in hybrid braking systems (HBS). The usage of HBS requires a more complex braking strategy to seize the most from each braking system and compensate for their weaknesses [5,6]. A braking strategy should define the braking force distribution and the coordination of the braking sources. The most common objectives of braking strategies are vehicle stability and energy recovery [5,6]. The integration of the anti-lock feature using RBS is a major concern when braking strategies are

World Electric Vehicle Journal 2020, 11, 45; doi:10.3390/wevj11020045 www.mdpi.com/journal/wevj World Electric Vehicle Journal 2020, 11, 45 2 of 20 defined [7]. In this last regard, electric machines’ torque response facilitates the implementation of the active control of braking force, which could improve the anti-lock performance in EVs [5]. Regenerative braking strategies have been studied for hybrid EVs (HEVs) and EVs in general. Ruan et al. proposed a comparison of three braking strategies for passenger cars in [8]. An eco strategy to increase energy recovery, a safety strategy for wet and snow roads, and a sport strategy for more aggressive braking have been proposed. The economic benefit of each regenerative braking strategy was highlighted using different driving cycles, including one designed by the authors to make the test more authentic and reliable. A similar approach was used in the study of Xiao et al. [9]. The ideal braking force distribution curve was used to define rear and front wheels braking forces. Also, a racing strategy based on a fuzzy controller was proposed to improve the dynamic response of the braking systems during hard braking conditions. This strategy considers the state of charge (SoC) of the battery bank, the speed of the electric machine, and the braking strength requested to select the proportion of frictional and electric braking force. In the study of Itani et al. [10], two braking control methods were compared. First, a sliding mode controller was used to control the wheels’ slip ratio during unknown road conditions. The second method was based on ECE R13H constraints for an M1 passenger vehicle. The distribution of braking forces was as biased as possible to the front wheels to increase the energy recovered during braking. However, the regenerative braking was not used at very high speeds to avoid stability problems in low adhesion roads. Xu, et al. proposed a model predictive control (MPC) together with a holistic control strategy to optimize the braking torque in a four in-wheel motor electric vehicle in [11]. The results obtained with the MPC were compared to a rule-based controller for dry, wet, and snow road conditions. Other works such as [12–15] presented different braking strategies with similar objectives and applications in EVs. The works mentioned above used different strategies to define the distribution of braking forces and the participation of regenerative braking in passenger EVs. However, it is not common to find braking strategies for rear wheels traction EVs, such as formula race cars. Kumar et al. presented a braking strategy for a series hybrid electric vehicle with a rear wheel power train in [16]. The strategy is focused on modifying the conventional parallel braking system to achieve higher energy recovery. A combined braking algorithm is introduced, and the braking force is mainly commanded using RBS. The FBS is used as an emergency system. However, the regenerative braking torque was only limited by the maximal torque that the electric machine can provide, and the charging capabilities of the storage system were not considered. Furthermore, the algorithm was tested in a highway and urban driving cycles. Sangtarash et al. presented a comparison of series and parallel braking strategies in a hybrid electric bus with a rear power train in [17]. The study was performed in AVL/CRUISE software, and the results showed that braking distribution biased to the rear wheels increases the energy recovered. In the study of Le Soliec et al. [18], a regenerative braking study was done focusing on the estimation of the road adherence conditions of the road and wheel slip control. Simulations were conducted for high and low friction conditions, evaluating the longitudinal slip during braking. Race cars have been a very interesting testbed for automotive industry development. Formula E and Formula SAE for electric cars have been used not only for entertainment and educational purposes, but also for research and development of EVs technologies such as regenerative braking and control strategies [7,19–21]. Some modeling and simulation works have been published within the framework of Formula SAE competition [22–24]. Furthermore, some other works have analyzed the energy recovery in HEV and EV race cars [7,21,25]. Other works presented for race cars included the EMR graphical formalism to organize the models. EMR is a powerful tool to represent power conversion systems and define their control structure [26]. In the study of Transi et al. [27], the study of a race car focused on a constant share strategy for regenerative braking was presented. The car model was described and then represented using EMR. An extension was presented in [28], where a hardware-in-the-loop (HIL) simulation was included and the simulation results were validated. Montesinos-Miracle et al., presented the simulation of a race car using EMR formalism in [29]. The car model included an electric differential, but regenerative braking was not considered. Lebel et al. World Electric Vehicle Journal 2020, 11, 45 3 of 20

World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 3 of 20 studied the benefits of regenerative braking for an electric superbike using EMR graphical formalism inin [ 30[30].]. TheirTheir resultsresults suggestedsuggested thatthat energyenergy recoveryrecovery increasesincreases thethe rangerange ofof thethe vehiclevehicle byby upup toto 14%.14%. GuoGuo etet al. al. proposedproposed aa methodmethod toto findfind the the optimal optimal distribution distribution ratioratio ofof the the regenerative regenerative braking brakingand and thethe friction friction braking braking torque torque to to maximize maximize energy energy recovery recovery inin [31[31].]. ToTo dodo that, that, a a genetic genetic algorithm algorithm waswas used,used, and and the the requested requested braking braking torque torque was was located located as muchas much as possibleas possible on theon the front front wheels wheels where where the electricthe electric machines machines were were located. located. MostMost of of the the previous previous studies studies are are focused focused on on the the range range extension extension due due to to the the recovery recovery energy. energy. In In a racea race context, context, the the distance distance is fixed, is fixed, and theand recovery the recovery energy energy during during braking braking can be consideredcan be considered to reduce to thereduce size ofthe the size battery of the and battery the FBS. and Such the aFBS. reduction Such a can reduction have an can impact have on an the impact velocity, on acceleration,the velocity, andacceleration, thus the general and thus performance the general ofperformance the car. of the car. ThisThis paper paper aimsaims toto determine determine thethe energy energy recovery recovery potential potential of of an an electric electric race race car car with with RBS RBS and and electricelectric didifferential.fferential. TheThe carcar modelmodel isis organizedorganized usingusing thethe EMREMR formalism,formalism, andand thethe brakingbraking strategystrategy isis presented. presented. TheThe proposedproposed brakingbraking strategystrategy isis comparedcompared toto otherother strategiesstrategies foundfound inin thethe literature.literature. TheThe potential potential energy energy recovery recovery and and braking braking stability stability of of each each strategy strategy are are highlighted, highlighted, and and the thepotential potential massmass reduction reduction of of the the battery battery is is quantified. quantified. SectionSection2 2presents presents the the model model of of the the car car studied studied using using the the EMR EMR formalism. formalism. The The control control structure structure isis introducedintroduced inin Section Section3 .3. Section Section4 deals4 deals with with the the investigated investigated braking braking strategy. strategy. The The results results are are presentedpresented and and discussed discussed in in Section Section5 .5.

2.2. ModelingModeling andand EMREMR ofof thethe StudiedStudied CarCar FigureFigure1 1shows shows the the structural structural description description of of the the studied studied car. car. The The power power train train includes includes two two electric electric machines,machines, one one for for each each rear rear wheel. wheel. The electricThe electric machines machines are permanent are permanent magnet synchronous magnet synchronous machines (PMSM)machines controlled (PMSM) controlled independently independently by two inverters by two inverters supplied supplied by a unique by a batteryunique battery pack. Thepack. shaft The ofshaft the of PMSM the PMSM is connected is connected to a fixed-ratioto a fixed-ratio gearbox, gearbox, connected connected to theto the rear rear wheels. wheels. With With this this power power traintrain configuration, configuration, a a mechanical mechanical di differentialfferential is is not not necessary, necessary, and and an an electrical electrical di ffdifferentialerential strategy strategy is includedis included to guarantee to guarantee stability stability during during a curve. a The cu environmentalrve. The environmental forces coming forces from coming the aerodynamic from the dragaerodynamic and rolling drag resistance and rolling are considered.resistance are considered.

ich1 iem1

DC/AC u Rear Front bat Converter axle left axle left

Tpmsm1 Twh1 ωgear1 ω ωwh1 wh3 y r e Fenv Batt ωgear2 ω vcar wh2 ω T wh4 pmsm2 Twh2 ich2

DC/AC Rear axle Front axle i u em2 bat Converter right right

FigureFigure 1.1.Structural Structural representationrepresentation ofof thethe race race car. car.

InIn anan energetic system, system, there there are are two two main main contro controll goals. goals. First, First, the dynamic the dynamic performance, performance, which whichimplies implies that the that system the system complies complies with with its itsfuncti functionalonal task task in in the the best best way.way. Second,Second, energyenergy consumption, which guarantees that the energy flow through the subsystems is safe and efficient. EMR is a graphical formalism developed to represent a system in such a way that both energetic World Electric Vehicle Journal 2020, 11, 45 4 of 20 consumption,World Electric Vehicle which Journal guarantees 2020, 11, x FOR that PEER the REVIEW energy flow through the subsystems is safe and effi 4cient. of 20 EMR is a graphical formalism developed to represent a system in such a way that both energetic interactions and dynamic performance control are clearly identified. Furthermore, EMR builds the interactions and dynamic performance control are clearly identified. Furthermore, EMR builds the control structure based on a system inversion approach [26]. control structure based on a system inversion approach [26]. The graphical elements in EMR include different colors and blocks that represent differently the The graphical elements in EMR include different colors and blocks that represent differently the interactions in the subsystems. The causality principle is at the core of this description: the physical interactions in the subsystems. The causality principle is at the core of this description: the physical integral causality is exclusively used and is related to energy accumulation elements [26]. The four integral causality is exclusively used and is related to energy accumulation elements [26]. The four basic basic elements used in EMR are described in Figure 2. Moreover, EMR leads to a systematic elements used in EMR are described in Figure2. Moreover, EMR leads to a systematic organization organization of control schemes from a systematic inversion philosophy (elements in light blue). The of control schemes from a systematic inversion philosophy (elements in light blue). The conversion conversion elements are directly inverted, the accumulation elements are inverted using closed-loop elements are directly inverted, the accumulation elements are inverted using closed-loop control, control, and the coupling elements are inverted using distribution inputs as a degree of freedom to and the coupling elements are inverted using distribution inputs as a degree of freedom to decompose decompose the energy flow. the energy flow.

Mono-physical Multi-physical x x1 y2 x1 y2

Source x1-ref y2-ref y y1 x2 y1 x2 x ref z12 z12

(a) (b)

x1 yp-ref x2-meas y2 xp-ref y1 y(p-1)-ref . x2 x x1 y2 y2-meas . . (m-1)-ref. . . . . x1-ref xm yp . xp x1-ref y1-ref y1 x2 y2-ref y p xk

(c) (d) Figure 2. EMR elements. (a) Source element; (b) Conversion element; (c) Accumulation element; (Figured) Coupling 2. EMR element. elements. (a) Source element; (b) Conversion element; (c) Accumulation element; (d) Coupling element. The detailed models for each subsystem and the complete EMR of the race car are presented as follows.The detailed models for each subsystem and the complete EMR of the race car are presented as follows.A lithium-iron-phosphate battery pack is used in the race car. In this case, a simpliﬁed model of theA battery lithium-iron-phosphate is considered. An battery ideal voltage pack is sourceused in in the series race withcar. In an this equivalent case, a simplified internal resistance model of representsthe battery the is voltageconsidered. and stateAn ideal of charge voltage of thesource battery in series during with operation an equivalent [32]. The internal battery isresistance a source inrepresents EMR. the voltage and state of charge of the battery during operation [32]. The battery is a source in EMR. u = uoc i Req (1) bat − bat 𝑢 =𝑢 Z−𝑖 𝑅 ti dt (1) SoC(t) = SoC bat (2) 0 − Cap3600 t0 𝑆𝑜𝐶(𝑡) =𝑆𝑜𝐶 − (2) where ubat is the voltage estimation, uoc is the open circuit voltage, ibat is the current, Req is the internal resistance,where 𝑢Cap is theis the voltage nominal estimation, capacity, 𝑢SoC isis the the open state ofcircuit charge voltage, of the battery,𝑖 is the and current,SoC0 is 𝑅 the initialis the valueinternal of SoCresistance,. 𝐶𝑎𝑝 is the nominal capacity, 𝑆𝑜𝐶 is the state of charge of the battery, and 𝑆𝑜𝐶 is theTo initial supply value both of PMSM𝑆𝑜𝐶. with one battery pack, a parallel connection is used. The current is split dependingTo supply on the both demand PMSM of with each one electrical battery machine pack, a (couplingparallel connection element in is EMR): used. The current is split depending on the demand of each electrical machine (coupling element in EMR): = + ibat ich1 ich2 (3) 𝑖 =𝑖 +𝑖 (3)

where 𝑖 and 𝑖 are the currents requested by each converter of machine 1 and 2. World Electric Vehicle Journal 2020, 11, 45 5 of 20

where ich1 and ich1 are the currents requested by each converter of machine 1 and 2. An equivalent static model of the PMSM and its inverter is considered for simplicity in an energetic study [33]. This model assumes that the torque produced by the machine (Tpmsm) is equal to the torque reference (Tpmsmre f ). The achievable torque in each PMSM is a function of the angular speed of the machine [34]. The current requested from the battery by each machine is estimated using the mechanical power and the eﬃciency of the PMSM and its inverter. The PMSM is represented by a multi-physical conversion element in EMR.

Tpmsm= Tpmsmre f (4)  Tpmsmωgear 1 Ppmsm(t) < 0 i = η qη q, where q =  (5) ch pmsm inv  ubat  1 Ppmsm(t) 0 − ≥ q where ηpmsm and ηinv are the efficiencies of the PMSM and its inverter respectively, q indicates the power flow direction, and ωgear is the angular velocity of the PMSM. Ppmsm is the instantaneous power of the PMSM, which is positive when the PMSM operates as a motor and negative when the PMSM operates as a generator (regenerative braking). A ratio kgear defines the conversion of the mechanical power coming from the PMSM that then supplies the wheel (mono-physical conversion element in EMR).  ( q T = kgearTpmsmηgear  1 Ppmsm(t) < 0 wh , where q = − (6)  ωgear = kgearωwh 1 Ppmsm(t) 0 ≥ where Twh is the resulting torque applied to the wheels, ηgear is the gearbox efficiency, and ωwh is the angular speed of the wheel. Gearboxes are described by mono-domain conversion elements. The torque and angular speed in the transmission are converted to longitudinal force and car velocity through the wheels as follows (mono-physical conversion element in EMR). ( T = Fpmsm R wh wh wh (7) vwh = ωwhRwh

where Fpmsmwh is the force applied to the wheel coming from the PMSM, Rwh is the radius of the wheel, and vwh is the velocity of the wheel. Furthermore, the force applied to each rear wheel (subscripts 1 and 2) is the result of the sum of the forces coming from the PMSM and the FBS (coupling element). On the other hand, the front wheels (subscripts 3 and 4) do not have an electric machine. So, they only have frictional braking forces associated. These forces applied by the FBS are assumed to be equal to their references (source element in EMR). = Fbrwh Fbrwh re f (8) − = + Fwh1,2 Fpmsmwh Fbrwh (9) = Fwh3,4 Fbrwh (10) where Fbrwh is the frictional braking force applied to the wheel and Fwh is the total force applied to the wheel. After the forces applied in each wheel are determined, the total force applied to the car and the velocity achieved by each wheel can be calculated as a function of the radius of the curvature. The radius of the curvature depends on the steering angle and should be included to guarantee stability during a curve [35]. This is represented as a coupling element in EMR. ! ! R l/2 R + l/2 = + curve + + curve Ftot Fwh1 Fwh3 − Fwh2 Fwh4 (11) Rcurve Rcurve World Electric Vehicle Journal 2020, 11, 45 6 of 20

 Rcurve l/2  = = −  vwh1 vwh3 vcar R World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW curve 6 of(12) 20  + l  v = v = v Rcurve /2  wh2 wh4 car Rcurve where 𝐹 is the total force applied to the wheels of the car (wheels 1 and 3 are in the left, and 2 and where Ftot is the total force applied to the wheels of the car (wheels 1 and 3 are in the left, and 2 and 4 4 are the right), 𝑅 is the radius of the curvature, 𝑣 is the velocity of the car, and 𝑙 is the width are the right), R is the radius of the curvature, v is the velocity of the car, and l is the width of of the car. curve car the car. Then, the total force applied to the wheels interacts with the forces imposed by the environment, Then, the total force applied to the wheels interacts with the forces imposed by the environment, 𝐹. The last is defined by the road conditions (slope, road, and tires material), car aerodynamics and F . The last is defined by the road conditions (slope, road, and tires material), car aerodynamics and velocity,env see Equation (14). In this case, the slope is neglected. This interaction defines the evolution velocity, see Equation (14). In this case, the slope is neglected. This interaction defines the evolution of of the velocity of the car (accumulation element in EMR). the velocity of the car (accumulation element in EMR). 𝐹 −𝐹 =𝑀 (13) dvcar Ftot Fenv = M (13) −1 dt 𝐹 = 𝜌𝐶 𝐴𝑣 +𝑀𝑔𝐶 (14) 2 1 2 Fenv = ρCxAvcar + MgCrr (14) where 𝐹 is the environment force, 𝑀 is the2 mass of the car, 𝜌 is the air density, 𝐶 is the drag coefficient,where Fenv 𝐴is theis the environment frontal area force, of theM iscar, the 𝑔 mass is the of gravity, the car, ρandis the𝐶 air is density,the rollingCx isresistance the drag coefficient.coefficient, A is the frontal area of the car, g is the gravity, and Crr is the rolling resistance coefficient. AfterAfter having having the the mathematical mathematical equations, equations, the the EMR EMR rules rules are are used used to to build build the the graphical graphical representationrepresentation starting starting from from the the battery battery pack pack to tothe the environment environment forces forces [26]. [ 26Figure]. Figure 3 presents3 presents the completethe complete EMR EMR of the of race the racecar. It car. should It should be mentione be mentionedd that thatthe inertias the inertias of the of transmission the transmission shafts shafts are neglectedare neglected compared compared to the to mass the mass of the of car. the car.

PMSM and wheels chassis inverter vwh1 (10) Parallel gearbox Fbr (4) (9) Brake wh1-ref connection (6) (7) Fpmsm wh1 (10) u (5) Tpmsm1 Twh1 wh1 vwh3 Battery bat Fbrwh1 Brake Fbrwh3-ref (3) wh3 (1) ich1 Fwh3 ubat ⍵gear1 ⍵wh1 vwh1 Fwh1 (11) Bat. Tpmsm1-ref (13) (4) (12) Ftot vcar ibat (6) (7) pmsm (5) Tpmsm2 Twh2 F wh2 (9) vwh1 ubat Env Fwh2 vcar Fenv ich2 ⍵gear2 ⍵wh2 vwh2 (10) Tpmsm2-ref (10) Fbrwh2 vwh2 vwh4 Fbr Fbrwh4-ref wh2-ref Brake Brake wh2 vwh2 wh4 Fwh4 FigureFigure 3. 3. CompleteComplete EMR EMR representation representation of of studied studied car. car. 3. Inversion-Based Control of the Studied Car 3. Inversion-Based Control of the Studied Car The EMR formalism is used to define the tuning path (Figure4) to control the car operation. The EMR formalism is used to define the tuning path (Figure 4) to control the car operation. The The controlled variable is the car velocity vcar. Following the causality principle, the force applied to each controlled variable is the car velocity 𝑣. Following the causality principle, the force applied to each wheel should be controlled during traction and braking to control vcar. The traction mode is controlled wheel should be controlled during traction and braking to control 𝑣 . The traction mode is using only the rear wheels, while the braking mode is controlled using all the wheels. Therefore, controlled using only the rear wheels, while the braking mode is controlled using all the wheels. the tuning variables are the torque references for each machine Tpmsmre f . Furthermore, the references Therefore, the tuning variables are the torque references for each machine 𝑇. Furthermore, the for the frictional braking forces in each wheel Fbrwh re f are also tuning variables during braking. references for the frictional braking forces in each −wheel 𝐹 are also tuning variables during For the inversion-based control, the starting point is the controlled variable. From here, the model braking.is followed backwards inverting each block found in the tuning path until each tuning variable is definedFor [the26]. inversion-based Therefore, the chassis control, is thethe firststarting element point to invert.is the controlled variable. From here, the model is followed backwards inverting each block found in the tuning path until each tuning variable is defined [26]. Therefore, the chassis is the first element to invert.

WorldWorld ElectricElectric VehicleVehicle JournalJournal 20202020,,11 11,, 45x FOR PEER REVIEW 7 of 20

Figure 4. Tuning path of the studied car. Figure 4. Tuning path of the studied car. The chassis is inverted through a closed-loop controller described in Equation (15). An IP controller The chassis is inverted through a closed-loop controller described in Equation (15). An IP can be used. Fenv is assumed to be measurable and is used as a feedforward signal. controller can be used. 𝐹 is assumed to be measurable and is used as a feedforward signal. Ftot re f = Cvcar vcarre f vcarmeas + Fenvmeas (15) 𝐹 =𝐶 𝑣− −𝑣+𝐹 (15) where Ftot is the reference of the total force applied to the car, Cv is the velocity controller, vcar is where 𝐹re f is the reference of the total force applied to the car,car 𝐶 is the velocity controller,meas the measured velocity of the car, vcarre f is the reference velocity, and Fenvmeas is the measured resistive 𝑣 is the measured velocity of the car, 𝑣 is the reference velocity, and 𝐹 is the force estimated using Equation (14). measured resistive force estimated using Equation (14). Then, Ftot re f is distributed into the reference forces of each wheel. First, the distribution between Then, 𝐹 is distributed into the reference forces of each wheel. First, the distribution rear and front wheels is defined by kα, which is zero during traction and determined by the braking between rear and front wheels is defined by 𝑘 , which is zero during traction and determined by the strategy during braking. Furthermore, the operation of the differential defines the forces applied to braking strategy during braking. Furthermore, the operation of the differential defines the forces the left and right wheels during straight line motion. This is described by the differential coefficient applied to the left and right wheels during straight line motion. This is described by the differential kd, which is usually defined as 0.5 for most of the traction applications [35]. If the car is moving in a coefficient 𝑘 , which is usually defined as 0.5 for most of the traction applications [35]. If the car is curved line, the electrical differential acts on the force distribution depending on the radius curvature moving in a curved line, the electrical differential acts on the force distribution depending on the to guarantee stability. This is represented by a coupling inversion in EMR. radius curvature to guarantee stability. This is represented by a coupling inversion in EMR.  Rcurve l/2  F = Ftot − (1 k )k  wh1re f re f Rcurve α d ⎧ 𝐹 =𝐹 (1−𝑘−)𝑘  l (  Rcurve + /2 k , braking ⎪  F = F (1 k )(1 k ) α  wh2re f totre f Rcurve α d k = ⎪𝐹  =𝐹 (1−𝑘−)(1 − 𝑘− ) α 𝑘 ,𝑏𝑟𝑎𝑘𝑖𝑛𝑔  l 0, traction (16)  Rcurve /2 𝑘 =  F = Ftot − kαk 0, 𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛  wh3re f re f Rcurve d kd = 0.5 (16) ⎨   𝐹 =𝐹 Rcurve + l/𝑘2 𝑘 𝑘 =0.5  F = F k (1 k ) ⎪ wh4re f totre f Rcurve α d − ⎪ 𝐹 =𝐹 𝑘(1 − 𝑘) ⎩ where Fwhre f is the reference for the force applied to each wheel. The contribution of the FBS and the PMSM to the braking force applied in the rear wheels is where 𝐹 is the reference for the force applied to each wheel. k k definedThe by contributionbr (coupling of inversion the FBS and in EMR). the PMSMbr value to th wille braking be defined force in applied Section 4.2in .the rear wheels is 𝑘  𝑘 defined by (coupling inversion in EMR).= value will be defined in Section 4.2.  Fbrwh re f kbrFwhre f  −  𝐹 =𝑘𝐹 , where 0 < kbr < 1 (17)  F = ( k )F pmsmwh re f 1 br whre f ,𝑤ℎ𝑒𝑟𝑒 0<𝑘 <1 (17) − ( − ) 𝐹 = 1−𝑘 𝐹 where Fbrwh re f is the reference for the frictional braking forces applied by the FBS, and Fpmsmwh re f is the where 𝐹 is the reference for the frictional braking forces applied by the FBS, and 𝐹− reference for− the electrical braking forces applied by the PMSM. is the reference for the electrical braking forces applied by the PMSM. The wheels’ inversion is simply an equation that calculates the reference torque in the rear wheels The wheels’ inversion is simply an equation that calculates the reference torque in the rear using the radius of the wheels. This is done only for rear wheels. The gearbox Equation (6) is then wheels using the radius of the wheels. This is done only for rear wheels. The gearbox Equation (6) is inverted using kgear. Both are represented by mono-physical conversion inversion in EMR. then inverted using 𝑘. Both are represented by mono-physical conversion inversion in EMR. = T𝑇whre f =𝑅RwhF𝐹pmsmwh re f (18) − (18)

Twh 𝑇 = = re f Tpmsmre f (19)(19) kgear World Electric Vehicle Journal 2020, 11, 45 8 of 20 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 8 of 20 where 𝑇 is the reference of the torque applied to each rear wheel, 𝑇 is the reference of where Twhre f is the reference of the torque applied to each rear wheel, Tpmsmre f is the reference of the thetorque torque produced produced by each by each PMSM. PMSM. Figure5 5 presents presents the the EMR EMR of theof maximalthe maximal control cont structurerol structure for the for race the car. race Moreover, car. Moreover, the strategy the strategyblock deﬁnes block the defines distribution the distribution inputs (k αinputs, kd, and (𝑘k,br 𝑘) as, and explained 𝑘) as inexplained the next section.in the next section.

PMSM and wheels inverter chassis gearbox (9) vwh (10) Parallel (4) connection Brake (6) (7) Fpmsmwh1 wh1 (10) ubat (5)T pmsm1 Twh1 vwh3 Fbrwh1 Battery Brake (3) wh3 Fbrwh3 (1) ubat ich1 ⍵gear1 ⍵wh1 vwh1 Fwh1 (11) Fbrwh3-ref Bat. (4) (12) (13) (15) Ftot vcar (5)T (6) T (7) Fpmsmwh2 (9) ibat ubat pmsm2 wh2 vwh1 Env Fwh2 vcar Fenv ich2 ⍵gear2 ⍵wh2 vwh2 (10) (10) Fbrwh2 vwh2 vwh4 Brake Brake wh2 vwh2 wh4 Fbrwh4 Fbrwh1-ref Fbrwh2-ref Fbrwh4-ref

(17) (16) (19) (18) (2) Fwh2-ref (14)

Tpmsm2-ref Twh2-ref Fpmsmwh2-ref (17)

Fwh1-ref Ftot-ref vcar-ref SoCest Tpmsm1-ref Twh1-ref Fpmsmwh1-ref (19) (18) kbr kα, kd strategy Ftot-ref Figure 5. MaximumMaximum control structure for studied car model. 4. Braking Strategy 4. Braking Strategy In the control strategy, the distribution inputs have to be defined in order to define the energy In the control strategy, the distribution inputs have to be defined in order to define the energy distribution between the wheels for both modes, tractions and braking. kα distributes the forces between distribution between the wheels for both modes, tractions and braking. 𝑘 distributes the forces the front and the rear part, kd distributes the forces between the left and right wheels (differential), between the front and the rear part, 𝑘 distributes the forces between the left and right wheels and kbr distributes the forces between the electric and mechanical brakes of the rear wheels. (differential), and 𝑘 distributes the forces between the electric and mechanical brakes of the rear wheels.4.1. Distribution of Braking Forces Between Front and Rear Wheels

4.1. DistributionThe braking of strategyBraking Forc mustes guaranteeBetween Front that and the Rear car stopsWheels. as quickly as the driver requests while the stability of the car is ensured. To guarantee this, the reference of the total braking force should The braking strategy must guarantee that the car stops as quickly as the driver requests while be well distributed into the rear and front wheels. This distribution is deﬁned by the variable kα the stability of the car is ensured. To guarantee this, the reference of the total braking force should be (see Equation (20)), which can be ﬁxed or variable, depending on the braking strategy [36]. well distributed into the rear and front wheels. This distribution is defined by the variable 𝑘 (see Equation (20)), which can be fixed or variable, depending on the braking strategy [36].  F f ront = Ftotre f kα  (20)  𝐹 =𝐹 𝑘  Frear= Ftot (1 kα) re f − (20) 𝐹 =𝐹(1 − 𝑘) where F f ront and Frear are the braking forces applied to the front and rear wheels, respectively. where 𝐹 and 𝐹 are the braking forces applied to the front and rear wheels, respectively. Regardless of the objective of the braking strategy, kα value must ensure the stability of the car duringRegardless braking, i.e.,of the wheels objective locking of the should braking be avoided. strategy, If 𝑘 the value rear wheels must ensure are locked, the stability the car willof the easily car duringenter in braking, oversteering i.e., condition,wheels locking leading should to instability be avoided. of the If car. the Onrear the wheels other hand,are locked, if the frontthe car wheels will easilyare locked, enter the in oversteering car will enter condition, in understeering leading and to inst its maneuverabilityability of the car. is On reduced the other [6]. Thehand, ideal if the braking front wheelsdistribution are locked, curve the delimits car will a safeenter braking in understeering distribution and ratio its maneuverability for any road condition. is reduced Its [6]. derivation The ideal is brakingsummarized distribution below. For curv moree delimits detailed a informationsafe braking refer distribution to [6]. ratio for any road condition. Its derivation is summarized below. For more detailed information refer to [6]. World Electric Vehicle Journal 2020, 11, 45 9 of 20

In general, the maximal braking force is deﬁned by the tires to road adhesive coeﬃcient and the normal force, which is usually determined by the weight of the car, see (21). Moreover, the weight distribution in each axle changes during acceleration and braking due to load transfer. The weight distributed in each axle can be estimated as a function of the deceleration j (j = dvcar/dt), using the − equilibrium of moments in the front and rear wheels, as shown in Equation (22).

Fbrmax = µW = µMg (21)   Mg j  W f ront = L Lb + Hg g  (22)  Mg j  Wrear = La Hg L − g where Fbrmax is the maximal braking force, µ is the adhesive coeﬃcient, W f ront and Wrear are the weights in front and rear axles respectively, L is the wheel base, La and Lb are the distances to the center of gravity (COG) from the front and rear axles respectively, Hg is the COG height, and j is the deceleration. Now, the ratio of the front and rear wheels braking forces can be derived from Equation (20). Similarly, the same ratio can be obtained from the weight distribution deﬁned in Equation (22) and the relation described in Equation (21). F f ront k = α (23) Frear (1 kα) − j F f ront W f ront Lb + Hg g = = (24) F W j rear rear La Hg − g Then, solving Equations (23) and (24) for kα, the braking force distribution ratio can be determined for any deceleration value. Some examples are presented in Figure6 (solid color lines).

j Lb + Hg g k = (25) α L

Next, it is important to deﬁne the maximal deceleration value jmax in terms of the road adhesive coeﬃcient. This can be done using the second Newton’s law, as shown in Equation (26).  dvcar Fbrmax  − = j = = gµ  dt max µ M  max (26)   Fbrmax = F f rontmax + Frearmax

where F f rontmax and Frearmax are the maximal forces that can be applied to the front and rear wheels before locking. Then, from Equation (26), the normalized maximal braking force in the rear can be defined in terms of the adhesive coefficient µ and the normalized maximal braking forces in the front. The dotted lines in Figure6 describe this distribution of braking forces considering maximal braking forces for different adhesive coefficients. Frear F f rontmax max = µ (27) Mg − Mg In Figure6, the intersection of the color lines (black dots) defines the distribution ratio where the maximal deceleration is achieved for different adhesive coefficient. If the black dots are joined, the ideal braking distribution curve is drawn. The shadowed area represents the braking distribution ratios where the front wheels will lock first. The area above the ideal braking distribution curve describes the distribution ratios where the rear wheels will lock first. World Electric Vehicle Journal 2020, 11, 45 10 of 20 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 10 of 20

FigureFigure 6. IdealIdeal braking braking distribution distribution curve. curve.

AlthoughAlthough the the ideal ideal braking braking distribution distribution curve curve can can be be used used as as safe distribution criteria, criteria, it it could place much much lower lower braking braking force force in in the the rear rear wheels wheels than than its its maximal maximal value value when when a a good good tire tire to to road road adhesionadhesion is achieved. Considering aa racingracing context,context, a a constant constant adhesive adhesive coe coﬃefficientcient can can be be assumed assumed as as0.9 0.9 (peak (peak value value for dry for asphalt dry asphalt [6]). Furthermore, [6]). Furthermore, the normal the force normal has anforce aerodynamic has an aerodynamic contribution contributionknown as aerodynamic known as downforce,aerodynamic which downforce, is deﬁned wh in Equationich is defined (28) and in canEquation be assumed (28) toand be can equally be assumeddistributed to inbe theequally rear anddistributed front axles. in the These rear conditions and front increaseaxles. These the maximalconditions braking increase force the that maximal can be brakingapplied force in a race that car.can Bearingbe applied this in in a mind, race car. the Bearing braking this strategy in mind, proposed the braking aims tostrategy recover proposed as much aimsenergy to asrecover possible as much by placing energy as as much possible braking by placing force in as the much rear wheels.braking force in the rear wheels. 1 1 2 DF𝐷𝐹= = C𝐶Av𝐴𝑣 (28)(28) 22 z car  𝐹 =𝜇(𝑊 +𝐷𝐹)  Frearmax= µ(Wrear + DFrear) (29)  (29)  𝐹 =𝜇𝑊 +𝐷𝐹  F f rontmax= µ W f ront + DF f ront

where DFrear and DF f ront are the downforce located in the rear and front axle, and Cz is the wheredownforce 𝐷𝐹 coe ffiandcient. 𝐷𝐹 are the downforce located in the rear and front axle, and 𝐶 is the downforce coefficient. The strategy described in Figure7 establishes the distribution coe fficient kα and the references The strategy described in Figure 7 establishes the distribution coefficient 𝑘 and the references for the rear and front braking forces. First, the strategy considers if Ftotre f can be supplied by the rear for the rear and front braking forces. First, the strategy considers if 𝐹can be supplied by the rear wheels. Then, if the total force must be distributed, the reference for the rear wheels (Frearre f ) is 95% of wheels. Then, if the total force must be distributed, the reference for the rear wheels (𝐹 ) is 95% its maximal value to avoid locking the wheels (Frearmax ). After this, Frearre f is compared to the maximal ofbraking its maximal force thatvalue could to avoid be produced locking the by thewheels RBS ( (𝐹FRBSmax ).). After If Frear this,re f can 𝐹 be produced is compared by the to RBS, the the strategy can decide the distribution of braking forces; if not, Frear is FRBS , and the remaining maximal braking force that could be produced by the RBS (𝐹). reIf f 𝐹max can be produced by braking force has to be provided by the front wheels (F f rontre f ). Finally, if this F f rontre f is lower than the RBS, the strategy can decide the distribution of braking forces; if not, 𝐹 is 𝐹, and the F f rontmax , the strategy can decide the distribution of braking forces. Otherwise, if F f rontre f is greater or remaining braking force has to be provided by the front wheels (𝐹). Finally, if this 𝐹 is equal to F f ront , the distribution of braking forces will be defined by the maximal forces in the front lower than 𝐹 max , the strategy can decide the distribution of braking forces. Otherwise, if 𝐹 and rear wheels. is greater or equal to 𝐹, the distribution of braking forces will be defined by the maximal forces in the front and rear wheels. World Electric Vehicle Journal 2020, 11, 45 11 of 20 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 11 of 20

Figure 7. Braking strategy proposed. Figure 7. Braking strategy proposed. 4.2. RBS and FBS Contribution in the Rear Wheels 4.2. RBS and FBS Contribution in the Rear Wheels The strategy must consider the maximal regenerative braking force (FRBSmax ), which is calculated The strategy must consider the maximal regenerative braking force (𝐹 ), which is calculated based on the SoC and charging capabilities of the battery, and the torque limitations of the PMSM. basedThis is on used the to SoC decide and the charging amount capabilities of braking forceof the produced battery, and by the the RBS torque and limitations the FBS in the of rearthe PMSM. wheels, This is used to decide the amount of braking force produced by the RBS and the FBS in the rear expressed by kbr. kbr is calculated based on the ﬂowchart presented in Figure8. The maximal braking wheels,force that expressed can be produced by 𝑘 . by 𝑘 the is PMSM calculated is limited based by theon maximalthe flowchart torque presented that the PMSM in Figure can produce 8. The maximal braking force that can be produced by the PMSM is limited by the maximal torque that the (Fpmsmmax (ωmeas)), and the maximal charging current that the battery can receive. Fpmsmmax (ωmeas) is PMSM can produce (𝐹 (𝜔)), and the maximal charging current that the battery can estimated based on the torque vs speed characteristic of the PMSM [33]. On the other hand, Fbatmax (ωmeas) receive. 𝐹 (𝜔 ) is estimated based on the torque vs speed characteristic of the PMSM [33]. is estimated based on the maximal charging current of the battery, as shown in Equation (30).

On the other hand, 𝐹(𝜔) is estimated based on the maximal charging current of the v i k η battery, as shown in Equation (30). bat batmax gear gear Fbatmax ωgear, ibatmax = (30) ωgear Rwh 𝐹 𝜔,𝑖= (30) Once FRBSmax is estimated, Kbr is selected and used in Equation (17) to deﬁne the contribution of the RBSOnce and 𝐹 the FBS is toestimated, the braking 𝐾 force. is selected and used in Equation (17) to define the contribution of the RBS and the FBS to the braking force.

World Electric Vehicle Journal 2020, 11, 45 12 of 20 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 12 of 20

Figure 8. Decision of RBS and FBS contribution coefficient Kbr. Figure 8. Decision of RBS and FBS contribution coefficient 𝐾. 5. Results 5. ResultsThe results of the model simulation are presented. First, the car model is implemented in Matlab-Simulink.The results of the Furthermore, model simulation different are braking presented. strategies First, arethe studiedcar model and is compared.implemented in Matlab- Simulink. Furthermore, different braking strategies are studied and compared. 5.1. Studied Race Car and Driving Cycle 5.1. StudiedThe study Race caseCar and of thisDriving work Cycle is a race car designed and built by a team of students at Coimbra PolytechnicThe study - ISEC, case Portugal.of this work The is aerodynamica race car desig parametersned and built are presented by a team in of Table students1. Furthermore, at Coimbra Polytechnicthe powertrain - ISEC, and Portugal. battery parameters The aerodynamic are presented parame inters Tables are presented2 and3. in Table 1. Furthermore, the powertrainThe software and battery Optimum parameters Lap is are used presented to obtain in Table the velocity 2 and 3. profile that the car can follow in a real competition [27,28,30,37]. In this case, a car is created based on the parameters of ISEC’s car. Furthermore, the racetrackTable selected 1. Aerodynamic is a real circuit parameters used in of SAE ISEC’s competition race car. 2012, which was available in the software database. The driving cycle and the racetrack map taken from Optimum Lap are presented in Figure9. Parameter Value Mass [kg] 375.00 Table 1. AerodynamicWidth [m] parameters of ISEC’s 1.35 race car. Drag coefficient (𝐶) 0.29 Parameter Value Downforce coefficient (𝐶) 1.20 FrontMass area [kg] [m2] 0.84 375.00 WheelWidth diameter [m] [m] 0.49 1.35 Drag coefficient (Cx) 0.29 Rolling resistance coefficient (𝐶) 0.03 Downforce coefficient (Cz) 1.20 FrontWheelbase area [m [m]2] 1.460.84 Wheel diameterLa [m] [m] 0.7 0.49 Rolling resistanceLb [m] coe fficient (Crr) 0.76 0.03 WheelbaseHg [m] [m] 0.34 1.46 La [m] 0.7 Lb [m] 0.76 Hg [m] 0.34 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 13 of 20

Table 2. Powertrain parameters of ISEC’s race car.

Parameter Value Rated torque [Nm] 47.70 Rated power [kW] 17.02 World Electric Vehicle Journal 2020Number, 11, 45 of electric machines 2 13 of 20 Total rated torque [Nm] 94.50 Total rated power [kW] 34.05 RatedTable angular 2. Powertrain speed parameters (rpm) of ISEC’s 3000 race car. Max. angularParameter speed (rpm) 6000 Value Gearbox ratio 50/14 Rated torque [Nm] 47.70 RatedTraction power mode [kW] Rear wheel 17.02 drive Number of electric machines 2 TableTotal 3. ratedBattery torque pack [Nm]parameters of ISEC’s 94.50race car. Total rated power [kW] 34.05 Rated angularParameter speed (rpm) Value 3000 Max. angularRated speedcell voltage (rpm) [V] 3.20 6000 GearboxMax. cell ratio voltage [V] 4.25 50/ 14 Traction mode Rear wheel drive Min. cell voltage [V] 2.50 Cells in series 30.0 Table 3. BatteryParallel pack arrays parameters of ISEC’s 1.0 race car. Maximal Parametercharging current [A] 80Value Rated Capacity [Ah] 90 Rated cell voltage [V] 3.20 InternalMax. cell resistance voltage [V] [Ω] 0.006 4.25 Min. cell voltage [V] 2.50 The software Optimum Lap is usedCells to obtain in series the velocity profile 30.0 that the car can follow in a real competition [27,28,30,37]. In this case,Parallel a car arrays is created based on 1.0 the parameters of ISEC’s car. Furthermore, the racetrack selectedMaximal is charginga real circuit current used [A] in SAE 80 competition 2012, which was available in the software database. RatedThe driving Capacity cycle [Ah] and the racetrack 90 map taken from Optimum Lap are presented in Figure 9. Internal resistance [Ω] 0.006

(a) (b)

Figure 9. (a)(a) velocity velocity profile profile for for one one lap lap (b) (b) racetrack racetrack map map [37]. [37]. 5.2. Simulation Using the Proposed Braking Strategy 5.2. Simulation Using the Proposed Braking Strategy In Figure 10a, the velocity of the car’s response is presented. The car can follow the overall In Figure 10a, the velocity of the car’s response is presented. The car can follow the overall behavior of the driving cycle. Figure 10b shows the velocity of the rear wheels (left and right), with behavior of the driving cycle. Figure 10b shows the velocity of the rear wheels (left and right), with differences when there is a curve. A zoom in the driving cycle is done for clarity. differences when there is a curve. A zoom in the driving cycle is done for clarity. The forces applied to the rear wheels are presented in Figure 11a. The left and right forces are The forces applied to the rear wheels are presented in Figure 11a. The left and right forces are equal when the car is moving in a straight line. Moreover, during curves, these forces applied are equal when the car is moving in a straight line. Moreover, during curves, these forces applied are different, showing the operation of the differential. Figure 11b shows the distribution of the braking forces in the front and rear wheels. The braking force in the rear wheels has a low variability in most braking events, showing that the strategy always tries to get the most of the rear braking forces. On the other hand, the variability of the forces of the front wheel is higher. The contribution of the front wheels is greater during hard braking but can be very low in light braking events. World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 14 of 20 World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 14 of 20 different, showing the operation of the differential. Figure 11b shows the distribution of the braking different,forces in theshowing front andthe operationrear wheels. of theThe differential braking forc. Figuree in the 11b rear shows wheels the has distribution a low variability of the braking in most forcesbraking in events,the front showing and rear that wheels. the strategy The braking always forc triee ins tothe get rear the wheels most of has the a lowrear variabilitybraking forces. in most On brakingthe other events, hand, showingthe variability that the of strategythe forces always of the triefronts to wheel get the is higher.most of The the contributionrear braking offorces. the front On thewheels other is hand, greater the during variability hard ofbraking the forces but canof the be frontvery lowwheel in islight higher. braking The events.contribution of the front Worldwheels Electric Vehicleis greater Journal during2020, hard11, 45 braking but can be very low in light braking events. 14 of 20

(a) (b) (a) (b) Figure 10. (a) Car velocity output; (b) Wheels velocity (electric differential effect). FigureFigure 10. 10.(a )(a) Car Car velocity velocity output;output; (b) (b) Wheels Wheels velo velocitycity (electric (electric differential diﬀerential effect). eﬀ ect).

(a) (b) World Electric Vehicle Journal 2020(a), 11, x FOR PEER REVIEW (b) 15 of 20 FigureFigure 11. Braking 11. Braking forces forces distribution distribution (a) Electric(a) Electric diff erentialdifferential effect effect on lefton left and and right right wheels; wheels; (b) Braking(b) appliedstrategyFigureBraking in etheff ect11. rearstrategy inBraking frontis lower effect andforces incompared rear front distribution wheels. and rearto the (a) wheels. oneElectric applied differential in the effectfront, on the left potential and right for wheels; energy (b) recovery Braking strategy effect in front and rear wheels. is lower compared to cars with front wheels’ traction. FigureFigure 12 shows 12 shows the estimationthe estimation of theof the maximal maximal braking braking force force that that avoids avoids lockinglocking in each axle axle and Since RBS is used, the battery of the car is charged during a braking operation. Figure 13a shows comparesand Figurecompares it with 12 the showsit with braking thethe estimationbraking force appliedforce of theapplied inmaximal the in rearthe braking rear and and front force front wheels. that wheels. avoids This This locking demonstrates demonstrates in each axle that that the the current and SoC of the battery. The current is positive during traction and negative during brakingandthe forcebrakingcompares distribution force it with distribution the of braking the of proposed the force proposed applied strategy stra in thetegy guarantees rear guarantees and front that that wheels. the the rear rear This and and demonstrates frontfront wheels that are are not braking. In the same way, the SoC decreases during traction and increases during braking. In Figure lockedthenot duringbraking locked during braking,force distribution braking, ensuring ensuring of the proposed stabilitythe stability stra of of thetegy the car guaranteescar in in thethe whole that the racetrack racetrack rear and while front while following wheels following are the the 13b,notvelocity a locked zoom profile. duringis presented braking, to ensuringbetter appreciate the stability the of effect the car of inthe the negative whole racetrack current inwhile the followingSoC. the velocity profile. velocityAs profile.it was mentioned in Section 4.2, the braking strategy defines the contribution of the RBS based on theAs SoC, it was charging mentioned capabilities in Section of 4.2, the the battery braking bank strategy and the defines braking the torque contribution limitation of the of theRBS PMSM. based onIn thethis SoC, case, charging the RBS capabilitiescan handle of the the total battery rear bank brak ingand force,the braking and the torque FBS islimitation used only of thein the PMSM. front Inwheels. this case, This the is possible RBS can due handle to the the considerable total rear brak reductioning force, in the and normal the FBS force is used in the only rear in axle, the whichfront wheels.limits the This braking is possible force due that to can the beconsiderable applied in re thductione rear wheels.in the normal However, force since in the the rear braking axle, which force limits the braking force that can be applied in the rear wheels. However, since the braking force

(a) (b)

FigureFigure 12. 12.Braking Braking forces forces andand maximalmaximal forces (a) (a) rear rear wheels; wheels; (b) (b )front front wheels. wheels.

(a) (b)

Figure 13. State of charge and battery current, (a) full racetrack; (b) zoom in t=14-17 s.

5.3. Comparison of Different Braking Strategies The proposed strategy is compared to three different braking strategies. Two fixed braking

distribution strategies are found in [36], with distribution variables 𝑘 =0.75 and 𝑘 =0.55. A third variable that considers the ideal braking distribution curve summarized in Section 4.1 is used to compare the results. Now, energy recovery and car stability are studied to highlight the importance of the braking strategy. Figure 14 shows the energy consumed by the car studied during an endurance test of the Formula SAE, which has 22 km length. The car performance is studied without RBS as a reference to compare the energy recovery potential of the different braking strategies. Assuming constant road conditions and an adhesive coefficient of 0.9, the braking strategy proposed has the lowest energy consumption. Of the other three, the strategy that uses the ideal World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 15 of 20 applied in the rear is lower compared to the one applied in the front, the potential for energy recovery is lower compared to cars with front wheels’ traction. Since RBS is used, the battery of the car is charged during a braking operation. Figure 13a shows the current and SoC of the battery. The current is positive during traction and negative during braking. In the same way, the SoC decreases during traction and increases during braking. In Figure 13b, a zoom is presented to better appreciate the effect of the negative current in the SoC.

World Electric Vehicle Journal 2020, 11, 45 15 of 20

As it was mentioned in Section 4.2, the braking strategy deﬁnes the contribution of the RBS based on the SoC, charging capabilities of the battery bank and the braking torque limitation of the PMSM. In this case, the RBS can handle the total rear braking force, and the FBS is used only in the front wheels. This is possible due to the considerable reduction in the normal force in the rear axle, which limits the braking force that can be applied in the rear wheels. However, since the braking force applied in the rear is lower compared to the one applied in the front, the potential for energy recovery is lower compared to cars with front wheels’ traction. Since RBS is used, the battery of the car is charged during a braking operation. Figure 13a shows the current and SoC of the(a) battery. The current is positive during traction and(b) negative during braking. In the same way,Figure the SoC 12. Braking decreases forces during and tractionmaximal andforces increases (a) rear wheels; during (b) braking. front wheels. In Figure 13b, a zoom is presented to better appreciate the eﬀect of the negative current in the SoC.

(a) (b)

FigureFigure 13.13. StateState ofof chargecharge andand batterybattery current,current, ((a)a) fullfull racetrack; racetrack; ( b(b)) zoomzoom inin tt=14-17= 14–17 s. s. 5.3. Comparison of Different Braking Strategies 5.3. Comparison of Different Braking Strategies The proposed strategy is compared to three different braking strategies. Two fixed braking The proposed strategy is compared to three different braking strategies. Two fixed braking distribution strategies are found in [36], with distribution variables k = 0.75 and k = 0.55. A third distribution strategies are found in [36], with distribution variables 𝑘 α=0.75 and 𝑘α =0.55. A third variable that considers the ideal braking distribution curve summarized in Section 4.1 is used to variable that considers the ideal braking distribution curve summarized in Section 4.1 is used to compare the results. compare the results. Now, energy recovery and car stability are studied to highlight the importance of the braking Now, energy recovery and car stability are studied to highlight the importance of the braking strategy. Figure 14 shows the energy consumed by the car studied during an endurance test of the strategy. Figure 14 shows the energy consumed by the car studied during an endurance test of the Formula SAE, which has 22 km length. The car performance is studied without RBS as a reference to Formula SAE, which has 22 km length. The car performance is studied without RBS as a reference to compare the energy recovery potential of the different braking strategies. compare the energy recovery potential of the different braking strategies. Assuming constant road conditions and an adhesive coefficient of 0.9, the braking strategy Assuming constant road conditions and an adhesive coefficient of 0.9, the braking strategy proposed has the lowest energy consumption. Of the other three, the strategy that uses the ideal proposed has the lowest energy consumption. Of the other three, the strategy that uses the ideal braking distribution curve guarantees safety for all road conditions but has the highest energy consumption. Furthermore, the strategies that use fixed braking distribution have higher energy consumption than the strategy proposed in this work. Moreover, these strategies might have stability issues. This will be shown later. The energy recovered during the endurance test simulation is estimated using Equation (31) and presented in Table4. The mass-saving potential is calculated as shown in Equation (32), assuming that the specific energy of LiFPO4 batteries is 100 Wh/kg [38]. The results are presented in Table5.

Erec = E Estr (31) noRBS − Erec Mred = (32) ebat World Electric Vehicle Journal 2020, 11, x FOR PEER REVIEW 16 of 20 braking distribution curve guarantees safety for all road conditions but has the highest energy consumption. Furthermore, the strategies that use fixed braking distribution have higher energy consumption than the strategy proposed in this work. Moreover, these strategies might have stability issues. This will be shown later. The energy recovered during the endurance test simulation is estimated using Equation (31) and presented in Table 4. The mass-saving potential is calculated as shown in Equation (32), assuming that the specific energy of LiFPO4 batteries is 100 Wh/kg [38]. The results are presented in Table 5.

𝐸 =𝐸 −𝐸 (31) World Electric Vehicle Journal 2020, 11, 45 16 of 20 𝑀 = (32) wherewhere E𝐸recis is the the energyenergy recovered,recovered, E𝐸noRBS andand E𝐸strare are the the energy energy consumed consumed byby thethe carcar withwith nono regenerationregeneration andand usingusing eacheach brakingbraking strategy,strategy, respectively, respectively,M 𝑀redis theis the mass mass reduction, reduction, and andebat 𝑒is the is speciﬁcthe specific energy energy of the of battery.the battery.

FigureFigure 14.14. Energy consumption comparison.

Table 4. Energy consumption results for endurance test. Table 4. Energy consumption results for endurance test. Test Energy Recovered [Wh] Test Energy recovered [Wh] NoNo RBS RBS 0.00 Proposed strategy 1264.3 Proposed strategy 1264.3 Ideal braking distribution 922.34 IdealFixed braking distribution distributionkα = 0.55 922.341129.6 FixedFixed distribution distribution 𝑘kα =0.55= 0.75 1129.6647.19 Fixed distribution 𝑘 =0.75 647.19 Table 5. Mass saving results. Table 5. Mass saving results. Test Mass Reduction [kg] Percentage of the Total Mass Mass reduction Percentage of the total NoTest RBS 0 0 Proposed strategy[kg] 12.64 mass 3.37 Ideal brakingNo RBS distribution 9.22 0 2.46 0 Fixed distribution kα = 0.55 11.29 3.01 Fixed distribution kα = 0.75 6.47 1.72

Proposed strategy 12.64 3.37 Ideal braking distribution 9.22 2.46 Fixed distribution 𝑘 = 11.29 3.01 0.55 Fixed distribution 𝑘 = 6.47 1.72 0.75

A reduction of 3.37% of the car mass could be achieved. The mass reduction could be higher considering that the FBS is used only in the front wheels. Therefore, the size of the calipers and rotors of the FBS in the rear could be greatly reduced, making the car even lighter. Reduced mass of the car can improve its performance in the different Formula SAE competitions. In fact, assuming that the carWorld mass Electric is reduced Vehicle Journal by 12.642020, 11 kg,, 45 the car should complete each lap 0.2s earlier, and the endurance17 test of 20 approximately 3.8 s earlier, according to Optimum Lap results. TheThe stability stability of of the the car car is is now now analyzed analyzed for for each each strategy strategy studied. studied. Figure Figure 12 12 showed showed that that the the brakingbraking strategy strategy proposed proposed avoids wheelswheels locking.locking. TheThe samesame characteristic characteristic is is studied studied for for the the rest rest of theof thebraking braking strategies. strategies. Figure Figure 15 shows15 shows the evolutionthe evolutio ofn the of braking the braking forces forces in the in rear the and rear front and wheels. front wheels.A section A section of the racetrack of the racetrack is selected is selected to display to display better the better possible the possible stability stability issues duringissues during hard braking hard brakingevents. events. The black The circumferencesblack circumferences highlight highlight the hard the braking hard braking events events where where the force the appliedforce applied in the inwheels the wheels exceeds exceeds the maximal the maximal braking braking force, force, which wh resultsich results in wheels in wheels locking. locking. This This happens happens to theto thestrategies strategies that that use use a ﬁxeda fixed braking braking distribution distribution ratio. ratio. TheThe rearrear wheels are locked locked when when the the braking braking distribution ratio 𝑘 =0.55 is used. For this strategy, instability issues might appear during hard distribution ratio kα = 0.55 is used. For this strategy, instability issues might appear during hard brakingbraking events, events, and and the the car car might might be be unsafe unsafe to to drive. drive. Furthermore, Furthermore, the the potential potential to to recover recover energy energy is is reducedreduced to to zero zero when when the the rear rear wheels wheels are are locked, locked, and and the the speed speed of of the the PMSM PMSM is is zero. zero. On On the the other other hand, the front wheels are locked when 𝑘 =0.75 is used. For this strategy, the maneuverability of hand, the front wheels are locked when kα = 0.75 is used. For this strategy, the maneuverability theof thecar is car reduced is reduced during during hard hard brak braking,ing, but the but stability the stability of the of car the ca carn be can ensured. be ensured. Furthermore, Furthermore, this strategythis strategy has the has lowest the lowest energy energy recovery recovery potential potential since sinceit always it always locates locates lower lower braking braking force forcein the in rearthe wheels. rear wheels. In both In bothcases, cases, the velocity the velocity of the of ca ther might car might not be not reduced be reduced as it asis expected it is expected since since the longitudinalthe longitudinal grip gripis highly is highly reduced reduced after after locking locking the thewheels. wheels. However, However, this this is isnot not studied studied here. here. Moreover,Moreover, the the strategy strategy where where the the ideal ideal braking braking distri distributionbution curve curve is is followed followed ensures ensures that that the the wheels wheels areare not not locked, locked, but but the the energy energy recovered recovered is is lower lower compared compared to to the the proposed proposed strategy. strategy.

(a) (b)

FigureFigure 15. 15. ComparisonComparison of of braking braking forces forces and and maximal maximal forces forces (a) (a )rear rear wheels; wheels; (b) (b )front front wheels. wheels. 6. Conclusions 6. Conclusion ISEC’s race car designed to participate in the Formula SAE competition has been modeled and ISEC’s race car designed to participate in the Formula SAE competition has been modeled and organized using the EMR formalism. The main objective of the model is to analyze braking strategies organized using the EMR formalism. The main objective of the model is to analyze braking strategies for a Formula SAE race car. Also, the electric differential operation is considered. The braking strategy for a Formula SAE race car. Also, the electric differential operation is considered. The braking strategy operation has been investigated focusing on the braking force distribution between front and rear wheels. The key points of the braking strategy are the energy recovery and stability issues that could appear. Four approaches defining the braking distribution factor have been studied. The strategy proposed has a rear-biased braking force distribution defined by the maximal braking forces, which are proportional to the adhesive coefficient and normal forces. Other strategies used to analyze the results consider the ideal braking distribution curve and fixed braking distribution between the front and rear wheels. The results obtained showed that the energy recovered using the proposed strategy is higher when is compared to the other strategies. On the other hand, the proposed strategy and the one that considers the ideal braking distribution curve guarantee stability during driving. Moreover, the strategies that considered a fixed braking distribution ratio showed that instability issues might appear. Furthermore, the potential for energy recovery during braking will be reduced if rear wheels are locked. World Electric Vehicle Journal 2020, 11, 45 18 of 20

Energy recovery during braking has been presented for the braking strategies studied and the impact of this feature in race cars was highlighted. A good braking strategy will allow for a better tradeoﬀ between car stability and eﬃciency. Regenerative braking allows for the use of less energy, which could help to increase the car’s autonomy. Additionally, RBS usage enables a mass reduction of the car, which is an important parameter in competitions such as Formula SAE.

Author Contributions: Conceptualization, A.C.H.-M., P.P. and A.B.; methodology, A.C.H.-M., P.P. and A.B.; software, A.C.H.-M., and A.B.; validation, A.C.H.-M., P.P. and A.B.; formal analysis, A.C.H.-M. and A.B.; investigation, A.C.H.-M.; resources, P.P. and A.B.; data curation, A.C.H.-M.; writing—original draft preparation, A.C.H.-M.; writing—review and editing, P.P. and A.B.; visualization, A.C.H.-M., P.P. and A.B.; supervision, P.P. and A.B.; project administration, P.P. and A.B.; funding acquisition, P.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Erasmus Mundus scholarship for the program Sustainable Transportation and Electrical Power Systems (STEPS). The APC was funded by FCT – Portuguese Foundation for Science and Technology under INESC Coimbra (DEEC Coimbra, Portugal) project UIDB/00308/2020 and by the European Regional Development Fund through the COMPETE 2020 Program and FCT – Portuguese Foundation for Science and Technology within project ESGRIDS (POCI-01-0145-FEDER-016434). Acknowledgments: This research is the continuation of the master thesis of A.C.H.-M. (STEPS Erasmus Mundus Joint Master Degree). The master thesis was developed in the Polytechnic of Coimbra and a short stay in University of Lille where the EMR summer school 2019 took part. This research was partially supported by project UIDB/Multi/00308/2020 and project ESGRIDS (POCI-01-0145-FEDER-016434). Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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